text
stringlengths 6
976k
| token_count
float64 677
677
| cluster_id
int64 1
1
|
|---|---|---|
Refine Your Search:
To present topics demonstrating the beauty and utility of mathematics to the general student population and to provide...
see more
To present topics demonstrating the beauty and utility of mathematics to the general student population and to provide knowledge and skills useful for college, life and career. The course will include topics related to patterns and reasoning, growth and symmetry, linear and exponential growth, and
Author:
Tim Dube
Date Added:
Jan 15, 2008 | 677.169 | 1 |
Mathematics Syllabus for West Bengal Joint Entrance Examination 2013
West Bengal Joint Entrance Examination Board releases WBJEE 2013 Notification for admission to undergraduate degree level Engineering & Technology courses including Pharmacy and Architecture in different Universities / Government Colleges as well as Self Financing Institutes of West Bengal.
Quadratic Equations: Quadratic equations with real coefficients; Relations between roots and coefficients; Nature of roots; Formation of a quadratic equation, sign and magnitude of the quadratic expression ax2+bx+c (where a, b, c are rational numbers and a ≠ 0).
Permutation and combination: Permutation of n different things taken r at a time (r ≤ n). Permutation of n things not all different. Permutation with repetitions (circular permutation excluded).Combinations of n different things taken r at a time (r ≤ n). Combination of n things not all different. Basic properties.Problems involving both permutations and combinations.
Principle of mathematical induction: Statement of the principle, proof by induction for the sum of squares, sum of cubes of first n natural numbers, divisibility properties like 22n— 1 is divisible by 3 (n ≥ 1), 7 divides 3 2n+1+2n+2(n≥ 1)
Matrices: Concepts of m x n (m ≤ 3, n ≤ 3) real matrices, operations of addition, scalar multiplication and multiplication of matrices. Transpose of a matrix. Determinant of a square matrix. Properties of determinants (statement only). Minor, cofactor and adjoint of a matrix. Non singular matrix. Inverse of a matrix. Finding area of a triangle. Solutions of system of linear equations. (Not more than 3 variables).
Sets, Relations and Mappings: Idea of sets, subsets, power set, complement, union, intersection and difference of sets, Venn diagram, De Morgan's Laws, Inclusion / Exclusion formula for two or three finite sets,Cartesian product of sets.Relation and its properties. Equivalence relation — definition and elementary examples, mappings, range and domain, injective, surjective and bijective mappings, composition of mappings, inverse of a mapping.
Coordinate geometry of two dimensions Basic Ideas: Distance formula, section formula, area of a triangle, condition of collinearity of three points in aplane. Polar coordinates, transformation from Cartesian to polar coordinates and vice versa. Parallel transformation of axes, concept of locus, elementary locus problems.
Straight line: Slope of a line. Equation of lines in different forms, angle between two lines. Condition of perpendicularity and parallelism of two lines. Distance of a point from a line. Distance between two parallel lines. Lines through the point of intersection of two lines.
Circle: Equation of a circle with a given center and radius. Condition that a general equation of second degree in x, y may represent a circle. Equation of a circle in terms of endpoints of a diameter . Parametric equation of a circle. Intersection of a line with a circle. Equation of common chord of two intersecting circles.
Parabola : Standard equation. Reduction of the form x = ay2+by+c or y = ax2+bx+c to the standard form y2= 4ax or x2= 4ay respectively. Elementary properties and parametric equation of a parabola.
Ellipse and Hyperbola: Reduction to standard form of general equation of second degree when xy term isabsent. Conjugate hyperbola. Simple properties. Parametric equations. Location of a point with respect toa conic.
Integral calculus: Integration as a reverse process of differentiation, indefinite integral of standard functions.Integration by parts. Integration by substitution and partial fraction.Definite integral as a limit of a sum with equal subdivisions. Fundamental theorem of integral calculus and its applications. Properties of definite integrals.
Application of Calculus: Tangents and normals, conditions of tangency. Determination of monotonicity, maxima and minima. Differential coefficient as a measure of rate.Motion in a straight line with constant acceleration.Geometric interpretation of definite integral as area, calculation of area bounded by elementary curves and Straight lines. Area of the region included between two elementary curves.
Candidates are informed to prepare the above mentioned syllabus of Mathematics to secure more marks to get Better Admissions in institutions of West Bengal. | 677.169 | 1 |
MATH 101
MATHEMATICAL THINKING
Course info & reviews
Presents mathematical topics and applications in a context designed to promote quantitative reasoning and the use of mathematics in solving problems and making decisions. Suitable for majors in humanities, education and others seeking a broad view of mathematics. No background in algebra required. This course earns three GEPs toward Go...
A
average
0
units
MATH 10114
3.53846 Stars
14 ratings
Reviews
12/22/2013
You just wrote this review 0 of 0 people found this review helpful.
Delete Review
12/19/2013
0 StarsSpring 2013-2014 | 0 of 0 people found this review helpful.
Content was difficult. Professor did not explain well.
12/17/2013
2 StarsFall 2013-2014 | 0 of 0 people found this review helpful.
This was not a hard class, but the professor that i had it with did not know how to teach, and therefore the class was pretty confusing. I ended up dropping the class.
12/17/2013
2 StarsFall 2013-2014 | 0 of 0 people found this review helpful.
This class was not a hard class, but the professor that i had it with did not know how to teach and therefore the class was confusing. I ended up dropping the class.
Students
Q&A
A) You perform an experiment in which you roll a fair die and flip a fair coin. As the result of... Show
more
A) You perform an experiment in which you roll a fair die and flip a fair coin. As the result of the experiment, you write down a single
integer which is obtained as follows: If the die comes up with a D and the coin shows a T, you write down D; if the die comes up with a D
and the coin shows a H, you write down 2*D. For example, if you roll a 5 on the die and the coin shows a tail, you write down 5, while if
you roll a 5 on the die and the coin shows a head, you write down 10. How big is the sample space for this experiment?
The yearly rate of return on the Standard & Poor's 500 (an index of 500 large-cap corporations) is approximately normal. From January 1, 1960
through December 31, 2009, the S&P had a mean yearly return of 10.98 percent, with a standard deviation of about 17.46 percent. Take this normal
distribution to be the distribution of yearly returns over a long period.
(a) In what interval do the middle 95 percent of all yearly returns lie?
(b) Stocks can go do as well as up. What are the worst 2.5 percent of annual returns?
(c) What is the interval of the middle 50 percent on annual returns on stocks, according to the distribution given in the previous
exercises? (Hint: What two numbers mark off the middle 50 percent of any distribution?)
1-Looking at your trophy collection more closely, you note that you have three ping-pong trophies... Show
more
1-Looking at your trophy collection more closely, you note that you have three ping-pong trophies, three volleyball trophies, and two math
trophies. You decide to put them on the shelf in such a way that all the trophies of the same type are together. In how many ways can this
be done?
2- After looking at the shelf for a while, you decide that it would look best if all the volleyball trophies came first, then the math
trophies, and then the ping-pong trophies. In how many ways can this be done?
1- You own a group of ten different pets and you need to pick three of them to send to your cousin M... Show more
1- You own a group of ten different pets and you need to pick three of them to send to your cousin Mort who is currently petless and miserable. In how
many different ways can you select the pets to be sent off?
2- On reflection, since all three pets are to be put in a single cage for shipping, you decide that it would not be a good idea to send both Cutty (a
bobcat) and Cheap (a parakeet) to Mort. With this restriction, in how many different ways can you select the pets to be sent off?
3- How many 4-digit numbers can you build if the number must be odd, odd digits can be repeated, and even digits cannot be repeated?
4-How many 4-digit numbers are there which consist of all different digits? | 677.169 | 1 |
-Hill's Top 50 Math Skills for GED Success
From making an appropriate estimate and solving for volume, this distinctive workbook features step-by-step instructions, example questions and an ...Show synopsisFrom making an appropriate estimate and solving for volume, this distinctive workbook features step-by-step instructions, example questions and an explanatory answer key, short concise lessons presented on double-page spreads, and more | 677.169 | 1 |
The main goal of this project is to improve student understanding of the geometric nature of multivariable calculus concepts,...
see more
The main goal of this project is to improve student understanding of the geometric nature of multivariable calculus concepts, i.e., to help them develop accurate geometric intuition about multivariable calculus concepts and the various relationships among them.To accomplish this goal, the project includes four parts:· Creating a Multivariable Calculus Visualization applet using Java and publishing it on a website: web.monroecc.edu/calcNSF· Creating a series of focused applets that demonstrate and explore particular 3D calculus concepts in a more dedicated way.· Developing a series of guided exploration/assessments to be used by students to explore calculus concepts visually on their own.· Dissemination of these materials through presentations and poster sessions at math conferences and through other publications.Intellectual Merit: This project provides dynamic visualization tools that enhance the teaching and learning of multivariable calculus. The visualization applets can be used in a number of ways:- Instructors can use them to visually demonstrate concepts and verify results during lectures.- Students can use them to explore the concepts visually outside of class, either using a guided activity or on their own.- Instructors can use the main applet (CalcPlot3D) to create colorful graphs for visual aids (color overheads), worksheets, notes/handouts, or tests. 3D graphs or 2D contour plots can be copied from the applet and pasted into a word processor like Microsoft Word.- Instructors will be able to use CalcPlot3D to create lecture demonstrations containing particular functions they specify and/or guided explorations for their own students using a scripting feature that is being integrated with this applet.The guided activities created for this project will provide a means for instructors to get their students to use these applets to actively explore and "play" with the calculus concepts.Paul Seeburger, the Principal Investigator (PI) for this grant project, has a lot of experience developing applets to bring calculus concepts to life. He has created 100+ Java applets supporting 5 major calculus textbooks (Anton, Thomas, Varberg, Salas, Hughes-Hallett). These applets essentially make textbook figures come to life. See examples of these applets at Impacts: This project will provide reliable visualization tools for educators to use to enhance their teaching in calculus and also in various Physics/Engineering classes. It is designed to promote student exploration and discovery, providing a way to truly "see" how the concepts work in motion and living color. The applets and support materials will be published and widely disseminated through the web and conference presentations.
An interactive multimedia tutorial for healthcare professionals wishing to refresh math skills and learn how to calculate...
see more
An interactive multimedia tutorial for healthcare professionals wishing to refresh math skills and learn how to calculate medication dosages and intravenous (IV) rates. The tutorial has three modules. Each module has includes a quiz. The first module covers information about fractions, decimals, ratios, proportions, and percentage. The second module covers information about conversions, medication administration, and dosage calculation. The third module covers information about intraveneous infusions including tubing calculation and intraveneous flow rates. Key words: Medication calculation; Intravenous flow rate calculation; Mathematics principles
From the website: "The Crump Institute for Molecular Imaging brings together faculty, students, and staff with a variety of...
see more
From the website: "The Crump Institute for Molecular Imaging brings together faculty, students, and staff with a variety of backgrounds - physics, mathematics, engineering, biology, chemistry, and medicine - to pursue innovative technologies and science to accelerate our understanding of biology and medicine.״
The resource contains many Flash physics animations covering topics such as chaos, mechanics, vectors, waves, relativity;...
see more
The resource contains many Flash physics animations covering topics such as chaos, mechanics, vectors, waves, relativity; includes a tutorial on using Flash with mathematical equations to create controlled animations.
This site contains a collection of fully developed high school curriculum modules that use the Internet in significant ways. ...
see more
This site contains a collection of fully developed high school curriculum modules that use the Internet in significant ways. There are currently 15 modules in Mathematics and 6 modules in Science; also, there are approximately two dozen additional modules that have been created by instructors and/or Education students.The learning modules here are web-based, technology intensive lessons focusing on mathematics and science in an applied context. They have been developed for teachers, by teachers, aligned with the Illinois State Learning Standards and the National Council for Teachers of Mathematics (NCTM) Standards. Some of the lessons are designed to last over several days, some only for a class period.
The MSTE lessons site contains a collection of excellent high school/lower college division math lessons (and a limited...
see more
The MSTE lessons site contains a collection of excellent high school/lower college division math lessons (and a limited number of science lessons) that use the Internet in significant ways. The lessons have been developed with descriptions of the problem, connections to standards, examples of use, references, and more. Java source codes are often available. The Office for Mathematics, Science, and Technology Education (MSTE) is a division of the College of Education at the University of Illinois at Urbana-Champaign.
A java application that lets you investigate the problem of creating a phylgenetic tree. there is some explanation of the...
see more
A java application that lets you investigate the problem of creating a phylgenetic tree. there is some explanation of the problem of drawing phylogenetic trees and a couple of versions of the applet. The applets let you randomly generate trees and also let you infer a phylogenetic tree with a couple of different methods. Number of species, mutation rate and the method for calculating the distance matrix can all be varied. | 677.169 | 1 |
Basic College Mathematics
Basic College Mathematics
Basic Mathematics
Basic Mathematics
Basic Mathematics
Basic Mathematics
Basic Mathematics
Basic Mathematics
BASIC MATHEMATICS
Basic Mathematics ( AIE )
Collaborative Learning Activities Manual
MathXL Tutorials on CD for Basic College Mathematics
Student Solutions Manual for Basic College Mathematics
Video Resources on DVD with Chapter Test Prep Videos for Basic College Mathematics
Summary
Worksheets for Classroom or Lab Practice offer extra practice exercises for every section of the text, with ample space for students to show their work. These lab- and classroom-friendly workbooks also list the learning objectives and key vocabulary terms for every text section, along with vocabulary practice problems. | 677.169 | 1 |
first-year calculus roughly in the order in which it was first discovered. The first two chapters show how the ancient calculations of practical problems led to infinite series, differential and integral calculus and to differential equations. The establishment of mathematical rigour for these subjects in the 19th century for one and several variables is treated in chapters III and IV. Many quotations are included to give the flavor of the history. The text is complemented by a large number of examples, calculations and mathematical pictures and will provide stimulating and enjoyable reading for students, teachers, as well as researchers. | 677.169 | 1 |
Algebraic Thinking
Moderator
Good afternoon and welcome to today's chat with NCTM President Cathy Seeley. Today's topic is algebraic thinking.
Here's our first question:
Question from:
Chico, California
One thing that might help develop some algebraic thinking is to present more problems that students have to be able to solve with any variation of the original given constraints. For example, if the question were "The pool manager Jim always keeps 100 gallons of water in his pool. At the end of each day he checks the pool to see how much water is in it. On Monday, only 82 gallons of water was left in the pool. How many gallons of water does Jim need to add?" The kids can figure this out with subtraction fairly easily. However, if there were more parts to the question like, "On Tuesday there was only 52 gallons left. How many gallons does Jim need to add now?" And then a final part to the question, "Write an equation that will tell Jim how many gallons to add at the end of any given day." This type of thinking will help them to develop the concept of variables and equations, and it is done in a fairly simple way.
Cathy Seeley:
This is exactly the kind of thinking that helps students develop the ability to use algebraic thinking to make generalizations. And this is just what we want students to do with algebra—to think beyond specific examples to more general cases. I prefer this idea to just learning recipes to solve problem 'types' (like coin problems, age problems, mixtures, etc.).
Question from
Missoula, Montana
Is algebra more than what most of us learned with x's and y's and many homework problems? You seem to imply that.
Cathy Seeley:
Algebra can be a powerful set of tools for representing situations, analyzing mathematical relationships, making generalizations and solving problems. It can extend well beyond the limited types of problems once filling traditional algebra texts to serving as a set of approaches in a student's mathematical toolkit. Today, algebra can be used to deal with data and make predictions. It can be used to model sophisticated situations from science, social studies or economics, to name a few. If we do our job well to develop algebraic thinking across the grades, American students will never be heard to say that they had no use for algebra. Rather, they will incorporate algebraic techniques into their broader mathematical thinking to deal with everyday life as well as advanced applications in mathematics and science.
Question from
Fayetteville North Carolina
One of the red flags I see in developing algebraic thinking is how we develop students' conceptual understanding of equality, especially in the early years. We must allow students to discuss and explore all aspects of what equality is and is not in order to break away from the idea that math is all about "the answer." Number relationships and balanced equations may not boil down to a nice little number for an answer, and it's important in the developing years for students to understand this.
Cathy Seeley:
I couldn't agree with you more. Equivalence/equality is undoubtedly one of the most important, connecting ideas in school mathematics. As you note, too many students see the equal sign as what precedes the answer. Developing this concept of equivalence calls for lots of experiences with materials as students are developing their conceptual understanding of numbers and operations. More important, it calls for teachers to help students connect their experiences with the mathematical idea(s) they are developing, in this case, equivalence or equality.
Question from
Lanham, Maryland
A major mistake is when we don't tell our first grade students the truth about the numbers. We should tell them the whole story about the numbers. We should tell them about positive and negative numbers both and also tell them that the real numbers are a part of all the numbers, and there are more numbers to learn (complex numbers). I have seen many students who think 2 minus 4 can't be done by asking, "If we have only two things, how can we take four out of them?" This line of thinking is because of not introducing students to negative numbers and algebraic thinking early. I hope we also show the power of algebra to our young students by letting them use it to solve problems. Thanks for asking.
Cathy Seeley:
Thanks for sharing this perspective. I recently visited a second-grade classroom where the (excellent) teacher was leading students in solving problems using subtraction. I heard her tell students to remember that you can't take a bigger number away from a smaller one. Unfortunately, this message, if sent often enough, can plant a future misunderstanding related to negative numbers and to the meaning of subtraction. Better to keep coming back to what the numbers represent and what the problem calls for in terms of a solution based on the particular situation. I'm not sure I'd advocate negative numbers with first-graders, but it's an interesting concept.
Question from
Omaha, Nebraska
I am a community college math teacher. Approximately 90 percent of the students who take our placement test (COMPASS, through ACT) assess into a developmental class in at least one of the areas: math, reading, English.
Yesterday I had a student wanting into my Intermediate Algebra class who assured me he had passed Algebra II in high school this spring. He could not combine like terms (he thought 5x - 5 = x), he could not graph a linear equation, he could not factor x2 - 5x -6, and so on. I don't really mind, since it does mean job security for me, but what is the disconnect between grades and knowledge?
Cathy Seeley:
This is a troubling phenomenon. My opinion is that the disconnect between grades and knowledge is more about how we teach than about how we grade. It is quite likely that this student, like others, passed a test at one time that called for these skills. I am guessing that this learning was temporary and superficial if it was based on lecture and memorization. Unless we connect algebraic skills to meaningful situations, build conceptual understanding and facilitate the use of algebraic thinking to solve problems, we risk perpetuating the study of algebra as a vast set of meaningless abstract rules and procedures that, once learned, may be quickly forgotten, never to return. The key, I believe, lies both in what content we choose to teach, and in how we engage students in their learning; watch for my next two President's Messages on this one-two punch. (By the way, in my recent teaching experience in a French system in West Africa, I noted what I had read elsewhere—that other countries do not use factoring as a general method for solving quadratic equations.) Thanks for bringing up this important and all too common problem.
Question from
Murfreesboro, Tennessee
I feel that students need to practice critical thinking skills in the elementary grades. This can be done throughout the curriculum, not just in the mathematics courses.
Cathy Seeley:
I couldn't have said it better myself. Not just starting earlier, but incorporating critical thinking across the curriculum. I believe that the different ways we teach problem solving and critical thinking in various content areas can help students access learning in different ways, based on their strengths.
Question from
Riverside, California
What is algebraic thinking?
We think of algebra as an art of symbol manipulation (e.g., let "x" be an unknown.) But I sense that teachers of algebra are not offering sufficient training in axiomatic thinking.
To prove the point, not a single algebra teacher in my experience could tell an inquisitive student, why (-a)x(-b) is PLUS (a)x(b).
Sure, there are hokey, plausibility arguments, like, "Why not?" or "Symmetry demands it." In fact, the axioms of distributivity, associativity, etc. come into play. Also, the real explanation may be too subtle for the classroom, which the teacher should acknowledge.
MORAL: Although the result may sound elementary, its justification may not be. The teacher should have the discernment to know how much to teach and how much to omit.
Cathy Seeley:
Your point brings up an important issue and one of NCTM's primary goals: providing high-quality professional development. This year's professional development focus of the year on algebraic thinking supports the kind of teacher understanding that you describe. Some universities and regional educational facilities offer high-quality professional development that gets to the heart of algebraic thinking. Unfortunately, such experiences are not accessible to all teachers. A rich set of resources to help teachers expand their deep understanding of algebraic concepts can be found throughout this year on NCTM's Web site. Look for the magnifying-glass icon on the home page and throughout the site to lead teachers to articles, books, and online resources. Teachers can use these individually, or, ideally, with colleagues as starting points for professional growth.
Question from
University Park, Pennsylvania
I think we need to clarify what we mean by "algebraic thinking." I believe too much emphasis is placed on patterns (which is an important idea). As I look at the Japanese curricular materials, what seems to be missing in the typical US approach is the emphasis on writing equations. We rarely discuss the importance of expressing our mathematical thinking using mathematical expressions and equations, nor trying to interpret mathematical thinking expressed as an equation or expression. Thus, for too many children expressions like 2x(5+3) simply means to calculate and find the answer.
This is just one simple example to illustrate the need for mathematics teachers to critically re-evaluate what we mean by "algebraic thinking." Representing our thinking through mathematical symbolisms must be an important part of algebraic thinking.
Cathy Seeley:
Your comments reinforce the importance of being able to use multiple types of representations for a situation, as called for in Principles and Standards for School Mathematics. The symbolic representation is an important part of a student's algebraic development. I hope you would agree that this representation needs to be well grounded in understanding what lies behind the symbols. By incorporating algebraic opportunities throughout the curriculum and across grades, we can help students develop all types of representational skills.
Question from
Archbald, Pennsylvania
When it comes to middle school math in the United States (specifically grade 8), what percent of the students do you think should be taking an Algebra 1 course as an elective instead of taking the traditional 8th grade math course, which includes some algebra concepts?
Cathy Seeley:
I don't think there is a set percent of students we can identify. Of more importance is the question of what is the nature of the middle school curriculum? A middle school program that addresses central features of Principles and Standards in School Mathematics, including such key ideas as proportionality and the transition to algebra, can clearly be beneficial to many students, much more so than what we might consider 'traditional' eighth-grade mathematics. We are no longer in a time, as we were when I started teaching middle school mathematics 35 years ago, when the middle school curriculum is simply a repeat of K-6 arithmetic. Now that we have so much to offer, there is far less reason to accelerate students into algebra. At the same time, if a student is to be able to take calculus in high school, or another course beyond the level of pre-calculus, they must either start the high school sequence in grade 8 or find a way to double up during high school. In summary, the major reason to accelerate: opening up options for advanced mathematics study to all students, especially those not typically represented in such courses. Cautions against accelerating: make sure the student is motivated to continue mathematics study every year in high school; make sure there are good course offerings for students in 11th and 12th grade; ensure that students have the benefit of the rich mathematics (especially proportionality) that should be part of the middle school program.
Question from
Honolulu, Hawaii
I think the ideas around equality or equivalence in the early grades are very important, as the question from North Carolina implied. In fact, young children can handle much more sophistication in ideas than we might have expected. This causes us to think about early mathematics in different ways.
Cathy Seeley:
Absolutely. As we rethink both priorities and opportunities at the elementary grades, we can likely not only build a foundation for algebraic thinking, but also strengthen students' development of concepts related to number, operations, and other mathematical strands.
Question from
Seattle
I attended the 6–8 and 9–12 Navigations E-workshops, which were very good in showing algebraic thinking across the grades and the development across the grades.
Cathy Seeley:
Great! This model for professional development is one we think holds a lot of promise, and the Council will continue to explore how we can use it to meet the needs of more teachers. In whatever form our professional development takes, this notion of developing this important thread of algebraic thinking across the grades is critical, starting with a student's earliest school experience.
Question from
Turners Falls, Massachusetts
Are we perpetuating an artificial segregation of math content (algebra, geometry, etc.) by emphasizing "algebraic thinking" as a concept? Why not place the emphasis on mathematical thinking?
Cathy Seeley:
Mathematical thinking is very appropriate for our broader attention. However, the term may be too broad. While all teachers might agree that they should help students learn to think, many elementary teachers, in particular, don't see where algebra fits in with their teaching. By focusing on what skills and knowledge build toward algebraic understanding, teachers can see more easily how these might fit with their teaching. For example, a teacher might agree that working with patterns, developing the notion of equivalence, and exploring relationships are important, even though they might not have thought of these as topics related to algebra. By making explicit what we mean by algebraic thinking, we can increase the likelihood that this type of experience can be part of what students do in school. But, as you observe, the bigger picture demands that we not teach algebraic thinking in isolation from other strands, but rather capitalize on our attention to algebraic thinking to find ways to connect it with other important parts of the curriculum toward a student's broader mathematical thinking.
Question from
Rochester, Michigan
I have two favorite problems that can help accomplish your goal. These are mathematical computations that are very down-to-earth and real-life, and they are things that everyone should have been empowered to solve by the time they leave elementary school. They are arithmetic but very much lead to algebraic thinking.
1. You and your best friend went on a weeklong vacation and agreed to split the costs down the middle. Over the week, each of you paid for various joint expenses (like gasoline, meal checks, motel bills) and kept track of what you had paid. At the end of the vacation, it turns out that you had paid A dollars and your friend had paid B dollars. Assume that A is less than B. How much money must you give to your friend at this point to even things up? This can be solved with algebra (in more than one way), by example, verbally, or with a picture, among other approaches. Multiple correct answers are possible.
2. (a) Exactly how many days old are you today? (b) Use your answer to (a) to figure out what day of the week you were born on. (The solution gets into the algebraic ideas of modular arithmetic.)
Problems like these make math be lively, relevant, and interesting, and they start leading students' thought patterns in the right directions.
Cathy Seeley:
Thanks for sharing these ideas. Adapted for appropriate grade levels, problems like these can be engaging tasks as part of a comprehensive approach to incorporating algebraic thinking in the elementary grades.
Question from
Hopewell, Virginia
As a first-year high school algebra teacher I am struggling to find ways to connect with students. Nearly 40 percent of my students are repeaters or students with learning disabilities, and keeping them from giving up on themselves is my main challenge. I have been moving around the room, asking students to stand up and act out parts of equations and mathematic properties, or even simply come up to write an answer on the board and then defend it. Still I hear the kids saying things like "I'm just dumb," or "I'll never learn this," or "Even if I knew how, it wouldn't do me any good." Are there any strategies out there to help engage students with limited motivation?
Cathy Seeley:
I think you have identified a critical factor—engaging students. I think that such engagement can precede motivation. By choosing tasks that allow students to tackle challenging and interesting problems, we can set up a classroom where students work in small groups to come up with solutions, discussing, justifying, and even arguing about approaches and consequences. Some of the National Science Foundation curriculum projects, both for middle school and high school, provide such tasks. This type of activity fits beautifully with the functions-based approach advocated in NCTM's Principles and Standards for School Mathematics.
Question from
Ontario, Canada
At what point do we introduce algebra to students who still have not mastered basic numeracy skills? Students who have troubles with operations using fractions are not likely going to understand or be successful with basic algebra. I have students in my grade 10 class that still use calculators to add and subtract fractions. They cannot even grasp the concept of solving for an unknown.
Cathy Seeley:
It would certainly be ideal to have all students be proficient in arithmetic before progressing to algebra. However, a few years ago I had an experience teaching a ninth-grade algebra class that caused me to re-examine my beliefs about necessary prerequisites for learning algebra. One particular student, whom I call Crystal, could not do fraction operations and asked if she could use a fraction calculator in the algebra class. I quickly discovered that, in spite of her arithmetic deficiency, Crystal was an outstanding algebraic thinker, as long as she had her fraction calculator to help her get answers to fraction problems. To make a long story a little shorter, eventually Crystal was motivated by her success in algebra to go back and learn fractions. She continued through precalculus in high school and went on to graduate from college and graduate school. We need to be careful not to let our own beliefs about how mathematics must be organized get in the way of allowing all students the opportunity to show us what they can do. Even though computational proficiency helps in higher-level mathematics, there is no evidence that students who are weak in some areas of computation cannot succeed in algebra or higher-level mathematics.
Question from
Athens, Georgia
What place does memorization of facts have in developing algebraic thinking? While we want to develop higher-order thinking and understanding of concepts, are there not certain facts students must simply "know?" And how do calculators in the classroom fit in to this?
Cathy Seeley:
This continues the discussion of the previous question. I absolutely agree that there are facts that students should know. I also recognize that there may be students who are otherwise ready to proceed to algebra or higher-level mathematics, even though they may not know all these facts (or skills). Technology offers us a way for both the teacher and student to see what students can do beyond what they have learned. And, as in the case of students like Crystal, sometimes success in more challenging mathematics can motivate learning of the things we wish they had known before. Engagement and opportunity can absolutely lead to motivation and success.
Question from
Raleigh, North Carolina
There are many well-respected theories out there on multiple intelligences. If a given student tends to think "geometrically" or visually, are we justified in forcing them to repeat algebra classes until they pass? If we are justified (I hope we are), then how do we get them to make this leap?
Cathy Seeley:
You've identified an important issue. In fact, students do think in different ways, and this might be one argument in support of a more integrated high school curriculum where algebra and geometry are not approached in isolation. But regardless of how you organize the curriculum, students should have opportunities to develop their thinking along different lines. When we teach algebraic problem-solving skills, this can increase students' repertoire of approaches beyond what may be their first line of attack. At the same time, a student who thinks geometrically or visually may benefit from a more visual approach to algebra. This is one of the most exciting advances in the teaching of secondary mathematics—that we can represent situations in multiple ways and can approach the solving of problems in different ways. Using the power of graphing technology and a functions-based approach to algebra can allow students to solve problems from either a symbolic or a graphical approach. There is no reason why any person who is reasonably successful in other content areas cannot be successful in mathematics if we offer multiple avenues toward mathematical understanding.
Question from
Newnan, Georgia
So which is better—a continuous mathematics course and integrated math course, or separate courses (Algebra I, Algebra II, etc)? How is the continuous different from integrated?
Cathy Seeley:
You folks in Georgia are certainly dealing with this issue. While many districts in the United States (and essentially all in Canada and elsewhere) have implemented an integrated approach to secondary mathematics, Georgia may be the first state to try to adopt a requirement for integrated high school mathematics in all schools (at least it's the only one that comes to mind). In any case, there are many apparent advantages to such an approach, not the least of which is the opportunity for students to connect otherwise isolated pieces of mathematical knowledge and skills. But the flip side is that this is a huge change for many schools, and the professional development and materials support needs are tremendous. If the state chooses to go this direction (or for schools outside of Georgia considering such a move), this professional development support and finding appropriate instructional materials are needs that must be addressed, as well as the need for ongoing support for teachers in terms of planning time, collaboration time, and so on.
The benefits may be significant and may be worth the effort, but it is important to recognize what is being asked of teachers and to acknowledge that making such a change on such a large scale may be quite challenging. For an individual teacher or school to adopt an integrated approach may be much more doable and may lead to more visible results more quickly. Regardless of whether you teach an integrated program or a course-by-course program, the essential things are what content you choose and how you actively engage students in their learning.
Moderator Thank you all for your stimulating participation this afternoon. We had far more questions submitted live during the hour (and in advance) than Cathy could answer today. She will review all questions submitted and add several answers for the final chat transcript, which will be posted on the NCTM Web site Monday or Tuesday.
Thank you again.
Cathy Seeley:
Thanks to all of you for your energetic participation. You've kept me typing as fast as I can! I've enjoyed this professional interchange, and I look forward to reflecting on as many other additional questions as I can. Be sure to check the NCTM Web site for resources on Algebraic Thinking, the Professional Development Focus of the Year. Look for the magnifying glass icon.
Thanks for your interest, and I'll see you at our next chat!
Moderator
The following questions are representative of those submitted in advance or during the hour of the online chat. Time restrictions prevented Cathy from answering all the questions submitted for this chat.
Question from 7. Jakarta, Indonesia
What is the best way to teach algebra in grades 1, 2, and 3 in elementary school?
Cathy Seeley:
This question is way too big for a short response. The best source of information NCTM has to offer on our Professional Development Focus of the Year can be found on the NCTM Web site, accessible from many paths. Look for the magnifying glass icon on the home page and throughout the Web site. You will be delighted at the wealth of resources on incorporating algebraic thinking across the grades. One resource you will find there is the algebra standard in Principles and Standards for School Mathematics, as well as the excellent series of Navigations publications, which includes algebra books for each grade band (Go to: ). We are also seeing an increasing number of online professional development programs related to algebraic thinking, although with so many available, you should choose any such program carefully.
Question from
Columbia, Maryland
Introducing mental math, mathematical properties, number patterns, representation of objects or numbers and generalization of the concept at an early age/grade.
Cathy Seeley:
These are important components of algebraic thinking. Mental math is tremendously important and helps students predict answers in thinking ahead about problems they are tackling. Patterns help students develop generalizations, which lie at the heart of thinking algebraically. And we now know that being able to represent situations in many ways is not only an important ability, but can help students deepen their understanding of the situation and develop mathematical ideas at the same time.
Question from
San Jose, California
It seems to me that schools are pushing the expectations for traditional Algebra I classes earlier and earlier. I have noticed children are having a harder time grasping concepts, and I feel that this is due partly to their cognitive immaturity. At what age is it cognitively appropriate to introduce these concepts?
Cathy Seeley:
This is an important and timely question. The issue of accelerating students into Algebra I earlier and earlier brings several problems, not the least of which is cognitive maturity. For example, what happens to the critical content of the middle grades, in particular proportional reasoning and increasingly sophisticated ways of dealing with data and statistics? I worry that when we accelerate students into algebra too soon, they may miss this. Also, unless we have good options at the eleventh and twelfth grades, why are we accelerating the students? In addition, if the student is not highly motivated to continue high school study through every year in high school, considering calculus, or possibly another advanced course, what is the benefit of accelerating the student? Furthermore, unless prevented by the school or district, some or even many of these students may stop their mathematics study before twelfth grade, which is a disaster for any student going to college. Finally, I think there may be questions of at what age students are developmentally ready to deal with the level of abstraction called for in a formal algebra course. While this may vary from student to student, we need to seriously question the value of pushing algebra ever further down into the middle school and elementary school. Rather, the NCTM focus on algebraic thinking gives us many ways to incorporate age-appropriate ideas that incorporate algebraic thinking from preschool on.
Question from
Dallas, Texas
I have taught math from grade 6 through AP Calculus. In my opinion, algebraic thinking should be taught at the lower grades with the use of manipulatives, such as use of the hands-on algebra that uses scales and colored pawns. Many of the students in high school still lack a sense of what equality means and hence are quick to violate it. Also, it would help a great deal if properties were introduced in the lower grades with their correct names. I have been accused of personally inventing the distributive property by a Pre-AP Geometry student who had never heard of it by the 9th grade, and of being a fool for not knowing what the "popcorn rule" is, as in her words, even the 4th grade teachers know what that is.
Further, a student who can spell Mississippi and knows its capital should also be able to state "subtract 5 from both sides" instead of talking about arbitrarily "moving it to the other side." Vertical consistency in notation beginning at the early grades would help a great deal also. Everyone who has taught Algebra knows the problems that come up with the use of "slashy fractions," that is, when 1/2 evolves into the ambiguous 1/2x or worse 1/2x + 3.
Another thing that would help is in the area of reduction of fractions and factoring. If the lower grades learned to factor the GCF from the numerator and denominator then cancel, their students would not be as confused about canceling binomial terms in a rational expression involving polynomials.
I think NCTM can help by promoting vertical team meetings that involve all levels and grades, not just the MS/HS Pre-AP/AP teachers.
Cathy Seeley:
Your comments address a range of issues. I'll start with the last: I think meeting across grade levels is one of the most important things that can happen within a school system. Such meetings are most constructive when all involved both share with and learn from each other. Often, I have found that secondary teachers benefit from seeing some of the powerful mathematical ideas addressed before they see students. And all teachers benefit from seeing what content their students may later deal with in school. I think the most important content directions we can give elementary teachers is to teach for a balance of understanding, skills, and applications and to do whatever is necessary to actively engage students in their own learning. I also think that we can do more toward maintaining mathematical precision in terms of definitions and language, but only if terms are attached to a sound understanding of what they represent. And by the way, I have never heard of the "popcorn rule" either.
Question from
Oakland, California
I wonder if you agree with trying to level the playing field by publicizing the resources available at zero or nominal cost over the Internet.
I am speaking of Web sites like AOL@School, Hotmath.com, Quickmath.com, etc. There are many others. These can relieve students of math anxiety when it comes to absorbing new concepts in algebra.
If teachers incorporate them into homework assignments, then more students might be able to keep up in class and "get it" more easily. Especially those without math help at home or tutors.
Thank you!
Cathy Seeley:
I think the best way to level the playing field is to provide all students with the opportunity to be actively engaged in learning mathematics that will serve them in the future. In looking at a couple of such Web sites, it appears that they provide help on doing homework exercises from common textbooks or answering questions. This kind of help may be quite useful for some purposes, especially if there is limited assistance at home. However, I also hope our mathematics teaching extends beyond this kind of assistance toward the rich classroom experiences that help students learn mathematics deeply in a way that will stay with them through their future mathematics study.
Question from Auckland, New Zealand
New Zealand has had a strand of Algebra in the mathematics curriculum since 1992, for students from age 5 up. This strand emphasizes algebra as relationships. For the first few years the equal sign and > and < are part of that focus. There is a continuing strand on patterning.
More recently a National Numeracy Project has been introduced for ages 5–14. This includes dealing with part-whole relationships in numbers for ease in mental calculation. For example 19 + 7 is done more easily as 20 + 6. We hold that the thinking behind this, which differs for different numbers and different operations, constitutes algebraic thinking.
My colleague Murray Britt and I have been doing research on the extent to which students can generalize from such examples to the correct use of algebraic notation to express this relationship as a variable.
Anyone interested in learning more about this can see Chapter 5 in my report available on
professional/2003Y7_9NPReport.pdf.
For a description of the algebraic thinking involved, write to me and I will send you a copy of a paper in press with Educational Studies in Mathematics.
[email protected]
Cathy Seeley:
Thanks for sharing this resource. I think the American mathematics curriculum can benefit a lot when we examine what is done outside the United States. I particularly like your early emphasis on the important ideas of equivalence and relationships. These are indeed central ideas to the development of algebraic thinking.
Question from
Archbald, Pennsylvania
Do you feel that calculators should be used on a daily basis in middle school math classes, specifically grades 7 and 8?
Cathy Seeley:
I think calculators should be available for students to use at these grades, understanding that the teacher helps students decide when and how to use them. It is critical that students learn to make these decisions. Teachers can help by identifying when calculators should not be used ("Put your calculators away; we're going to do some mental math.") and when they can be helpful (as in solving complex problems). We also need to make sure that we capitalize on the availability of calculators by giving students challenging problems that go beyond the limitations of what we can do with students when they do not have such access. Simply giving students long lists of computational exercises and then giving them calculators to do them defeats the purpose of having this tool available.
Question from
Oak Park, Illinois
This is not terribly profound, but it is a part of the puzzle of developing algebraic thinking preK–12: Students need lots of experiences in varied real-world contexts of creating data tables (especially 2-variable T-charts or T-tables) and thoroughly examining the patterns in the data. Then they need to graph the data and again thoroughly analyze those patterns. Then they should relate the patterns in the graph to the patterns they saw in the table.
I have done many such activities with students of varying backgrounds in grades 3 through 8. With those students who are ready, we examine the patterns again and use the two variables to create expressions and then equations.
Cathy Seeley:
This is a wonderful summary of the power of learning how to represent situations in multiple ways. It provides students with the opportunity to see algebra as the study of patterns and relationships, which sets the stage beautifully for their increasingly sophisticated development of algebraic thinking and, eventually, symbolic procedures.
Question from
Chicago
There is a challenge with students coming into high school and understanding very few concepts of algebra. There is a program in CPS to have students who scored lower than 50 percent on the Iowa Basic Skills test to take a double Algebra course as 9th graders. Part of this is Algebra Problem Solving (IMP or Mathscape). These programs do not seem to emphasize any real algebra equation work that the future tests call for. Are problem solving curricula really beneficial if the goal is to pass tests that requires students to do equations quickly?
Cathy Seeley:
Problem-solving curricula are absolutely essential to prepare our students well for success in algebra and the courses that follow it. However, the flip side is not true. Teaching only equation-solving skills without adequate attention to understanding and problem solving is short-sighted and likely to backfire on students as they move deeper into the secondary mathematics program. Of course, balance is the key, and an ideal algebra program will include not only problem solving, but developing conceptual understanding and skills development as well.
Question from
Ft. Lauderdale, Florida
The first order of business is to ensure that the K–5 teachers have enough content knowledge to prepare the youngsters. Content knowledge is lacking in most K–5 teachers. Mathematics has evolved over the last decade, however, some teachers are still teaching for the industrial era. They believe that computation is the basis of elementary education. Some elementary teachers are not comfortable teaching mathematics. NCTM could offer online courses to help elementary teachers acquire the needed content knowledge with which to help the K–5 students develop a strong solid base on which middle and high school teachers can build.
Cathy Seeley:
Teaching algebraic thinking beginning at the elementary level presents increased demands on elementary teachers' understanding of mathematics at a deep level. Professional development is a priority for NCTM. Online courses offer great potential for delivering professional development to teachers who might not otherwise have access to it.
Question from
Fairburn, Georgia
Many teachers don't know what "algebraic thinking" looks like in the elementary classroom. If they did, they would feel more comfortable when they are told to include it in their curriculum.
Cathy Seeley:
This is another reason why professional development plays such an important role. Teachers of mathematics at all levels, not just elementary, need to make a lifelong commitment to their professional growth. For elementary teachers, experiencing the kind of algebra that is conceptual and relevant can be a liberating experience, not to mention the benefits for guiding their students' learning.
Question from
Marion, New York
We can help elementary students look for patterns through the combined creation of tables, graphs, and equations. Even young children can extend patterns to create tables, learn to graph the results, and describe the pattern of the table and the graph in words. Students can think and question critically in an algebraic context when they are given the opportunity to extend a problem through patterns and see it through a visual model. For example: My third graders create tables and graph the story of the Gingerbread Man. If the Gingerbread Man begins to run as soon as the oven is opened, and runs 2 feet per second, the wife begins to run 2 seconds after the oven is opened and she runs 2 feet per second, the wolf begins to run 5 seconds after the oven is opened and he runs 4 ft per second does the Gingerbread Man ever get eaten? They pull amazing stuff off the tables and graphs.
Cathy Seeley:
This is a nice way to develop informal ideas of algebra. Thanks for sharing your example!
Question from
Tempe, Arizona
People who share these views are usually "burned at the stake," but several years of my 39 years of teaching math I taught with the Saxon series. Students liked it and did better on college entrance exams! It was an incremental development and geometry was integrated throughout. Students could go at their own pace. Independent study was much easier. It de-emphasizes the sacred role of the teacher. Students wanted their own copy of the text to use in college. (Most students would like to burn their own copy of the math text they had to use!)
And yet it was banned from use in some areas because it didn't have colored pictures and it supposedly did follow the standards. If students do better in college because of it and like it much better, then maybe the standards need to be revised or replaced.
Cathy Seeley:
Thanks for your honest and direct comments. You are correct that the Saxon text has not been adopted in many states and systems. Teachers tend to have strong feelings either for or against the program, and, frankly, results are rather mixed. I think that if students are to take responsibility for their own learning and become independent algebraic/mathematical thinkers, they need to come to rely more on themselves than either the textbook or the teacher. I continue to believe in the importance of the teacher, not for telling students things, but for structuring a classroom where learning is likely to happen. A textbook is only a tool, and it cannot address the wide range of problems students will encounter. It may help students deal with certain types of problems, and that's great. My preferred model of teaching would have students engaged in a variety of activities, including a good dose of small-group work on solving engaging problems that draw students into figuring out how they will use mathematics to find solutions. The role of the teacher is critical not only in structuring and facilitating their work, but in asking good questions that push students' thinking farther than they thought they could go. One of the strengths of NCTM as an organization is the diversity of views of its members. When we openly discuss our different points of view, we can all become better educators.
Question from San Francisco
I am a Math Learning Specialist. I help learning-disabled students really understand math. Why do so few math teachers understand the ways in which various learning disabilities hamper students' abilities to master math?
Cathy Seeley:
Unfortunately, many teacher education programs are limited in the amount of time they can spend learning about special needs students of all kinds. One of the roles you can play is to connect classroom teachers with resources, including yourself, on how they can better meet the needs of their students. It's great to have someone in a resource role who understands both mathematics (at a very deep level) and also special needs students.
Question from
Centreville, Virginia
I love this topic! I am constantly trying to find ways to incorporate algebraic thinking in my classroom both within formal mathematics discussions and in the thinking that is essential when students are solving practical applications of algebraic concepts to real life. I think that weaving the strands throughout mathematics is very important in grade school and middle school. I do, however, support the structure of separate courses that exist in many of the schools throughout the United States. This approach allows students to concentrate their attention on one strand so that the topic can be studied in depth. I taught mathematics for 15 years in California, and the schools in which I taught had this form of "traditional mathematics." However, I tutored students who were attending schools that had adopted "integrated mathematics." Their understanding of mathematics was shallow and they felt scattered. One student in particular, who was actually naturally very talented in mathematics, was very frustrated. Her conclusion was that she was just dumb and unable to understand math. She gave up somewhere in her sophomore year and couldn't wait to unload that course. I kept trying to assure her that she was actually very talented in mathematics, but her test and quiz scores were lower than she wanted and only furthered her self-assessment in this field. I was very sad for her. Her brother, on the other hand, went to a different school that had a traditional approach. He was also talented in mathematics but was lazier than his sister. He still excelled in the subject and continues to be confident that he is good in mathematics. What was sad for me was that both students were extremely capable, but only one believed it. I attribute that, in part, to the way they experienced mathematics in the classroom. The girl's understanding was a mile wide but an inch deep. The boy's was narrower, but he had a deep understanding of algebra and algebraic thinking that allowed him to solve complex problems and feel successful in mathematics. The breadth of his knowledge would undoubtedly grow as he took more courses.
These are not the only students with whom I have had this experience. I have actually tutored many students in both integrated mathematics courses and in traditional courses and have seen the same results across the board. It is, to me, striking.
I actually had many students come to our high school having been in an integrated program with a similar experience. They had very little exposure to in-depth algebraic thinking and a smattering of knowledge about different strands of mathematics. They generally demonstrated an inadequate knowledge of Algebra when tested for placement in Geometry and would be enrolled in Algebra I for the school year.
Perhaps with an excellent textbook and an excellent instructor, integrated mathematics would be able to allow ALL students to be successful. I don't know. Certainly there will always be those brilliant students who could learn mathematics on their own without regard to the methodology implemented. What I have seen generally, though, is that otherwise talented students feel frustrated and inadequate. It seems like a well-intentioned program is failing them in this country.
Cathy Seeley:
Thanks for your participation in this chat. You raise an important issue. I honestly think that we don't have adequate information to determine whether an integrated program works better than a traditional one at the secondary level. The rest of the world uses such an approach. Perhaps one of the issues is getting clarity about what outcomes we want to see. If we are evaluating students on traditional equation-solving skills, then it may well be that students coming from a non-traditional program might not perform the skills at the same level at the same time as more traditionally prepared students. However, if we evaluate the ability to use algebra, for example, to solve diverse problems, I think we would all agree that many of our students over the years have not understood how to apply the skills they learned to the problems they encountered. I found this over and over again in teaching and tutoring students at this level. And this was the case when algebra was taught only to the more successful students. If we are now to help more/all students learn algebra and higher-level mathematics, I think we must be open to different ways of teaching and even different priorities in terms of the content we address and the organization of the curriculum.
Question from
West Liberty, Ohio
I've long been a believer in an integrated approach to mathematics, combining algebra, geometry, statistics, and data analysis. We have not taught a formal geometry class at my high school for at least the past 25 years. The state of Ohio seems to be moving in this direction also with the state standards, indicators, and the Ohio Graduation Test.
Cathy Seeley:
There continue to be success stories where school systems, and now states, may be moving in this direction which is widely used outside of the United States.
Question from
Oakland, Maine
I think there is a misunderstanding of what "algebra" is with teachers, students, and parents. Many teachers are reluctant to use the term algebra in the early development of mathematical thinking, and children begin to think of algebra as a monster that is too scary to conquer. Parents are always willing to jump on the "My child needs to be taking algebra" bandwagon, without looking at the development of the child's mathematical thinking and realizing the background is rich with algebra.
I feel teachers need to use terminology and make connections to algebra at an earlier stage of a student's mathematical development. If this happens kids will begin to work with concepts more fluidly, and parents will realize that algebra is not a separate subject but part of understanding of what we call "math."
Cathy Seeley:
It would be great if students didn't think of their mathematics experience as a set of isolated topics. That's just the goal of incorporating algebraic thinking as a strand within a balanced mathematics program PK-12.
Question from
Towson, Maryland
What can we do about the biggest textbook publishing houses that continue to produce and peddle those books? How do we "push" school districts to consider alternative programs such as IMP or CORE or Connected Mathematics?
Cathy Seeley:
I think we may be focusing on the wrong battle. If we pay attention to what mathematics we want to teach and help teachers grow through professional development that lets them learn how to engage students in learning, then I trust teachers to demand tools that support that learning. Publishing is all about supply and demand.
Question from
Culpeper, Virginia
The current math standards of many of our states create barriers to offering integrated secondary math courses. For example, in Virginia, we were beginning to offer math this way, but in most cases have backed away, since our state standards are fairly traditional and require tests in Algebra I, Geometry, and Algebra II, each separately.
Cathy Seeley:
Our accountability systems do influence what we teach. But I would argue that if we teach a rich, balanced problem-solving-focused program that includes skills, concepts and applications, our students can do fine, even on low-level skills-based tests. (See our chat from last month on accountability)
Question from Jefferson, Ohio
Our guidance counselor says that the colleges and universities are not on the same page. They want to know whether a student has had Algebra II, Trig, Calculus, etc. How can you change everyone's thinking?
Cathy Seeley:
The state of New York, for one, has offered integrated mathematics for many years with no negative impact on students entering college. I have found guidance counselors generally quite open to meeting the needs of students, but often uninformed about how to do that in mathematics. As mathematics educators, we have a responsibility to work together with counselors, providing accurate information from the universities and colleges most often selected by your students. It may be appropriate to engage in conversation with mathematics faculty from these post-secondary institutions to get a clear message of what will work. If there are, in fact, barriers at that level, solutions will start by connecting and communicating.
Question from
Princeton, New Jersey
How can assessment be modified to encourage the integration of topics and early attention to algebraic thinking? Do assessment strands such as Numbers, Measurement, Geometry, Data, and Algebra help or hinder?
Cathy Seeley:
I think the strands do not in and of themselves hinder learning. I will return to the notion that if we teach a good, rich, balanced mathematics program, our students can do fine on pretty much any test they encounter.
Question from:
36. Taichung, Taiwan
I'm currently teaching in an American school in Taiwan and we still use the Algebra I, Geometry, and Algebra II courses. My own children, however, went to school in Arizona where those courses were integrated. My son seemed to pick up on the various concepts great and had a great score on his SAT while my daughter, on the other hand, struggles to understand math and had a very hard time with the SAT test. She reviewed for the test quite a bit and was unable to have any concrete knowledge or comprehension of the algebra concepts that were still Algebra I concepts but the more advanced ones.
My first year here, I taught the middle school math where the math is integrated, streaming up from elementary as well as the 8th graders that were placed in Algebra I. Having done both at the same time, I did note that it was very beneficial for my 8th graders to learn the algebra concepts—although they had a hard time understanding them—and then see them all the way throughout the book for the rest of the year. I continually heard comments such as, "Oh, I get it now. I never understood that before." It was easier to teach Algebra as a coherent class and much less time consuming in preparation. I also found that the more advanced 8th graders who were in the Algebra I class did fine on the formula memorizing and plugging in numbers, but they had a difficult time with problem solving and the pace we needed to keep in that class. And our block program didn't allow us to delve into the problem solving very much. I've moved into the 5th grade classroom this year, and they were just introduced to the concept o =
"n." You should have heard my room; you'd have thought Martians had landed. Once they calmed down, they could do the math, they just had to learn to rearrange the equation so they'd be ready later on.
Previously, I taught kindergarten. They were quite open to problem solving because they didn't have any preconceived notions of how things had to work. I had my kids really doing multiplication and division before the year was over as well as carrying and borrowing, but they weren't afraid of it because it was encased in games.
I've noticed that we tend to want the kids to know more per grade level as the years progress so we can do more with them in the later years, but if they don't spend enough time with some very concrete things at a young age, they have no real concept of mathematics for the more abstract things. Unfortunately, we are in a time crunch that makes it very difficult to spend the time with the concrete manipulative—the way I would like to at least.
Cathy Seeley:
Students learn in different ways, and there are many variables that affect their learning. As teachers, we must continue to offer a variety of learning experiences and give students opportunities to deal with the mathematics themselves, not just watching the teacher present the "rule du jour." I agree with you that our curriculum is still too crowded. We need to take the time to talk with teachers above and below our grade level to identify where we should spend more time and where we can spend less. We certainly need to free ourselves from the belief that we have to teach every page, or even every chapter in a textbook. Developing lasting understanding, as you observe, takes time. And I would argue that this time is not only worth it, but necessary.
Question from
Vilas, North Carolina
What kinds of staff development is offered to help teachers gain a deeper understanding of the content in algebra? Where can teachers find extensions/projects to differentiate instruction in the algebra strands in all grades?
Cathy Seeley:
I would refer you to your local and regional educational centers, as well as the universities in your state. North Carolina has done some nice work in assessment and mathematics over the years. In terms of NCTM, check out the Algebraic Thinking Focus on the Web site. Look for the magnifying glass icon, and you will discover a wealth of resources. The Navigations and Illuminations examples are among many others.
Question from
Naperville, Illinois
There is increasing pressure for all students at the eighth grade to take algebra. I put the emphasis on all, because there appears to be little or no concern on ability or prep for the course. It is just implied that students do better on tests when they have had algebra. The concern tends to be with math test scores and not with the content or subject matter that should be presented to the students at any level. What are your thoughts on this aspect of the mathematics curriculum?
Cathy Seeley:
I think we must be careful if we move all students into a formal algebra course. Whether we do this or offer a rich middle school curriculum that develops a strong base of algebraic thinking, we should not shortcut the development that leads to symbolic understanding. For example, one of the most important concepts that leads to success in algebra, in my opinion, is a solid understanding of proportionality well beyond the simple study of ratios, proportions, and percent. This critical connecting concept needs to be well developed, and often, when the middle school curriculum is compressed, students do not develop this understanding. I have shared in other responses my concerns about what happens to these students in high school. The main thing is to prepare all students well for success in four years of academic mathematics so that they can succeed in college.
Question from
Ridgefield, New Jersey
What are your thoughts on how much algebra should be taught at the middle school level and also being from an urban area where many students are low level? How do we catch them up?
Cathy Seeley:
There is increasing evidence that many students thought to be low-level, especially when they come from a background of poverty, are in fact victims of lack of access to educational opportunities. The best thing we can do is to hook them into engaging tasks where they get to show us what they can do. I have found in situations like this that we can often discover new stars among students that nobody thought could succeed.
Question from
Mobile, Alabama
I have children that are very weak in fundamental skills such as facts and procedures. As we all are aware, high-stakes state tests are trying to push me to objectives that I feel won't be fully understood if that groundwork is not laid down properly. When administration comes in and sees a lot of broad framework and understanding being done, they get nervous. They do not see the DIRECT OBJECTIVES being covered. An incentive bonus at the end of the year is at stake for test grades for all the teachers here. After the long-winded introduction—Is there a point at which you can feel safe that the students will be motivated to go back and work on facts and operations while you're pushing ahead to stuff—like simplifying fractions for instance—or do you feel that something that low level has to be tackled before moving on? I read the Crystal story, but this is more elementary.
Cathy Seeley:
This is a real and challenging problem. First, I continue to believe we must teach a mathematics program that has integrity. We cannot be pulled down to teaching in ways that we think do not serve students well for their future. If we teach a sound mathematics program, grounded in understanding, driven by challenging and engaging problems and inclusive of computational development, our students will do fine on the tests, even low-level tests. And do not underestimate the potential of a calculator to allow students access to problems that would otherwise be beyond their skill level. If motivated, students can learn these higher-level skills. A comment Crystal made could be made by an elementary student with this kind of success as well—when the opportunity arose for her to learn fraction operations AFTER succeeding in algebra, she observed: "The time has come; I'm wasting too much time using my calculator to do fractions." I'm not suggesting giving up and handing out calculators for all computation. But we must allow students the opportunity to solve problems. The biggest difference in test performance among groups of high-performing and low-performing students is not found in low-level facts and procedures. Rather, the differences show up in the more complex problem-solving situations that many students have never experienced.
Question from
North Chicago area, Illinois
I think one of the major issues we are faced with as a 'Nation at Risk' is how to present mathematics problems in a way that encourages students (not discourages them) to think and interpret what is being asked for themselves—and then validate their non-linear thinking. Too often, teachers approach a problem with a particular agenda and do not encourage creative solutions to problems. Here's an example of what I mean:
In a 3rd grade lesson about place value, the opening question for individual reflection looks like this "With the follow numbers, what is the largest number that can be made and what is the smallest? (given 8, 5, 3, 0, 1, 9) What teachers sometimes do is put so many constraints on problems that students are encouraged not to think creatively? (The teacher WANTS 985,310 and 103,589 so she/he adds statements to the original – carefully crafted – question, like, since there are 6 digits you need a 6-digit number in your answer … whereas students might come up with all sorts of other creative notions (exponents or decimals or mathematical symbols).
Or
Talking about patterns 2,000 1,200 800 600 500 ___ ___ and having a student describe how we divide the difference by 2 each time (which is clever but since the teacher has a particular agenda, the answer is not accepted and dismissed in search of the 'correct' answer that was being sought (1/2 of the difference is being subtracted). Both are correct and furthermore making the connection between should be celebrated and emphasized!!!
I think this does a disservice to students' development of mathematical thinking. I don't think there is ever a moment where a teacher should not take every opportunity to point out connections between concepts. I am keenly aware that not all teachers of math are acutely aware of all these connections but shouldn't that be a priority of ours to help make meaningful connections and/or set up meaningful connections in the years to come? Do you have any thoughts on the relative importance of this notion?
Cathy Seeley:
Increasingly, I believe that the role of the teacher is to ask good questions, not tell students answers (or even hints). Far better than minute coaching are questions like: How do you know? Why do you think so? Would that still be true if you had twice as many? And so on. I think you are correct that students need opportunities to express their developing algebraic/mathematical thinking, even if that goes in a different direction than the teacher expected. The key is for the teacher to help the student connect what the student has done with the appropriate mathematics being used so that the student has the possibility of using what he or she has learned in another situation.
Question from
Savannah, Georgia
I teach honors Algebra II and a couple different levels of precalculus. Though I have the students use their calculator often in class to enhance their understanding of many different concepts, I still mix in non-calculator quizzes and portions of tests that require good computational skills and a command of operations on fractions. Is this overkill or a necessity for students who should be preparing for calculus courses and need a strong algebraic foundation?
Cathy Seeley:
I think it is helpful to reinforce skills previously taught, and I definitely think some non-calculator work is appropriate, especially for mental math. However, I also think we need to put in perspective the level of computational proficiency that is actually necessary for success in higher-level mathematics. It may not be a useful way to spend time in algebra II to have students do pencil-and-paper long division, for example, when they are unlikely to need this algorithm in the future, and when there are more important concepts they need to practice. But the teacher can choose how to balance the program so that students' needs are best addressed.
Question from
Hopkinsville, Kentucky
Has the NCTM studied the types of courses and curriculum used in England, Spain, or even South American countries. I often had foreign students, and their problem solving skills left my students in the dust.
Cathy Seeley:
There is mixed evidence about problem-solving expertise in other countries. One of the complaints of some international mathematics educators is that their students are better at skills than at creatively solving problems. However, there are definitely some interesting programs in problem solving, especially from England and the Netherlands. (I'll have to check out Spain and South America a bit more…) In the Unite States, our most traditional skills-based secondary programs probably did leave many students ill equipped to handle more complex problems. Consequently, many of the newest mathematics curriculum programs have incorporated the more innovative aspects of some of these programs in their focus on problem solving.
Question from
Eldersburg, Maryland
I am struggling with helping teachers understand the importance of teaching alternative algorithms as a way of facilitating an understanding of algebra. In order to be able to manipulate an algebraic equation, very young children can think about and discover many ways to solve a regrouping with addition problem or subtraction, multiplication, and division. They learn to think about pure computation in a multitude of ways. How can we help elementary teachers see this as a valuable part of their instruction?
Cathy Seeley:
I think that seeing other approaches to solving problems is valuable, as is experiencing other ways to perform computational skills. Other countries often use different computational algorithms than we do. Among other benefits, students can see that mathematics is not magic, but rather, efficient ways to deal with numbers and problem situations. Professional development can help teachers themselves experience alternative algorithms, which can provide them with insights into student thinking.
Question from
Saint George, South Carolina
How do you get students out of a find-the-answer mode and into understanding that problems can have multiple solutions?
Cathy Seeley:
We have conditioned students through our teaching techniques to look for quick answers. The best way to combat this kind of student thinking is by teaching differently. When we engage students in interactive solving of problems that may not easily be solved and/or that may have more than one possible correct solution, over time they can learn to trust their own thinking, rather than trying to guess which rule to use.
Question from
Cleveland
How should algebraic thinking best be developed for middle school students? Many people believe that a traditional algebra course, focused on rules and manipulations, should simply be taught earlier. Shouldn't middle school algebra look different than that?
Cathy Seeley:
YES!!! Middle school mathematics should be a time of rich exploration of new ways of thinking. It should also be a time of powerful connections between elementary work with numbers and operations leading to symbolic awareness and the ability to make generalizations. The most important connecting idea at this level, in my opinion, is the development of proportional reasoning. If this is done well, students can see how quantities can grow proportionally, leading naturally into the study of linear relationships. Whether we offer algebra as a course in middle school, or whether we develop algebraic thinking that sets students up for success in an algebra course later, middle school mathematics needs to be a rich, balanced program, with an emphasis on representing and solving problems.
Question from
Macomb, Illinois
Can you say some more about "early algebra," especially the need to think in abstractions and understand the varied meaning of the equal sign? For example, in the 70s the program Developing Mathematical Processes had young children working with equations such as W + D = B + C. Perhaps we should try to work with these ideas even before the push to use numbers and counting to verify answers.
Cathy Seeley:
There are many ways for students to explore patterns at an early age, especially patterns dealing with equivalence, such as the one you suggest. Students can explore equivalence with balance scales (I believe these were an important tool in the DMP program; they used a nifty two-piece blue plastic balance that was nearly indestructible), representing what they find with pictures and, possibly, basic symbols. The key idea is that this kind of mathematical exploration can plant important seeds that can develop into algebraic understanding if nurtured well.
Question from
Reston, Virginia
Robert Gagne offered a hierarchy of learning that progressed from verbal skills through problem solving - with intermediate steps such as concept learning, analysis, synthesis and principle learning in between. Similarly, wouldn't we do well to pursue a generally acceptable definition of algebraic thinking? The more precisely we can define our objectives the better our chances of developing strategies for achieving them.
Cathy Seeley:
It is often helpful to rely on a theory of learning that lays out developmental or instructional stages. In the case of algebraic thinking, there have been several articles and other publications that address this development. One excellent source is Principles and Standards for School Mathematics. Other NCTM resources can be found through the NCTM Web site, especially through the articles and links included as part of the Algebraic Thinking Focus (look for the magnifying-glass icon).
Question from Fairfield, Connecticut
At what grade level can you start to involve students in critical thinking?
Cathy Seeley:
If you ask my friend, Denise, an outstanding kindergarten teacher, she would tell you that you can involve students of any age in appropriate critical thinking. Asking students to justify their thinking and make predictions are just a couple of ways we can do that.
Question from
Lexington, Kentucky
You wrote, "The rest of the world, including our colleagues in Canada, teach mathematics, not as separate courses, but as a continuous program from elementary through secondary school. In the United States, some schools offer an alternative, such as an integrated program that incorporates algebra as a strand blended with geometry and other advanced topics." I would be interested in knowing about comparisons of the "successes" of such approaches in developing algebraic reasoning and conceptual understanding.
Cathy Seeley:
In terms of international comparisons, the TIMSS studies are the best source of information. For the integrated secondary programs in the United States supported by the National Science Foundation, you can visit the Web site for the dissemination of information about these five programs at: The biggest challenge in answering your question is that we don't have widely accepted measures for algebraic reasoning and conceptual understanding to use in comparing various approaches.
Question from
Acton, Indiana
As we strive to achieve the goals of "No Child Left Behind" and raising standards/expectations of our students in the middle school environment, it seems that these two ideas do not always mesh very well. Many schools have either increased the number of students enrolled in Algebra I courses in the middle grades and/or have begun teaching Algebra I and geometry courses. Unfortunately, some of these courses have led to algorithmic teaching of basic algebra skills instead of an understanding of a variety of mathematical concepts such as those outlined in the Principles and Standards for School Mathematics.
Would many students in the middle grades be better served if an emphasis were placed on developing understanding a wide range of concepts and standards instead of offering a traditional algebra course?
There area several factors that influence the necessity to offer an algebra course and/or a geometry course in the middle grades, including demands from the local school board and parents as well as restrictions of coursework due to limitations of textbooks. Should there be another course offering developed for the middle grades, which would include the development of student understanding in the areas of algebra, geometry, data analysis, number sense, and measurement?
Cathy Seeley:
This seems like an important balance for all students to experience during the middle school years. Even if we teach an algebra course, I would hope that these elements would precede the course, and also would be incorporated appropriately in the algebra course itself.
Acton, Indiana
What is the best practice to identify students that would be best served by a traditional algebra course?
Cathy Seeley:
Increasingly, I am coming to question whether any students are well served by a traditional algebra course, if by that you mean a course focused on learning abstract rules and procedures. Even students who have been successful in such courses in the past could well have benefited from experiences with graphical and other representations, as well as focusing on using algebra to solve complex problems, rather than primarily solving certain types of story problems.
Question from
Besancon, France
What are the main weaknesses that have been identified in student achievement in the algebra field, and at which levels? What does NCTM suggest to try to overcome these problems?
Cathy Seeley:
For many years, students who study algebra have had difficulty applying what they learn to solve problems. Teachers have searched for ways to 'teach story problems.' As we try to teach algebra to more students, this problem is exacerbated, especially in traditional algebra classrooms focused on abstract rules and procedures. Principles and Standards for School Mathematics calls for a much richer vision of algebra, building on the use of functions to represent and analyze relationships. In such a program, many or all students can learn a range of problem-solving approaches that serve them well regardless of the type of problem.
Question from
Sigourney, Iowa
I use an integrated curriculum and use a lot of explorations. The students do very well with it, but I am having difficulties getting that knowledge to transfer to textbook problems. Any ideas about how to ease this transfer?
Cathy Seeley:
If your students are successful in a good integrated program, they are likely learning important knowledge and skills. In terms of transferring to textbook problems, I'm not sure which types of problems you mean. If you mean routine equations to solve, this is an important skill that may call for more practice than they have had, connected to what they know about how equations help in solving problems. If you mean typical types of word problems (coins, age, etc.), you will need to decide how much time to spend on such problems based on your curriculum and whatever test is used in your accountability system. Otherwise, it may be reasonable to question to what extent traditional textbook problems are useful. But I continue to believe that a strong program that balances skills, conceptual understanding, and problem solving is the best preparation we can give students for whatever types of tests they face.
Question from
Accra, Ghana
What will be Algebra III, Algebra IV, Algebra V, etc.? I think algebraic concepts pervade the mathematics curriculum from K to university. So segmentation might not be useful. Rather, we must find ways to teach algebraic concepts appropriately to different groups of students and help them to make sense of those concepts as they relate to the students' daily lives. For example, commutative law remains commutative at all levels K to university. How can we teach it to K students so that they can recognize it in terms of a + b = b + a at the middle school level? Can they also apply the concept in real-life situations? I think these are some of the fundamental issues to be concerned with.
Greetings from Ghana.
Cathy Seeley:
Hello to Ghana. I agree with you that segmenting the curriculum in this way does not serve students well. There are many ways to connect mathematics, including algebra, to students' lives both in and outside of school. Many elementary teachers give students experiences dealing with commutativity as a concept, even if they may not give it a name. Developing many properties and characteristics of equivalence is definitely worth the investment at the elementary level.
Question from
Springfield, Missouri
When parents ask, how do we respond to the question: What is so important about algebra anyway?
Cathy Seeley:
Algebra has long been a gatekeeper, and the evidence is solid that it is critical as a first step toward college. Since many students don't know themselves whether they are likely to attend college, we need to start them along this path. But this in itself is not enough to drive us to call for algebra for all students. Algebra, as described by today's vision in Principles and Standards for School Mathematics, is a powerful set of tools that helps students represent relationships, make generalizations, analyze situations, make predictions, and, most importantly, solve many kinds of problems in mathematics, other disciplines, and life outside of school. Today we know how to connect algebra to engaging and useful situations. Invite parents to experience this kind of problem experience in your classroom (ideally) or through materials you send home.
Question from
Kingfield, Maine
As a high school mathematics teacher, I see many students that come in to my classroom without basic number sense. Would you think that before we can work on algebraic thinking, we need to work on number sense first, or do you think they can (or even should), be emphasized simultaneously?
Along those lines, it would seem that part of the answer to that question depends on the resources of the school and district itself. If developing algebraic thinking at the elementary and middle grades requires the purchase of manipulatives and other items not normally in a math class budget, then it would seem to be prohibitive for smaller and more rural districts.
Thoughts?
Cathy Seeley:
Absolutely, number sense is critical as a primary goal of elementary mathematics. I believe that incorporating algebraic thinking can support this number sense. For example, as we develop the idea of equivalence, students can learn multiple ways to represent a number. As students explore patterns with numbers, they are laying the foundation for more sophisticated algebraic patterns later. These goals can and should be connected. I might argue a bit with your statement that manipulatives and other tools might not normally be in a math class budget. These are as essential to the teaching of mathematics as maps and globes to the teaching of social studies or science equipment to the teaching of science. There are ways to economize, but basic counting, place-value and geometric manipulatives, to name a few, should be a priority even in small schools.
Question from
Indianapolis, Indiana
Do you have some data indicating how parents can support their children in fostering algebraic thinking?
Cathy Seeley:
There are many ways for parents to help their students developing algebraic skills. Asking students to notice patterns, formulate generalizations and make predictions is a great way to do this throughout elementary and middle school. This can start with simple situations like the sequence of lights in a traffic light and can extend to noticing patterns of house numbers. Asking students to give many names for a number and talking about equivalence can also help. Having students justify their thinking any time they are working with a mathematical situation is helpful. Questions like "How do you know?" "Why do you think so?" "What else can you say about the situation?" can also be useful.
Question from
Naperville, Illinois
Mathematics programs: Where does one find an unbiased evaluation of mathematic programs? That is, one that looks at the aspects of the curriculum and gives an honest refection of how the programs meet the goals and objectives of NCTM.
Cathy Seeley:
This is the question of the hour (and day and year). Gathering and packaging this type of information is a priority of many efforts right now, both within and outside of NCTM. Recent efforts by the U.S. Department of Education and the Mathematical Sciences Education Board have told us that there simply is not enough information to make such judgments at this point. Individual programs are often accompanied by related data, but it is important to look at studies across districts and schools. Instead, we can look at recommendations such as those presented in Principles and Standards for School Mathematics that are based on more focused studies of effective practices | 677.169 | 1 |
Problem Solving Approach to Mathematics - With CD - 10th edition
Summary: The new edition of this best-selling text includes a new focus on active and collaborative learning, while maintaining its emphasis on developing skills and concepts. With a wealth of pedagogical tools, as well as relevant discussions of standard curricula and assessments, this book will be a valuable textbook and reference for future teachers. With this revision, two new chapters are included to address the needs of future middle school teachers, in accordance to the NCTM Focal Poin...show morets document. ...show less
With CD!VeryGood
South Beach Bookstore Miami Beach, FL
0321570545 | 677.169 | 1 |
Curriculum & Instruction: Functions&Algebra for Teachers
EDCI 200 Z05 (CRN: 60951)
3 Credit Hours
Jump Navigation
About EDCI 200 Z05
This course builds on the prior course Mathematics as a Second Language. Participants will obtain deep understanding of the concept of a function, utilize functions in problem solving, appreciate the pervasiveness of the function idea in the K-8 mathematics curriculum as well as everyday life, and engage in a variety of problem-solving activities that relate directly to the K-8 mathematics classroom. Topics include functions, graphs, inverse functions, linear functions, the algebra and geometry of straight lines, solving linear equations and inequalities, and an introduction to nonlinear functions. | 677.169 | 1 |
A software to calculate expression, roots, extremum, derivateve, integral, etc.Features: Math Calculator is an expression calculator. You can input an expression including variable x, for example, log(x), then input a valueof x; You can also input an expression such as log(20) directly.Math Calculator is an equation solver Math Calculator can be used to solve equations with one variable, for example, sin(x)=0. Math Calculator is a function analyzer Math Calculator has the abilities of finding maximum and minimum.Math Calculator is a derivative calculator and calculus calculator. You can use this program to calculate derivative and 2 level derivate of a given function.Math Calculator is an integral calculator. Math Calculator has the ability of calculating definite integral. | 677.169 | 1 |
Offering a uniquely modern, balanced approach, Tussy/Gustafson/Koenig's DEVELOPMENTAL MATHEMATICS FOR COLLEGE STUDENTS, Third Edition, integrates the best of traditional drill and practice with the best elements of the reform movement. To many developmental math students, algebra is like a foreign language. They have difficulty translating the words, their meanings, and how they apply to problem solving. Emphasizing the "language of algebra," the text's fully integrated learning process is designed to expand students' reasoning abilities and teach them how to read, write, and think mathematically. It blends instructional approaches that include vocabulary, practice, and well-defined pedagogy with an emphasis on reasoning, modeling, communication, and technology skills. The text's resource package--anchored by Enhanced WebAssign, an online homework management tool--saves instructors time while also providing additional help and skill-building practice for students outside of class.
N. Tuchina A way to Success: English for University Students ( Uploaded - Ryushare ) | 677.169 | 1 |
About:
Basic Properties of Real Numbers: Symbols and Notations
Metadata
Name:
Basic Properties of Real Numbers: Symbols and Notations
ID:
m18872Topics covered in this module: understand the difference between variables and constants, be familiar with the symbols of operation, equality, and inequality, be familiar with grouping symbols, be able to correctly use the order of operations. | 677.169 | 1 |
Course Description (From UGa Bulletin): To appropriately select and use technology in mathematics instruction with an emphasis on the organization and design of materials for secondary mathematics courses.
Prerequisites for EMAT 4700/6700:
Textbook:
None. Readings as posted . . .
Time:
Place:
Course outline
The following software will be used:
1. Geometer's Sketchpad 4.07 or 5.0
2. Graphing Calculator 3.5 or 4.0
3. Excel
4. Microsoft Word
5. Firefox
6. Dreamweaver
7. FTP tools if needed
Projects/Course Requirements.
Objectives
• To use application software and technological tools to solve mathematical problems, engage in mathematical investigations, create mathematical demonstrations, and construct new ideas of mathematics for yourself.
• To analyze the affordances of software applications and its connections to the mathematics and how to take this into account when planning activities and lessons.
• To design mathematical activities and lessons that capitalize on the affordances of technology.
• To communicate mathematical ideas that arise from computer investigations using word processing and web technologies.
• To communicate mathematical ideas via the computer applications.
• To become familiar with recent issues in the literature regarding the use of technology in mathematics education.
The University of Georgia seeks to promote and ensure academic
honesty and personal integrity among students and other members
of the University Community. A policy on academic honesty has
been developed to serve these goals. All members of the academic
community are responsible for knowing the policy and procedures
on academic honesty. | 677.169 | 1 |
for Dummies
A plain-English guide to the basics of trig Trigonometry deals with the relationship between the sides and angles of triangles... mostly right ...Show synopsisA plain-English guide to the basics of trig Trigonometry deals with the relationship between the sides and angles of triangles... mostly right triangles. In practical use, trigonometry is a friend to astronomers who use triangulation to measure the distance between stars. Trig also has applications in fields as broad as financial analysis, music theory, biology, medical imaging, cryptology, game development, and seismology. From sines and cosines to logarithms, conic sections, and polynomials, this friendly guide takes the torture out of trigonometry, explaining basic concepts in plain English and offering lots of easy-to-grasp example problems. It also explains the "why" of trigonometry, using real-world examples that illustrate the value of trigonometry in a variety of careers.Tracks to a typical Trigonometry course at the high school or college levelPacked with example trig problemsFrom the author of "Trigonometry Workbook For Dummies" "Trigonometry For Dummies" is for any student who needs an introduction to, or better understanding of, high-school to college-level trigonometryVery good. No dust jacket as issued. Trade paperback (US). Glued...Very good. No dust jacket as issued. Trade paperback (US). Glued binding. 368 p. Contains: Illustrations. For Dummies (Lifestyles Paperback). Audience: General/trade. BOX# 071010A: This book is in Very Good condition, and the CD (is not) includes: This book has (0) pages contain HIGHLIGHT and (0) pages with UNDER-LINES, NOTES, and ANSWERS () pages are Creases. Hard Cover tips or Paperback Covers and Stem or binding is (damaged); The Jacket Cover is (N/A). This is an Ex-Library s: (N/A). SPECIAL NOTES (). If this book is not as the above condition, return at 4045 NW 185 ST. Miami Gardens, FL 33055. For full refund no need for email or call.
Description:New, Publisher overstock, may have small remainder mark....New, Publisher overstock, may have small remainder mark. Excellent condition, never read, purchased from publisher as excess inventory.
Description:New. A plain-English guide to the basics of trigFrom sines and...New. A plain-English guide to the basics of trigFrom sines and cosines to logarithms, conic sections, and polynomials, this friendly guide takes the torture out of trigonometry, explaining basic concepts in plain English, offering lots of easy-to-gras | 677.169 | 1 |
Find a Glen EllynUse of a graphing calculator can be a blessing and a curse, and it's essential that students recognize how calculators can both help and hinder them in their quest to understand mathematics, the true universal language.For most of my adult life, I've played guitar. While I can strum or finger-pi... | 677.169 | 1 |
College Mathematics for Business, Economics, Life Sciences, and Social Sciences - With Mymathlab - 11th edition
Summary: Designed to be accessible, this book develops a thorough, functional understanding of mathematical concepts in preparation for its application in other areas. Concentrates on developing concepts and ideas followed immediately by developing computational skills and problem solving. Features a collection of important topics from mathematics of finance, algebra, linear programming, probability, and descriptive statistics, with an emphasis on cross-discipline principles ...show moreand practices. For the professional who wants to acquire essential mathematical tools for application in business, economics, and the life and social sciences. ...show less
Only lightly used. Book has minimal wear to cover and binding. A few pages may have small creases and minimal underlining. Book selection as BIG as Texas.
$34.50 +$3.99 s/h
Good
Books Revisited Chatham, NJ
Possible retired library copy, some have markings or writing.
$141 | 677.169 | 1 |
...
More About
This Book
the 2007 QTS Standards.
Table of Contents
Mathematics background
Interest in mathematics
Perceived competence and confidence in mathematics
Mathematics test
Answers to test questions
Targets for further development
Revision and further | 677.169 | 1 |
8th Grade Math Curriculum
Posted: October 10, 2001
Last Updated: May 3, 2006
Rationale
The eighth grade mathematics curriculum is designed to expose and facilitate learning of the basic mathematical
concepts, knowledge, and skills necessary for the successful participation and application in the secondary level
math classes.
Course Description
The eighth grade student's mathematics course is designed to review and acquire a solid foundation, which includes
the knowledge of numerical operations, mathematical systems, geometry, number theory, probability and statistics,
measurements, and algebraic expressions.
Grade Classification: 8
Duration: 1 Year
Performance Required Alignment to Instructional Strategies Assessment Level of
Show-Me Standards Performance
Required
General Objective # NO,1,A Performance Content
The student will read, write,
and compare numbers 3.3
MA 5
1. Specific Objectives:
The student will: MA, NO, 1, In cooperative learning groups students Students will complete a
A. compare and order A, 8 will discuss the values of rationals and constructed response
rationals and percents percents and plot them on a number question taken from the
including finding their line. Assessment Annotations for
approximate location on a the Curriculum
number line. Real Numbers by Richard Powers Frameworks.p.195
Benchmark B problem #2
Ordering Rational Numbers by Richard
Page 1 of 15
Last Updated: 4/17/2010
Powers
General Objective #NO, 1, B Performance Content
The student will represent 3.4
and use rational numbers. MA 1
2. Specific Objectives:
The student will: MA, NO, 1, Discuss methods of fractional Students will create a
B, 8 representation and computation performance event in the
1. use fractions to solve including shortcuts. Practice with form of a take home
problems. selected response quizzes. test(including an answer
key) that compares the cost
MAP Released Items by Richard of items from several
Powers stores. The performance
event will include topics
such as fraction off,
percentage off, sales tax,
etc.
2. use decimals to solve MA, NO, 1, Discuss methods of decimal Students will create a
problems. B, 8 representation and computation performance event in the
including shortcuts. Practice with form of a take home
selected response quizzes. test(including an answer
key) that compares the cost
of items from several
stores. The performance
event will include topics
such as fraction off,
percentage off, sales tax,
etc.
3. use percents to solve MA, NO, 1, Analyze the use of proportions and Students will create a
problems. B, 8 equations calculating percents including performance event in the
but not limited to Commission, form of a take home
Discount/Sale Price, and Percent of a test(including an answer
number. Practice with selected response key) that compares the cost
quizzes. of items from several
stores. The performance
event will include topics
such as fraction off,
percentage off, sales tax,
etc.
Page 2 of 15
Last Updated: 4/17/2010
General Objective # NO, 1, Performance Content
C
The student will compose 3.6 MA 1
and decompose numbers.
3. Specific Objectives:
The student will: MA, NO, 1, Interpret among the equivalent Students will take a free
1. make conversions among C, 8 representations of fractions, decimals and constructed response
fractions, decimals,and and percents, and make conversions. exam.
percents.
2. make conversions among MA, NO, 1, Re-introduce students to expanded Students will take a free
expanded notation, scientific C, 8 notation and exponential notation. Also, and constructed response
notation and standard form. engage the students in the actual exam.
multiplication of large and small
numbers and promote the discovery of
patterns to aid in the computation.
General Objective # NO, 1, Performance Content
D
The student will classify and 1.10 MA 5
describe numeric
relationships.
4. Specific Objectives:
The student will: MA, NO, 1, Analyze the rules of divisibility and Students will complete a
1. use factors and multiples D, 8 apply them in the evaluation of the selected response quiz.
to describe relationships LCM and GCF.
between and among numbers
including but not limited to MAP Released Items by Richard
LCM, GCF, and divisibility Powers
rules.
2. use factors and multiples MA, NO, 1, Using multiple methods, provide Students will complete a
to justify characteristics of D, 8 extensive examples of how to create the selected response quiz.
numbers including but not factorization using only prime numbers
limited to prime and after facilitating a conversation on the
composite numbers, and nature of prime and composite numbers.
prime factorization.
MAP Released Items by Richard
Powers
General Objective # NO,2,B Performance Content
The student will describe
effects of operations 3.4, 4.1 MA 1
5. Specific Objectives:
Page 3 of 15
Last Updated: 4/17/2010
The student will: MA, NO, 2, Using a variety of techniques such as Given problems with errors
1. describe the effects of B, 8 rules, money, temperature, etc., students the student will be expected
multiplication and division will calculate and analyze products and to find the error and
on integers. quotients. describe the effect the
wrong process had on the
solution.
General Objective # NO, 2, Performance Content
C
The student will apply 1.6, 1.10 MA 5
properties of operations.
6. Specific Objectives:
The student will: MA, NO, 2, Review the order of operations for Students will complete a
1. apply properties of C, 8 rational numbers, and design problems selected response quiz. The
operations to rational to solve including problems with quiz will contain problems
numbers, including order of inverse operations. including order of
operations and inverse operations and inverse
operations. Order of Operations by Richard Powers operations.
General Objective # NO, 2, Performance Content
D
The student will apply 1.6, 3.4
operations on real numbers. MA 5
7. Specific Objectives:
The student will: MA, NO, 2, Given a list of perfect squares and Via a selected response
1. apply the relationship D, 8 exam students will evaluate
square roots, the teacher will facilitate a
between squares and square discussion and provide examples of how squares and square roots of
roots to solve a problem. to approximate the values of nonperfect perfect squares and
squares. approximate the values of
square roots of non perfect
Squares, Square Root, Cubes , and Cube squares, and solve problems
Roots by Richard Powers involving them.
2. apply the relationship MA, NO, 2, Given a list of perfect cubes and cube Via a selected response
between cubes and cube D, 8 roots, the teacher will facilitate a exam students will evaluate
roots to solve a problem. discussion and provide examples of how cube and cube roots of
to approximate the values of nonperfect perfect squares and
squares. approximate the values of
cube roots of non-perfect
Squares, Square Root, Cubes , and Cube cubes, and solve problems
Roots by Richard Powers involving them.
Page 4 of 15
Last Updated: 4/17/2010
General Objective # NO, 3, Performance Content
C
The student will compute 1.10, 3.3 MA 1
problems.
8. Specific Objectives:
The student will: MA, NO, 3, Using manipulatives the students will Students will complete a
1. apply all operations on C, 8 discover the rules of operations on all selected response test over
rational numbers. rational numbers and make applications the applications of all
with this knowledge. operations on rational
numbers.
Adding Integers by Richard Powers
General Objective # NO, 3, Performance Content
D
The student will estimate and 3.3, 4.1
justify solutions. MA 1
9. Specific Objectives:
The student will: MA, NO, 3, Based on previously answered questions Given problems in a
D, 8 the students will be presented with selected response format
1. estimate the results of all estimation strategies that help them find students will determine the
operations on rational the reasonableness of an answer. solution through estimation.
numbers and percents.
2. justify the results of all MA, NO, 3, Based on estimates the students will Students will pick a
operations on rational D, 8 defend the estimates and provide three problem and its solution
numbers and percents. reasons. and defend the
reasonableness of the
solution.
General Objective #NO, 3, E Performance Content
The student will use
proportional reasoning. 3.3
MA 1
10. Specific Objectives:
The student will: MA, NO, 3, Facilitate a discussion on rates and Students will be able to
1. solve problems involving E,8 provide example of how to convert rates determine a rate based on a
rates into unit rates including unit pricing. table or graph, and use this
rate to calculate an amount.
2. solve problems involving MA, NO, 3, Assist students in measuring on a scale Using a variety of
proportions, such as scaling E,8 drawing and making conversions, as assessments (selected and
and finding equivalent ratios. well as using proportions to solve for a closed ended constructed
missing value. response), the students will
Page 5 of 15
Last Updated: 4/17/2010
solve problems using
proportions including but
not limited to scale
drawing.
General Objective # AR,1,B Performance Content
The student will create and
analyze patterns. 1.6, 3.6 MA 4
11. Specific Objectives:
The student will: MA, AR, 1, Teacher will model the use of patterns Students will design and
B, 8 and how to determine the next terms in justify pictorial and
1. generalize patterns a numerical pattern as well as graphical representations of
represented graphically using discovering the rule to determine the nth patterns including writing
words or symbolic rules, term. algebraic expressions or
including recursive notation. number sentences.
Patterns and Sequences by Richard
Powers
General Objective # AR, 1, Performance Content
C
The student will classify 1.6
objects and representations. MA 4
12. Specific Objectives:
The student will: MA, AR, 1, Given multiple representations of Students will compare and
1. compare and contrast C, 8 patterns students will analyze the contrast patterns based on
various forms of similarities and differences. data given in multiple
representations of patterns. formats such as tables,
graphs etc.
General Objective # AR, 1,D Performance Content
The student will identify and
compare functions. 1.6, 3.6 MA 4
13. Specific Objectives:
The student will: MA, AR, 1, Facilitate a discussion on the definition Using tables, graphs and
1. compare properties of D, 8 and characteristics of linear functions. equations, the student will
linear functions between or verify whether a linear
among tables, graphs, and function is represented. The
equations. student will also compare
intercepts, slopes, etc.
General Objective # AR, 2, Performance Content
A
Page 6 of 15
Last Updated: 4/17/2010
The student will represent 1.6, 3.1 MA 4
mathematical situations.
14. Specific Objectives:
The student will: MA, AR, 2, Analyze patterns represented in various The student will complete
1. use symbolic algebra to A, 8 forms, and discuss the development of a a constructed response quiz
represent and solve problems symbolic rule to find any term in the in which he/she will
that involve linear pattern. generate patterns and their
relationships including symbolic rule.
recursive relationships. Understanding and Writing Equations
by Richard Powers
MAP Released Items by Richard
Powers
General Objective # AR, 2,B Performance Content
The student will describe and 3.6 MA 4
use mathematical
manipulation.
15. Specific Objectives:
The student will: MA, AR, 2, Based on previously learned skills Students will complete an
1. generate equivalent forms B, 8 students will create and simplify linear exam in which they
for linear expressions. expressions by applying properties of simplify expressions and
operations and combining like terms. solve equations including
but not limited to changing
between slope intercept and
standard form
General Objective #AR, 3, A Performance Content
The student will use
mathematical models. 1.6, 3.6 MA 4
16. Specific Objectives:
The student will: MA, AR, 3, Teacher will model different ways to Students will choose
1. model and solve problems, A,8 represent and solve problems. another appropriate
using multiple representation of a problem
representations such as as well as solve, and
graphs, tables, equations or analyze the process,
inequalities. solution and graph of an
equation and inequality.
Page 7 of 15
Last Updated: 4/17/2010
General Objective #AR, 4, A Performance Content
The student will analyze
change. 1.6, 4.1 MA 2,
MA 4
17. Specific Objectives:
The student will: MA, AR, 4, Teacher will facilitate a discussion that Students will analyze the
1. analyze the nature of A,8 analyzes the nature of change in linear result of changes in slope
changes (including slope and relationships. and intercepts including
intercepts) in quantities in negative slope, and rate of
linear relationships. change.
General Objective #GSR, 1, Performance Content
A
1.6, 3.6 MA 2
The student will describe and
use geometric relationships.
18. Specific Objectives:
The student will: MA, GSR, 1, Using concept circles students will Students will take a series
1. describe, classify and A, 8 analyze triangles and their relationships. of selected and constructed
generate relationships Also students will solve problems by response questions.
between and among triangles writing and solving equations including
using their defining Pythagorean Theorem.
properties including Classifying Polygons by Richard
Pythagorean theorem. Powers
2. describe, classify and MA, GSR, 1, Using concept circles students will Students will take a series
generate relationships A, 8 analyze quadrilaterals and their of selected and constructed
between and among relationships. Also students will solve response questions.
quadrilaterals using their problems by writing and solving
defining properties. equations
Classifying Polygons by Richard
Powers
3. describe, classify and MA, GSR, 1, Using concept circles students will .Students will take a series
generate relationships A, 8 analyze polygons and their of selected and constructed
between and among relationships. Also students will solve response questions.
polygons using their defining problems by writing and solving
properties. equations
Classifying Polygons by Richard
Powers
4. describe, classify and MA, GSR, 1, Using concept circles students will Students will take a series
generate relationships A, 8 analyze 3-dimensional objects and their of selected and constructed
Page 8 of 15
Last Updated: 4/17/2010
between and among types of relationships. response questions
3-dimensional objects using
their defining properties Cross sections by Richard Powers
including a)Pythagorean
Theorem b) cross section of
a 3-dimensional object
results in what 2-
dimensional shape.
General Objective #GSR, 1, Performance Content
B
1.6, 3.6 MA 2
The student will apply
geometric relationships.
19. Specific Objectives:
The student will: MA, GSR, 1, Students will create, analyze and solve Students will take a series
1. apply relationships B, 8 problems using similar polygons of selected and constructed
between corresponding sides including solving for a missing side or response questions
and corresponding areas of angle.
similar polygons to solve
problems.
General Objective #GSR, 2, Performance Content
A
3.6 MA 2
The student will use
coordinate systems.
20. Specific Objectives:
The student will: MA, GSR, 2, Students will create similar triangles Students will participate in
1. use coordinate geometry A, 8 and put in vertices to make right a performance event in
to analyze properties of right triangles when given the hypotenuse. which they will graph the
triangles. Also students will calculate area. missing vertex to form a
right triangle, and solve for
The Coordinate System by Richard the missing side and/or
Powers perimeter and area.
2. use coordinate geometry MA, GSR, 2, Students will graph points and
to analyze properties of A, 8 determine which quadrilateral results.
quadrilaterals. Also students will discuss attributes and
calculate areas.
Page 9 of 15
Last Updated: 4/17/2010
Area and Perimeter of Rectangles by
Richard Powers
General Objective #GSR, 3, Performance Content
A
3.6 MA 2
The student will use
transformations on objects.
21. Specific Objectives:
The student will: MA, GSR, 3, Following teacher led discussion and Students will take a series
1. reposition shapes under A, 8 examples; students will complete of selected and constructed
formal transformations such guided and independent practice. response questions
as reflection, rotation, and Students will also analyze
translation. transformations and identify which has
occurred in an example.
Transformations by Richard Powers
General Objective #GSR, 3, Performance Content
B
3.6 MA 2
The student will use
transformations on functions.
22. Specific Objectives:
The student will: MA, GSR, 3, After completing dilations students will Students will take a series
1. describe the relationship B, 8 hypothesize about and evaluate the of selected and constructed
between the scale factor and areas based on scale factor including the response questions
the areas of the image using use of a table of values for linear, area,
a dilation and volume.
(stretching/shrinking).
Transformations by Richard Powers
General Objective #GSR, 3, Performance Content
C
1.6 MA 2
The student will use
symmetry.
23. Specific Objectives:
The student will: MA, GSR, 3, Students will determine if a regular Students will take a series
1. identify the number of C, 8 polygon has rotational symmetry and of selected and constructed
Page 10 of 15
Last Updated: 4/17/2010
rotational symmetries of describe the symmetry. response questions
regular polygons.
Rotational Symmetry by Richard
Powers
General Objective #GSR, 4, Performance Content
A
3.3 MA 2
The student will recognize
and draw 3-dimensional
representations.
24. Specific Objectives:
The student will: MA, GSR, 4, Students will compare an isometric Students will create an
1. create isometric drawings A, 8 drawing with a mat plan. isometric drawing from a
from a given mat plan. given mat plan.
Isometric Drawings and Mat Plans by
Richard Powers
General Objective #GSR, 4, Performance Content
B
3.1 MA 2
The student will draw and
use visual modes.
25. Specific Objectives:
The student will: MA, GSR, 4, Teacher will model the use of perimeter Students will manipulate
1. draw or use visual models B, 8 and area of complex polygons with shapes to create a new
to represent and solve manipulatives and discuss the shape or change perimeter
problems. similarities and differences. or area. Also students will
view a visual model and
pick the corresponding
result. (tent or cube)
General Objective #M, 1, B Performance Content
The student will identify 1.6 MA 2
equivalent measures.
26. Specific Objectives:
The student will: MA, M, 1, B, Students will convert within cubic units Students will take a quiz of
1. identify the equivalent 8 including but not limited to yd to ft and selected and constructed
volume of measures within a m to cm. response questions.
system of measurement (m ³
to cm ³)
Page 11 of 15
Last Updated: 4/17/2010
General Objective #M, 2, B Performance Content
The student will use angle 1.4, 3.2 MA 2
measurement.
27. Specific Objectives:
The student will: MA, M, 2, B, Based on discussion of reflex angles Students will perform
1. use tools to determine the 8 and prior knowledge of protractors, measurements of reflex
measure of reflex angles to students will measure angles that are angles.
the nearest degree. greater than 180 degrees
Angle Measure by Richard Powers
General Objective #M, 2, C Performance Content
The student will apply 3.4, 4.1 MA 2
geometric measurements.
28. Specific Objectives:
The student will: MA, M, 2, C, Teacher will demonstrate the use of Student will pick a three-
1. describe how to solve 8 formulas and how to evaluate them to dimensional object and
problems involving surface find surface area, volume, or any part of explain how to find surface
area and/or volume of a the formula including finding errors of area and volume.
rectangular or triangular computation.
prism, or cylinder.
Surface Area by Richard Powers
General Objective #M, 2, D Performance Content
The student will analyze 1.7, 3.8 MA 2
precision.
29. Specific Objectives:
The student will: MA, M, 2, D, Following a discussion on the precision Students will have to
1. analyze precision and 8 (rounding to the smallest place), the choose the most precise
accuracy in measurement students will determine the number of measurement and determine
situations and determine significant digits. the number of significant
number of significant digits. digits.
Using Significant Digits by Richard
Powers
General Objective #M, 2, E Performance Content
The student will use 1.6, 1.10 MA 2
relationships within a
Page 12 of 15
Last Updated: 4/17/2010
measurement system.
30. Specific Objectives:
The student will: MA, M, 2, E, Students will make conversion Students will take a quiz of
1. convert square or cubic 8 including but not limited to ft to yd selected and constructed
units to equivalent square or squared. response questions.
cubic units within the same
system of measurement.
General Objective #DP, 1, A Performance Content
The student will formulate 1.2 MA 3
questions.
31. Specific Objectives:
The student will: MA, DP, 1, Students will create and analyze a Teacher observation of
1. formulate questions, A, 8 survey and determine if it is biased or survey the student created
design studies, and collect not. and analyzed.(Social
data about a characteristic. Studies)
Displaying Data by Richard Powers
General Objective #DP, 1, C Performance Content
The student will represent 1.8, 3.6 MA 3
and interpret data.
32. Specific Objectives:
The student will: MA, DP, 1, C, Students will use provided data to Students will take an exam
1. select, create, and use 8 choose and create a representation of selected and constructed
appropriate graphical including but not limited to all types of response questions.
representation of data graphs, stem and leaf, histogram, Venn
(including scatter plots). diagram with which the students will
create 5 questions that can be answered
by using the representation.
Displaying Data by Richard Powers
General Objective #DP, 2, A Performance Content
The student will describe and 3.4 MA 3
analyze data.
33. Specific Objectives:
The student will: MA, DP, 2, Using appropriate data representations Students will take an exam
1. find, use and interpret A, 8 such as box and whisker, students will of selected and constructed
measures of center, outliers, find, use and interpret values including response questions.
and spread, including range mean without outliers, median, mode,
Page 13 of 15
Last Updated: 4/17/2010
and inter quartile range. spread, outliers, and most typical.
Displaying Data by Richard Powers
MAP Released Items by Richard
Powers
General Objective #DP, 2, B Performance Content
The student will compare 3.6 MA 3
data representations.
34. Specific Objectives:
The student will: MA, DP, 2, B, Determine the best choice for Given a set of data the
1. compare different 8 representing data and which students will choose and
representations of the same representation is misleading. construct an approprite
data and evaluate how well representation along with a
each representation shows Displaying Data by Richard Powers representation that is
important aspects of the data. misleading.
General Objective #DP, 3, A Performance Content
The student will develop and 3.5 MA 3
evaluate inferences.
35. Specific Objectives:
The student will: MA, DP, 3, Determine whether a correlation is Students will graph two sets
1. make conjectures about A, 8 positive, negative, or none. Also of data and determine the
possible relationships students will approximate a fitted line. correlation. If appropriate
between 2 characteristics of the students will put in a
a sample on the basis of Displaying Data by Richard Powers fitted line and make
scatter plots of the data and predictions.
approximate lines of fit.
General Objective #DP, 4, A Performance Content
The student will apply basic 3.5 MA 3
concepts of probability.
36. Specific Objectives:
The student will: MA, DP, 4, Students will describe how theoretical Students will take a quiz of
1. make conjectures (based A, 8 and experimental probability are selected and constructed
on theoretical probability) different as well as how they can work response questions.
about the results of together to help us find the solution to
experiments. problems. These problems range from
simple probability to dependent events
Page 14 of 15
Last Updated: 4/17/2010
to finding the probability of an
experiment based on theoretical
probability.
Experimental and Theoretical
Probability by Richard Powers
Page 15 of 15
Last Updated: 4/17 | 677.169 | 1 |
Basic Math For Management Professionals ...
(Paperback)
The lack of mathematical knowledge is a major obstacle for many marketing and management professionals. Without a solid foundation in accounting, finance, mathematics or economics, these people often become confused and frustrated, leading to situations where they are unable to play a strategic role in the higher echelons of a company.This is a simple and fun to read book which provides an introduction to the underlying mathematical concepts in marketing and management, in easy to understand terms. Written in a conversational style, this delightful book provides the tools in order to fully understand mathematical concepts, using realistic approaches, real life examples and illustrations to gently introduce these concepts to the reader. The book also includes relevant non-mathematical issues such as price sensitivity, product distribution and sales estimates | 677.169 | 1 |
Identify the essential content and process readiness indicators for student success in Algebra I
Identify virtual manipulatives and interactive applets that target the essential skills and knowledge aligned with each of the content readiness indicators
Analyze virtual manipulatives and interactive applets according to given criteria including: alignment with mathematics learning goals, instructional strengths and limitations, ease of use, and availability of support materials
Use virtual manipulatives and interactive applets in activities that target the essential skills and knowledge required to meet the essential algebra readiness indicators
Develop activities that use virtual manipulatives and interactive applets to target the essential skills and knowledge required to meet the essential algebra readiness indicators
*This course is open to all certified WV educators who are teaching mathematics as well as school administrators. **This course is recognized as a technology course and will meet the technology coursework requirments for recertification. ***This course is approved for one course of the three course General Math bundle for Special Education teachers who are working toward becoming "Highly Qualified."
Course Syllabus
Getting Ready for Algebra Using Virtual Manipulatives
Course Description
There is substantial evidence to suggest that a solid foundation in algebra provides a gateway to the higher levels of mathematics necessary for success in higher education, technological or scientific occupations, and business applications. Given this reality, as well as the increased focus on accountability and high academic standards, many schools and districts have instituted policies that require all students to complete algebra as a requirement for high school graduation.
In response to the accountability measures outlined in the No Child Left Behind Act of 2001, the Southern Regional Education Board (SREB) worked with a panel of teachers and experts from the Educational Testing Service (ETS) to develop 17 Algebra I readiness indicators, including the 5 "process" indicators and the 12 "content and skills" indicators. This course is structured around the 12 content and skills readiness indicators and will introduce a collection of virtual manipulatives that will help curriculum planners and classroom teachers meet the demand to prepare students for Algebra I.
Prerequisites
This is an introductory workshop for teachers, technology specialists, curriculum specialists, professional development specialists, and other school personnel who integrate technology into mathematics instruction. Participants are expected to have a set of baseline skills in both mathematics and technology. The prerequisite skills and knowledge are as follows:
Technological
Participants are expected to have basic technology skills and regular access to computers. Specifically, participants should be proficient with browsing the Internet, using email, and saving and accessing computer files.
Mathematics Content/Standards
This online workshop addresses the mathematics skills and knowledge that are necessary for students to be successful in algebra as described in the SREB report, Getting Students Ready for Algebra I, and the National Council of Teachers of Mathematics' (NCTM's) Principles and Standards for School Mathematics (PSSM 2000).
Participants should have a working knowledge of the expectations outlined in the NCTM Algebra Standard, which states:
"Instructional programs from pre-kindergarten through grade 12 should enable all students to:
understand patterns, relations, and functions,
represent and analyze mathematical situations and structures using algebraic symbols,
use mathematical models to represent and understand quantitative relationships, and
analyze change in various contexts" (PSSM p. 37).
Additionally, participants should have specific understanding of the algebra goals and expectations for students in grades 6-8 as outlined in NCTM's Principles and Standards for School Mathematics (PSSM 2000) on pages 222-231.
Goals
identify virtual manipulatives and interactive applets that target the essential skills and knowledge aligned with each of the content readiness indicators,
analyze virtual manipulatives and interactive applets according to given criteria including: alignment with mathematics learning goals, instructional strengths and limitations, ease of use, and availability of support materials,
use virtual manipulatives and interactive applets in activities that target the essential skills and knowledge required to meet the essential algebra readiness indicator,s
develop activities that use virtual manipulatives and interactive applets to target the essential skills and knowledge required to meet the essential algebra readiness indicators.
Assessment and Course Requirements
Each session includes readings, an activity, and a discussion assignment, which participants are required to complete.
Course Products
As a final product, participants will create a lesson plan that incorporates a virtual manipulative or online tool into the curriculum.
Discussion Participation
Students will be evaluated on the frequency and quality of their discussion board participation. Students are required to post a minimum of three substantial postings each session, including one that begins a new thread and one that responds to an existing thread. Postings that begin new threads will be reviewed based on their relevance, demonstrated understanding of course concepts, examples cited, and overall quality. Postings that respond to other students will be evaluated on relevance, degree to which they extend the discussion, and tone.
Session Two: Number and Operations Indicators
The activities in this and all other sessions will help you make connections between readiness indicators, instructional strategies, and virtual manipulatives. You will first engage in the activities as learners and then discuss the activities from both the learning and teaching perspectives.
Download the "Visualizing Fractions" activity and complete the assignment.
Session Five: Algebra and Functions Indicators
For this activity and the remaining activities in this session, you may want to refer to the attached list called "Notation for Functions," which contains the proper notation for identifying various functions within the online tools.
Session Six: Summary and Final Project
Read "Mathematically Appropriate Uses of Technology." This reading discusses some of the issues that mathematics educators face in deciding which technology tools can improve student achievement and learning.
Participants will also complete their final project which is a plan, where they are going to select a virtual manipulative and describe a plan for using it to address one of the SREB algebra readiness indicators with middle school students. | 677.169 | 1 |
Precal ... MORE8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients
8.6 De Moivre's Theorem; Powers and Roots of Complex Numbers
8.7 Polar Equations and Graphs
8.8 Parametic Equations, Graphs, and Applications
9. Systems and Matrices
9.1 Systems of Linear Equations
9.2 Matrix Solution of Linear Systems
9.3 Determinant Solution of Linear Systems
9.4 Partial Fractions
9.5 Nonlinear Systems of Equations
9.6 Systems of Inequalities and Linear Programming
9.7 Properties of Matrices
9.8 Matrix Inverses
10. Analytic Geometry
10.1 Parabolas
10.2 Ellipses
10.3 Hyperbolas
10.4 Summary of the Conic Sections
11. Further Topics in Algebra
11.1 Sequences and Series
11.2 Arithmetic Sequences and Series
11.3 Geometric Sequences and Series
11.4 The Binomial Theorem
11.5 Mathematical Induction
11.6 Counting Theory
11.7 Basics of Probability
Appendices
Appendix A. Polar Form of Conic Sections
Appendix B. Rotation of Axes
Appendix C. Geometry Formulas
Glossary
Solutions to Selected Exercises
Answers to Selected Exercises
Index of Applications
Index
Photo Credits
Marge Lial has always been nowDavid Schneider has taught mathematics at universities for over 34 years and has authored 36 books. He has an undergraduate degree in mathematics from Oberlin College and a PhD in mathematics from MIT. During most of his professional career, he was on the faculty of the University of Maryland--College Park. His hobbies include travel, dancing, bicycling, and hiking.
Callie Daniels has always had a passion for learning mathematics and brings that passion into the classroom with her students. She attended the University of the Ozarks on an athletic scholarship, playing both basketball and tennis. While there, she earned a bachelor's degree in Secondary Mathematics Education as well as the NAIA Academic All-American Award. She has two master's degrees: one in Applied Mathematics and Statistics from the University of Missouri-Rolla, the second in Adult Education from the University of Missouri- St. Louis. Her hobbies include watching her sons play sports, riding horses, fishing, shooting photographs, and playing guitar. Her professional interests include improving success in the community college mathematics sequence, using technology to enhance students' understanding of mathematics, and creating materials that support classroom teaching and student understanding. | 677.169 | 1 |
Class 7 Mathematics Made EasyDesigned for the students in Class 7, Class 7 Mathematics Made Easy comprehensively covers 15 chapters in 15 LIVE interactive classes. In addition, you get access to comprehensive courseware: Videos + 10 PPTs + 10 PDFs.
Animated presentations and worksheets are included for every topic to give you a better understanding. Each chapter consists of more than 50 slides of Presentation and 10 worksheets for every chapter in word doc or jpg. You can download the courseware at anytime and can learn it at own pace of time.
Innovatively created content consists the following:
Animated PPTs are made for each chapter
Many examples from web based solutions
Includes lesson plan and game based instructions also
Educational articles for Maths
New creative ideas
Worksheets for every topic for practice exercise
Online multiple choice questions
Note - You decide which chapters you want to start with!
Class 7 Mathematics Made Easy online course package:
15 LIVE interactive online classes + Access to class recordings
Course timings: Monday-Wednesday-Friday between 9 AM to 12 PM (IST)
Courseware: Videos + 10 PPTs +10 PDFs + Docs
15 online tests to assess your performance
Course outline:
Chapter No.
Topic
Chapter 1
Integers 1
Chapter 2
Fractions and Decimals 29
Chapter 3
Data Handling 57
Chapter 4
Simple Equations 77
Chapter 5
Lines and Angles 93
Chapter 6
The Triangle and its Properties 113
Chapter 7
Congruence of Triangles 133
Chapter 8
Comparing Quantities 153
Chapter 9
Rational Numbers 173
Chapter 10
Practical Geometry 193
Chapter 11
Perimeter and Area 205
Chapter 12
Algebraic Expressions 229
Chapter 13
Exponents and Powers 249
Chapter 14
Symmetry 265
Chapter 15
Visualising Solid Shapes 277
About the Instructor
Bhakti Bhanushali Mumbai, India
Bhakti Mange has been involved in both private tuitions and online learning for last 10 years. She is experienced in use of e- teaching and educational software. She is a dedicated and dynamic e-teaching instructor who believes in creating and nurturing a lifelong love for knowledge in children. | 677.169 | 1 |
Geometric Progression
A progression is another way of saying sequence thus a Geometric Progression is also known as a Geometric Sequence.
A Geometric Progression is a special sequence defined by the special property that the ratio of two consecutive terms is the same for all the terms in the sequence. Whereas in Arithmetic Progression we talked of difference, here we talk of ratios...
read more
Sequences and Series
Sequences
A sequence in mathematics is defined as an ordered list of elements (usually numbers) whose order defines some underlying property of the list. The order of the elements is very important and changing even one element would change the meaning of the entire
sequence.
The elements in a sequence are separated by commas and the length of a sequence is...
read more
Arithmetic Progression
A progression is another term for
sequence. Therefore, Arithmetic Progressions (also known as Arithmetic Sequences) are special sequences defined by the property that the difference between any two consecutive terms of the sequence are constant. Whereas the rule for regular sequences is
that the difference between consecutive terms has to have some kind of...
read more
Sets
A set is one of the most fundamental concepts in mathematics. Sets can be taught at an elementary level all the way through higher level mathematics.
A set is defined as a group or collection of distinct objects. The elements of a set can be anything: numbers, people, letters, etc. The way we usual denote sets is by giving them capital letters for a name.
Given set A and B
A...
read more
Radical Functions
Radical Functions contain
functions involving roots. Most examples deal with
square roots. Graphing radical functions can be difficult because the domain almost always must be considered.
Let's graph the following function:
First we have to consider the domain of the function. We must note that we cannot have a negative value under the square root...
read more | 677.169 | 1 |
Preface-- 1 Elements of Group Theory-- 2 Some Related Algebraic Structures-- 3 Linear Vector Space-- 4 Elements of Representation Theory-- 5 Representations of Finite Groups-- 6 Representations of Linear Associative Algebras-- 7 Representations of the Symmetric Group-- 8 The Rotation Group and its Representations-- 9 The Crystallographic Point Groups-- 10 The Lorentz Group and its Representations-- 11 Introduction to the Classification of Lie Groups - Dynkin Diagram-- Index.
(source: Nielsen Book Data)
Publisher's Summary:
Professor Srinivasa Rao's text on Linear Algebra and Group Theory is directed to undergraduate and graduate students who wish to acquire a solid theoretical foundation in these mathematical topics which find extensive use in physics. Based on courses delivered during Professor Srinivasa Rao's long career at the University of Mysore, this text is remarkable for its clear exposition of the subject. Advanced students will find a range of topics such as the Representation theory of Linear Associative Algebras, a complete analysis of Dirac and Kemmer algebras, Representations of the Symmetric group via Young Tableaux, a systematic derivation of the Crystallographic point groups, a comprehensive and unified discussion of the Rotation and Lorentz groups and their representations, and an introduction to Dynkin diagrams in the classification of Lie groups. In addition, the first few chapters on Elementary Group Theory and Vector Spaces also provide useful instructional material even at an introductory level. An authority on diverse aspects of mathematical physics, Professor K N Srinivasa Rao taught at the University of Mysore until 1982 and was subsequently at the Indian Institute of Science, Bangalore. He has authored a number of texts, among them being "The Rotation and Lorentz Groups and their Representations for Physicists" (Wiley, 1988) and "Classical Mechanics" (Universities Press, 2003). The first edition of "Linear Algebra and Group Theory for Physicists" was co-published in 1996 by New Age International, and Wiley, New York. (source: Nielsen Book Data) | 677.169 | 1 |
It can make the math easier to do-my head hurts less because of it-but if you don't know how to use it, you my end up banging your head against the wall in frustration at trying to get it to work. That little book they send with it doesn't help much either when you can't understand it. | 677.169 | 1 |
Intermediate Algebra - 2nd edition
Summary: This student-focused text addresses individual learning styles through the use of a complete study system that starts with a learning styles inventory and presents targeted learning strategies designed to guide students toward success in this and future college-level courses.
Students who approach math with trepidation will find that Intermediate Algebra, Second Edition, builds competence and confidence. The study system, introduced at the outset and used c...show moreonsistently throughout the text, transforms the student experience by applying time-tested strategies to the study of mathematics. Learning strategies dovetail nicely into the overall system and build on individual learning styles by addressing students' unique strengths. The authors talk to students in their own language and walk them through the concepts, showing students both how to do the math and the reasoning behind it. Tying it all together, the use of the Algebra Pyramid as an overarching theme relates specific chapter topics to the 'big picture' of algebra. ...show less
Nice condition with minor indications of previous handling85 +$3.99 s/h
VeryGood
Wonder Book Frederick, MD
Very Good condition. 2nd ed. Issued without dust jacket.
$2.93 | 677.169 | 1 |
As part of the market-leading Graphing Approach SeriesEnhanced accessibility to students is achieved through careful writing and design, including same-page examples and solutions, which maximize the readability of the text. Similarly, side-by-side solutions show algebraic, visual, and numeric representations of the mathematics to support students' various learning styles.
The Library of Functions thread throughout the text provides a definition and list of characteristics for each elementary function and compares newly introduced functions to those already presented to increase students' understanding of these important concepts. A Library of Functions Summary also appears inside the front cover for quick reference.
Technology Support notes provided at point-of-use throughout the text guide students to the Technology Support Appendix, where they can learn how to use specific graphing calculator features to enhance their understanding of the concepts presented. These notes also direct students to the Graphing Technology Guide on the textbook web site for keystroke support.
Houghton Mifflin's Eduspace online classroom management tool offers instructors the option to assign homework and tests online, provides tutorial support for students needing additional help, and includes the ability to grade any of these assignments automatically.Book Description:McDougal-Littell. Hardcover. Book Condition: New. 0618394788 Premium Books are Brand New books direct from the publisher sometimes at a discount. These books are NOT available for expedited shipping and may take up to 14 business days to receive. Bookseller Inventory # Z0618394788ZN | 677.169 | 1 |
Math labs are used to help students to develop their mathematical abilities. They are usually staffed with tutors who encourage active participation of students in solving mathematical problems. The labs also have textbooks and solution manuals, computer access loaded with software used in math classes. Here, is an example of a college math lab | 677.169 | 1 |
What you will study
Stage 1
We recommend that you begin your studies with Mathematical thinking in schools (ME620)Mathematical thinking in schools::This course is designed to help you develop your knowledge and understanding of the teaching of mathematics. It is suitable for any Key Stage, and will broaden your ideas about how people learn and use mathematics. There is no formal examination: assessment is based on two tutor-marked assignments and an end-of-module assessment. In order to complete the assessments, you will need access to learners of mathematics. Students on this course have worked with a variety of learners from Key Stage 2 pupils to adults. Places are allocated on a 'first come, first served' basis, so you should register as early as you can.undergraduate.qualification.pathways.V14-1,module,ME620,,1. This 30-credit module will develop your knowledge and understanding of the teaching of mathematics, with an emphasis on Key Stage 3, and broaden your ideas about how people learn and use mathematics.
For the other 90 credits you can choose from the following:
Developing algebraic thinking (ME625)Developing algebraic thinking::This course is for you if you are interested in developing your knowledge and understanding of the learning of algebra particularly at Key Stages 2–4. It integrates development of the core ideas of algebra with relevant pedagogical constructs and principles, and will extend your awareness of how people learn and use algebra. There is no formal examination: assessment is based on three tutor-marked assignments and an end-of-module assessment. In order to complete the course assessments, you will need access to learners of algebra at Key Stages 2–4, which could include adult learners.undergraduate.qualification.pathways.V14-1,module,ME625,,1 – this 30-credit module will develop your understanding of how people learn and use algebra at Key Stages 2–4, and of different teaching constructs and principles.
Developing geometric thinking (ME627)Developing geometric thinking::Develop your knowledge and understanding of the learning of geometry particularly at Key Stages 2–4. This course integrates development of the core ideas of geometry with relevant pedagogical constructs and principles, and will extend your awareness of how people learn and use geometry. There is no formal examination: assessment is based on three tutor-marked assignments and an end-of-module assessment. To complete these assessments, you'll need access to learners of geometry at Key Stages 2–4, which could include adult learners.undergraduate.qualification.pathways.V14-1,module,ME627,,1 –this 30-credit module will develop your understanding of the learning of geometry at Key Stages 2-4, and explore a range of different teaching approaches and develop your geometric thinking.
Developing statistical thinking (ME626)Developing statistical thinking::This course will help you develop your knowledge, appreciation and understanding of the learning of statistics particularly at Key Stages 2 to 4. As well as improving your statistical thinking, you'll learn about different teaching approaches, including use of ICT tools such as scientific calculators and computers. There is no formal examination: assessment is based on three tutor-marked assignments and an end-of-module assessment. To complete these assessments, you'll need access to learners of statistics at Key Stages 2–4, which could include adult learners.undergraduate.qualification.pathways.V14-1,module,ME626,,1 – this 30-credit module explores the learning of statistics and data handling at Key Stages 2–4, and investigates different teaching approaches, including the use of ICT tools.
Researching mathematics learning (ME825)Researching mathematics learning::This course is designed for the professional development of anyone working with learners of mathematics – whether as a teacher, classroom assistant, lecturer, adviser or parent. You'll develop ways of exploring mathematics teaching and learning; interpret current thinking on the subject; and investigate aspects of the social and political context. You will consider tasks to generate pupils' activity and explore the impact of different tasks on learners. You'll also reflect and work on you own mathematics, develop your mathematical autobiography and explore further the readings and ideas that influence you most.undergraduate.qualification.pathways.V14-1,module,ME825,,1 – this 60-credit postgraduate module offers a valuable insight into the teaching and learning of mathematics for anyone involved – from classroom assistants and teachers to lecturers and parents two years part-time study to complete this qualification, but you can take anything from one (full-time study equivalent) to seven years.
Register
If you have already completed some successful study at higher education level at another institution you may be able to transfer credit for this study and count it towards an Open University qualification. If you wish to apply to transfer credit you must do so as soon as possible, and before you register for your chosen qualification. If you are awarded credit for study completed elsewhere, you may find that you need to study fewer OU modules to complete your qualification with us.
Visit our Credit Transfer site for more information and details of how to apply for credit transfer | 677.169 | 1 |
This summer, Oglethorpe will offer an opportunity for entering students to jumpstart their college mathematics sequencing!
Who: New students for fall 2013 who plan to major in science, mathematics or engineering.
What: Condensed summer offerings of Intro to Functions (MAT-120) and Advanced Functions (MAT-130). When: July 1-16 (MAT 120), and/or July 17-31 (MAT-120 and MAT-130). Where: On campus (commuting only; no residential option). Why: To get a head start on the pre-/co-requisites for fall courses in science/math majors. Supplies: Textbook (included), laptop or computer, graphing calculator, pencil and notebook. Cost: $800/course (includes lunch and required textbook).
Why would I do the IMO?
These courses are for students who want to "get the math juices flowing" in a supportive yet academically intensive environment. You might wish to get a head start on your first semester by completing a mathematics course that is a pre-/co-requisite for a major-area course that you aspire to take in the fall (e.g. BIO-101). You might wish to be a step ahead as you adjust to a college classroom, meet faculty members, and prepare for your intended course of study.
Is it open to everyone?
These courses are only for new students whose first semester at Oglethorpe will be fall 2013, and who intend to major in science, pre-engineering or mathematics. Students must be able to commute or secure their own off-campus accommodations, as the residence halls will not be an option.
Do I need prior experience with pre-calculus or trigonometry?
The IMO is ideal for students who have already been exposed to coursework in these areas, and who wish to brush up on or expand their knowledge before the fall semester. The IMO will not be a leisurely overview of functions; students will begin solving problems on Day 1. If you are confident in your abilities but surprised by your placement score, the IMO may be for you. This is a good chance to "get the math juices flowing" and build on what you have already learned.
What if I decide not to do it, or I can't come on these dates?
That's okay. You are well-prepared to begin at Oglethorpe with or without the IMO sessions. Oglethorpe offers support for any student to finish any degree in four years, and you can always contact the professional advisors in the Academic Success Center to discuss your academic plan and consider your options. Remember, the IMO is simply for jumpstarting the course sequencing. It is exactly equivalent in content to what the mathematics professors will cover in the fall and spring semesters.
What if my mom says I should, but I'm unsure?
Given the intensive nature of the accelerated format, the IMO is only worthwhile if the student is self-motivated to participate. Your parent, coach, teacher or peers cannot provide the momentum for you to succeed, since you will need to focus, manage your time, and study hard in order to pass and demonstrate your mathematical proficiency. During the IMO, you will not have much time and energy to spare, so it's important that YOU want to invest that time and energy! You should also consider whether you have the mathematical skill level to feel comfortable in such an all-day, every-day mathematics setting.
Can I take both MAT-120 and MAT-130 this summer, before the semester starts?
Yes, providing that you earn a "C-" or higher in the first offering of IMO MAT-120. Keep in mind that these sessions are back to back: you would take a final exam in MAT-120 on Tuesday, and then begin MAT-130 on Wednesday! If your schedule, stamina, and placement grade allow, you are welcome to do both IMO sessions.
So these are real college courses, worth 4 credits? Yes. Although the syllabus is compressed into a short time period, the IMO sessions are equivalent to full semester courses. One full day of an IMO class is comparable to a week and a half of a semester course. The courses will appear on your transcript like any other 4-credit course, with no qualifying designation of "IMO" or "summer".
What if I haven't taken the mathematics placement test yet? Although MAT-120 is open to any new student, it is generally a good idea to find out your placement before investing your time in the intensive course format. Your AP, IB or college-level transcripts may suggest placement at Oglethorpe. If you do not have scores or transcripts yielding mathematics placement then you may request to take the Accuplacer placement test. The admission office will offer proctoring. Call the front desk at 404.364.8307 for more information and upcoming testing dates, or you may sign up online for placement testing.
How do I sign up for the IMO? Click here to be directed to the online registration form!
What material is covered?
MAT-120 and MAT-130 explore questions such as, "What is a function? How can I tell? How do they behave, and how do I work with them? What special categories of functions often show up?" In MAT-120, students will encounter linear, polynomial, rational, exponential and logarithmic functions. In MAT-130, you will work with more advanced topics such as trigonometry, polar and parametric functions, and vector analysis.
Who are the instructors? Dr. John Merkel, Associate Professor of Mathematics, will teach one of the MAT-120 offerings and the only MAT-130 offering. Dr. Brian Patterson, Assistant Professor of Computer Science and Mathematics, will teach the other MAT-120 offering. (Click to view faculty website.) Both courses will also feature Supplemental Instruction (SI), with an upperclassman peer teacher who sits in on all class time and helps lead small-group work, answers questions, etc.
Is there tutoring available? Tutoring is available by request during the IMO. Your SI peer teacher will also be available during daytime group sessions to help you review challenging material.
Am I allowed to be absent?
No. Attendance is mandatory. You should not miss part or all of any IMO class day—not for work, a doctor's appointment, a carpooling delay, a prior commitment or any other reason. Because one day of IMO is equal to a week and half of a regular semester, it will be extremely detrimental to your learning process if you are absent for any portion of it.
Can I drop the course after I start?
Yes, you will have until 5 p.m. on the second day of class to drop the course. Absolutely no drops will be allowed from the third day on. Dropping a class means that you are completely removed from the roster, and no record of having started the course will appear on your transcript.
What if I start, but after a couple of days I know I am really (really) struggling. What do I do?
College courses also have an option called "withdrawal." Withdrawal means that the course will still be listed on your transcript, but instead of a letter grade (A-F), you will have a W. If you withdraw from a course, it does not become part of your college GPA. It is not a dishonor to cut one's losses and withdraw from a course, and it is not unusual to see a W on an undergraduate transcript. The IMO withdrawal deadlines are 5 p.m. on the sixth day of the course.
What will the format be like? The course will meet in the Academic Success Center area, which is a large, comfortable room that allows for both classroom instruction and space for small groups to work together or for individual students to concentrate. The course assumes that students will be using a laptop or a computer, and your professor will use lots of collaborative learning techniques through the WebAssign program (part of your textbook). WebAssign includes tools such as example problems, interactive practice, and video tutorials. Only about 15% of his time will be spent lecturing. The rest will be for small groups to work together and ask questions or for individuals to practice concepts on their own.
Is there homework, too? Yes. For example, in MAT-120 you will have daily homework in WebAssign. The WebAssign format is adaptive, meaning that you can work a problem as many times as you wish until you get it right (and understand why you did). You will have the opportunity to do much of your homework in class while you are at OU. The better use you make of your class time, the less you will have to do later at home.
How is it graded? You will receive a letter grade A-F. Each course will have a midterm and a final exam, along with daily individual work, homework and frequent quizzes.
What is the textbook? Where can I get it?
The required textbook for both courses is Precalculus w/ WebAssign Code, edition 6, by Stewart (ISBN0-8400-6807-7). The price of the textbook ($248.99) is included in your $800 IMO fee. You will be provided with your copy of the book on the first day of class. Your copy will include a unique WebAssign code for setting up an account. Because of this unique code, you should not purchase used copies of the book.
Can't I just get a used copy on the Internet?
No. You must have a new (not used) copy of the book in order to obtain your WebAssign code. You will be unable to participate in the IMO sessions without one. Remember, if you are starting in MAT-120, you will use the same textbook and code in MAT-130, so your investment in a new book will serve you well for at least two courses.
Do I need a graphing calculator? Yes, you must have a graphing calculator for both IMO courses (and, looking ahead, for numerous other OU courses). If you need to obtain a graphing calculator for college courswork, the instructors suggest a TI-83 or TI-84.
You say I need to bring a laptop (or request computer access at OU). Why? The IMO courses will rely on computer-based practice sets, examples, videos and interactive learning tools. It is ideal for each student to bring his or her own laptop, or to arrange to borrow one from a family member or friend for the duration of the IMO session. If this is strictly impossible, let us know and we will pursue arrangements for computer access at OU.
Can I use an iPad?
No. WebAssign relies on a Flash feature, which Apple products do not support. Your laptop must be able to use Flash.
What should I bring with me on the first day? Before arriving on the first day, you should obtain and bring a laptop, a graphing calculator, a pencil and a notebook. You should be in class a little before 9:30 a.m. with these items ready. You should also bring excitement to bond with your new classmates while doing a lot of math!
What if I live out of town? The IMO is a non-residential program. Oglethorpe is unable to offer residence hall accommodations to any new students. All participants must be able to drive in or take transit in order to attend. We recognize that participation may not be possible for students who live outside the metro area. However, the IMO is open to any new student planning to pursue a science/mathematics major, as long as the student can make their own arrangements for transportation and a place to stay.
Why can't I stay on campus?
The summer is
a busy period for the residence life staff and the campus facilities team, who rely on this time to clean and prepare the residence halls for the fall semester. Because the staff needs time to prepare for fall move-in; and because there are many shifts in space happening as OU finalizes the construction of the new campus center, it is not possible to offer housing to IMO students this summer.
What about meals?
Oglethorpe will provide lunch in the dining hall on campus for each day of the IMO. Participants should indicate any requests for dietary accommodations when registering for the program.
Will there be a break in the program? Can I leave campus?
Yes, there will be a midday break each day for lunch. There will be an approximately 1 hour break between noon and 1:00 p.m. when class resumes. Students may spend this time as they choose.
Where will the course meet?
The courses will meet in the Academic Success Center. The ASC is on the ground floor of Weltner Library, which is located in Lowry Hall at the base of the academic quad. When you enter, take the stairs or elevator down one level, and make two right turns to get to the ASC. Parking is available in the lots near the baseball field and tennis courts.
What if I don't have access to a laptop?
If this is the case, Oglethorpe will arrange for you to use a computer terminal in the ASC while you are in class. Please register for the IMO at least one business day before the start date, so that the OU staff has adequate time to assist you. Remember that you will also need access to a computer to complete daily quizzes and homework after class, so you should consult the OU library summer hours or make arrangements within your household or neighborhood for "after hours" work.
What happens next after the IMO? If you pass the IMO session of MAT-120, you will be eligible to register for MAT-130 either in the next IMO session or in the 2013-2014 academic year. You will also become eligible for MAT-121, BIO-101 and CHM-101 in the fall. If you pass the MAT-130 session, you will be eligible to register for MAT-131, PHY-101 and PHY-201 as well as the courses listed above. This may mean that you need to complete a drop/add form to modify your course registration from Passport. A professional advisor will be available to guide you through your options.
Does the fee become part of my fall tuition?
No. The IMO is billed as a summer session course, so you will receive a separate charge. The IMO will not affect your fall balance, payment plan or financial aid.
Is there financial aid available for the IMO?
No, Oglethorpe is not planning to offer financial assistance for the IMO.
How and when do I submit the $800 fee? You must submit the IMO fee before the session begins, or by 9 a.m. on the first day. You may call the business office at (404) 364-8302 to pay by credit card, or you may visit the business office in person (ground floor of Lupton Hall). All credit card transactions incur a 2.99% service fee.
What if I forget to pay the $800 beforehand?
You must submit your fee on or before the first day of class. If you have not submitted payment by 5 p.m. on the first day of the session, you will be dropped from the IMO course session. The payment deadline for MAT-120 is 5 p.m. on 7/1 (or 7/17 for the second MAT-120 offering), and for MAT-130 it is 5 p.m. on 7/17.
What is the total expense? What else do I need to budget for?
In addition to the $800 course fee, if you do not own a graphing calculator, you will need to borrow or purchase one (a new TI-83 is approximately $100). You may also wish to plan on fuel or transit costs to and from Oglethorpe.
What if I have more questions? You can email [email protected] with additional questions about the IMO. You will receive a reply from a professional advisor in the Academic Success Center who will help you obtain information or consider your options. | 677.169 | 1 |
practice exercise, you will answer just a few questions and you won't receive a ...
There is an answer key at the end. ... numerical skills/prealgebra, algebra,
college algebra, geometry, and trigonometry.
McDougal Littell Pre-Algebra will give you a strong foundation in algebra while
also ... This book will also help you become better at taking notes and taking tests
. .... Practicing Test-Taking Skills, 162.
with skills and concepts taught in pre-algebra classes. ... through the first problem
with your students, showing where the answers are to go, etc. An ... Book A
Operations with whole numbers, basic facts, ... Practice activities for first-year
algebra. | 677.169 | 1 |
Math in Everyday Life
9780825142581
ISBN:
082514258X
Edition: 3 Pub Date: 2002 Publisher: Walch Education
Summary: Newton, David E. is the author of Math in Everyday Life, published 2002 under ISBN 9780825142581 and 082514258X. Two hundred twenty six Math in Everyday Life textbooks are available for sale on ValoreBooks.com, one hundred nine used from the cheapest price of $0.52, or buy new starting at $11.74 Math in Everyday Life, 1.[less] | 677.169 | 1 |
DO PROSPECTIVE ELEMENTARY AND MIDDLE SCHOOL
TEACHERS UNDERSTAND THE STRUCTURE OF ALGEBRAIC EXPRESSIONS?
L. Pomerantsev and O. Korosteleva
Department of Mathematics and Statistics
California State University, Long Beach
1250 Bellflower Blvd.
Long Beach, CA 90840-1001
[email protected] [email protected]
Abstract
A large number of students' mistakes in algebra are due to their inability to see the
structure of a mathematical expression. This study analyzes and compares the typical
mistakes made by prospective elementary and middle school teachers as these students
progress through the courses at California State University at Long Beach. The study
shows that the students have difficulties recognizing structures of algebraic expressions
not only at the introductory level but also later as the students take calculus and senior
level courses.
The need for a thorough understanding of the structure of an algebraic expression when
performing mathematical operations has been recognized by a number of authors. Kirshner
(1989) suggests that "the ability to comprehend the syntactic structure of an algebraic expression
is fundamental to competent performance in algebra." In Yerushalmy (1992), one can find that
"the ability to transform involves mastering of algebraic rules as well as analyzing structures of
expressions."
Much research (Booth, 1989; Booth, 1999; Herscovics et al., 1995; Kieran, 1989; Kieran,
1999; Lodholz, 1999; Sfard, 1991; Sfard et al., 1994; Wagner et al., 1999b) shows that the
difficulties in recognizing the structure of mathematical expressions are due to the different
treatment of expressions in algebra and arithmetic. "In algebra, [students] are required to
recognize and use the structure that they have been able to avoid in arithmetic" (Kieran, 1989).
In arithmetic, mathematical expressions are treated from the operational point of view, as a
command to perform operations, whereas in algebra, mathematical expressions are treated from
the structural viewpoint, as an object of algebraic manipulation. "Abstract notions can be
approached in two fundamentally different ways: structurally as objects, & operationally – as
processes" (Sfard, 1991).
For instance, in arithmetic, the expression "3 + 4 " is a "sum" of two numbers perceived by
students as a command to perform addition of the two numbers (operational approach), while in
algebra, the "sum" is the "name" of the expression (the structural approach). Thus, students
should be aware of the importance of the treatment of mathematical expressions from both points
of view (operational and structural), "… certain mathematical notions should be regarded as fully
developed only if they can be conceived both operationally and structurally" (Sfard, 1991).
In mathematical textbooks, the operational meaning of the word "sum" is explained
precisely. However, the structural meaning of it is explained only for expressions involving a
L. Pomerantsev, O. Korosteleva : Do Prospective Elementary and Middle School Teachers…
single operation (for example, " x + y " is a sum). For expressions with more than one operation,
there is no explanation. Students are not being taught that, for instance, "2 • 3 + 5 " is a sum as
well.
According to Sfard (1991), the structural understanding of an abstract notion can be
automatically acquired through much operational practice. However, at this time, students do not
have enough operational practice with evaluating mathematical expressions because of the
intensive use of calculators. Possibly, this is one of the reasons that at the present time students
cannot "see" the structure of a mathematical expression, and have enormous difficulties with
understanding symbolic language.
If students are to be adequately prepared for algebra – to transition successfully from
arithmetic to algebra – they must have a foundation in elementary school that prepares them for
conceiving a given arithmetic expression as a mathematical object as well as an operational
process. It is anticipated that students who understand this concept in the elementary grades
would approach algebra more confidently and perform more successfully.
In order to teach successfully algebraic language and algebraic thinking processes to
beginners, teachers should have a full command of the subject themselves. If an elementary
school teacher does not understand the structure of an expression in depth, his or her ability to
communicate the concept to students is severely impaired.
The intent of the study is to test whether the structural representation of mathematical
expression is understood by pre-service elementary and middle school teachers.
Methodology
In the study conducted at California State University, Long Beach (CSULB) in the Spring
term, 2002, a diagnostic test was administered to 366 students. The test assessed the students'
understanding of the terminology related to the structure of mathematical expressions and the
syntax of algebraic language. The students were given one multiple-choice question and four
free-response questions. In addition, the students were asked to explain their answers. The
questions are as follows:
Question 1. What is the name of the expression 4 x 2 − 9 y 2 ?
Choices: (a) difference of squares, (b) difference of products, (c) square of difference.
2+ x
Question 2. If possible, cancel out the common factor in the expression .
2
2x + 2
Question 3. If possible, simplify the expression .
2x
Question 4. Use the statement "If x 2 = 25 , then x = ±5 " to solve the equation ( x + 1) 2 = 25 .
Question 5. Use the statement " If 2 x + 3 = 5 , then x = 1 " to solve the equation 2( y + 1) + 3 = 5.
The first question aimed at finding out whether the students understood the structure of the
expression. The choices (a) or (b), difference of squares or difference of products, were
considered the acceptable answers. In the second and third questions, students were supposed to
ac a
use the fundamental property of fractions = , b, c ≠ 0, to cancel the common factor in the
bc b
2
Issues in the Undergraduate Mathematics Preparation of School Teachers
numerator and the denominator. Acceptable answers for the second question were "impossible"
x x+1 1
and "1 + ." For the third question, " ," and "1+ " were regarded as correct. The last two
2 x x
questions dealt with students' ability to recognize a similarity in the structures of the equations
and their ability to use the given statement wisely. The "ideal" solutions in these cases were
" x + 1 = ±5 ⇒ x = 4 or − 6 ," and " y + 1 = 1 ⇒ y = 0, " respectively.
The experiment was held in eight different classrooms. The responses were then combined
into seven major groups according to the level of the courses the participants were taking. The
purpose of administering the test to the different groups was to investigate whether the students'
skills in performing simple algebraic manipulations improve as they take more mathematics
courses towards their degrees. The choice of these particular courses was motivated primarily by
the large sizes of the classes and the willingness of the instructors to conduct the testing. The
total sizes and the description of the groups are summarized as follows:
Table 1. Description and size of the participating groups of students
Group Name Description Size
1 Beginning MATH 001 Elementary 28
Algebra Algebra and Geometry
2 Intermediate MATH 010 Intermediate 70
Algebra Algebra
3 Introductory MTED 110 (Math Education) 47
Real Numbers The Real Number System for
Elementary and Middle
School Teachers
4 Finite Math MATH 114 Finite Math 57
5 (Pre)calculus MATH 117 Precalculus 57
MATH 122 Calculus I
6 Calculus MATH 123 Calculus II 35
7 Advanced Education MTED 402 Problem Solving 72
Course Applications in Mathematics
for Elementary and Middle
School Teachers
The Liberal Studies Program at CSU, Long Beach, offers an Integrated Teacher Education
Program (ITEP) that prepares K-8 multiple subject teachers. To fulfill the concentration in
mathematics requirement, students must take the following core courses: Probability and
Activities-Based Statistics (MTED 105), Real Numbers (MTED 110), Geometry and
Measurements (MTED 312), and Problem Solving Applications (MTED 402).
The Department of Mathematics and Statistics offers a B.S. in Mathematics degree with
Option in Mathematics Education. This option is for students preparing to teach mathematics at
the secondary school level. The math course sequence required for this degree includes Calculus
I,II and III (MATH 122, 124, and 224), Introduction to Linear Algebra (MATH 247), Number
Theory (MATH 341), College Geometry (MATH 355), Ordinary Differential Equations I
3
L. Pomerantsev, O. Korosteleva : Do Prospective Elementary and Middle School Teachers…
(MATH 364A), Probability Theory (MATH 380), Statistics (MATH 381), and Introduction to
Abstract Algebra (MATH 444).
An appropriate score on the Entry-Level Math (ELM) requirement is a prerequisite for MATH
001 and MATH 010. About 20% of the students in these courses are future K-8 elementary
teachers (Groups 1 and 2).
As mentioned in the above paragraph, MTED 110 and MTED 402 are the capstone courses for
the K-8 pre-service teachers, who entirely populate these courses (Groups 3 and 7). MTED 110
serves as a prerequisite for MTED 402. Three years of high school mathematics is required for
MTED 110.
MATH 114 is offered primarily to Business majors, so the percentage of math education students
taking this course is not more than 5% (Group 4).
Calculus courses are mandatory for students preparing to teach mathematics at the secondary
school level. However, the majority of students in MATH 117, 122, 123 are Engineering or
Computer Science majors. Only about 20% of the course body is comprised of math ed students
(Groups 5 and 6).
Results
Overall, the study reveals that students have a strong conceptual misunderstanding of the
structures underlying the mathematical symbols. The table below gives the percentages of
students who answered the questions correctly in each of the seven groups.
Table 2. Percentages of correct responses
Question
Group 1 2 3 4 5
1 78.5%* 25.0% 0% 0% 14.30%
2 92.9% 28.6% 10.0% 2.9% 5.7%
3 95.8% 44.7% 29.8% 10.7% 21.3%
4 91.2% 63.1% 42.1% 29.8% 33.3%
5 94.8% 75.4% 36.8% 26.3%* 26.3%
6 97.2% 82.9% 71.4%* 42.9%** 34.3%
7 87.5% 44.4% 22.2% 5.6% 18.1%
* ** Percentages differ significantly from the other percentages in the same column (according to Duncan's multiple range test
with type I error a=0.1).
As shown in Table 2, the students performed uniformly poorly regardless of the question and
the group they belonged to.
Next, we will consider each question separately, analyzing the answers the students gave and
the types of mistakes they made.
Question 1 was a multiple-choice question, so the possible mistakes were picking (c), square of
difference, or failing to answer. The summary of the percentages of mistakes made is given in
4
Issues in the Undergraduate Mathematics Preparation of School Teachers
the table below. For comparison, the table also contains the percentages of correct answers (in
boldface).
Table 3. Percentages of mistakes made in answering Question 1
Answers
Groups (a) (b) (c) left blank
1 57.1%* 21.4% 17.9%* 3.5%
2 80.0% 12.9% 2.9% 4.2%
3 76.6% 19.2% 2.1% 2.1%
4 77.2% 14.0% 5.3% 3.5%
5 79.0% 15.8% 5.3% 0%
6 82.9% 14.3% 0% 2.8%
7 50.0%* 37.5% 5.6% 6.9%
* See the footnote after Table 2
As seen from the table, the majority of students in each group chose the difference of squares
as the name of the expression "4 x 2 − 9 y 2 ." One might think that the reason for this choice is
that the students noticed that 4x 2 and 9 y 2 are equal to ( 2 x ) 2 and ( 3 y ) 2 , respectively.
Therefore, as one might think, the students have skipped one step in their heads and come up
with difference of squares. Indeed, some of them did. They wrote that they picked the difference
the
of squares because " square root of 4 is 2 and the square root of 9 is 3" (Group 4, Finite
Math), or "both squares can be ( ) as well as their numbers" (Group 4), or "it is a subtraction
of two perfect squares" (Group 5, Precalculus). On the other hand, there were explanations of the
following type "both variables, x and y , are squared" (Group 1, Beginning Algebra, and Group
4), or "they both have squares and a subtraction sign" (Group 2, Intermediate Algebra), or "it is
x 2 − y 2 , just has coefficients" (Group 5). The message is clear here: the students see only the
squares of x and y , and ignore the fact that they are multiplied by coefficients. There w ere
other types of responses that showed that students are unfamiliar with the notion of the structure
of an expression. For instance, they circled the difference of squares because "it is two different
variables" (Group 5), or "I have always heard that terminology" (Group 6, Calculus), or "it is the
only phrase I have heard of " (Group 6), or "don't know why just sounds right" (Group 6).
Moreover, not all the students chose the correct answer because they understood the structures of
expressions well, but because "they have the same square and the products are different" (Group
2), or "they are two different variables" (Group 5). The person who selected the difference of
products as the answer, explained "it can't be the difference of two squares because x and y are
not the same number and therefore cannot be subtracted from each other while written in this
form" (Group 7, Advanced Education Course).
Students' mistakes on Question 2 can be divided into four categories: (1) Cancelled the twos
/
2+ x 2+ x
to get x or 1 + x or x / 2 ; (2) Made an equation = 0 and solved it getting x = −2 ;
/
2 2
2
(3) Multiplied by but ended up multiplying by 2 only the numerator obtaining 4 + 2 x
2
5
L. Pomerantsev, O. Korosteleva : Do Prospective Elementary and Middle School Teachers…
(or, erroneously, x + 4 or 2 + x or 2( x + 1) ); (4) Didn't attempt to do the problem or got unusual
x+1
answers like or 1 or 3/2, etc. Notice that the mistakes indicate a poor knowledge of the
x
structure of an algebraic expression and a poor command of the algebraic rules. Table 4 presents
the percentages of responses in the four categories (along with the percentage of the correct
responses).
Table 4. Percentages of mistakes made in answering Question 2
Categories of Mistakes
Group (1) (2) (3) (4) correct
1 71.4%* 0% 0% 3.6% 25.0%
2 54.3% 1.4% 1.4% 14.3%* 28.6%
3 38.3% 6.4% 10.6% 0% 44.7%
4 26.3% 0% 5.3% 5.3% 63.1%
5 15.8% 0% 3.5% 5.3% 75.4%
6 8.6% 2.9% 2.8% 2.8% 82.9%
7 41.7% 4.2% 4.2% 5.5% 44.4%
* See the footnote after Table 2
From the table is it apparent that an overwhelming majority of students in all the groups have
made a mistake in category (1) that shows their conceptual misunderstanding of the structural
form of the expression. The students in Group 1, Beginning Algebra, had the most trouble with
the question. This group had the lowest percentage who got the question right, and the largest
percentage who made a mistake in category (1). However, a high percentage (82.9%) of the
students in the Calculus group, Group 6, answered the problem correctly and only 8.6% made a
mistake in category (1). This indicates that more practice in higher level mathematics brings
understanding of algebraic language.
Some of the students who wrote the correct answer "impossible" also wrote correct
explanations like "no common factors" (Group 5, Precalculus) or "2 is not a common factor of
the numerator" (Group 5) or "the 2 does not factor in ( 2 + x) " (Group 5), or "it is already
simplified to the lowest terms" (Group 2, Intermediate Algebra). Some explanations, however,
were wrong and made it plain that students have difficulties with symbolic language. For
instance, some chose the correct answer "impossible" because "you don't know what x is"
(Group 2, and Group 7, Advanced Education Course), or "the solution to the problem is not
2+ x
defined (ex., = ? )" (Group 7).
x
Question 3, even though, seemingly analogous to Question 2, turned out to be
insurmountably difficult for some of the participants. Typical mistakes can be classified as
6
Issues in the Undergraduate Mathematics Preparation of School Teachers
2 2x
(1) Cancelled 2 x 's or 2's to obtain 2 or 3 or
or ; (2) Said "impossible"; (3) Left blank; or
2x 2
(4) Wrote incomprehensible answers (totaling 28 varieties). The percentages of each type of
mistakes are summed up below.
Table 5. Percentages of mistakes made in answering Question 3
Categories of Mistakes
Group (1) (2) (3) (4) correct
1 60.7%* 25.0%* 0% 14.3% 0%
2 58.6%* 2.9% 7.1% 21.4%* 10.0%
3 31.9% 2.1% 4.3% 31.9% 29.8%
4 24.6% 8.8% 3.5% 21.0% 42.1%
5 21.1% 14.0% 8.8% 19.3% 36.8%
6 5.7%** 0% 2.9% 20.0% 71.4%*
7 38.9% 11.1% 5.6% 22.2% 22.2%
* ** See the footnote after Table 2
The poorest performance was observed in Group 1, Beginning Algebra (none of the students
answered the question correctly, and 60.7% used the cancellation rule improperly), while the
students in the Calculus course (Group 6) did the best on this question. A significantly higher
percentage (71.4%) of the Calculus students came up with the right answer and only 5.7% of
them cancelled improperly.
As for Question 4, there were three kinds of typical mistakes: (1) Plugged x = ±5 into the
equation ( x + 1) 2 = 25 to get contradictions 36=25 and 16=25; (2) Tried to solve the quadratic
equation ignoring the given statement (sad to say, only two students managed to get the correct
answer this way); (3) did not do the problem or wrote "impossible" or something else. The
results are given below.
Table 6. Percentages of mistakes made in answering Question 4
Categories of Mistakes
Group (1) (2) (3) correct
1 32.1% 35.7% 32.2% 0%
2 34.3% 22.8% 40.0% 2.9%
3 31.9% 46.8% 10.6% 10.7%
4 21.1% 35.1% 14.0% 29.8%
5 17.5% 42.1% 14.1% 26.3%
6 11.4% 37.1% 8.6% 42.9%
7 26.4% 43.0% 25.0% 5.6%
Notice that none of the learners in Group 1, Beginning Algebra, did the problem correctly,
while the Calculus students, Group 6, got the highest percentage of correct answers (42.9%). A
7
L. Pomerantsev, O. Korosteleva : Do Prospective Elementary and Middle School Teachers…
large proportion of students in all the groups have committed the mistake of the first type, which
shows their understanding of the need to use the symbolic pattern but a misunderstanding of the
structure of the equation. Instructive was the reasoning behind the answers. For example, as an
explanation for why it was impossible, in his opinion, to solve the problem at hand, one of the
students in Group 7, Advanced Education Course, wrote "This is impossible to solve. If
x 2 = 25 , then ( x + 1) 2 = 25 is not solvable because no matter what x equals, positive or
negative, then if you add one to it and square it, it would not equal 25." This stresses once again
the students' lack of understanding when they are dealing with symbolic language.
Question 5 was easier for the participants but some of them were taken aback by the fact that
two different variables, x and y , were used. They explained "Can't solve because no x in
equation." (Group 3, Finite Math) or "I'm confused because the variable was changed from x to
y." ( Group 2, Intermediate Algebra). When working on this question, the students either
(1) Ignored the given statement and solved the equation in y directly; or (2) Left the space blank
or wrote something incoherent. The results are below.
Table 7. Percentages of mistakes made in answering Question 5
Categories of Mistakes
Group (1) (2) correct
1 32.1%* 53.6%* 14.3%
2 57.1% 37.2%** 5.7%
3 61.7% 17.0% 21.3%
4 52.6% 14.1% 33.3%
5 54.4% 19.3% 26.3%
6 57.1% 8.6% 34.3%
7 59.7% 22.2% 18.1%
* ** See the footnote after Table 2
A really low percentage of correct answers was observed in Groups 1 and 2 (Beginning and
Intermediate Algebra) (14.3% and 5.7%). The most mathematically advanced Calculus Group
(Group 6) did better (34.3%), even though the differences are not statistically significant.
Notably, the students in Group 1 have made the smallest proportion of mistakes of the first type
(trying to solve the equation in y ), and the largest proportion of untypical errors.
Discussion
The conducted study has shown that college students perform unsatisfactorily in the
manipulation of algebraic expressions. The questions on the test dealt with recognizing the
structure of an algebraic expression (Questions 1, 4 and 5), and applying rules for cancellation of
a common factor (Questions 2 and 3). The problem of conceptual misunderstanding of both
topics is most severe at the novices' level and is still substantial in Calculus classes. From Table
2, the poorest performance is shown by Groups 1 and 2, which is not surprising since these
students have failed the ELM. The future K-8 multiple subject teachers (Groups 3 and 7) did
8
Issues in the Undergraduate Mathematics Preparation of School Teachers
show slightly better results but not significantly better. Group 4 (predominantly Business majors)
gave noticeably more correct responses than the previously-considered groups. Finally, for the
pre-calculus and calculus students (Groups 5 and 6), the percentage of correct answers was the
highest but still quite low. Consequently, if students are not taught algebraic structures and the
language of algebra properly in elementary courses, the non-understanding will persevere
throughout their studies and later in their careers.
Researchers recognize the need of a powerful method to teach the subject to elementary
school teachers (Carpenter et al. 2000; CBMS, 2001; Kaput, 1995; Wagner et al, 1999a; Kieran,
1999). Unfortunately, researchers agree, no such method exists yet. As Kieran (1999) points out,
"… it is not obvious how the use of symbol manipulators in the early stages of learning algebra
can help students develop a structural conception of algebraic expressions. This is the question
for future research." The current study once again underscores the need for this research.
References
Booth, L. (1989). A question of structure. In S. Wagner & C. Kieran (Eds.), Research Issues in
the Learning and Teaching of Algebra (pp. 57-59). Reston, VA: National Council of Teachers of
Mathematics; Hillsdale, NJ: Lawrence Erlbaum.
Booth, L. (1999). Children's difficulties in beginning algebra. In B. Moses (Ed.), Algebraic
Thinking, Grades K-12 (pp. 299-307). Reston, VA: National Council of Teachers of
Mathematics.
Carpenter, T., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary
grades. Research report 00-2, University of Wisconsin-Madison, Madison, WI: National Center
for Improving Student Learning and Achievement in Mathematics and Science.
Conference Board of the Mathematical Sciences. (2001). The mathematical education of
teachers. Issues in mathematics education, v. 11. AMS, Providence, RI: CBMS.
Herscovic, N., & Linchevski, L. (1994). A cognitive gap between arithmetic and algebra.
Educational Studies in Mathematics, 27, 59-78.
Kaput, J. (1995). Transforming algebra from and engine of inequity to an engine of mathematical
power by "algebrafying" the K-12 curriculum. 73rd annual meeting of the National Council of
Teachers of Mathematics, Boston, MA.
Kieran, C. (1989). The early learning of algebra: a structural perspective. In S. Wagner & C.
Kieran (Eds.), Research Issues in the Learning and Teaching of Algebra} (pp. 33-56). Reston,
VA: National Council of Teachers of Mathematics; Hillsdale, NJ: Lawrence Erlbaum.
Kieran, C. (1999). The learning and teaching of school algebra. In B. Moses (Ed.), Algebraic
Thinking, Grades K-12 (pp. 341-361). Reston, VA: National Council of Teachers of
Mathematics.
9
L. Pomerantsev, O. Korosteleva : Do Prospective Elementary and Middle School Teachers…
Krishner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics
Education, 20, 274-287.
Lodholz, R. (1999). The transition from arithmetic to algebra. In B. Moses (Ed.), Algebraic
Thinking, Grades K-12 (pp. 52-58). Reston, VA: National Council of Teachers of Mathematics.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and
objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36.
Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification -- the case of algebra.
Educational Studies in Mathematics}, 26, 191-228.
Wagner, S., & Kieran, C. (1999a). An agenda for research on the learning and teaching of
algebra. In B. Moses (Ed.), Algebraic Thinking, Grades K-12 (pp. 362-372). Reston, VA:
National Council of Teachers of Mathematics.
Wagner, S., & Parker, S. (1999b). Advancing algebra. In B. Moses (Ed.), Algebraic Thinking,
Grades K-12 (pp. 328-340). Reston, VA: National Council of Teachers of Mathematics.
Yerushalmy, M. (1992). Syntactic manipulations and semantic interpretations in algebra: the
effect of graphic representation. Learning and Instruction, 2, 303-319 | 677.169 | 1 |
Our users:
It's good news for any school, teacher or student when such a fantastic software is specifically designed for Algebra. Great job! Joseph K., MN
The Algebrator could replace teachers, sometime in the future. It is more detailed and more patient than my current math teacher. I, personally, understand algebra better. Thank you for creating it! Michael, OH
Thank you for the responses. You actually make learning Algebra sort of fun. James Mathew, CA
Students struggling with all kinds of algebra problems find out that our software is a life-saver. Here are the search phrases that today's searchers used to find our site. Can you find yours among them? | 677.169 | 1 |
Book DescriptionThe Yaglom geometric transformation books are aimed at a mathematical Olympiad audience. The books are not primarily about transformations, they are mainly about how to use transformations to solve problems. Yaglom wrote the book in Russian back in the fifties, presenting geometrical transformations techniques to geometric problem solving. He introduces transformation types and then shows how they can be used to solve problems that otherwise would sometimes be almost unsolvable using elementary (synthetic) methods. Then he presents a set of problems that vary from difficult to fiendish. Full solutions are provided, you will need them. The Russian book has chapters on congruence, similarity and projective transformations that are contained in the three English translations published by the MAA. It continues with sections on inversion transformations that are not (yet?) translated into English.
This is the third of three books, it presents projective transformations and how to use them to solve geometric problems in the euclidean domain. Although dealing with projective transformations it is definitely not a book on projective geometry. It does contain a section on non euclidean geometry but that deals mainly with hyperbolic geometry. | 677.169 | 1 |
Algebra and Trigonometry
Description
Beecher, Penna, and Bittinger's Algebra and Trigonometry is known for enabling students to "see the math" through its focus on visualization and early introduction to functions. With the Fourth Edition, the authors continue to innovate by incorporating more ongoing review to help students develop their understanding and study effectively.
Mid-chapter Mixed Review exercise sets have been added to give students practice in synthesizing the concepts, and new Study Guide summaries provide built-in tools to help them prepare for tests. MyMathLab has been expanded so that the online content is even more integrated with the text's approach, with the addition of Vocabulary, Synthesis, and Mid-chapter Mixed Review exercises from the text, as well as example-based videos created by the authors.
Features
Functions appear early and integrated, reflecting the authors' belief that functions are best taught as a theme of the course, not as an isolated topic.
Functions are introduced in Chapter 1, so that students to start the course with a new topic rather than a review of equation-solving that was covered in previous math courses.
Students will come to understand the concept of a function by being exposed repeatedly to thelanguage, notation, and use of functions throughout the text.
The authors take a visual approach to the course. The early introduction to functions allows for the use of graphs to provide a visual aspect to solving equations and inequalities. In addition, specific features enable students to "see the math" and make connections between concepts.
Algebraic/Graphical Side-by-Side Examples present the solutions in a two-column format to help students understand the connection between algebraic manipulation and the graphical interpretation.
Visualizing the Graph exercises help develop students' ability to make the mental link between different types of equations and their corresponding graphs.
Connecting the Concepts, a hallmark feature of the text, invites the student to stop and check their understanding of how concepts work together in one section or several sections. Concepts are summarized visually-using graphs, outlines, or charts-so that students deepen their understanding and make connections.
Ongoing review features throughout the text reinforce the concepts and help students build understanding.
NEW! Mid-chapter Review exercises are one-page mixed review sets at logical breaks in the chapter, helping students to reinforce their understanding of the concepts. These exercises are assignable in MyMathLab.
NEW! Study Summaries have been added to the Chapter Review, giving students a built-in study aid when reviewing and preparing for tests. In MyMathLab, these Study Summaries are accompanied by new videos to reinforce the key concepts and ideas.
Enhanced! Vocabulary Review exercises appear in the last section of each chapter, and check students' understanding of the language of mathematics. These are now assignable in MyMathLab and can serve as reading quizzes.
Enhanced! Synthesis exercises, included at the end of each exercise set, encourage critical thinking by asking students to apply multiple skills or concepts within a single exercise. For the Fourth Edition, these are assignable in MyMathLab.
Classify the Function exercises, appearing in the Skill Maintenance section of the exercise sets, ask students to identify a number of functions by their type (linear, quadratic, rational, etc.). Throughout the text, the variety of functions increases and these exercises become more challenging.
Review Icons refer students to an earlier, related section where they can go to review prerequisite concepts that are needed for the current section.
Study Tips are occasional, brief reminders in the margin, to promote effective study habits such as good note taking and exam preparation.
Technology Connections are optional sections that guide students in the use of the graphing calculator as another way to check problems.
Zeros, Solutions, andx-Intercepts are a theme of the text. The authors aim to help students see the connection between the real zeros of the function, the solutions of the associated equation, and the first coordinates of the x-intercepts of the graph. When students develop their understanding of these connections, their probability of success increases for this course.
New to this Edition
Additional ongoing review features have been integrated throughout, to help students reinforce their understanding and improve their success in the course.
Mid-chapter Review exercises are one-page mixed review sets at logical breaks in the chapter, helping students reinforce their understanding of the concepts.
Study Summaries have been added to the Chapter Review, giving students a built-in study aid when reviewing and preparing for tests.
MyMathLab is more closely integrated with the text and now offers new question types, for a more robust online experience that mirrors the authors' approach.
Example-based videos, created by the authors themselves, walk students through the detailed solution process for key examples in the textbook. Videos have optional subtitles.
Vocabulary exercises have been added, which can serve as reading quizzes.
Mid-chapter Reviews are new to the text and are assignable online, helping students to reinforce their understanding of the concepts.
Study Summaries are new to the text, and in MyMathLab theseare accompanied bysectionsummary videos, which cover key definition and procedures from the text.
Sample homework assignments are pre-selected by the authors for each section. These are indicated in the Annotated Instructor Edition by a blue underline within each end-of-section exercise set. These homework sets are assignable in MyMathLab.
The first four chapters of the text have been reorganized, to make the material easier to teach and learn. By presenting this material in four chapters rather than three, the level of difficulty is more balanced in this new edition.
Table of Contents
R. Basic Concepts of Algebra
R.1 The Real-Number Systemeal Numbers
R.2 Integer Exponents, Scientific Notation, and Order of Operations
R.3 Addition, Subtraction, and Multiplication of Polynomials
R.4 Factoring Terms with Common Factors
R.5 The Basics of Equation Solving
R.6 Rational Expressions
R.7 Radical Notation and Rational Exponents
Study Guide
Review Exercises
Chapter Test
¿
1. Graphs; Linear Functions and Models
1.1 Introduction to Graphing
Visualizing the Graph
1.2 Functions and Graphs
1.3 Linear Functions, Slope, and Applications
Visualizing the Graph
Mid-Chapter Mixed Review
1.4 Equations of Lines and Modeling
1.5 Linear Equations, Functions, Zeros, and Applications
1.6 Solving Linear Inequalities
Study Guide
Review Exercises
Chapter Test
¿
2. More on Functions
2.1 Increasing, Decreasing, and Piecewise Functions; Applications
2.2 The Algebra of Functions
2.3 The Composition of Functions
Mid-Chapter Mixed Review
2.4 Symmetry and Transformations
Visualizing the Graph
2.5 Variation and Applications
Study Guide
Review Exercises
Chapter Test
¿
3. Quadratic Functions and Equations; Inequalities
3.1 The Complex Numbers
3.2 Quadratic Equations, Functions, Zeros, and Models
3.3 Analyzing Graphs of Quadratic Functions
Visualizing the Graph
Mid-Chapter Mixed Review
3.4 Solving Rational Equations and Radical Equations
3.5 Solving Linear Inequalities
Study Guide
Review Exercises
Chapter Test
¿
4. Polynomial and Rational Functions
4.1 Polynomial Functions and Modeling
4.2 Graphing Polynomial Functions
Visualizing the Graph
4.3 Polynomial Division; The Remainder and Factor Theorems
Mid-Chapter Mixed Review
4.4 Theorems about Zeros of Polynomial Functions
4.5 Rational Functions
Visualizing the Graph
4.6 Polynomial and Rational Inequalities
Study Guide
Review Exercises
Chapter Test
¿
5. Exponential and Logarithmic Functions
5.1 Inverse Functions
5.2 Exponential Functions and Graphs
5.3 Logarithmic Functions and Graphs
Mid-Chapter Mixed Review
5.4 Properties of Logarithmic Functions
5.5 Solving Exponential Equations and Logarithmic Equations
5.6 Applications and Models: Growth and Decay; Compound Interest
Study Guide
Review Exercises
Chapter Test
¿
6. The Trigonometric Functions
6.1 Trigonometric Functions of Acute Angles
6.2 Applications of Right Triangles
6.3 Trigonometric Functions of Any Angle
¿ Mid-Chapter Mixed Review
6.4 Radians, Arc Length, and Angular Speed
6.5 Circular Functions: Graphs and Properties
6.6 Graphs of Transformed Sine and Cosine Functions
Visualizing the Graph
Study Guide
Review Exercises
Chapter Test
¿
7. Trigonometric Identities, Inverse Functions, and Equations
7.1 Identities: Pythagorean and Sum and Difference
7.2 Identities: Cofunction, Double-Angle, and Half-Angle
7.3 Proving Trigonometric Identities
Mid-Chapter Mixed Review
7.4 Inverses of the Trigonometric Functions
7.5 Solving Trigonometric Equations
Visualizing the Graph
Study Guide
Review Exercises
Chapter Test
¿
8. Applications of Trigonometry
8.1 The Law of Sines
8.2 The Law of Cosines
8.3 Complex Numbers: Trigonometric Form
Mid-Chapter Mixed Review
8.4 Polar Coordinates and Graphs
Visualizing the Graph
8.5 Vectors and Applications
8.6 Vector Operations
Study Guide
Review Exercises
Chapter Test
¿
9. Systems of Equations and Matrices
9.1 Systems of Equations in Two Variables
Visualizing the Graph
9.2 Systems of Equations in Three Variables
9.3 Matrices and Systems of Equations
9.4 Matrix Operations
9.5 Inverses of Matrices
9.6 Determinants and Cramer's Rule
9.7 Systems of Inequalities and Linear Programming
9.8 Partial Fractions
Study Guide
Review Exercises
Chapter Test
¿
10. Analytic Geometry Topics
10.1 The Parabola
10.2 The Circle and the Ellipse
10.3 The Hyperbola
10.4 Nonlinear Systems of Equations and Inequalities
Visualizing the Graph
Mid-Chapter Mixed Review
10.5 Rotation of Axes
10.6 Polar Equations of Conics
10.7 Parametric Equations
Study Guide
Review Exercises
Chapter Test
¿
Photo Credits
Answers
Index
Index of Applications
Author
Judy Beecher has an undergraduate degree in mathematics from Indiana University and a graduate degree in mathematics from Purdue University. She has taught at both the high school and college levels with many years of developmental math and precalculus teaching experience at Indiana University-Purdue University Indianapolis (IUPUI). In addition to her career in textbook publishing, she enjoys traveling, spending time with her grandchildren, and promoting charity projects for a children's camp.
Judy Penna received her undergraduate degree in mathematics from Kansas State University and her graduate degree in mathematics from the University of Illinois. Since then, she has taught at Indiana University-Purdue University Indianapolis (IUPUI) and at Butler University, and continues to focus on writing quality textbooks for undergraduate mathematics students. In her free time she likes to travel, read, knit and spend time with her children.
Marvin Bittinger has been teaching math at the university level for more than thirty-eight years. Since 1968, he has been employed at Indiana University-Purdue University Indianapolis (IUPUI), Professor Bittinger currently lives in Carmel, Indiana with his wife Elaine. He has two grown and married sons, Lowell and Chris, and four granddaughters. | 677.169 | 1 |
Vedic Mathematics is the name given to the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). The aim of such sutra is to simplify the entire calculations and arithmetical operations.
Vedic Maths Course
There are 16 sutras in vedic mathematics as per Sri Bharati Krsna Tirthaji Maharaj. The below mentioned list of topics (course content) is how I look at it.
Cube Roots (by seeing only & not by working on it). Different methods for perfect cube and non-perfect cube
Dates and calendar
Magic Square
Special mathematical tricks and techniques
Application of vedic mathematics in trigonometry, geometry, algebra and calculus
Value (finding and remembering) of fractions
Checking the accuracy of different mathematical operations
Differential calculus
Solving an algebraic equation of one variable
"Transpose and adjust (the coefficient)"
Solving simultaneous equation and quadratic equation
"If the Samuccaya is the same (on both sides of the equation, then) that Samuccaya is (equal to) zero"
By the Paravartya rule
"If one is in ratio, the other one is zero."
Solving cubic equation
Solving fractional equation
"The ultimate (binomial) and twice the penultimate (binomial) (equals zero),"
Through (QM) I'll try to cover all the above vedic maths related topics. Lot of other topics, techniques, shortcuts will be part of QM. Mathematical games, puzzles, riddles, etc. are also included to make every kid sharp.
Hello , thanks admin for writing so many methods , tricks , techniques for solving Additions ,Subtractions , Multiplications & Divisions in Quicker math"s ! Interested students are requested to Learn one by one to know Basic Math's & Vedic Maths !! I am also one of the Tutor to teach Basic Math's & Vedic Maths as Home Tuition Classes & through Mail id for Outstation Students .To know more details please do write on my mail id is [email protected] thanks once again Quicker Math's * I have passed M.Sc. ( STAT ) in Ist class from Nagpur University in 1981 .
Sir,
I wanted to solve data interpretation questions (Bank Po Exam) fast.
I understand all the questions but get failure at the time of solving. It takes too much of time. Please tell me some tricks to solve such questions quickly.
Thanking You,
Regards | 677.169 | 1 |
Burleson CalculusThis means that I will explain fundamentals with the aid of practical examples. I think the best way to understand a subject is to solve a large number of example problems. I consider myself a very patient and painstaking person | 677.169 | 1 |
The best Collection of Math cheat sheetsMany Cheat Sheets in your mobile to have the formulas wherever and whenever you want.With this application you can use your travel time to study, or just have it as a quick reference when needed.NOTE: App can be moved to the SD Card!!Contents:=========Algebra. Elementary techniques for factoring binomials and trinomialsAlgebra. Exponent laws and factoring tipsAlgebra. Solving quadratic equations by completing the squareAlgebra. College Algebra quick referenceAlgebra. Solution of the 3rd degree polynomial equationAlgebra. Solution of the 4th degreee polynomial equationTrig. Basic trig identitiesTrig. Law of sines cosines etc and other triangle formulasTrig. Graphs of the trig functionsTrig. Inverse trig functionsTrig. Power reducing formulas for powers of sines and cosinesTrig. Graph paper for plotting in polar coordinatesTrig. Two unit circles with trig funcion valuesTrig. Single unit circle with trig function valuesCalculus. Basic differentiation formulas and some useful trig identitiesCalculus. Basic differentiation and integration formulasCalculus. Definitions and theorems pertaining to Riemann sums and definite integralsCalculus. A quick reference sheet on Taylor polynomials and seriesCalculus. A summary of convergence testsCalculus. Guidelines for evaluating integrals involving powers of sines and cosinesCalculus. Guidelines for evaluating integrals involving powers of secants and tangentsCalculus. Standard forms for conic sectionsCalculus. Common infinite seriesCalculus. Trigonometric substitutionCalculus. Cylindrical coordinatesCalculus. Spherical coordinatesCalculus. Hyperbolic functionsCalculus. Applications of integralsCalculus. Applications of integralsCalculus. Common ordinary differential equationsCalculus. Common ordinary differential equationsCalculus. Common ordinary differential equationsCalculus. Undetermined coefficients and variation of parametersCalculus. Vector formulasCalculus. Simple summary of cylindrical and spherical coordinatesMisc. Some prime and composite numbersMisc. Sets. Functions lines and sequencesStatistics formula sheet Page 1Statistics Page 2Statistics Page 3 | 677.169 | 1 |
Ohio Graduation Test Mathematics Review
Author:
Unknown
ISBN-13:
9781932410303
ISBN:
1932410309
Pub Date: 2002 Publisher: American Book Company
Summary: REA's new Mathematics test prep for the Ohio Graduation Test (OGT) provides all the instruction and practice that students need to excel. Passing this exam is required to receive a high school diploma. The book's review covers the areas articulated in Ohio's Academic Content Standards for Mathematics: Number, Number Sense, and Operations; Measurement; Geometry and Spatial Sense; Patterns, Functions, and Algebra; and ...Data Analysis and Probability . Includes two full-length practice tests and complete explanations of all answers. Details: - All materials in this book are aligned with Ohio's Academic Content Standards - Two full-length practice tests- Lessons enhance all skills necessary for the exam- Confidence-building tips reduce test anxiety and boost test-day readiness"REA ... Real review, Real practice, Real results."[read more] | 677.169 | 1 |
Algebra.com is a free online algebra study guide and problem solver designed to supplement any algebra course. There are hundreds of solved problems, video solutions, sample test questions, worksheets, and interactives.
Discussion for Algebra Study Guide with Videos
John Redden
(Faculty)
I originally I wrote this study guide to benefit our online algebra students. Online students seldom see problems worked out by hand. In addition, the videos were designed to supplement regular course instructional materials. It soon became clear that my traditional in-class students were benefiting from the material as well. With that I decided to open it up to everyone on the web as an open educational resource. I truly hope it helps.
Technical Remarks:
This is a Blogger website linking to YouTube videos. It is mobile friendly. | 677.169 | 1 |
books.google.com thirteen books of Euclid's Elements. The works of Archimedes, including The method. Introduction to arithmetic
User ratings | 677.169 | 1 |
9780495389613
ISBN:
0495389617
Edition: 4 Pub Date: 2008 Publisher: Cengage Learning
Summary: Algebra can be like a foreign language. But one text delivers an interpretation you can fully understand. Building a conceptual foundation in the "language of algebra," approac...hes
Tussy, Alan S. is the author of Elementary and Intermediate Algebra (with CengageNOW Printed Access Card), published 2008 under ISBN 9780495389613 and 0495389617. Three hundred seventy seven Elementary and Intermediate Algebra (with CengageNOW Printed Access Card) textbooks are available for sale on ValoreBooks.com, one hundred twenty four used from the cheapest price of $23.62, or buy new starting at $163.49ISBN-13:9780495389613
ISBN:0495389617
Edition:4th
Pub Date:2008 Publisher:Cengage Learning
Valore Books is the top book store for cheap Elementary and Intermediate Algebra (with CengageNOW Printed Access Card) rentals, or used and new condition books available to purchase and have shipped quickly. | 677.169 | 1 |
Intermediate Algebra - With 2 CDs - 10th edition
Summary: This concise and cumulative guide shows students the art of technical writing for a variety of contexts and institutions. Using examples from the business and non-corporate world, the book emphasizes transactional writing through practical explanations, real-world examples, and a variety of ''role-playing'' exercises. Each section builds on the next as readers learn a variety of models of style and format. This edition features a stronger emphasis on electronic commu...show morenication, integrated coverage of ethics, and more explanation of how to create technical documents that produce concrete results. ...show less
3.1 The Rectangular Coordinate System 3.2 The Slope of a Line 3.3 Linear Equations in Two Variables Summary Exercises on Slopes and Equations of Lines 3.4 Linear Inequalities in Two Variables 3.5 Introduction to Functions
Chapter 4: Systems of Linear Equations
4.1 Systems of Linear Equations in Two Variables 4.2 Systems of Linear Equations in Three Variables 4.3 Applications of Systems of Linear Equations 4.4 Solving Systems of Linear Equations by Matrix Methods
9.1 The Square Root Property and Completing the Square 9.2 The Quadratic Formula 9.3 Equations Quadratic in Form Summary Exercises on Solving Quadratic Equations 9.4 Formulas and Further Applications 9.5 Graphs of Quadratic Functions 9.6 More about Parabolas and Their Applications 9.7 Quadratic and Rational Inequalities
11.1 Additional Graphs of Functions 11.2 The Circle and the Ellipse 11.3 The Hyperbola and Functions Defined by Radicals 11.4 Nonlinear Systems of Equations 11.5 Second-Degree Inequalities and Systems of Inequalities14436240321443624 | 677.169 | 1 |
Introduction: High school graduation requires a minimum of 20 units of Math, and must include passing Algebra 1 or its equivalent. The paths below are models to be used as guidelines only in creating your own unique program. Consult your SHS Course Catalog for ideas and ask your parents, teachers and counselor for suggestions in developing a meaningful four-year plan which will best prepare you for your particular post-graduation plans. The future is yours; plan for it!
Course Descriptions
ALGEBRA 1*
Grade
Duration
Credits
Repeat Status
9-12
Year
5/5
No
Fulfills Requirements: Math for SHS, for UC/CSU, c Prerequisite: Algebra Readiness or teacher recommendation Course Description: This course is designed to meet the California state requirement for Algebra 1. Its topics include operations of real numbers, equations and their applications, graphing, systems of equations, exponents and radicals, polynomials and factoring, quadratic functions and equations, rational expressions. Students must pass Algebra 1 (or its equivalent in Algebra 2) to graduate. * Also Sheltered Algebra 1
Fulfills Requirements: Math for SHS, for UC/CSU, c Prerequisite: "B" or better in Algebra I. Freshmen must have an excellent score on the SHS placement test as well as a teacher recommendation and a satisfactory GPA. Course Description: This course will study proofs and applications of angle relationships, perpendicular lines, parallel lines and planes, component triangles, similar polygons, constructions, loci, coordinate geometry, areas and volumes. Trigonometry and symbolic logic may also be introduced. Recommended for math and physical science majors.
ALGEBRA 2
Grade
Duration
Credits
Repeat Status
10-12
Year
5/5
No
Fulfills Requirements: Math for SHS, for UC/CSU, c Prerequisite: A "C" or better in Algebra and Geometry Course Description: This course is designed for college-prep students who would like to continue their study in algebra but who do not intend to pursue a math or physical science major. Topics of study include systems of numbers, polynomials and rational expressions, linear equations and inequalities, coordinate geometry, relations and functions, quadratic functions, conic sections and trigonometry. Sequences and series may be included if time permits.
Fulfills Requirements: Math for SHS, for UC/CSU, c Prerequisite: Previous teacher recommendation. Freshmen must have an excellent score on the SHS Placement Test, a satisfactory GPA and a teacher recommendation. Course Description: This course is open to freshman students with an exceptionally strong history of high math achievement who are also motivated to accelerate their math education. Topics will include systems of numbers, polynomials, rational expressions, linear equations and inequalities, coordinate geometry, relations and functions, quadratic functions, systems of sentences, real exponents, logarithmic functions, conic sections, sequences and series. May also include probability, statistics, trigonometry, matrices, determinants and vectors. Recommended for math and science majors. Much self-discipline is required in this course.
Fulfills Requirements: Math for SHS, for UC/CSU, c, g Prerequisite: Previous teacher recommendation Course Description: This is a very challenging, fast-paced course with major emphasis on an introduction to calculus.
AP CALCULUS (B/C)
Grade
Duration
Credits
Repeat Status
10-12
Year
5/5
No
Fulfills Requirements: Math for SHS, for UC/CSU, c, g Prerequisite: Pre-Calculus Course Description: AP Calculus is a university-level calculus course intended for those who may wish to continue to advanced work in mathematics, the sciences, engineering or business at the college level. The course content and expectations will conform to the Advanced Placement Calculus BC curriculum as described in the current College Board "acorn" book. These topics include: functions and graphs; derivatives (concept of derivative, techniques for finding derivatives, and application of derivative); integrals (interpretations applications of integrals, applications of integral, techniques of antidifferentiation); polynomial approximations and aeries. Students are expected to take the AP Calculus BC Exam in May.
Fulfills Requirements: Math for SHS, for UC/CSU, c, g Prerequisite: Algebra 2 with a grade of A or B both terms or consent of instructor Course Description: The purpose of this course is to introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. Students are exposed to four broad conceptual themes:
The advanced placement statistics course curriculum will be covered in two high school semesters. Students who successfully complete the course and examination may receive credit and/or advanced placement for a one-semester introductory college statistics course.
Fulfills Requirements: Math for SHS Prerequisite: None Course Description: This course is designed to prepare students for Algebra 1. Its topics include preliminary mathematical and arithmetic concepts and skills. Additionally, it introduces students to 3 Algebraic concepts that students will then build upon in Algebra 1.
TUTORING If your student is struggling with his/her homework, check out hotmath.com. It is a free resource for our students which shows the step-by-step process for solving math problems that have been assigned as homework. Contact your student's math teacher for the password.
Contact your teacher for lists of private tutors in the Santa Cruz area. | 677.169 | 1 |
Beginning Algebra
9780321769527
ISBN:
032176952X
Edition: 8 Pub Date: 2011 Publisher: Prentice Hall PTR
Summary: Tobey, John Jr, Jr. is the author of Beginning Algebra, published 2011 under ISBN 9780321769527 and 032176952X. Six hundred fifty Beginning Algebra textbooks are available for sale on ValoreBooks.com, one hundred sixteen used from the cheapest price of $119.35, or buy new starting at $166the primary subject of this book is math. The book is effective. It helped me to not only go over what I already knew and review it but also take my time to understand what I didn't know or already forgot.
The most interesting thing I learned in this book was shortcuts with fractions. I am one of many people who does not like fractions. This book has ways to simplify those fractions which makes it easier to do them.
I didn't really find anything in this book to be not helpful. I like that it has an online part to compete homework, and to help explain different parts of algebra problems so you would understand better
I think this book is closely related to other algebra books I've had to use, mostly in high school. | 677.169 | 1 |
Guys, I am in need of help on factoring, cramer's rule, inverse matrices and triangle similarity. Since I am a newbie to Remedial Algebra, I really want to understand the basics of Intermediate algebra completely. Can anyone recommend the best place from where I can begin learning the fundamental principles? I have an exam next week.
Hi, Algebrator available at the site can be of great aid to you. I am a math tutor who give private math classes to students and I recommend Algebrator to my pupils since that aids them a lot when they sit to solve their homework by themselves at home.
That's true, a good software can do miracles . I used a few but Algebrator is the best. It doesn't make a difference what class you are in, I myself used it in Basic Math and Algebra 1 too, so you don't have to worry that it's not on your level. If you never had a program before I can assure you it's not complicated, you don't need to know anything about the computer to use it. You just have to type in the keywords of the exercise, and then the software solves it step by step, so you get more than just the answer.
I remember having difficulties with simplifying expressions, decimals and graphing lines. Algebrator is a really great piece of algebra software. I have used it through several algebra classes - Intermediate algebra, Algebra 2 and Algebra 2. I would simply type in the problem and by clicking on Solve, step by step solution would appear. The program is highly recommended. | 677.169 | 1 |
innovative book, two experienced educators present a fresh and engaging approach to mathematics learning in the middle grades with the transition from arithmetic to algebra. The authors provide a collection of balanced, multi-dimensional assessment tasks designed to evaluate students' ability to work with mathematical objects and perform mathematical actions. Assisting teachers in their efforts to put into practice the NCTM and Common Core State Standards, these assessments were carefully developed and tested to make them as revealing and adaptable as possible, suitable for incorporation into any curriculum. Teachers will appreciate the explicit and illustrative material the authors include to specifically help assess the mathematical understanding of students in grades 58. The text features a teachers' guide to each task, reproducible student tasks, and solutions and rubrics. | 677.169 | 1 |
You are here
Math for Teachers: An Exploratory Approach
Edition:
2
Publisher:
Kendall Hunt
Number of Pages:
645
Price:
0.00
ISBN:
9780757581069
At my current institution, "Mathematics for Elementary Teachers" is a one-semester course that meets for 6 hours per week, ostensibly divided into three hours of lecture and three of laboratory weekly. I have been teaching that course for many years — indeed, no one else currently in my department has taught it — and in that time, I have looked at many textbooks for that audience and that course. As is the case with many standard service courses, there seems to be considerable agreement on most of the topics to be covered, so I have developed my own core list of criteria for evaluating these books.
First, I hope that a math-for-elementary-teachers textbook will be a resource for future teachers — something they can keep with them as they move out of my class and into their first teaching position. On that score, Stein and Wallace have written a fine text. The emphasis is on the mathematics, and while the students' goal to teach is not far from the surface, the content manages to dominate. Indeed, there is no laundry list of NCTM Standards to detract from the primacy of the mathematics. (I accept that others may regard this as a flaw.)
I also hope that students will find the mathematics they will use as professionals in their textbook, and so I look carefully for a full section explaining the normal distribution and the mathematics behind percentiles, which teachers will need when trying to interpret their students' standardized test results. Unfortunately, no such section is present here, though there is a very brief mention of percentiles. While that is a flaw in my opinion, it's one that can be easily filled in by those who feel it's important.
That, however, is the only concern I have about this book. The standard topics are all here and covered in an unusual level of detail — which is to be expected when the book includes more than a year's worth of material. A student armed with this book and with the experience of learning from it will be well-prepared, mathematically, for a career as an elementary school teacher.
Mark Bollman ([email protected]) is associate professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. His claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.
Comments
This book provides a wonderful range of topics across all grades. In addition to the content typically covered in an elementary and middle school mathematics class you can find star polygons, Fermat's last theorem, semi-regular tessellations, perspective drawings, line designs, box puzzles and many more. Many chapters also contain interesting historical connections or explorations.
I like the balance between short descriptions and examples and the wealth of interesting explorations. In working with school teachers I often find it challenging to convey to them what I consider a "good problem" and how to find one. In my opinion a good problem motivates students to explore a mathematical question for which they want to know the solution without having to be told how to do it. Well, this book really provides a school teacher with many good problems to run or enrich a mathematics classroom. The material is challenging enough to cover part of high school mathematics as well.
Having taught classes for future teachers and offered professional development for many years now, I strongly believe that it is necessary to focus on inquiry-based learning and teaching instead of a mostly lecture-centered classroom. It is very difficult for College students and school teachers to shift their beliefs towards teaching mathematics in an inquiry-based way - mostly because their own experiences in lecture based mathematics classrooms have left them to expect that a problem cannot be solved unless an expert first explains how to do it. For that reason I find it essential to use books like Robert Stein's that support the inquiry of mathematics.
The focus of the book is clearly the mathematics (and not the methods of teaching it) but the book does provide much more: It covers in detail the different models for the operations, it repeatedly emphasizes the need to understand rather than memorize, and gives many helpful suggestions and connections for the use of the problems in a classroom.
I highly recommend Robert Stein's book as a resource for content and methods classes for future teachers as well as a support for in service school teachers. | 677.169 | 1 |
Here are a bunch of programs that I've written. All of these are for the TI 83 series calculators, which means that they work for the TI 83 plus, TI 83 plus silver edition, TI84 plus, and TI 84 plus silver edition calculators. These are all written in the TI Basic language, on my TI 84 plus silver edition. To put these programs on your calculator, you need a TI graph link cable and the TI connect software that comes with the calculator. To get more detailed instructions, visit the TI website. There are several categories of programs - games, utilities, animations, and miscellaneous fun.
To download the programs, click the blue names of the categories to download a zip file with all of those programs.
Calculator Dance Party
These are programs that make your calculator more useful. Some solve common equations or formulas and make it quicker for you to do simple mathematical functions, others help make the use of your calculator easier or better.
Ambiguous Case
This is a simple utility for solving the ambiguous case - a part of the pre-calculus curriculum where you determine the number of solutions a triangle could have based on one side and two angles. The ambiguous case involves comparing the sine of several numbers. This program just needs side A and angle a "little a" and angle b "little b" and it solves it for you.
Chemistry Help
This program is possibly the most useful one I've written. To find the conversion factor between one mole of a molecule and the number of grams, you have to multiply the atomic mass of each element by the number of that element and add those numbers to get the total. This program does that and then stores the result for X so that you can easily use it in a calculation.
Chemistry Help 2
This program helps you finds the percent composition of a substance from the atomic mass and number of atoms of each element in a molecule.
Chemistry Help 3
This program helps you find the molecular formula. It takes the atomic mass and number of atoms of each element in a compound, and then the molecular mass, and tells you the number you need to multiply the empirical formula by.
Distance Formula
This is a simple program that finds the distance between two points using the formula d = squareroot((x1-y1)squared + (x2-y2)squared). It saves a few keystrokes and helps if you forget the formula.
Graph Edit
This program allows you to set the window to one of several different preset views. The default view can be difficult to use because the pixels are irrational numbers, so this utility has views that make each pixel .25 or 1, making it much easier to find points on a line.
Key Find
This is mostly for writing other programs. It simply tells you the getkey value of the button you press. The getkey function is used for receiving input when writing programs.
Quadratic Formula
This is a simple utility to solve the quadratic formula.
Simplify Squareroot
This utility helps you simplify complex squareroots, or even roots to other powers, like cube roots.
Stats
This utility does some basic statistical functions like mean and standard deviation with the numbers in list 1.
Stick Figure Dance
Chuck
This is one of several programs built along the same idea. Whenever you type something in, instead of getting the answer, you get something else. In this one whenever you hit "enter", the calculator says "Chuck Norris is watching you".
IM Bored
And so I was when I wrote this program. This simply makes a little asterisk bounce around the screen.
Keypad Arrows
This program simply makes an asterisk move around the screen. It is different from the many other programs that do this because it uses the keypad (numbers 8, 4, 6, 2) as arrows instead of the arrow keys.
Move 0
Yep. It moves a 0 around the screen. This is a good program to learn to write if you are trying to learn to program calculators.
Not Quite
This change any answer by a random number between -3 and 3. So you cold type in 1+1 and get randomly -1, 0, 1, 2, 3, 4, or 5. Great for stumping friends.
Wrong Answer
Instead of giving you the answer the calculator says "Im not telling you".
The purpose of these is to provide a basis for writing other programs. I keep these on my calculator so I can copy them into other programs to save me time writing them. To copy a program into another one, be in the program editing screen and then press [2nd] [RCL] (above the STO-> button) and hit the left arrow. Select a program and it will copy into the one you are currently editing.
Basic Menu
This is a 5 option dynamic menu written on the graph screen instead of using the default menu option.
Direction Input
This is an empty frame for programs where you are using the arrows to move something around the screen. | 677.169 | 1 |
More About
This Textbook
Overview to help students reach different results. A variety of fundamental proofs demonstrate the basic steps in the construction of a proof and numerous examples illustrate the method and detail necessary to prove various kinds of theorems.
Jumps right in with the needed vocabulary-gets students thinking like mathematicians from the beginning
Offers a large variety of examples and problems with solutions for students to work through on their own
Includes a collection of exercises without solutions to help instructors prepare assignments
Editorial Reviews
From the Publisher
"I really enjoyed the "Collection of Proofs." It is a great addition. These exercises will really stretch a student's imagination, and go a long way to impressing on them the standards for a believable proof and the necessity of understanding a proposition before embarking on its proof (or the search for a counter-example).
It should be clear that I find NBP an indispensable adjunct to my linear algebra course and to my efforts to increase my students' mathematical maturity. The new material only makes a great book even greater." Robert Beezer, University of Puget Sound.
Meet the Author
Antonella Cupillari is an associate professor of mathematics at Pennsylvania State Erie in Behrend College. She received her Laurea in Mathematics in Italy, and her M.A. and Ph.D. at the State University of New York at Albany. She has been a participant in the Mathematical Association of America/National Science Foundation Institute on the "History of Mathematics and Its Use in Teaching." Cupillari is the author of several papers in analysis, mathematics education, and the history of mathematics. She is also the author of the first edition of The Nuts and Bolts of Proof | 677.169 | 1 |
Purpose: To introduce the learner to the behaviour and analysis of nonlinear systems, in particular nonlinear and forced oscillations. Solutions to linear differential equations can only behave in a fairly limited number of ways, but the presence of nonlinear elements may introduce totally new phenomena. Seemingly simple nonlinear differential equations can lead to unexpectedly complex solution structures. This module introduces analytical approximation methods as well as qualitative methods for analysing the behaviour of solutions to the nonlinear systems. Contents: Perturbation methods, forced oscillations, harmonic and subharmonic response, stability of periodic solutions, bifurcation, structural stability, chaos. | 677.169 | 1 |
Another major theme in Algebra II is groups of equations, all rolled into one. This might seem complicated at first, but there are ways to organize the chaos. One way is using matrices. Yes—the singular is matrix. And yes—we are in it.
In addition to matrix notation, we will also introduce the concept of series and sequences. This section might go on…and on…and on…but it's another helpful way to keep track of lots of numbers and equations. | 677.169 | 1 |
What Do Parents Say?
I LOVE Time4Learning! It holds the attention of my kids, plus I can keep track of their learning without hovering over their shoulders.Online Algebra Lessons Scope & Sequence of Activities
Time4Learning's integrated algebra curriculum combines pre-algebra & algebra into one course that allows students to start the sequence at many different entry points, to progress at their own pace, and to move ahead or back up at any time. Algebra is available by request at no additional cost.
Time4Learning's online algebra program combines engaging lessons in a logical sequence to build a sound foundation for algebra. Each unit begins with multimedia lessons followed by interactive practice exercises. Printable worksheets offer additional practice and online assessments are available to parents in order to track progress.
For more information, view the algebra overview or learn how Time4Learning can help to conquer math anxiety. Time4Learning's educational content is provided primarily by CompassLearning Odyssey®.
Math - Algebra Lesson Plans Total Activities: 313
Non-Scored
Scored
Worksheet
Answer Key
Quiz
Test
Chapter 1: "Algebra Tool Tutorials"
Lesson
Activity Name
Type
LA#
Worksheet
Odyssey
Writer
Algebra Tool Tutorials:
Algebra Balance Tutorial
AL178
Algebra Tiles Tutorial
AL179
Calculator Tutorial
AL180
Equation Writer Tutorial
AL181
Grapher Tutorial
AL182
Test
Chapter 2: "Arithmetic with Letters"
Lesson
Activity Name
Type
LA#
Worksheet
Odyssey
Writer
Arithmetic and Algebra: This lesson introduces true, false, and open statements.
Range, Mean, Median, and Mode: This lesson defines range, mean, median, and mode and explains how to calculate each for a given set of data.
Range, Mean, Median, and Mode
AL068
Lesson Quiz: Range, Mean, Median, and Mode
Box-and-Whiskers Plot: This lesson explains how to construct a box-and-whiskers plot to illustrate the concentration and spread of data in a set
Box-and-Whiskers Plot
AL069
Lesson Quiz: Box-and-Whiskers Plot
The Probability Fraction: This lesson explains probability and how to calculate it.
The Probability Fraction
AL070
Lesson Quiz: The Probability Fraction
Probability and Complementary Events: This lesson explains how to classify probabilities as not likely or very likely, more likely or less likely. It then shows how to determine the complement of a probability event.
Probability and Complementary Events
AL071
Lesson Quiz: Probability and Complementary Events
Tree Diagrams and Sample Spaces: This lesson models the use of a tree diagram to show possible combinations in a sample space.
Tree Diagrams and Sample Spaces
AL072
Lesson Quiz: Tree Diagrams and Sample Spaces
Dependent and Independent Events: This lesson explains how to determine the probability of events that are dependent or independent of one another.
Dependent and Independent Events
AL073
Lesson Quiz: Dependent and Independent Events
The Fundamental Principle of Counting: This lesson explains how to use the fundamental principle of counting to find the number of possible outcomes.
The Fundamental Principle of Counting
AL074
Lesson Quiz: The Fundamental Principle of Counting
Application: Multistage Experiments: This lesson illustrates an application of calculating probability for experiments with multiple stages.
Powers and Roots: Calculator: Explore powers and roots using a calculator
Powers and Roots: Calculator
AL172
Odyssey Writer: Powers and Roots: Calculator
AL173
Chapter Test: Irrational Numbers and Radical Expressions:
Test
Chapter 14: "Geometry"
Lesson
Activity Name
Type
LA#
Worksheet
Odyssey
Writer
Angle and Angle Measure: This lesson defines types of angles and line segments and demonstrates computation of angle measures.
Angle and Angle Measure
AL118
Lesson Quiz: Angle and Angle Measure
Pairs of Lines in Plane and in Space: This lesson contrasts relationships of pairs of lines in a plane and in space and explains the relationships among angles formed in a plane.
Pairs of Lines in Plane and in Space
AL119
Lesson Quiz: Pairs of Lines in Plane and in Space
Angle Measures in a Triangle: This lesson demonstrates the proofs of the theorems that the sum of the measures of the angles of any triangle is 180 degrees and the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
Angle Measures in a Triangle
AL120
Lesson Quiz: Angle Measures in a Triangle
Naming Triangles: This lesson classifies triangles by their angles and by their sides.
Naming Triangles
AL121
Lesson Quiz: Naming Triangles
Quadrilaterals: This lesson presents five types of quadrilaterals and the measurement of their angles.
Quadrilaterals
AL122
Lesson Quiz: Quadrilaterals
Congruent and Similar Triangles: This lesson introduces congruent and similar triangles and the theorems that prove congruence.
Congruent and Similar Triangles
AL123
Lesson Quiz: Congruent and Similar Triangles
Trigonometric Ratios: This lesson explains trigonometric ratios and the formulas for calculating sine, cosine, and tangent.
Trigonometric Ratios
AL124
Lesson Quiz: Trigonometric Ratios
Application: Using Geometric Shapes: This application explores the number of diagonals that can exist in a geometric shape, as a predictable pattern, related to the number of sides.
Sign up for Time4Learning and gain access to a variety of educational materials, which will engage and challenge your child to succeed. Make Time4Learning a part of your children's homeschool resources. | 677.169 | 1 |
Multiplication with Exponents Division with Exponents Operations with Monomials Addition and Subtraction of Polynomials Multiplication with Polynomials Binomial Squares and Other Special Products Dividing a Polynomial by a Monomial Dividing a Polynomial by a Polynomial Summary Review Cumulative Review Test Projects
5. FACTORING
The Greatest Common Factor and Factoring by Grouping Factoring Trinomials More Trinomials to Factor The Difference of Two Squares Factoring: A General Review Solving Equations by Factoring Applications Summary Review Cumulative Review Test Projects
Review of Solving Equations Equations with Absolute Value Compound Inequalities and Interval Notation Inequalities Involving Absolute Value Factoring the Sum and Difference of Two Cubes Review of Systems of Equations in Two Variables Systems of Equations in Three Variables Summary Review Cumulative Review Test Projects
8. EQUATIONS AND INEQUALITIES IN TWO VARIABLES
The Slope of a Line The Equation of a Line Linear Inequalities in Two Variables Introduction to Functions Function Notation Algebra with Functions Variation Summary Review Cumulative Review Test Projects53 +$3.99 s/h
Good
One Stop Text Books Store Sherman Oaks, CA
2003-02-26 Paperback Good
$30.95 +$3.99 s/h
Good
Books Revisited Chatham, NJ
Possible retired library copy, some have markings or writing.
$52.00 +$3.99 s/h
Good
a2zbooks Burgin, KY
The text has some marking, the cover has wear with lower front corner curl, creases and has clear tape on the upper lower spine. The title page is missing. Quantity Available: 1. ISBN: 0534398790. I...show moreSBN/EAN: 9780534398798. Inventory No: 1560805222. ...show less
$52.00 +$3.99 s/h
Good
A2ZBooks Ky Burgin, KY
Pacific Grove, CA 2004 Softcover Good Condition The text has some marking, the cover has wear with lower front corner curl, creases and has clear tape on the upper lower spine. The title page is mis...show moresing. Quantity Available: 1. ISBN: 0534398790. ISBN/EAN: 9780534398798. Inventory No: 1560805222 NO CD. Has a name inside the front cover and a school stamp on the title...show more a "pink" stain on the lower corner outer page edges. Has NO CD. Quanti...show morety Available: 1. ISBN: 0534398790. ISBN/EAN: 9780534398798. Inventory No: 1560780740 NO CD. Has a name insi...show morede the front cover and a school stamp on the title a "pink" stain on the ...show morelower corner outer page edges. Has NO CD. Quantity Available: 1. ISBN: 0534398790. ISBN/EAN: 9780534398798. Inventory No: 1560780740 | 677.169 | 1 |
Barry McQuarrie's Math 1001 Archive
Math 1001 Survey of Math
Course Prerequisites
To succeed in this course you will need to have had two years of high school math.
Goals
This course provides an overview of mathematics as used in our society.
A student who successfully completes this course will
gain proficiency with mathematical models relating to a wide spectrum of real life situations,
including scheduling, the traveling salesman problem, and personal finance.
be able to critically assess these models, the assumptions inherent in the models, and their
applicability to different situations,
understand basic statistics and probability,
understand symmetry, and identify symmetry in the world around them,
understand tiling, and construct simple tilings,
use a spreadsheet to analyze data and understand personal finance and other mathematical ideas.
Textbook
NOTE: TEXTBOOK EDITION IS ACCURATE FOR SPRING 2010.
The textbook for the course is
For All Practical Purposes, 8th Ed., COMAP.
The bookstore will have the latest edition, and
the course calendar below is based on the 8th Edition.
The differences
between the editions is usually minimal, but if you use an earlier edition be aware that some of the
sections may be numbered differently, content may be slightly different, and problems listed as
practice below may not line up with your older edition.
This is a very good book, in my opinion, but it certainly contains far more material than we
will cover in this class. It should prove to be an excellent resource for you in the future.
To be prepared for the lectures you should read the section the lecture is on
before the lecture is given. I will typically not be able to cover everything from the
section in the lecture, but I will indicate what material you are responsible for
from each section.
Time Commitment
To succeed in this course you will need to be willing to spend,
per week, nine hours outside of
class reading the textbook and working problems (UMM policy is
that one credit is defined as three hours of learning effort per
week for an average student to earn an average grade in the course:
4 credits times 3 hours/week/credit - 3 hours/week in
lecture = 9 hours/week outside class).
Course Components
Practice.
On the course webpage I suggest practice homework
problems for each lecture.
You should do as much extra practice as you deem necessary to
enhance your understanding of a topic.
Falling behind in
this course,
as in any university course,
can lead to disaster,
so it is important that you keep up with the material. Practice problems are not graded.
Brain Builders.
In class I will hand out short Brain Builders, which are exercises based on some of the
concepts we are studying. Sometimes these Brain Builders will be completed and turned in during class,
sometimes I will let you take them home and turn them in the following class.
The Brain Builders should take about half an hour to complete. The course
calendar below list tentative dates for the Brain Builders.
Assignments.
Assignments will involve more complex problems than on the Brain Builders.
Assignments will be handed out in class, and collected in class (the
due dates are listed on the calendar below).
Assignments will be handed in at the beginning of class on the day they are due,
unless you have spoken to me beforehand and I have granted an extension.
Putting assignments in my mailbox or under my office door while I am teaching another course is
severely frowned upon unless we have agreed that you will be doing this.
If this is done when I am teaching your class I will not accept the work--believe it or not,
people have actually done this!
I am demanding that solutions be written up well.
This means solutions
should be a self-contained document. They should be written legibly,
contain diagrams or tables where appropriate, and
should state the problem and explain the solution. Interspersing
English sentences which explain what you are
doing can help in this regard.
With its worked-out examples, the book provides many examples of a
good solution. To say it a different way, solutions with
totally correct computations lacking in necessary good explanations
will tend to receive 85%, not 100%.
It is OK to collaborate on assignments, and I anticipate many of you will work with other
students in the class, however, every student turns in their
own solutions to all the problems on each assignment.
Collaboration does not mean that others do your thinking for you.
Collaboration in this course means there is a good back and forth conversation among study
partners, but never direct copying of another's work.
For example, if a study partner gets
stuck on a problem, you should help them get unstuck by telling them in words what it was that
you did to get past the part they are stuck on.
Using words instead of showing them your work is important, since they
will then have been provided a hint but will still need to do the work themselves.
This facilitates learning, which simply copying your work will not. If in helping a classmate
you get to the point where you think you need to show them your work for them to
be able to answer the question, don't show them your work--it is time for them to come
visit office hours.
Excel.
Excel is a component of some assignments, and each student will create their own Excel-based solutions when
these are asked for. Basically, you should not work two people to one computer--if two people are working on
separate computers they can talk with each other if they get stuck, but each person creates their own solution,
and that is what I want. Do not leave copies of your assignments on public computers! Copy them to your own disk
and then delete them from the Recycle Bin before you leave a public computer.
Exams.
You will not be allowed any outside material on
your desks during these exams.
You will need a calculator that can do exponents (23=8 for example) for some of the problems on
the tests and final.
Debriefing after tests should be done during office hours, after you have had a chance to reflect
on the exam.
Grading
Here is the University-wide uniform grading policy.
A Represents achievement that is outstanding relative to the level
necessary to meet course requirements.
C Represents achievement that meets the course requirements in every
respect.
D Represents achievement that is worthy of credit even though it
fails to fully meet the course requirements.
F Represents failure and indicates that the coursework was completed
but at a level unworthy of credit, or was not completed and there was
no agreement between the instructor and student that the student would
be temporarily given an incomplete.
A few of you may be taking the course S-N. In this case, you need to
earn a C- to receive an S. An incomplete grade (I) is only given under truly extraordinary circumstances (falling behind
in the course is not a sufficient reason for an I to be granted).
The grade
for the course will be calculated by the following formula:
Brain Builders
20%
Assignments
45%
Tests
35%
Your numerical grades will be converted to letter grades and
finally Grade Points via the following cutoffs
(see the UMM Catalog for more on
Grades and Grading Policy):
Numerical
95%
90%
87%
83%
80%
77%
73%
70%
65%
60%
Below 60%
Letter
A
A-
B+
B
B-
C+
C
C-
D+
D
F
Grade Point
4.00
3.67
3.33
3.00
2.67
2.33
2.00
1.67
1.33
1.00
0.00
Respectful Classroom
Be in class on time. I nor you fellow classmates enjoy
the disruption late arrival causes. I know that situations crop
up that will entail late arrival (please come even if you are
late!) but try to ensure it is the exception and not the rule.
If you need to leave class early, let me know before class and slip out
as unobtrusively as possible.
During class, cell phones and music devices should be turned off,
and headphones removed from ears.
To ask a question during class, you can get my attention by saying my name
(``Barry, could you explain how you know the graph has an Euler circuit?") or raise your hand.
As a student you may experience a range of issues that can cause barriers to
learning, such as strained relationships, increased anxiety, alcohol/drug problems,
feeling down, difficulty concentrating, and/or lack of motivation. These mental health
concerns or stressful events may lead to diminished academic performance or reduce a
student`s ability to participate in daily activities.
If you have any special needs or requirements to
help you succeed in the class, come and talk to me as soon as
possible, or visit the appropriate University service yourself.
You can learn more about the range of services available on campus by visiting
the websites:
Cooperation is vital to your future success,
which ever path you take. I encourage cooperation amongst
students where ever possible, but the act of copying or other
forms of cheating will not be tolerated.
Academic dishonesty in any portion of the academic work for a course is grounds for
awarding a grade of F or N for the entire course.
Any act of plagiarism
that is detected will result in a mark of zero on the entire assignment
or test for both parties.
If you are in any way unclear about what constitutes
academic dishonesty, reread the earlier section on Assignments where I discuss
collaboration, and please come and talk to me if you have any questions.
UMM's Academic Integrity policy and procedures can be found at
Since the assignments are handed out days in advance,
only under exceptional circumstances (which can be officially documented) will I accept late work.
You will receive a mark of zero if an assignment is submitted late. However, please talk with me
asap (do not wait until the next class) if you missed turning
something in, even if it is after the deadline.
If you are going to miss an exam, let me know in
advance so we can work out alternate plans. Taking an exam early can usually
be arranged.
Lecture Preparation
The majority of your learning will take place outside of lectures, as you
work problems and read the text. You will not learn everything you need to learn in this
course simply by coming to lectures, nor if you miss lectures. You must come to lectures and
put in the time outside of class to master the material.
To get the most out of the course you should
work on homework
for the course every day.
I can not stress enough how
important it is that you work problems! The homework
identifies the types of problems from the text that
should be mastered.
read the section before the lecture
and do not fall behind.
Make sure when you are reading that you are reading for comprehension. This means you are
thinking about what you are reading, rereading paragraphs when necessary, and pausing to work through
examples to ensure you understand them completely. Make notes about the material, especially
anything you don't fully understand. Then come see me, your study group, or a tutor to discuss
these concepts so that you do understand them. Reading for
comprehension takes practice, but it is an essential skill to develop to help you succeed at
university. As you read, try to focus on understanding rather than memorization.
discuss any difficulties with me during office hours, or by appointment,.
Please make the most of my Office Hours! When you come to office hours, come prepared: for conceptual
questions bring your notes on the topic and any problems you have
done relating to that topic; for homework questions, bring the work you have done on
that problem.
form a study group.
Exam Preparation
Here are some suggestions to guide your preparation for tests.
If you have a technique which works for you and isn't listed
here, please let me know so I can pass it on to your peers!
Review assignments and homework solutions.
This will provide you with an overview of the material you need to be studying.
Review the concepts and vocabulary in the text.
Can you talk about the concepts? Do you know the basic results from the concept review?
Make notes on the topics you are studying as you review.
Write short sentences to describe how to solve
problems. Describe verbally to a friend how you would solve a
particular type of problem. This verbal description will help you remember the process
of solving particular problems during the test.
Do problems from the text which have solutions that are similar to problems
seen in class or assigned as homework.
Branch out and do other types of problems that appeared less frequently
throughout the section.
Studying in many short sessions is more effective than one or
two marathon studying sessions.
Consider making a time schedule which maps out when and what you will study.
You might choose a long term time frame (Friday Morning: History,
Friday Afternoon: Precalculus, etc),
and a short term time frame for each day that lists what exactly you will focus on.
The short term time frame
can be created every day and be more flexible.
Create goals which you can reasonably be expected to meet.
Get as much sleep as possible while you study for tests.
Come to your exams well rested,
and mentally sharp.
Study in an environment that mimics the environment the test will take place in.
It should be quiet and clear of clutter.
Use the practice tests questions provided on the course webpage as a practice test,
maybe only doing a selection of the problems so it is a bit shorter. Answer these questions as if you
are taking a test, without the textbook or any other resources that will not be provided on the
test.
For a given chapter (or section), create practice "tests" for yourself,
maybe three or four questions which you have the solution to,
and then answer them without reference to the text. Correct your test yourself,
or work with a friend and have them correct your test
and you correct theirs. Do not move on to other questions until
you have mastered these ones. You might consider
imposing a time limit on these mini-tests.
If you do study in groups, also study alone so you can focus on the
types of questions you need to work on.
Come and talk with me (email me to set an appointment if necessary)
if there are questions you have.
Course Calendar
Here is the tentative lecture schedule. You are responsible for any changes
to this schedule which are announced in class. | 677.169 | 1 |
From differentiation to integration - solve problems with ease
Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or, worse yet, not know where to begin? Have no fear! This hands-on guide focuses on helping you solve the many types of calculus problems you encounter in a focused, step-by-step manner. With just enough refresher explanations before each set of problems, you'll sharpen your skills and improve your performance. You'll see how to work with limits, continuity, curve-sketching, natural logarithms, derivatives, integrals, infinite series, and more!
100s of Problems!
Step-by-step answer sets clearly identify where you went wrong (or right) with a problem
The inside scoop on calculus shortcuts and strategies
Know where to begin and how to solve the most common problems
Use calculus in practical applications with confidence
We do not deliver the extra material sometimes included in printed books (CDs or DVDs). | 677.169 | 1 |
Glencoe Secondary Mathematics to the Common Core State Standards, Algebra 2 SE Supplement
Mastering the Achieve ADP Algebra II EOC Exam
Math Triumphs--Foundations for Algebra 2
Summary
TheStudy Guide & Intervention Workbookcontains two worksheets for every lesson in the Student Edition. Helps students: Preview the concepts of the lesson, Practice the skills of the lesson, and catch up if they miss a class. Tier 2 RtI (Response to Intervention) addresses students' needs up to one year below grade level. | 677.169 | 1 |
2nd Edition
Today's mathematics classrooms increasingly include students for whom English is a second language. Teaching Mathematics to English Language Learners provides readers a comprehensive understanding of both the challenges that face English language learners (ELLs) and ways in which educators might | 677.169 | 1 |
MERLOT Search - category=2513&materialType=Online%20Course
A search of MERLOT materialsCopyright 1997-2013 MERLOT. All rights reserved.Fri, 6 Dec 2013 10:39:54 PSTFri, 6 Dec 2013 10:39:54 PSTMERLOT Search - category=2513&materialType=Online%20Course
4434Algebra2go: An Online Supplemental Instruction Tool Array
Algebra2go is a free unrestricted collection of pre-algebra and algebra related study materials designed to address the affective dimensions of student learning.Algebra I Online
This course contains both content that reviews or extends concepts and skills learned in previous grades and new, more abstract concepts in algebra. Students will gain proficiency in computation with rational numbers (positive and negative fractions, positive and negative decimals, whole numbers, and integers) and algebraic properties. New concepts include solving two-step equations and inequalities, graphing linear equations, simplifying algebraic expressions with exponents, i.e. monomials and polynomials, factoring, solving systems of equations, and using matrices to organize and interpret dataAn Introduction to Complex Numbers
This is a free, online textbook/course that teaches about complex numbers. It is a workbook that has exercises through-out, with some of the answers provided as audio files.ASCII Art
cool site on ascii artCalculusCalculus for Beginners and Artists
This complete course in Calculus for beginners is one of MIT's OpenCourseWare offerings. It includes nearly a dozen Java applets to illustrate some of the concepts covered; there is a corresponding set of Flash applets with accompanying audio.Calculus I
This free and open online course in Calculus 1 was produced by the WA State Board for Community & Technical Colleges [ is the mathematics of CHANGE and almost everything in our world is changing.CalBut II
This free and open online course in Calculus II.YouCalculus III
This free and open online course in Calculus III Causal and Statistical Reasoning
This online course comes from the Open Learning Initiative (OLI) by Carnegie Mellon. "The course includes self-guiding materials and activities, and is ideal for independent learners, or instructors trying out this course package."״Does excessive exposure to violent video games cause violent behavior? Does increased gun availability cause more crime or less? This course examines the nature of causal claims and the statistical sorts of evidence used to support them." 'Our material is delivered in three forms - Concept modules, Case studies, and the Causality Lab. The Case Studies are a collection of over one hundred short news pieces (1-3 pages) - that each deal with some study concerned with a causal claim. They are listed in the Syllabus as part of the Appendix - and they can viewed alphabetically or in hierarchy by topical area (e.g. Health, Social Sciences, etc.). You will read several of them as part of the concept modules, but they are interesting in their own right and we urge you to explore them and find your own over the web. If you find a particularly interesting study we have not included, please send us the URL by email <[email protected]>, and we will try to incorporate the study into the repository.The Causality Lab is a virtual environment to simulate the science of causal discovery. The lab contains a "true" causal model behind the scenes that was created by the instructor (or another student), and your job is to set-up experiments, collect data, create hypotheses, and compare the predicitons from your hypotheses against the data to find the truth. Causality Lab exercises are included as a regular part of the course, but they are also available as a series of stand alone lessons accessible from the Syllabus in Unit 7: Causality Lab Lessons.The "Concept modules" are meant to present the basic concepts and terminology behind Causal and Statistical Reasoning. Each is meant to cover about the same amount of material delivered in a textbook chapter. Each includes text, pictures, movies, simulations, questions for you to answer, and a quiz at the end of the module that you might be assigned to take for credit. They take anywhere between one to five hours to complete. We have grouped the modules into five topical areas: •Area 1: Causal Theories •Area 2: Statistical Evidence: Association and Independence •Area 3: Causal Theories --> Statistical Evidence •Area 4: Statistical Evidence -->Causal Theories: Problems •Area 5: Statistical Evidence -->Causal Theories: Strategies' | 677.169 | 1 |
Here is an update to Algebra/Calculus Assistant, version 0.2. I've added some more mathematical functions.
Spoiler for README:
Algebra/Calculus Assistant Version 0.2
Coded in lua by andyauff
----------------------------------------------------
CHANGELOG:
Version 0.2
--Got rid of the find slope and y-intercept option and replaced it with the equation conversions (standard to y-intercept
and vice versa). The standard to y-intercept still says the slope and y-intercept, though.
--Added function to find the inverse of a 2x2 matrix
--Added function to simplify any fraction into its simplest form
Version 0.1 (Initial Release)
-Features:
--Find slope and y-intercept of a line in standard form
--Find determinant of a 2x2 matrix
--Find determinant of a 3x3 matrix
--Find the solution to a system of 2 equations with 2 variables
-----------------------------------------------------
WHAT IS IT?
This is a simple lua application for the PSP that has 7 useful mathematical functions built into it.
WHAT DOES IT DO?
These are the 7 functions built into the program:
1) Convert equations from standard form to y-intercept form
2) Convert equations from y-intercept form to standard form
3) Find the determinant of a 2x2 matrix
4) Find the determinant of a 3x3 matrix
5) Find the solution to a system of 2 equations with 2 variables*
6) Find the inverse of a 2x2 matrix
7) Simplify a fraction
* This means that it finds the intersection of two lines in standard form
That's just about it. The controls are simple and there are some in-application controls instructions too. | 677.169 | 1 |
Introduction to Closure notion of closure pervades mathematics, especially in the fields of topology and projective geometry. Demonstrating this pervasiveness in the field, this graduate-level book provides a complete introduction to closure systems. With an emphasis on finite spaces and algebraic closures, the text covers graph theory, ordered sets, lattices, projective geometry, and formal logic as they apply to the study of closures. Each chapter presents a vignette to illustrate the topic covered. The author also includes numerous exercises as well as concrete examples to support the material discussed. | 677.169 | 1 |
Trigonometry - 6th edition
Summary: This easy-to-understand trigonometry text makes learning trigonometry an engaging, simple process. The book contains many examples that parallel most problems in the problem sets. There are many application problems that show how the concepts can be applied to the world around you, and review problems in every problem set after Chapter 1, which make review part of your daily schedule. If you have been away from mathematics for awhile, study skills listed at the beginning of the first...show more six chapters give you a path to success in the course. Finally, the authors have included some historical notes in case you are interested in the story behind the mathematics you are learning. This text will leave you with a well-rounded understanding of the subject and help you feel better prepared for future mathematics108359 Premium Books are Brand New books direct from the publisher sometimes at a discount. These books are NOT available for expedited shipping and may take up to 14 business day...show mores to receive. ...show less
$290.34 +$3.99 s/h
New
Bookeyez2 snellville, GA
Brand New Title. We're a Power Distributor; Your satisfaction is our guarantee!
$296.70 +$3.99 s/h
New
textbook_rebellion2 Troy, MI
0495108359317.93 +$3.99 s/h
New
PROFESSIONAL & ACADEMIC BOOKSTORE Dundee, MI
0495108 | 677.169 | 1 |
... lines of code around spatial geometry and linear algebra, you will also find a multitude of mathematical features beyond geometry. All methods are comprehensively documented. We offer a royalty-free full ...
Digital Challenge is a set of interactive activities for use in teaching basic digital concepts. The activities give students immediate feedback to reinforce correct responses. All student responses are corrected and ...
... >Very easy exciting and fast to use 2. Algebra : > >Can store up to three algeriac ... add, subtract, multiply, and divide of any two algebraic equations algebraically and produce an algebraic result, it ...
... allows the application of the construction concept in Algebra, Calculus, etc. Use its full potential to create interactive and dynamic math calculations and visualizations and you will discover new ways ...
... The libraries include numerical and analytical calculations, linear algebra operations, equation solving algorithms. Many libraries are based on the JAIDA classes for data manipulation, construction of histograms and functions. jHepWork ...
... populations in models described in the Markovian process algebra PEPA. GPA efficiently implements these techniques and extensions allowing calculation of accumulated rewards and computation of passage time probabilities. The ...
With Linear Algebra, you can solve systems of linear equations using the LU factorization of the matrix of coefficients. You can also perform different operations (add, subtract, multiply), finding the ...
It is a calculator for algebra.The inputs and outputs are in algebraic format.It can do the operations of add, subtract, ... factorized. It records the history of operations on algebraic functions.You can copy one function to another which ...
... in a very short time. Included areas: Arithmetic, Algebra, Geometry, Trigonometry, Analytic Geometry and miscellaneous.Visual Mathematics, a member of the VirtualDynamics Math Virtual Lab, is an Intuitively-Easy-To-Use software.Visual Mathematics modules ...
... Calculator An extensive equation library is included covering algebra, finance, geometry, physics, trig and business applications containing over 80 equations You can add, edit and save your own equations in ... | 677.169 | 1 |
This is a
college-level mathematics course for students who have been highly
successful in Algebra 2. The purpose is to introduce students to the
major concepts and tools of elementary statistics as they collect,
analyze, and draw conclusions from data. Students will have
extensive opportunities to explore data, plan studies, anticipate
patterns, and use statistical inference. Students enrolled in the
course are required to take the AP Statistics Exam in May. | 677.169 | 1 |
All the concepts,terminologies related to real functions have been covered in this book. more than 150 solved examples are there in the book.Graphs of basic functions are discussed in detail. A must have book for JEE aspirants Magazine Description: A must have book for all engineering aspirants. The topic REAL FUNCTIONS has been explained in detail. All the basic terminologies related to FUNCTIONS have been explained with good number of solved examples. Coverage : Number Chart,Intervals,Functions,Testing For a Function by Vertical Line Test,Domain and Range of Some Standard Functions,Value of a Function,Classification of Functions,Bijective Function,Number of bijection (One-One onto),Composite Functions,Inverse of a Function,Odd and Even Function,Periodic Function | 677.169 | 1 |
Books
Matrices
An engaging introduction to vectors and matrices and the algorithms that operate on them, intended for the student who knows how to program. Mathematical concepts and computational problems are motivated by applications in computer science. The reader learns by doing, writing programs to implement the mathematical concepts and using them to carry out tasks and explore the applications. Examples include: error-correcting codes, transformations in graphics, face detection, encryption and secret-sharing, integer factoring, removing perspective from an image, PageRank (Google's ranking algorithm), and cancer detection from cell features. A companion web site,
codingthematrix.com
provides data and support code. Most of the assignments can be auto-graded online. Over two hundred illustrations, including a selection of relevant xkcd comics.
Chapters: The Function, The Field, The Vector, The Vector Space, The Matrix, The Basis, Dimension, Gaussian Elimination, The Inner Product, Special Bases, The Singular Value Decomposition, The Eigenvector, The Linear Program
Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This new edition of the acclaimed text presents results of both classic and recent matrix analysis using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications. The authors have thoroughly revised, updated, and expanded on the first edition. The book opens with an extended summary of useful concepts and facts and includes numerous new topics and features, such as: - New sections on the singular value and CS decompositions - New applications of the Jordan canonical form - A new section on the Weyr canonical form - Expanded treatments of inverse problems and of block matrices - A central role for the Von Neumann trace theorem - A new appendix with a modern list of canonical forms for a pair of Hermitian matrices and for a symmetric-skew symmetric pair - Expanded index with more than 3,500 entries for easy reference - More than 1,100 problems and exercises, many with hints, to reinforce understanding and develop auxiliary themes such as finite-dimensional quantum systems, the compound and adjugate matrices, and the Loewner ellipsoid - A new appendix provides a collection of problem-solving hints.
Along with new and updated examples, the Third Edition features:
A novel approach to Francis' QR algorithm that explains its properties without reference to the basic QR algorithm
Application of classical Gram-Schmidt with reorthogonalization
A revised approach to the derivation of the Golub-Reinsch SVD algorithm
New coverage on solving product eigenvalue problems
Expanded treatment of the Jacobi-Davidson method
A new discussion on stopping criteria for iterative methods for solving linear equations
Throughout the book, numerous new and updated exercises—ranging from routine computations and verifications to challenging programming and proofs—are provided, allowing readers to immediately engage in applying the presented concepts. The new edition also incorporates MATLAB to solve real-world problems in electrical circuits, mass-spring systems, and simple partial differential equations, and an index of MATLAB terms assists readers with understanding the basic concepts related to the software.
The field of random matrix theory has seen an explosion of activity in recent years, with connections to many areas of mathematics and physics. However, this makes the current state of the field almost too large to survey in a single book. In this graduate text, we focus on one specific sector of the field, namely the spectral distribution of random Wigner matrix ensembles (such as the Gaussian Unitary Ensemble), as well as iid matrix ensembles. The text is largely self-contained and starts with a review of relevant aspects of probability theory and linear algebra. With over 200 exercises, the book is suitable as an introductory text for beginning graduate students seeking to enter the field., and scientists.
"This book is intended to teach useful matrix algebra to 'students, teachers, consultants, researchers, and practitioners' in 'statistics and other quantitative methods'.The author concentrates on practical matters, and writes in a friendly and informal style . . . this is a useful and enjoyable book to have at hand." -Biometrics
This book is an easy-to-understand guide to matrix algebra and its uses in statistical analysis. The material is presented in an explanatory style rather than the formal theorem-proof format. This self-contained text includes numerous applied illustrations, numerical examples, and exercises.
Ever since the Irish mathematician William Rowan Hamilton introduced quaternions in the nineteenth century--a feat he celebrated by carving the founding equations into a stone bridge--mathematicians and engineers have been fascinated by these mathematical objects. Today, they are used in applications as various as describing the geometry of spacetime, guiding the Space Shuttle, and developing computer applications in virtual reality. In this book, J. B. Kuipers introduces quaternions for scientists and engineers who have not encountered them before and shows how they can be used in a variety of practical situations.
The book is primarily an exposition of the quaternion, a 4-tuple, and its primary application in a rotation operator. But Kuipers also presents the more conventional and familiar 3 x 3 (9-element) matrix rotation operator. These parallel presentations allow the reader to judge which approaches are preferable for specific applications. The volume is divided into three main parts. The opening chapters present introductory material and establish the book's terminology and notation. The next part presents the mathematical properties of quaternions, including quaternion algebra and geometry. It includes more advanced special topics in spherical trigonometry, along with an introduction to quaternion calculus and perturbation theory, required in many situations involving dynamics and kinematics. In the final section, Kuipers discusses state-of-the-art applications. He presents a six degree-of-freedom electromagnetic position and orientation transducer and concludes by discussing the computer graphics necessary for the development of applications in virtual reality.
Numerical Methods provides a clear and concise exploration of standard numerical analysis topics, as well as nontraditional ones, including mathematical modeling, Monte Carlo methods, Markov chains, and fractals. Filled with appealing examples that will motivate students, the textbook considers modern application areas, such as information retrieval and animation, and classical topics from physics and engineering. Exercises use MATLAB and promote understanding of computational results.
The book gives instructors the flexibility to emphasize different aspects--design, analysis, or computer implementation--of numerical algorithms, depending on the background and interests of students. Designed for upper-division undergraduates in mathematics or computer science classes, the textbook assumes that students have prior knowledge of linear algebra and calculus, although these topics are reviewed in the text. Short discussions of the history of numerical methods are interspersed throughout the chapters. The book also includes polynomial interpolation at Chebyshev points, use of the MATLAB package Chebfun, and a section on the fast Fourier transform. Supplementary materials are available online. | 677.169 | 1 |
Beginning and Intermediate Algebra Activities - 2nd edition
Summary: "This collection of activities facilitates discovery learning, collaborative learning, graphing technology, connections with other areas of mathematics and other disciplines, oral and written communication, real data collection, and active learning. Designed as a stand-alone supplement for any beginning or intermediate algebra text, it incorporates the recommendations from NCTM in "Curriculum and Evaluation Standards for School Mathematics" and from AM...show moreATYC in "Crossroads in Mathematics Standards for Introductory College Mathematics Before Calculus." Providing maximum flexibility, the activities can be used during class or in a laboratory setting to introduce, teach, or reinforce a topic." ...show less
0534998739 Good Condition! Used texts may NOT contain supplemental materials such as CD40 +$3.99 s/h
New
Directtext4u El Monte, CA
PAPERBACK New 0534998739 New book may have school stamps but never issued. 100% guaranteed fast shipping! !
$15.29 +$3.99 s/h
LikeNew
Little Italy Books Marietta, GA
2004-02-16 Paperback Like New Book appears unused. covers have light wear. Pages are clean and unmarked/I will ship promptly with free delivery/tracking confirmation. Why wait, for a few dollars mo...show morere choose expedited shipping and receive your order in a couple of days. ...show less
$22.95 +$3.99 s/h
Good
Big Planet Books Burbank, CA
2004-02 | 677.169 | 1 |
Practice makes perfect-and helps deepen your understanding of 1,001 I Practice Problems For Dummies, with free access to online practice problems, takes you beyond the instruction and guidance offered in I For Dummies, giving you 1,001 opportunities to practice solving problems from the major topics in . You start with some basic operations, move on to ic properties, polynomials, and quadratic equations, and finish up with graphing.
He author defines "Geometric Computing" as the geometrically intuitive development of algorithms using geometric with a focus on their efficient implementation, and the goal of this book is to lay the foundations for the widespread use of geometric as a powerful, intuitive mathematical language for engineering applications in academia and industry. | 677.169 | 1 |
Linear Algebra. Ideas and Applications. 3rd Edition
This expanded new edition presents a thorough and up-to-date introduction to the study of linear algebra
Linear Algebra, Third Edition provides a unified introduction to linear algebra while reinforcing and emphasizing a conceptual and hands-on understanding of the essential ideas. Promoting the development of intuition rather than the simple application of methods, the book successfully helps readers to understand not only how to implement a technique, but why its use is important.
The book outlines an analytical, algebraic, and geometric discussion of the provided definitions, theorems, and proofs. For each concept, an abstract foundation is presented together with its computational output, and this parallel structure clearly and immediately illustrates the relationship between the theory and its appropriate applications. The Third Edition also features: - A new chapter on generalized eigenvectors and chain bases with coverage of the Jordan form and the Cayley-Hamilton theorem - A new chapter on numerical techniques, including a discussion of the condition number - A new section on Hermitian symmetric and unitary matrices - An exploration of computational
approaches to finding eigenvalues, such as the forward iteration, reverse iteration, and the QR method - Additional exercises that consist of application, numerical, and conceptual questions as well as true-false questions
Illuminating applications of linear algebra are provided throughout most parts of the book along with self-study questions that allow the reader to replicate the treatments independently of the book. Each chapter concludes with a summary of key points, and most topics are accompanied by a "Computer Projects" section, which contains worked-out exercises that utilize the most up-to-date version of MATLAB(r). A related Web site features Maple translations of these exercises as well as additional supplemental material.
Linear Algebra, Third Edition is an excellent undergraduate-level textbook for courses in linear algebra. It is also a valuable self-study guide for professionals and researchers who would like a basic introduction to linear algebra with applications in science, engineering, and computer science.
SHOW LESS
READ MORE >
Preface.
Features of the Text.
1. Systems of Linear Equations.
1.1 The Vector Space of m x n Matrices.
The Space Rn.
Linear Combinations and Linear Dependence.
What Is a Vector Space?
Why Prove Anything?
True-False Questions.
Exercises.
1.1.1 Computer Projects.
Exercises.
1.1.2 Applications to Graph Theory I.
Self-Study Questions.
Exercises.
1.2 Systems.
Rank: The Maximum Number of Linearly Independent Equations.
True-False Questions.
Exercises.
1.2.1 Computer Projects.
Exercises.
1.2.2 Applications to Circuit Theory.
Self-Study Questions.
Exercises.
1.3 Gaussian Elimination.
Spanning in Polynomial Spaces.
Computational Issues: Pivoting.
True-False Questions.
Exercises.
Computational Issues: Flops.
1.3.1 Computer Projects.
Exercises.
1.3.2 Applications to Traffic Flow.
Self-Study Questions.
Exercises.
1.4 Column Space and Nullspace.
Subspaces.
Subspaces of Functions.
True-False Questions.
Exercises.
1.4.1 Computer Projects.
Exercises.
1.4.2 Applications to Predator-Prey Problems.
Self-Study Questions.
Exercises.
Chapter Summary.
2. Linear Independence and Dimension.
2.1 The Test for Linear Independence.
Bases for the Column Space.
Testing Functions for Independence.
True-False Questions.
Exercises.
2.1.1 Computer Projects.
2.2 Dimension.
True-False Questions.
Exercises.
2.2.1 Computer Projects.
Exercises.
2.2.2 Applications to Calculus.
Self-Study Questions.
Exercises.
2.2.3 Applications to Differential Equations.
Self-Study Questions.
Exercises.
2.2.4 Applications to the Harmonic Oscillator.
Self-Study Questions.
Exercises.
2.2.5 Computer Projects.
Exercises.
2.3 Row Space and the Rank-Nullity Theorem.
Bases for the Row Space.
Rank-Nullity Theorem.
Computational Issues: Computing Rank.
True-False Questions.
Exercises.
2.3.1 Computer Projects.
Chapter Summary.
3. Linear Transformations.
3.1 The Linearity Properties.
True-False Questions.
Exercises.
3.1.1 Computer Projects.
3.1.2 Applications to Control Theory.
Self-Study Questions.
Exercises.
3.2 Matrix Multiplication (Composition).
Partitioned Matrices.
Computational Issues: Parallel Computing.
True-False Questions.
Exercises.
3.2.1 Computer Projects.
3.2.2 Applications to Graph Theory II.
Self-Study Questions.
Exercises.
3.3 Inverses.
Computational Issues: Reduction vs. Inverses.
True-False Questions.
Exercises.
Ill Conditioned Systems.
3.3.1 Computer Projects.
Exercises.
3.3.2 Applications to Economics.
Self-Study Questions.
Exercises.
3.4 The LU Factorization.
Exercises.
3.4.1 Computer Projects.
Exercises.
3.5 The Matrix of a Linear Transformation.
Coordinates.
Application to Differential Equations.
Isomorphism.
Invertible Linear Transformations.
True-False Questions.
Exercises.
3.5.1 Computer Projects.
Chapter Summary.
4. Determinants.
4.1 Definition of the Determinant.
4.1.1 The Rest of the Proofs.
True-False Questions.
Exercises.
4.1.2 Computer Projects.
4.2 Reduction and Determinants.
Uniqueness of the Determinant.
True-False Questions.
Exercises.
4.2.1 Application to Volume.
Self-Study Questions.
Exercises.
4.3 A Formula for Inverses.
Cramer's Rule.
True-False Questions.
Exercises 273.
Chapter Summary.
5. Eigenvectors and Eigenvalues.
5.1 Eigenvectors.
True-False Questions.
Exercises.
5.1.1 Computer Projects.
5.1.2 Application to Markov Processes.
Exercises.
5.2 Diagonalization.
Powers of Matrices.
True-False Questions.
Exercises.
5.2.1 Computer Projects.
5.2.2 Application to Systems of Differential Equations.
Self-Study Questions.
Exercises.
5.3 Complex Eigenvectors.
Complex Vector Spaces.
Exercises.
5.3.1 Computer Projects.
Exercises.
Chapter Summary.
6. Orthogonality.
6.1 The Scalar Product in Rn.
Orthogonal/Orthonormal Bases and Coordinates.
True-False Questions.
Exercises.
6.1.1 Application to Statistics.
Self-Study Questions.
Exercises.
6.2 Projections: The Gram-Schmidt Process.
The QR Decomposition 334.
Uniqueness of the QR-factoriaition.
True-False Questions.
Exercises.
6.2.1 Computer Projects.
Exercises.
6.3 Fourier Series: Scalar Product Spaces.
Exercises.
6.3.1 Computer Projects.
Exercises.
6.4 Orthogonal Matrices.
Householder Matrices.
True-False Questions.
Exercises.
6.4.1 Computer Projects.
Exercises.
6.5 Least Squares.
Exercises.
6.5.1 Computer Projects.
Exercises.
6.6 Quadratic Forms: Orthogonal Diagonalization.
The Spectral Theorem.
The Principal Axis Theorem.
True-False Questions.
Exercises.
6.6.1 Computer Projects.
Exercises.
6.7 The Singular Value Decomposition (SVD).
Application of the SVD to Least-Squares Problems.
True-False Questions.
Exercises.
Computing the SVD Using Householder Matrices.
Diagonalizing Symmetric Matrices Using Householder Matrices.
6.8 Hermitian Symmetric and Unitary Matrices.
True-False Questions.
Exercises.
Chapter Summary.
7. Generalized Eigenvectors.
7.1 Generalized Eigenvectors.
Exercises.
7.2 Chain Bases.
Jordan Form.
True-False Questions.
Exercises.
The Cayley-Hamilton Theorem.
Chapter Summary.
8. Numerical Techniques.
8.1 Condition Number.
Norms.
Condition Number.
Least Squares.
Exercises.
8.2 Computing Eigenvalues.
Iteration.
The QR Method.
Exercises.
Chapter Summary.
Answers and Hints.
Index.
"Linear Algebra (third edition) is an excellent undergraduate-level textbook for courses in linear algebra. It is also valuable self-study guide for professionals and researches who would like a basic introduction to linear algebra with applications in science, engineering, and computer science." (Mathematical Review, Issue 2009e)
"This volume is ground-breaking in terms of mathematical texts in that it does not teach from a detached perspective, but instead, looks to show students that competent mathematicians bring an intuitive understanding to the subject rather than just a master of applications." (Electric Review, November 2008)
"This book should make a good text for introductory courses." (Computing Reviews, September 30, 2008) | 677.169 | 1 |
Binomial Theorem Teacher Resources
Find Binomial Theorem educational ideas and activities
Title
Resource Type
Views
Grade
Rating
A week's worth of teaching on the Binomial Theorem. Lesson examples and a plethora of worksheets included. Learners find coefficients of specific terms within binomial expansions using notation of factorials and then apply these skills in using the Binomial Theorem to find solutions to practical applications.
A comprehensive lesson that explores and researches Pascal's triangle and relates its properties to the Binomial Theorem through a variety of lessons. Have the class practice expanding polynomials using the theorem. A few other formulas and functions related to this theorem will be explored.
In this Algebra II worksheet, 11th graders apply the binomial theorem to expand a binomial and determine a specific term of the expansion. The one page worksheet contains four problems. Answers are provided.
In this binomial expansions practice instructional activity, students utilize the theorem to determine the nth term in the expansion for 10 problems, then expand it completely for an additional 6 problems.
In this binomial theorem learning exercise, students identify the coefficients of a binomial expansion. They determine the designated term of an expansion. This two-page learning exercise contains 13 problems.
Continuing his discussion of the Poisson Distribution (or Process) from the previous video, Sal takes students through the derivation of the traffic problem he had begun. The math gets gritty in this video as Sal takes out the graphic calculator to solve the problem.
In this system of equations worksheet, 11th graders solve and complete 23 various types of problems. First, they graph each system of inequalities shown. Then, students write a polynomial function of least degree with integral coefficients that has the given zeros. They also determine the equation for each conic, name the conic and state the center.
For this difference of equations worksheet, students solve and complete 49 various types of problems. First, they obtain the solution of any linear homogeneous second order difference equation. Then, students apply the method of solution to contextual problems. In addition, they use generating functions to solve non-homogeneous equations.
Eleventh graders investigate the way credit cards work when collecting interest. In this algebra activity, 11th graders investigate the growth of interest exponentially when using a credit card. They calculate what the cheapest rate is to repay a $2000 loan using different interest rates.
In this Algebra II worksheet, 11th graders solve problems in which they find the indicated term of the binomial expansion. The two page worksheet contains a combination of ten multiple choice and free response questions. Answers are included. | 677.169 | 1 |
Prealgebra - 5th edition
Summary: Prealgebra, 5/e, is a consumable worktext that helps students make the transition from the concrete world of arithmetic to the symbolic world of algebra. The Aufmann team achieves this by introducing variables in Chapter 1 and integrating them throughout the text. This text's strength lies in the Aufmann Interactive Method, which enables students to work with math concepts as they're being introduced. Each set of matched-pair examples is organized around an objective...show more and includes a worked example and a You Try It example for students. In addition, the program emphasizes AMATYC standards, with a special focus on real-sourced data. The Fifth Edition incorporates the hallmarks that make Aufmann developmental texts ideal for students and instructors: an interactive approach in an objective-based framework; a clear writing style; and an emphasis on problem solving strategies, offering guided learning for both lecture-based and self-paced courses. The authors introduce two new exercises designed to foster conceptual understanding: Interactive Exercises and Think About It Very good BOOK IS VERY CLEAN INSIDE. NO WRITING OR HIGHLIGHTING. BOOK HAS TAPE AND USED BOOK STICKERS ON THE COVER. GREAT SHAPE.
$99.97 +$3.99 s/h
New
Balkanika Online WA Woodinville, WA
PAPERBACK New 0618956883 Brand new book. STUDENT US EDITION. Never used. Nice gift. Best buy. Shipped promptly and packaged carefully.
$99.99 +$3.99 s/h
Good
Texts Direct Lexington, KY
2010 Paperback Good This listing is for a paperback book not a DVD set. This is an AIE copy and includes answers. Perfect condition. Great dicounted text. Ships same or next business day. NO INTERN...show moreATIONAL ORDERS PLEASE | 677.169 | 1 |
The unique feature of this compact students introduction is that it presents concepts in an order that closely follows a standard mathematics curriculum, rather than structure the book along features of the software. As a result, the book provides a brief introduction to those aspects of the Mathematica software program most useful to students. The second edition of this well loved book is
completely rewritten for Mathematica 6 including coverage of the new
dynamic interface elements, several hundred exercises and a new chapter
on programming. This book can be used in a variety of courses, from precalculus to linear algebra. Used as a supplementary text it will aid in bridging the gap between the mathematics
in the course and Mathematica. In addition to its course use, this book
will serve as an excellent tutorial for those wishing to learn
Mathematica and brush up on their mathematics at the same time. Download linksA Handbook for Precalculus, Calculus, and Linear Algebra (2nd edition)
Can't Download? Please search mirrors if you can't find download links for "A Handbook for Precalculus, Calculus, and Linear Algebra (2nd edition | 677.169 | 1 |
Calculus Early Transcendentals Single Variable - 9th edition
Summary: The ninth edition continues to provide engineers with an accessible resource for learning calculus. The book includes carefully worked examples and special problem types that help improve comprehension. New applied exercises demonstrate the usefulness of the mathematics. Additional summary tables with step-by-step details are also incorporated into the chapters to make the concepts easier to understand. The Quick Check and Focus on Concepts exercises have been updated as well. Engine...show moreers become engaged in the material because of the easy-to-read style and real-world examples. ...show less
Chapter 5 Integration 5.1 An Overview of the Area Problem 5.2 The Indefinite Integral 5.3 Integration by Substitution 5.4 The Definition of Area as a Limit; Sigma Notation 5.5 The Definite Integral 5.6 The Fundamental Theorem of Calculus 5.7 Rectilinear Motion Revisited Using Integration 5.8 Average Value of a Function and its Applications 5.9 Evaluating Definite Integrals by Substitution 5.10 Logarithmic and Other Functions Defined by Integrals
Chapter 6 Applications of the Definite Integral in Geometry, Science, and Engineering 6.1 Area Between Two Curves 6.2 Volumes by Slicing; Disks and Washers 6.3 Volumes by Cylindrical Shells 6.4 Length of a Plane Curve 6.5 Area of a Surface of Revolution 6.6 Work 6.7 Moments, Centers of Gravity, and Centroids 6.8 Fluid Pressure and Force 6.9 Hyperbolic Functions and Hanging Cables | 677.169 | 1 |
Essentials of Scientific Computing: Numerical Methods for Science and Engineering
Written for students in the fields of engineering and science, this book provides comprehensive coverage of classical numerical methods. Different numerical techniques and the interrelation between them are covered, including iterative process, extrapolation, and matrix factorization. Topics covered include solving systems of linear equations, computational eigenvalue problems, and numerical integration.
An introduction to MATLAB is included in Chapter 9 and an appendix contains a brief review of functions used in numerical analysis.
Free Mathematical Modeling Technical Kit
Learn how you can quickly build accurate mathematical models based on data or scientific principles. | 677.169 | 1 |
More About
This Textbook
Overview
Calculus teachers recognize Calculus as the leading resource among the "reform" projects that employ the rule of four and streamline the curriculum in order to deepen conceptual understanding. The Sixth Edition uses all strands of the 'Rule of Four' - graphical, numeric, symbolic/algebraic, and verbal/applied presentations - to make concepts easier to understand. The book focuses on exploring fundamental ideas rather than comprehensive coverage of multiple similar cases that are not fundamentally unique.
Editorial Reviews
Booknews
Calculus can be taught as nothing but rules and procedures--losing sight of both the mathematics and its inherent practical value. In 1989, the Calculus Consortium based at Harvard was formed to create a completely new calculus curriculum. A part of their endeavor is this textbook, which presents a radically different approach to the teaching and learning of the subject. The two guiding principles: 1) every topic should be presented geometrically, numerically, and algebraically; and 2) formal definitions and procedures evolve from the investigation of practical problems (the way of Archimedes 18, 2002
College book
This book is used to teach Calculus at its very core and not simply mere basics. It offers questions which should be given thought to, and which are challenging in some aspects. This is very important as many degree programs like engineering where there is a need of a very strong base of maths.
1 out of 1 people found this review helpful.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Anonymous
Posted November 19, 2001
Horrible!!
This book is horrible!! Very confusing and vague. For a first time calculus student, one would find this text to be very difficult to study from. I do not recommend this book to anyone interested in studying calculus.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Anonymous
Posted February 7, 2001
The WORST
Focused on theory rather than the math involved in doing the problems. Impossible to self learn with this book. The examples did not help at all with the problems.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Anonymous
Posted December 26, 2000
DON'T BUY IT!
This textbook is very vague and not clear on the explanation of concepts. The preface says that the authors wished to create a book where students could not derive the answers to exercises from looking at worked out examples. Well golly how convenient. It seems to me it's just another lame brained excuse to hand out a worthless book. My advice don't waste your money purchasing this book. Instead go to a used bookstore and buy another calculus book.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Anonymous
Posted January 2, 2000
Terrible Terrible Book!!
Worst math book I have ever encountered...As a non-traditional student going back to school to retrain due to defense cut-backs, I have to say I have never had such a confusing text book before. I am lucky I have had calculus before and this is a refresher course for me. I am a math/science major and will never recommend this book to anyone...try instead Caluculus 9th edition by Thomas Finney...that is a great book to remove the mystery of calculus.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. | 677.169 | 1 |
Descriptions and Ratings (1)
Date
Contributor
Description
Rating
1 Feb 2013
MathWorks Classroom Resources Team
Freshmen course taught by Rajeevan Amirtharajah at the University of California Davis.
The goal of the course is to teach engineering problem solving using sustainable engineering as an example. Topics covered are 1D, 2D vectors and manipulations, mathematical and logical operations, loops, flow control, custom function, structures, object-oriented programming, string regular expression, Graphical User Interface design, and plotting.
Students will also gain hands-on experience working with hardware (Developed based on Arduino UNO) to gather sunlight data. The course emphasizes topics in solar cell technology. | 677.169 | 1 |
From Kant to Hilbert Volume 1
A Source Book in the Foundations of Mathematics
William Bragg Ewald
This two-volume work provides an overview of this important era of mathematical research through a carefully chosen selection of articles. They provide an insight into the foundations of each of the main branches of mathematics - algebra, geometry, number theory, analysis, logic, and set theory - with narratives to show how they are linked. | 677.169 | 1 |
I have a number of problems based on mathematics mcq I have tried a lot to solve them myself but in vain. Our teacher has asked us to solve them out ourselves and then explain them to the entire class. I reckon that I will be chosen to do so. Please help me!
You don't need to ask anyone to solve any sample questions for you; in fact all you need is Algebrator. I've tried many such algebra simulation software but Algebrator is way slope and multiplying matrices, but this software really helped me get over those.
When I was in school, I faced similar problems with linear algebra, quadratic inequalities and triangle similarity. But this superb Algebrator helped me through all my Intermediate algebra, Pre Algebra, and Intermediate algebra. I only typed in a problem from my workbook, and step by step solution to my math homework would appear on the screen by simply clicking on Solve. I highly recommend the Algebrator.
Thanks for the details. I have purchased the Algebrator from and I happened to go through equivalent fractions this evening. It is pretty cool and easily understandable. I was impressed by the descriptive explanations offered on subtracting exponents. Rather than being exam oriented, the Algebrator aims at educating you with the fundamental principles of Basic Math. The payment guarantee and the unimaginable rebates that they are currently offering makes the purchase particularly appealing.
I remember having difficulties with adding exponents, graphing and binomials. Algebrator is a really great piece of algebra software. I have used it through several math classes - Pre Algebra, Basic Math and Algebra 1. I would simply type in the problem from a workbook and by clicking on Solve, step by step solution would appear. The program is highly recommended. | 677.169 | 1 |
Product Description
From Amazon.ca
Microsoft Student with Encarta Premium 2008 is a productivity suite of homework tools and trusted information designed to help students quickly and easily complete assignments in a variety of subjects and excel in school.
Microsoft Student with Encarta Premium 2008 gives students a full set of tools to help them succeed and get homework done right the first time.
Make research entertaining. Find information on just about any topic. Thousands of articles, photos, sound clips, and links to related information from trusted sources(2)-plus political, climatic, and topographical maps of the world-make discovering new information fun!
Equation Solver. Enter a problem into the Equation Solver and get step-by-step help on how to solve it. And with free online access to Hotmath(3), students can quickly find their textbook and select the problem they're solving. They'll instantly receive step-by-step guidance on how to solve the problem and what the correct answer should be.
Microsoft Student with Encarta Premium 2008 gives students a full set of tools to help them succeed and get homework done right the first time.
Full-featured graphing calculator A full-featured graphing calculator that's simple to learn and use (similar to those costing more than $100!) helps visualize and solve math and science problems.
Equation Solver Stuck on a math problem? Input your own problems into the Equation Solver and get a quick answer and step-by-step help on how to solve it.
Triangle Solver and Equation Library Interact with a library of more than 100 equations and explore the relationships of triangles and their parts while viewing associated rules or graphs.
Foreign language helpBook summaries Need to know more about that literary story? Includes more than 1,000 book summaries that help students understand some of the most commonly studied literary works from middle school to college level.
The easy, all-in-one homework assistant that helps students excel in school. Designed to be easy to use and simple to learn, Microsoft Student with Encarta Premium 2008 makes learning fun. Whether it's a math, research, or foreign language assignment, students can find the right tools and information to get homework started quickly, get fast answers to academic questions, and complete assignments that can help earn higher grades.
Find trusted content when you need it. Complete with Encarta Premium 2008, the #1 best-selling encyclopedia software brand(1), it's content you can trust. With editorially approved content, students can use Web Links to more than 25,000 Web sites, preselected by Encarta editors for relevant and age-appropriate research material(2). Through multimedia content, Encarta Premium 2008 provides engaging visual tools to help explore and discover historical events and places. It even includes Encarta Kids, a separate encyclopedia geared to young learners ages 7 to 12!
Get questions answered fast. With time pressures to complete assignments, it's frustrating to get stuck without knowing where to go for answers. Whether they're stumped on a math, science, or a foreign language problem, students can quickly get the answers they need. They also get helpful ideas on how to solve problems, so the next time they're more likely to figure out the answer on their own!
Includes a full suite of homework tools to help students get homework done right the first time.
Encarta Premium 2008 Students can fi nd the information they're looking for-quickly and easily-with Encarta Premium 2008. With trusted content that's accurate and up-to-date, Encarta has been the #1 best-selling encyclopedia software brand for the past 8 years!(1) You'll be amazed at how much time your student will spend researching-compelled by the sights and sounds of one fascinating presentation after another.
Students can easily-and quickly-access trusted information provided by world experts. And since they can automatically download updates from the Encarta Web site, they'll have up-to-date and accurate reference software when they need it.(2)
Microsoft Math Microsoft Math features a large collection of tools, tutorials, and instructions designed to help students learn mathematical concepts while quickly solving math problems. In an instant, they'll see how to solve problems-step by step! Works for many grade levels-from basic math, pre-algebra, algebra (including logarithms and exponents) to trigonometry!
Need advanced help? A full-featured graphing calculator that's simple to learn and use (similar to those costing more than $100!) helps visualize and solve math and science problems.
Triangle Solver and Equation Library. Interact with a library of more than 100 equations and explore the relationships of triangles and their parts while viewing associated rules or graphs.
Templates and Tutorials Having trouble getting started on class projects? Sometimes the hardest part about completing a project is getting started. Microsoft Student with Encarta Premium 2008 includes the latest version of Learning Essentials.
Get past the blank page. Easy-to-use tools, templates, and tutorials help students get past formatting questions to the core learning in assignments. Learning Essentials includes tips and tutorials for managing projects and creating high-quality reports, presentations, science projects, and other assignments.
Foreign LanguageLiterature Need to know more about that literary story? Microsoft Student with Encarta Premium 2008 includes more than 1,000 book summaries that help students understand some of the most commonly studied literary works from middle school to college level.
Note:
1. The NPD Group/NPD Techworld, January 2000 to February 2007. Based on total U.S. retail sales.
2. An active Internet connection is required for Math Online Help, Update Encarta, Web Links, Weather and Radio Links and Encarta Premium Online. Update Encarta and Math Online Help are available through October 2008. Access to Encarta Online Premium requires a Microsoft .NET Passport (Windows Live ID) and an Internet connection. You must be 13 years or older to create a Windows Live ID.
3. Hotmath contains primarily US-based textbook problems.
Product Description
Microsoft Student with Encarta Premium 2008 Win32 English US Only DVD Mini Box.
Microsoft Student has lots of useful resources that is aimed from middle school to high school. Encarta dictionary was also very helpful. I was a big fan of it. Encarta is compatible with Windows Vista. If you own Microsoft Office (preferably a newer version) it adds many many helpful features for typing essays, stories and a hundred other things.
I've been using Encarta along with Britannica for years. I once believed that for overall depth of content, the higher mark would go to EB. But with regard to software ease of use and organization of material, Encarta would always get the nod. Not so anymore. This new version is a step backward compared to earlier years and seems to be plagued by its own desire to be both a great encyclopedia/dictionary and a homework aid for students. In the end, it achieves only a mediocre showing in both categories.
Just as an example: the new dictionary looks better and has two new tabs for translations and verb conjugations, but performs poorly compared to the 2006 version, in my opinion. It is now almost incapable of recognizing certain word inflections as typed. As an example, take the word "intoning", a present participle of "intone". Well, if you type it in to the dictionary, it will not be recognized as a word...even though if you look up "intone", you'll find the present participle form listed there. So in order to get a match, you must type in the basic form of the word in most cases.
However...some words, such as "intoxicate", have separate entries for other forms (in this case, the present participle "intoxicating"). I don't recall this idiosyncrasy in the 2006 version of Encarta dictionary. In that version, any inflection you typed in would lead you back to the basic form (e.g., present indicative) definition. This new dictionary is actually fairly annoying after a few days of working with it.
It also seems that some words were completely dropped. For example, "reenact" isn't a word from Encarta's perspective. You won't find it under that listing, or the hyphenated "re-enact". But you can find it in Merriam-Webster and practically every other dictionary on the planet. It's almost like they abridged what Encarta had in earlier versions. The weird thing about it is that you can find the word "reenact" in the Thesaurus and Verb Conjugation tabs...but not in the dictionary. What's up with that? Did Microsoft lay off their software testing team?
Perhaps there's some logic to it all. But in my opinion, it comes across as sloppy and not very helpful. I once used the little Encarta icon on my task bar every day. Now I'm opting for Britannica 2008's Merriam-Webster component instead. Oh well, it seems that some good things come to an end through unnecessary tinkering.
23 of 25 people found the following review helpful
2.0 out of 5 starsnot for adultsMarch 10 2008
By Michael A. Johnson - Published on Amazon.com
Amazon Verified Purchase
Unfortunately this is not an encyclopedia and dictionary with student aids, but student programs with the encyclopedia added on. You must wade through all kinds of homework, math, etc. programs to reach the dictionary and encyclopedia. Microsoft apparently thinks that after you leave school you can never have need of an encyclopedia or dictionary again. Also once you load it on, you cannot turn it off. It will run forever on the grounds that you might have to look up something. Once you do get in, the interface is harder to move around in than in the old 2004 edition.
26 of 30 people found the following review helpful
2.0 out of 5 starsEach recent version gets worseSept. 2 2007
By Rod Walsh - Published on Amazon.com
Amazon Verified Purchase
I've used Encarta for many years. However, the 2004 version was the last version worthy of carrying the Encarta name.
Navigation used to be extremely simple. Now, you need a PhD to get around. The use of the "find" box in 2004 is so much more helpful than the "search" box in this 2008 version.
This version crashes regularly (running Vista). By regularly, I'd say it crashes 20% of the time. On the plus side, they have all been soft landings and I can re-open without a re-boot. | 677.169 | 1 |
You are here
Calculus: Modeling and Application 2nd edition
David A. Smith and Lawrence C. Moore
Calculus: Modeling and Application, 2nd Edition, responds to advances in technology that permit the integration of text and student activities into a unified whole. In this approach, students can use mathematics to structure their understanding of and investigate questions in the world around them, to formulate problems and find solutions, then to communicate their results to others.
Calculus: Modeling and Application covers two semesters of single-variable calculus. Its features include
use of real-world contexts for motivation,
guided discovery learning,
hands-on activities (including built-in applets),
a problem-solving orientation,
encouragement of teamwork,
written responses to questions,
tools for self-checking of results,
intelligent use of technology, and
high expectations of students.
It is important to note that this textbook is a website. When purchasing this textbook you will have access to two versions:
Computer/CAS Version. This version requires the Firefox browser for proper display of mathematics and one of the commercial computer algebra systems (Maple, Mathematica, or Mathcad) for most of the interactivity in the text. Some activities require access to a printer.
Tablet/Sage Version. This version was written for the iPad but will run on other tablets or computers in either Safari or Firefox (but not Internet Explorer or Chrome). It uses the free computer algebra system Sage to provide interactivity through remotely-processed interacts. Sage interacts require only calculator-form entry of functions and numbers. You need not have Sage installed on your machine, and no printer is required.
At the end of each section there are pages labeled Exercises and Problems. The Exercise pages each have a WebWork button that links to MAA's WebWork courses page. Adopters who set up WebWork courses (at MAA or on their local servers) get access to the Calculus: Modeling and Application Library (in addition to the national library), and it has problems that are more or less like the Exercises in Calculus: Modeling and Application. Instructors can make up homework sets from both the Calculus: Modeling and Application and national libraries. The authors plan to continue to add to the Calculus: Modeling and Application collection, so it will get stronger over time. All of the Exercises are potentially machine-gradable (suitable for WebWork), whereas the Problems need human responses.
Access to Calculus: Modeling and Application can be purchased at maa.pinnaclecart.com. Instructors wishing to adopt the text for their course should contact the MAA Service Center (1-800-331-1MAA) about obtaining desk access.
Note to professors:Some problems do not work in both versions (due to differing technologies), so if your students are using both versions, be sure to check problem numbers in both places when making homework assignments.
Note to purchasers of the text: To gain easy access to the book website in the future
The page that comes up will look like a receipt page and it will have a link to the text (Access Calculus ebook). BOOKMARK this page so that in the future all you have to do is click on the bookmark, the login window will pop up, you log in, and then the page with the link pops up. | 677.169 | 1 |
This site provides resource materials pertaining to areas such as arithmetic, algebra, geometry, and precalculus. Topics...
see more
This site provides resource materials pertaining to areas such as arithmetic, algebra, geometry, and precalculus. Topics featured include sequences and series, Euclidean and non-Euclidean geometry, complex numbers, fractals, chaos, and number theory.
An introduction to cartography emphasizing map projections, their properties, applications and basic mathematics. Concepts...
see more
An introduction to cartography emphasizing map projections, their properties, applications and basic mathematics. Concepts addressed can be presented at a basic level or expanded to explore the use of more advanced mathematics.This site offers terms and formulas in a mathematical dictionary appropriate for students taking algebra and calculus...
see more
This site offers terms and formulas in a mathematical dictionary appropriate for students taking algebra and calculus courses. The main page contains an A to Z clickable menu displaying all of the terms in the site's dictionary that begin with that particular letter. This site is an interactive math dictionary with enough math words, math terms, math formulas, pictures, diagrams, tables, and examples from beginning algebra to calculus. | 677.169 | 1 |
SME 3023
TRENDS AND ISSUES IN EDUCATION FOR
MATHEMATICAL SCIENCES
TOPIC:
MATHEMATICS ANXIETY
NAMA : KAMARUDDIN BIN HARIJAMAN
NO MATRIK : D20081033178
SEMESTER :8
PROGRAM : SAINS MATEMATIK DENGAN PENDIDIKAN
PENSYARAH : PROF. DR MARZITA BT PUTEH
What is mathematics anxiety?
Mathematics anxiety defined as feelings of tension and anxiety that interfere with the
manipulation of numbers and the solving of mathematical problems in a wide verify of ordinary
life and academic situations. Mathematics anxiety can cause one to forget and lose one's self-
confidence (Tobias, 1978). People feel that they are incapable of doing activities and classes that
involve mathematics. So, many of them take the major that little mathematics is required.
Mathematics anxiety also is emotional rather than the intellectual problem.
What does this phenomenon was happen?
Mathematics anxiety does not have a single cause. From the research was conducted
(Puteh. 1998), it was found the causes of mathematics anxiety such as teacher personality and
their style of teaching, public examination and their effect and feel worry and difficulty of
mathematics. Besides, poor instruction and poorly designed mathematics circular was argued by
Lazarus (1974) contributes the mathematics anxiety. Bush (1991) commented that mathematics
anxiety arises from a climate in which negative attitudes and anxiety are transmitted from adults
to children. Often mathematics anxiety is the result of a student's negative or embarrassing
experience with math or a math teacher in previous years. Such an experience can leave a student
believing him or herself deficient in math ability. This belief can actually result in poor
performance, which serves as confirming evidence to the student. This phenomenon is known as
the self-fulfilling prophecy. Mathematics anxiety results in poor performance rather than the
reverse.
Who is involved in mathematics anxiety?
This mathematics anxiety is problem to many school students, college students and also
teachers. Individual that related to the mathematics anxiety is among the teacher and student.
There are some perception among the teacher trainees that the phobia of mathematics.
They think mathematics are numbers, symbols and calculations. Besides, mathematics have to
solve problems is seen as something difficult and perceived as a burden. They express their
frustrations and helplessness when there were unable to solve their problem. They believed that
they are not capable of solving mathematics problems that take more than a few minutes to
complete and thus they give up on any problem they cannot solve. Therefore this situations is
creating mathematics anxiety.
All students learn mathematics at greatly different speeds. Mathematics is a difficult
subject to teach and learn. Therefore teacher must build a good perception of student to their
knowledge. There are many thing that contribute mathematics anxiety among student such as
experienced a variety experiences in the learning of mathematics, student also get great pressure
from their parents and etc. therefore student feels stress to learn mathematics so this contribute
mathematics anxiety
When mathematics anxiety happen?
Individual that have mathematics anxiety has negative attitude towards mathematics.
Five aspect of the trainees' self-image with regard to mathematics were identified from the
interviews (Puteh, 1998) such as dislike being challenged, low confidence, slow learner and low
self esteem, easy giving-up and self blaming for poor mathematics performance.
Mathematics anxiety also occurs when student not remembering what they learn in
mathematics. This especially to student that take the examination like UPSR, PMR and SPM.
Therefore, if they still fail in do a task constantly, they will tend to give up in mathematics.
Student who has this mathopobhia, they will turn away from mathematics problem. It will lead
them to keep away their interest to mathematics subject.
Who and what created mathematics anxiety?
There is a factor that caused concern of mathematics anxiety is a relationship between
student and teacher. The teacher-student relationship seemed to have affected their attitudes
negatively. Almost of the teacher or trainees have a problem to fear of asking for help or asking
the question. This attitudes caused a large gap between teacher and student. Besides, there a
several teacher very strictness and fierceness towards student. The teacher attitude to the
learners affects the way the learners respond to the subject. Actually the classes is interesting
and stimulating but the student feel stress and uncomfortable in class because scare to teacher.
Besides, the teaching skill also affecting towards mathematics anxiety. The most
prominent issue raised by the trainees was that their teacher were using an old fashioned way or
traditional way of teaching (Puteh, 2002). From Kogelman and Warren (1978) commented that
the neglect of the individual personality which often occurs in traditional learning situations
often leads to feelings of frustrations, discouragement and general mathematics avoidance. It is
hard to deny that the amount of anxiety that was created in the traditional approach adopted by
the teacher of these student.
How to reduce mathematics anxiety?
We must admit the mathematics anxiety exists. The mathematics anxiety problem can be
reduce in many way. One of the way is as teacher need more strengthen of content knowledge
about mathematics and pedagogy content. In addition, teacher also need to improve teaching
skills and make the teaching and learning more interesting. This can avoid student feel bored in
the class while the learning process conducted. Besides, relationship between student and
teacher can provided confidence to student to continue learning mathematics. This relationship
causes student having fun in study mathematics and can avoid the stress during learning
mathematics in the class.
Another way to reduce mathematics anxiety is student think that examination is indicator
to measure the performance with health competition among the peers. In this situations, parents
need play an important role to build motivation to student.
How to eliminate mathematics anxiety?
To eliminate the mathematics anxiety directly or immediately are impossible but it can be
happen. There are way to overcome mathematics anxiety like shifts the mindset of mathematics
and maintain a positive attitude towards mathematics itself and our ability to do a mathematics
problems. Once this mindset is in place, students can focus on the math skills with an open mind,
exuding confidence, hope, and interest, which will raise academic achievement and self-
confidence takes the energy off the anxiety, and turns it to the positive, which is highly
motivating.
Besides, parents also can help like talk about past experiences of overcoming adversity
and succeeding. In doing so, a shift in attitude will occur, as the student feel more confident. In
addition, parents also help their children channel these feelings onto.
When we recognize the anxiety before it takes over and can transform it, they can be
successful in mathematics | 677.169 | 1 |
Mathematical Modeling: Models, Analysis and Applications
9781439854518
Buy New Textbook
Not Yet Printed. Place an order and we will ship it as soon as it arrives.
$872 how-to guide presents tools for mathematical modeling and analysis. It offers a wide-ranging overview of mathematical ideas and techniques that provide a number of effective approaches to problem solving. Topics covered include spatial and stochastic modeling. The text provides real-life examples of discrete and continuous mathematical modeling scenarios. MATLAB ® , Mathematica ® , and MatCont are incorporated throughout the text. The examples and exercises in each chapter can be used as problems in a project. | 677.169 | 1 |
Overview of Curriculum
Mathematics and statistics are among the great achievements of human intellect and at the same time powerful tools. As Galileo said, the book of the universe "is written in the language of mathematics." The goal of the department is to enable students to appreciate these achievements and use their power. To that end, majors and minors in the department receive a firm foundation in pure mathematics and the opportunity to apply it—to statistics, physical science, biological science, computer science, social science, operations research, education, and finance—the list grows.
Students typically enter our department with strong skills, but there is always room for improvement and new knowledge. Majors and minors grow in:
Through core courses, students learn fundamental concepts, results, and methods. Through elective courses, they pursue special interests. In the process, students develop a further appreciation for the scope and beauty of our discipline.
Graduates of the department follow many careers paths, leading them to graduate school, in mathematics, statistics, or other fields, to professional schools, or to the workplace.
Introductory Courses
Most first-year students entering Swarthmore have had calculus while in high school and place out of at least one semester of Swarthmore's calculus courses, whether they continue with calculus or decide, as is often best, to try other sorts of mathematics. See the discussion of placement later. However, some entering students have not had the opportunity to take calculus or need to begin again. Therefore, Swarthmore offers a beginning calculus course (MATH 015) and several courses that do not require calculus or other sophisticated mathematics experiences. These courses are STAT 001 (Statistical Thinking, both semesters), MATH 003 (Introduction to Mathematical Thinking, spring semester), and STAT 011 (Statistical Methods, both semesters). MATH 003 is a writing course. MATH 029 (Discrete Mathematics, both semesters) also does not require any calculus but is a more sophisticated course; thus, some calculus is a useful background for it in an indirect way. Once one has had or placed out of two semesters of calculus, many other courses are available, especially in linear algebra and several-variable calculus.
Placement and Credit on Entrance to Swarthmore
Placement Procedure
To gain entrance to mathematics or statistics courses at any time during one's Swarthmore years, students are expected to take at least one of the following exams: the Advanced Placement (AP) or International Baccalaureate (IB) exams, Swarthmore's Calculus Placement Exam, or Swarthmore's Math/Stat Readiness Exam. Students who do take AP or IB exams may be required to take the departmental exams as well, or parts thereof. In particular, students intending to take either MATH 15 or MATH 28 must take Swarthmore's Calculus Placement Exam. Versions of the Calculus Placement Exam and the Readiness Exam are sent to entering first-year students over the summer, along with detailed information about the rules for placement and credit.
Advanced Placement/International Baccalaureate Credit
Placement and credit mean different things. Placement allows students to skip material they have learned well already by starting at Swarthmore in more advanced courses. Credit confers placement as well but also is recorded on the student's Swarthmore transcript and counts toward the 32 credits needed for graduation. The Swarthmore Calculus Placement Exam is used for placement only, not credit. Credit is awarded on the basis of the AP and the IB exams, as follows:
1 credit (for STAT 011) for a score of 4 or 5 on the Statistics AP Test of the College Board.
1 credit (for MATH 015) for a score of 4 on the AB or BC Calculus AP Test of the College Board (or for an AB subscore of 4 on the BC Test) or for a score of 5 on the Higher Level Mathematics Test of the IB.
1.5 credits (for MATH 015 and the first half of MATH 025) for a score of 5 on the AB Calculus AP Test (or for an AB subscore of 5 on the BC Test) or a score of 6 or 7 on the higher-level IB. Students who receive this credit and want to continue calculus take MATH 026.
2 credits (for MATH 015 and 025) for a main score of 5 on the BC Calculus AP Test.
Alternatively, any entering student who places out of MATH 015 or 025 may receive credit for those courses by passing the final exams in these courses with a grade of straight C or better. These exams must normally be taken during the student's first semester at Swarthmore, at the time when the final exam is given for the course. Students who wish to take these exams must arrange to do so with the departmental placement coordinator and should do so during their first semester at Swarthmore.
Students who are eligible on entrance for credit for a course, but who take the course anyway, will lose the entrance credit.
First-year students seeking advanced placement and/or credit for calculus taken at another college or university must normally validate their work by taking the appropriate external or Swarthmore placement examination, as described earlier. The department does not grant credit directly for college courses taken while a student is in high school. For work beyond calculus completed before entering Swarthmore, students should consult the departmental placement coordinator to determine the Swarthmore courses into which they may be placed and additional materials they may need to present for this placement. The department will not normally award credit for work above the first-year calculus level completed before entering Swarthmore.
The Academic Program
Major and Minor Application Process
Students apply for a major in the middle of the second semester of the sophomore year. Before all the usual steps of the College's Sophomore Plan process, applicants to the Mathematics and Statistics Department should begin by completing our online Major/Minor Application Form, available at After the Sophomore Plan process is over, students may apply to add or change a major or minor at any time, but applications will normally be held until the next time that sophomore applications are considered (around March 1).
Course Major
Acceptance into the Major
The normal preparation for a major in mathematics is to have obtained credit for, or placement out of, at least four of the following five course groups by the end of the sophomore year: Calculus I (MATH 015), Calculus II (MATH 025 or 026), Discrete Mathematics (MATH 029), Linear Algebra (MATH 027 or any flavor of 028), and Several Variable Calculus (MATH 033, 034, or 035). In any event, all majors must complete the Linear Algebra and Several Variable Calculus requirement by the end of the first semester of the junior year.
To be accepted as a major or a minor, a candidate normally should have a grade point average of at least C+ in courses taken in the department to date, including courses in the fall term of the first year, for which we have shadow grades. A candidate should have at least one grade at the B level. Students should be aware that upper-level courses in mathematics are typically more demanding and more theoretical than the first- and second-year courses. This is an important factor in considering borderline cases. In some cases, applicants may be deferred pending successful work in courses to be designated by the department.
Basic Requirements
By graduation, a mathematics major must have at least 10 credits in mathematics and statistics courses. At least 5 of the credits counted in the 10 must be for courses numbered over 040. (Courses numbered under 10 do not count toward the major in any event.) Furthermore, every major is required to obtain credit for, or place out of, each of the following course groups: MATH 015; MATH 025, or 026; MATH 027, 028, or 028S; MATH 033, 034, or 035; MATH 063; and MATH 067. The two upper-level core courses, MATH 063 (Introduction to Real Analysis) and MATH 067 (Introduction to Modern Algebra), will be offered at least every fall semester. At least one of these two should be taken no later than the fall semester of the junior year. Majors are expected to complete both MATH 063 and 067 before the spring semester of the senior year; permission to delay taking either course until the senior spring must be requested in writing as early as possible but in any event no later than the beginning of the fall semester of the senior year. Finally, course majors must satisfy the departmental comprehensive requirement by passing MATH 097, Senior Conference. Normally, at least 3 of the 5 credits for courses numbered over 040 must be taken at Swarthmore, including MATH 097 and at least one of the core courses MATH 063 and 067. Note that MATH 097 is given in the fall only.
Note that placement counts for satisfying the requirements but not for the 10-credit rule. Those students who are placed out of courses without credit must take other courses to obtain 10 credits. If you believe you are eligible for credit for courses taken before Swarthmore (because of AP or IB scores) but these credits are not showing on your transcript, please see the registrar.
The two required core courses, Introduction to Real Analysis (MATH 063) and Introduction to Modern Algebra (MATH 067), are offered every fall semester, and we try to create enough sections to keep them relatively small and seminar-like. We hope, but cannot promise, to offer one or the other of 063 and 067 each spring as well.
Mathematics majors are encouraged to study in some depth an additional discipline that makes use of mathematics. We also recommend that they acquire some facility with computers. Students bound for graduate work should obtain a reading knowledge of French, German, or Russian.
Special Emphases
The preceding requirements allow room to choose an optional special emphasis within the mathematics major. For instance: A student may major in mathematics with an emphasis on statistics by taking the following courses at the advanced level: (1) the core analysis course (MATH 063); (2) Mathematical Statistics I (STAT 061); (3) Probability (MATH 105) or Mathematical Statistics II (STAT 111); (4) Data Analysis and Visualization (STAT 031); (5) the Senior Conference (MATH 097); and (6) another mathematics course numbered over 040. Students are encouraged but not required to select the core algebra course (MATH 067) if they choose this emphasis. When a student does an emphasis in statistics, STAT 031 counts as if it were numbered over 040. Students interested in mathematics and computer science should consider a mathematics major with a minor in computer science or an Honors Program with a mathematics major and a computer science minor. Details on these options are in the catalog under computer science.
Students thinking of graduate work in social or management science, or a master's in business administration, should consider the following options.
Basic courses: single-variable calculus (two semesters), one or more practical statistics courses (STAT 061 and 031), linear algebra, discrete math, several-variable calculus, and introductory computer science; advanced courses: (1) Modeling (MATH 056); (2) at least one of Probability (MATH 105), Mathematical Statistics I (STAT 061), and possibly Mathematical Statistics II (STAT 111); (3) at least one of Combinatorics (MATH 069) or Operations Research (ENGR 057); (4) the three required core courses (MATH 063, MATH 067 and MATH 097); and (5) Differential Equations (MATH 043 or 044). Because this program is heavy (one who hopes to use mathematics in another field must have a good grasp both of the relevant mathematics and of the intended applications), one of the core course requirements may be waived with permission of the department.
Students thinking of graduate work in operations research should consider the following options. Basic courses: same as previous paragraph. Advanced courses: (1) the three required core courses (MATH 063, MATH 067 and MATH 097); (2) Combinatorics (MATH 069) and Topics in Discrete Mathematics (MATH 059 or 079); (3) Mathematical Statistics (STAT 061); and (4) at least one of Number Theory (MATH 058), Modeling (MATH 056), or Probability (MATH 105).
Students interested in quantitative economics, mathematical finance, or similar fields should consider a double major in mathematics and economics, or a major in mathematics with a minor in economics. Students thinking of graduate work in quantitative economics or mathematical finance should consider a math major with a program including at least MATH 43, MATH 54, MATH 63 and STAT 61 together with appropriate additional coursework to round out a mathematics major or a mathematics major with emphasis in statistics.
Course Minor
Acceptance into the minors
The requirements for acceptance into either course minor, such as prerequisite courses and grade average, are the same as for acceptance into the major. Students may not minor in both mathematics and statistics.
By graduation, a mathematics course minor must have 6 credits in mathematics or statistics, at least 3 of which must be for courses numbered 045 or higher. Also, at least 1 of these 3 credits must be for MATH 063 or 067. Also, at least 2 of these 3 credits must be taken at Swarthmore.
Basic requirements of the statistics course minor
By graduation, a statistics course minor must have 6 credits in mathematics or statistics. Every statistics course minor must obtain credit for, or place out of, CPSC 21, STAT 031, and STAT 061. At least one of STAT 031 and STAT 061 must be taken at Swarthmore. Students are advised to take CPSC 21 as early as possible, as it can be difficult to add the course in junior and senior years.
Honors Major
All current sophomores who wish to apply for Honors should indicate this in their Sophomore Plan, should work out a tentative Honors Program with their departmental adviser, and should submit the College's Honors Program Application along with their Sophomore Plan. (All Sophomore Plan forms and Honors forms are available from the registrar or the registrar's website.) Honors applications are also accepted at the end of the sophomore year or during the junior year. Students, in consultation with their advisers, often change their Honors Programs anyway as time goes on.
Basic requirements
To be accepted as an Honors major in mathematics, a student should have a grade point average in mathematics and statistics courses to date of at least B+.
An Honors math major program consists of three preparations of two credits each, for a total of six distinct credits. One preparation must be in algebra and one in analysis (real or complex). The student must also satisfy all requirements of the mathematics major with the exception of the comprehensive requirement (MATH 097, Senior Conference).
Preparations
The department offers preparations in the fields listed below. Each preparation is subject to External Examination, including a 3-hour written examination and a 45-minute oral examination. Each preparation consists of a specified pair of credits. The specified credits are listed after each field. Algebra (067 and 102) Real Analysis (063 and 101) Complex Analysis (063 and 103) Discrete Mathematics (069 and either 059 or 079) Geometry (either 055 or 075, and 106) Probability (061 and 105) Statistics (061 and 111) Topology (104, a 2-credit seminar) Since no course is allowed to count in two honors preparations, it is not possible for a student to offer both Real Analysis and Complex Analysis as fields. Similarly, one may take only one of Probability and Statistics as fields. The external examination component of the program is meant to prompt students to learn their core subjects really well and to show the examiners that they have done so—that is, show that they deserve Honors. However, no three fields cover everything a strong student would ideally learn as an undergraduate. Honors majors should consider including in their studies a number of advanced courses and seminars beyond what they present for Honors. Senior Honors Study/Portfolio None is required or offered.
Honors Minor
For the honors portion of their program, minors must complete one preparation chosen from those in the previous section.
Transfer Credit
Courses taken elsewhere may count for the major. However, the number of upper-level transfer credits for the major is limited. Normally, at least 3 of the 5 upper-level courses used to fulfill the major must be taken at Swarthmore, including at least one of the core courses MATH 063 and MATH 067. Exceptions should be proposed and approved during the Sophomore Plan process, not after the fact. Also, the usual College rules for transfer credit apply: students must see the professor in charge of transfer twice: in advance to obtain authorization, and afterwards to get final approval and a determination of credit. In particular, for MATH 063 and 067, students are responsible for the syllabus we use. If a course taken elsewhere turns out not to cover it all, the student will not get full credit (even though the transfer course was authorized beforehand) and the student will not complete the major until he or she has demonstrated knowledge of the missing topics. Similarly, for honors preparations students are responsible for the syllabi we use; we will not offer special honors exams based on work done at other institutions.
Off-Campus Study
Students planning to study abroad should obtain information well in advance about the courses available at the institution they plan to attend and check with the department about selecting appropriate courses. It may be difficult to find courses abroad equivalent to our core upper-level courses, or to our honors preparations, since curricula in other countries are often organized differently.
Teacher Certification
Swarthmore offers teacher certification in mathematics through a program approved by the state of Pennsylvania and administered by the College's Educational Studies Department. For further information about the relevant set of requirements, please refer to the Educational Studies section of the Bulletin. One can obtain certification either through a mathematics major or through a Special Major in Mathematics and Education, in either case if taken with appropriate electives.
Courses
Note 1: For courses numbered under 100, the ones digit indicates the subject matter, and the other digit indicates the level. In most cases, a ones digit of 1 or 2 means statistics, 3 to 6 means continuous mathematics, and 7 to 9 means noncontinuous mathematics (algebra, number theory, and discrete math). Courses below 10 do not count for the major, from 10 to 39 are first- and second-year courses, from 40 to 59 are intermediate, in the 60s are core upper-level courses; from 70 to 89 are courses that have one or more core courses as prerequisites, and in the 90s are independent reading courses. Note 2: There are several sets of courses below where a student may not take more than one of them for credit. For instance, see the descriptions of MATH 033, 034 and 035. In such cases, if a student does take more than one of them, each group is treated for the purpose of college regulations as if they have the same course number. See the Repeated Course Rule in section 8.2.4.
STAT 001. Statistical Thinking
Statistics provides methods for collecting and analyzing data and generalizing from their results. Statistics is used in a wide variety of fields, and this course provides an understanding of the role of statistics in these fields and in everyday life. It is intended for students who want an appreciation of statistics, including the ability to interpret and evaluate statistical claims critically but who do not imagine they will ever need to carry out statistical analyses themselves. (Those who may need to carry out statistical analyses should take STAT 011.) This course cannot be counted toward a major in mathematics, is not a prerequisite for any other course, and cannot be taken for credit after or simultaneously with any other statistics course, including AP Statistics and ECON 031. Prerequisite: Four years of traditional high school mathematics (precalculus). 1 credit. Each semester. Fall 2013. Schofield. Spring 2014. Schofield.
MATH 003. Introduction to Mathematical Thinking
Students will explore the world of mathematical ideas by sampling logic, number theory, geometry, infinity, topology, probability, and fractals, while we emphasize the thinking and problem-solving skills these ideas stimulate. Class meetings will involve presentation of new material; group work on problems and puzzles; and lively, maybe even passionate discussions about mathematics. This course is intended for students with little background in mathematics or those who may have struggled with math in the past. It is not open to students who already have received credit on their Swarthmore transcripts for mathematics, Advanced Placement credit included, or who concurrently are taking another mathematics course, or who have placed out of any Swarthmore mathematics course. (See "Placement Procedure" earlier.) Students planning to go on to calculus should consult with the instructor. This course does not count toward a major in mathematics. Writing course. 1 credit. Spring 2014. Gomez.
MATH 007. Elementary Topics in Mathematics in Applied Contexts
This course is offered occasionally and is interdisciplinary in nature. It provides an introduction to some area of mathematics in the context of its use in another discipline. In fall 2010 this was a course in biomathematics. 1 credit. Not offered 2013–2014.
STAT 011. Statistical Methods
STAT 011 prepares students to carry out basic statistical analyses with the aid of computer software. Topics include basic summary statistics and graphics, design of surveys and experiments, one and two-sample t-tests and tests of proportions, chi-square tests, and an introduction to linear regression and analysis of variance. The course is intended for students who want a practical introduction to statistical methods and who intend to do, or think they may eventually do, statistical analysis, especially in the biological and social sciences. Students who receive credit on entrance for the Statistics AP Exam should not take this course; they have placed out of it and will lose their AP credit if they take it. Note that STAT 011 overlaps considerably with ECON 031; both courses cover similar topics, although ECON 031 focuses more on economic applications while STAT 011 draws examples from a variety of disciplines. Eligible for Cognitive Science credit. Prerequisite: Four years of traditional high school mathematics (precalculus). 1 credit. Each semester. Fall 2013. Sedlock. Spring 2014. Cook, Everson.
MATH 015. Elementary Single-Variable Calculus
A first-semester calculus course with emphasis on an intuitive understanding of the concepts, methods, and applications. Graphical and symbolic methods will be used. The course will mostly cover differential calculus, with an introduction to integral calculus at the end. Prerequisite: Four years of traditional high school mathematics (precalculus) and placement into this course through Swarthmore's Math/Stat Readiness Examination. Students with prior calculus experience must also take Swarthmore's Calculus Placement Examination (see "Placement Procedure" section earlier). 1 credit. Fall 2013. Mavinga, Shimamoto.
MATH 015HA. Calculus Workshop
An honors-level workshop designed to support MATH 015 students who plan to take at least four other STEM courses during their time at Swarthmore. During class, students work in small groups on challenging problems designed to promote deep understanding and mastery of the material. Prerequisite: Students must apply for admission to this attachment. Admission will be determined by a commitment to both hard work and excellence, rather than by high school GPA, math SAT scores, or past performance in math classes. 0.5 credit. Graded credit/no credit. Not offered 2013–2014.
STAT 021. Elementary Topics in Statistics: Quantitative Paleontology
This course will explore current areas of research in paleobiology and macroevolution. For instance, does evolutionary change generally occur gradually or in short bursts? How reliably does the fossil record preserve information about ecosystems? What factors make species more likely to go extinct? To answer these and other questions, paleobiologists use a range of statistical and mathematical techniques. We will emphasize conceptual understanding and applications of such quantitative methods, rather than their underlying theory or proofs. Class meetings will include a combination of lectures, discussion of journal articles, and conversations with leading paleontologists via Skype. Prerequisite: BIOL 002, or STAT 011 or equivalent. 1 credit. Not offered 2013–2014.
MATH 026. Advanced Topics in Single-Variable Calculus
For students who place out of the first half of MATH 025. This course goes into more depth on sequences, series, and differential equations than does MATH 025 and includes power series and convergence tests. This course, or MATH 025, is required of all students majoring in mathematics, physics, chemistry, or engineering. Students may not take MATH 026 for credit after MATH 025 without special permission. Prerequisite: Placement by examination (see "Advanced Placement and Credit Policy" section). 1 credit. Fall 2013. Grinstead.
MATH 027. Linear Algebra
This course covers systems of linear equations, matrices, vector spaces, linear transformations, determinants, and eigenvalues. Applications to other disciplines are presented. This course is a step up from calculus: It includes more abstract reasoning and structures. Formal proofs are discussed in class and are part of the homework. Students may take only one of MATH 027, MATH 028, and MATH 028S for credit. Prerequisite: A grade of C or better in some math course numbered 025 or higher or placement by examination (see "Advanced Placement and Credit Policy" section). 1 credit. Each semester. Fall 2013. Campbell, Cook. Spring 2014. Campbell, Cook.
MATH 028. Linear Algebra Honors Course
More theoretical, abstract, and rigorous than MATH 027. The subject matter will be equally as valuable in applied situations, but applications will be emphasized less. MATH 028 is intended for students with exceptionally strong mathematical skills, especially if they are thinking of a mathematics major. Students may take only one of MATH 027, MATH 028, and MATH 028S for credit. Prerequisite: A grade of B or better in some math course numbered 025 or higher, or placement by examination, including both placement out of calculus and placement into this course via Part IV of Swarthmore's Calculus Placement Exam (see "Placement Procedure" section). 1 credit. Fall 2013. Bergstrand. Spring 2014. Johnson.
MATH 028S. First-Year Seminar: Linear Algebra Honors Seminar
MATH 028S covers the same material as the lecture-based MATH 028 but uses a seminar format (maximum 12 students) with additional meetings. Hands-on student participation takes the place of most lectures. Students may take only one of MATH 027, MATH 028, and MATH 028S for credit. Prerequisite: Placement by examination, including both placement out of calculus and placement into this course via Part IV of Swarthmore's Calculus Placement Exam (see "Placement Procedure" section). 1 credit. Fall 2013. Maurer.
MATH 029. Discrete Mathematics
An introduction to noncontinuous mathematics. The key theme is how induction, iteration, and recursion can help one discover, compute, and prove solutions to various problems—often problems of interest in computer science, social science, or management. Topics will include mathematical induction and other methods of proof, recurrence relations, counting, and graph theory. Additional topics may include algorithms, and probability. There is a strong emphasis on good mathematical writing, especially proofs. While it does not use any calculus, MATH 029 is a more sophisticated course than MATH 15 or MATH 25; thus success in a calculus course demonstrates the mathematical maturity needed for MATH 29. Prerequisite: Strong knowledge of at least precalculus, as evidenced by taking another mathematics course numbered 15 or above, or through our placement examinations (see "Placement Procedure" section). Familiarity with some computer language is helpful but not necessary. Eligible for Cognitive Science credit. Writing course. 1 credit. Fall 2013. Maurer. Spring 2014. Bergstrand.
STAT 031. Data Analysis and Visualization
This course will study methods for exploring and modeling relationships in data. We introduce modern techniques for visualizing trends and formulating hypotheses. We will also discuss methods for modeling structure and patterns in data, particularly using multiple regression and related methods. The format of the course emphasizes writing assignments and interactive problem solving using real datasets. Statistics Prerequisites: Credit for AP Statistics, STAT 011, STAT 061, or ECON 031; or STAT 001 and permission of the instructor. Eligible for Cognitive Science credit. Writing course. 1 credit. Fall 2013. Schofield. Spring 2014. Sedlock.
MATH 033. Basic Several-Variable Calculus
This course considers differentiation and integration of functions of several variables with special emphasis on two and three dimensions. Topics include partial differentiation, extreme value problems, Lagrange multipliers, multiple integrals, line and surface integrals, Green's, Stokes', and Gauss' theorems. The department strongly recommends that students take MATH 034 instead, which is offered every semester and provides a richer understanding of this material by requiring linear algebra (MATH 027 or 028) as a prerequisite. Students may take only one of MATH 033, MATH 034, and MATH 035 for credit. Prerequisite: MATH 025, or 026 or placement by examination (see "Advanced Placement and Credit Policy" section). Students who have taken linear algebra at Swarthmore or elsewhere may not take MATH 033 without the instructor's permission. 1 credit. Fall 2013. Mavinga.
MATH 034. Several-Variable Calculus
Same topics as MATH 033 except in more depth using the concepts of linear algebra. The department strongly recommends that students take linear algebra first so that they are eligible for this course. Students may take only one of MATH 033, MATH 034, and MATH 035 for credit. Eligible for Cognitive Science credit. Prerequisite: MATH 025, or 026; and MATH 027, 028, or 028S. 1 credit. Each semester. Fall 2013. Epstein. Spring 2014. Epstein.
MATH 035. Several-Variable Calculus Honors Course
This version of MATH 034 will be more theoretical, abstract, and rigorous than its standard counterpart. The subject matter will be equally as valuable in applied situations, but applications will be emphasized less. It is intended for students with exceptionally strong mathematical skills and primarily for those who have completed MATH 028 or 028S successfully. Students may take only one of MATH 033, MATH 034, and MATH 035 for credit. Prerequisite: A grade of C or better in MATH 028 or 028S, or permission of the instructor, or in the fall for entering students who have placed out of linear algebra, permission of the departmental placement coordinator. 1 credit. Fall 2013. Gomez. Spring 2014. Grinstead.
STAT 032. Topics in Statistics: Data Analysis Projects in Public and Social Policy
This course is offered occasionally, when it was last offered in spring 2011 it was a Community-Based Learning project course in data analysis. Students worked in teams on a semester-long data analysis problem. Projects were drawn from data from local organizations in order to attempt to answer questions of direct importance to them. A key objective of the course is to expose students to the variety of challenges faced by the data analyst. Topics may include multiple regression, analysis of variance, analysis of covariance, and other related methods. Students research the scientific background of their problem and consult with the local organizations from which their data came. Prerequisite: STAT 011, or permission of the instructor. 1 credit. Not offered 2013–2014.
MATH 043. Basic Differential Equations
This course emphasizes the standard techniques used to solve differential equations. It will cover the basic theory of the field with an eye toward practical applications. Standard topics include first-order equations, linear differential equations, series solutions, first-order systems of equations, Laplace transforms, approximation methods, and some partial differential equations. Compare with MATH 044. Students may not take both MATH 043 and 044 for credit. The department prefers majors to take MATH 044. Prerequisites: Several-variable calculus or permission of the instructor. 1 credit. Spring 2014. Johnson.
MATH 044. Differential Equations
An introduction to differential equations that has a more theoretical flavor than MATH 043 and is intended for students who enjoy delving into the mathematics behind the techniques. Problems are considered from analytical, qualitative, and numerical viewpoints, with an emphasis on the formulation of differential equations and the interpretations of their solutions. This course does not place as strong an emphasis on solution techniques as MATH 043 and thus may not be as useful to the more applied student. Students may not take both MATH 043 and 044 for credit. The department prefers majors to take MATH 044. Eligible for Cognitive Science credit. Prerequisites: Linear algebra and several-variable calculus or permission of the instructor. 1 credit. Spring 2014. Mavinga.
MATH 046. Theory of Computation
MATH 053. Topics in Analysis
Course content varies from year to year depending on student and faculty interest. Recent topics have included financial mathematics, dynamical systems, and Fourier analysis. Prerequisites: Linear algebra and several-variable calculus. 1 credit. Alternate years. Not offered 2013–2014.
MATH 054. Partial Differential Equations
The first part of the course consists of an introduction to linear partial differential equations of elliptic, parabolic, and hyperbolic type via the Laplace equation, the heat equation, and the wave equation. The second part of the course is an introduction to the calculus of variations. Additional topics depend on the interests of the students and instructor. Prerequisites: Linear algebra, several-variable calculus, and either MATH 043, MATH 044, PHYS 050, or permission of the instructor. 1 credit. Alternate years. Spring 2014. Mavinga.
MATH 055. Topics in Geometry
Course content varies from year to year. In recent years, the emphasis has been on introductory differential geometry. See also MATH 075. Prerequisites: Linear algebra and several-variable calculus or permission of the instructor. 1 credit. Alternate years. Not offered 2013–2014.
MATH 056. Modeling
An introduction to the methods and attitudes of mathematical modeling. Course content varies from year to year depending on student and faculty interest. Because modeling in physical science and engineering is already taught in courses in those disciplines, applications in this course will be primarily to social and biological sciences. Various standard methods used in modeling will be introduced. These may include differential equations, Markov chains, game theory, graph theory, and computer simulation. The course will balance theory with how to apply these subjects to specific modeling problems coming from a variety of disciplines. The format of the course will include projects as well as lectures and problem sets with the hope that those enrolling will have the opportunity to apply what they have learned to appropriate problems within their own area of interest. Prerequisites: Linear algebra and several-variable calculus or permission of the instructor. 1 credit. Alternate years. Fall 2013. Campbell.
MATH 058. Number Theory
The theory of primes, divisibility concepts, and multiplicative number theory will be developed. Prerequisites: Linear algebra and several-variable calculus or permission of the instructor. 1 credit. Alternate years. Not offered 2013–2014.
STAT 061. Probability and Mathematical Statistics I
This course introduces the mathematical theory of probability, including density functions and distribution functions, joint and marginal distributions, conditional probability, and expected value and variance. It then develops the theory of statistics, including parameter estimation and hypothesis testing. The emphasis is on proving results in mathematical statistics rather than on applying statistical methods. Students needing to learn applied statistics and data analysis should consider STAT 011 or 031 in addition to or instead of this course. Prerequisites: MATH 033 or 034 or permission of the instructor. STAT 011 or the equivalent is strongly recommended. 1 credit. Fall 2013. Everson.
MATH 067. Introduction to Modern Algebra
This course is an introduction to abstract algebra and will survey basic algebraic systems—groups, rings, and fields. Although these concepts will be illustrated by concrete examples, the emphasis will be on abstract theorems, proofs, and rigorous mathematical reasoning. Required additional meetings. Prerequisite: Linear algebra or permission of the instructor. Writing course. 1 credit. Fall 2013. Johnson.
MATH 069. Combinatorics
This course continues the study of material begun in MATH 029. The primary topics are enumeration and graph theory. The first area includes, among other things, a study of generating functions and Polya counting. The second area is concerned with relations between certain graphical invariants. Additional topics may include one or more of the following topics: design theory, extremal graph theory, Ramsey theory, matroids, matchings, codes, and Latin squares. Prerequisites: Grades of C or better in MATH 029 and at least one other course in mathematics numbered 27 or higher, or permission of the instructor. 1 credit. Alternate years. Not offered 2013–2014.
MATH 073. Advanced Topics in Analysis
MATH 075. Advanced Topics in Geometry
An advanced version of MATH 055, sometimes given instead, and typically requiring MATH 063, 067, or both. The topic for 2013–2014 is computational geometry and topology. This version of the course may not be used as part of the Honors preparation in Geometry. Prerequisites: At least one of MATH 055, MATH 063, MATH 067, or MATH 069. MATH 063 recommended especially. 1 credit. Fall 2013. Shimamoto.
MATH 077. Advanced Topics in Algebra
An advanced version of MATH 057, sometimes given instead, and requiring the core course in algebra. (In 2013–2014 MATH 057 will be offered instead.) Prerequisites: Linear algebra and MATH 067. 1 credit. Not offered 2013–2014.
MATH 079. Advanced Topics in Discrete Mathematics
MATH 093/STAT 093. Directed Reading
MATH 096/STAT 096. Thesis
MATH 097. Senior Conference
This course is required of all senior mathematics majors in the Course Program and must be taken at Swarthmore. It provides an opportunity to delve more deeply into a particular topic agreed on by the student and the instructor. This focus is accomplished through a written paper and either an oral presentation or participation in a poster session. 0.5 credit. Fall 2013. Talvacchia.
MATH 102. Modern Algebra II
This seminar is a continuation of Introduction to Modern Algebra (MATH 067). Topics covered usually include field theory, Galois theory (including the insolvability of the quintic), the structure theorem for modules over principal ideal domains, and a theoretical development of linear algebra. Other topics may be studied depending on the interests of students and instructor. Eligible for Cognitive Science credit. Prerequisite: MATH 067. 1 credit. Fall 2013. Bergstrand. Spring 2014. Staff.
MATH 103. Complex Analysis
A brief study of the geometry of complex numbers is followed by a detailed treatment of the Cauchy theory of analytic functions of a complex variable: integration and Cauchy's theorem, power series, residue calculus, conformal mapping, and harmonic functions. Various applications are given, and other topics—such as elliptic functions, analytic continuation, and the theory of Weierstrass—may be discussed. Prerequisite: MATH 063. 1 credit. Alternate years. Fall 2013. Grinstead.
MATH 106. Advanced Topics in Geometry
The course content varies from year to year among differential geometry, differential topology, and algebraic geometry. In 2013, the topic is expected to be advanced differential geometry. Prerequisites: MATH 055 and 063 or permission of the instructor. 1 credit. Alternate years. Not offered 2013–2014.
STAT 111. Mathematical Statistics II
This seminar is a continuation of STAT 061. It deals mainly with statistical models for the relationships between variables. The general linear model, which includes regression, variance, and covariance analysis, is examined in detail. Topics may also include nonparametric statistics, sampling theory, and Bayesian statistical inference. Eligible for Cognitive Science credit. Prerequisites: Linear algebra and a grade of C+ or better in STAT 061; CPSC 021. 1 credit. Spring 2014. Sedlock. | 677.169 | 1 |
First Year Academics: Calculus
Mathematics is the common language of science and engineering and is essential for understanding many aspects of the physical world. To provide a solid mathematical foundation, MIT has a two-subject General Institute Requirement in calculus.
If you take Calculus at MIT, which version of 18.01 and 18.02 should you take?
The choice of subjects will depend on your background, preparation, and interests.
18.01 Calculus I: Prerequisite: High school algebra and trigonometry. For students who have a year or less of high school calculus and no AP credit.
18.014 and 18.024 Calculus with Theory:Prerequisite: Strong interest and ability in mathematics, interest in rigorous proofs. This is intended as a two-semester sequence, but students completing 18.014 will be well-prepared for all versions of 18.02. Students choosing 18.014 should be familiar with the computational aspects of single-variable calculus, though these aspects may be reviewed during the term.
If you have credit for 18.01 via AP or other exam score, ASE, or Transfer Credit, you may register for 18.02 or any of its variants. See the Math Department's Calculus page for descriptions of classes. | 677.169 | 1 |
Mathematics For ElementaryMathematics for Elementary School Teachers," 3/e, offers pre-service teachers a comprehensive mathematics course designed to foster concept development through examples, investigations, and explorations. Visual icons throughout the main text allow instructors to easily connect the text to the hands-on activities in the corresponding Explorations Manual."Classroom Connections" in both the exposition and the exercises guide students to connect the mathematics being taught with effective teaching strategies. Students must analyze educational math... MOREematics research, evaluate common student errors, and see alternative solution methods, enabling them to better prepare for their future teaching careers."Investigations" encourage students to think about a topic before discussing the math or viewing examples. These can be used as classroom discussion questions, for independent reading, or as review."Multiple Strategies" presented throughout the examples and exposition of the text allow students to analyze numerous approaches to solving problems.
Note: Each chapter concludes with an Investigation, Exercise, a Chapter Summary and Review Exercises | 677.169 | 1 |
Center for Applied Probability - Georgia Tech
Formed in the Spring of 1995, run by a group of faculty members in the School of Mathematics and the School of Industrial and Systems Engineering (ISyE) at Georgia Tech. Members, events, mailing list, links to other sites.
...more>>
CMP Unit Organizers - Art Mabbott
To introduce students to new units from the Connected Mathematics Project (CMP), Art Mabbott has developed a set of Unit Organizers. Adapting work done by B. Keith Lenz, of the University of Nebraska, Mabbott has also shared his MS Word documents with
...more>>
Cool School Tools - Tim Fahlberg
Shockwave whiteboard movies on algebra, geometry, probability, statistics, the mathematics of finance, and more. A whiteboard movie (WM) is a multimedia screen recording of writing on an electronic whiteboard (real or virtual) with or without voice and/or
...more>>
CPO Online - Cambridge Physics Outlet
A company founded by teachers and scientists that creates hands-on equipment and curriculum for teaching science, math, and technology from grades 4-12 and beyond, and provides effective professional development in science and math that is both content
...more>>
CTAP Middle School Math Project - CTAP Region IV
A database of online technology resources supporting California middle school math content standards for grades 6, 7, and algebra. Designed for use in classroom instruction or as homework helpers for students, the matrices resources align to state standards Bruce Wilson
David Bruce Wilson researches probability, combinatorics, and theoretical computer science. Abstracts of his articles on these subjects are available on the web and may be downloaded in PostScript or .dvi formats. Software available for download includes
...more>> | 677.169 | 1 |
...
Show More algebraic manipulations. Students of computer science whose curriculum may not allow the study of many ancillary mathematics courses will find it particularly useful. Mathematics students seeking a first approach to courses such as graph theory, combinatorics, number theory, coding theory, combinatorial optimization, and abstract algebra will also enjoy a clear introduction to these more specialized fields. The main changes to this new edition are to present descriptions of numerous algorithms on a form close to that of a real programming language. The aim is to enable students to develop practical programs from the design of algorithms. Students of mathematics and computer science seeking an eloquent introduction to discrete mathematics will be pleased by this work | 677.169 | 1 |
Descriptions and Ratings (1)
Date
Contributor
Description
Rating
18 Jun 2013
MathWorks Classroom Resources Team
Software Carpentry helps researchers be more productive by teaching them basic computing skills. We run boot camps at dozens of sites around the world, and also provide open access material online for self-paced instruction. The benefits are more reliable results and higher productivity: a day a week is common, and a ten-fold improvement isn't rare. | 677.169 | 1 |
Algebra has been developing through the interaction between the investigation of its own algebraic structures and its applications to different areas of Mathematics and other branches of Science. This informative research volume consists of survey and original articles by reputed algebraists which are refereed by the experts in the relevant fields. The survey articles provide an excellent overview of the various areas of research in Algebra. The original articles by reputed algebraists in Ring Theory, Module Theory, Semigroup Theory, Lattice Theory, Category Theory, Derivations, Hyper and Fuzzy Structures etc. exhibit new ideas, tools needed for the successful applications and discuss newMore... techniques and methodologies for current research in different branches of Algebra. Over 300 bibliographic references make Algebra and its Applications: Recent Developments an indispensable resource book for the beginners and advanced experts in Algebra | 677.169 | 1 |
Course Content includes the following:
• Creating a worksheet and an embedded chart
• Working with formulas, functions and formatting – such as entering formulas in a worksheet
Using, average, max and min functions
• What-if analysis – Making decisions using the IF Function
• Financial functions still use PowerPoint for presentations and still teach the uses of Microsoft Applications to others. I had an extraordinary understanding of prealgebra in 6th grade, and would often tutor my classmates to help them better understand it. In terms of teaching pre-algebra, I've taught the subject to various students in my high school who asked for assistance | 677.169 | 1 |
The history of computing could be told as the story of hardware and software, or the story of the Internet, or the story of ďsmartĒ hand-held devices, with subplots involving IBM, Microsoft, Apple, Facebook, and TwitterStudents will save time and master non-calculus-based probability and statistics with this powerful study guide. It simplifies difficult theories and focuses on making clear the areas students typically find hardest to understandSolving the problems in this book will help demonstrate mastery of the mathematical concepts taught in elementary school. They are represented in a visually engaging manner in order to make mathematics more fun and interesting. More emphasis is paid to problem solving abilities rather than the usual mathematics grind. | 677.169 | 1 |
New Scientist full online access is exclusive to subscribers. Registered users are given limited access to content, find out more. To read the full article, log in or subscribe to New Scientist.
Physics tool makes students miss the point
Software designed to help physicists tackle complicated mathematics seems to be encouraging students to focus on the wrong aspects of scientific problems.
Interested in how students use computer programs to solve problems, physicists Thomas Bing and Edward Redish of the University of Maryland, College Park, analysed videos of teams of students as they worked on their assignments. Among other tools, the students used Mathematica, a program that crunches not only numbers but also symbols, enabling it to do algebra and calculus.
By solving equations that might take days to solve with a pencil and paper, Mathematica frees up researchers to explore larger questions and to explore more problems. But this comes at a cost, Bing and Redish warn.
Using Mathematica for physics involves two stages: choosing a strategy for solving the problem, and then implementing that strategy by typing in a few lines of computer code. Although | 677.169 | 1 |
Excursions in Modern Mathematics - 5th edition
Summary: For undergraduate courses in Liberal Arts Mathematics, Quantitative Literacy, and General Education.
This collection of "excursions" into modern mathematics is organized into four independent parts, each consisting of four chapters--1) Social Choice, 2) Management Science, 3) Growth and Symmetry, and 4) Statistics. The book is written in an informal, very readable style, with pedagogical features that make the material both interesting and clear....show more Coverage centers on an assortment of real-world examples and applications, demonstrating the usefulness, relevance, and attractiveness of liberal arts mathematics.
Features
NEW--"Profiles" --Incorporates biographical profiles of featured mathematicians at the end of each chapter.
Humanizes the materials in each chapter and introduces some of the key mathematicians in these fields.
NEW--"Projects and Papers" --Found at the end of chapter problem sets, offers topics for projects and papers appealing to instructors who want their students to do cooperative learning or research papers.
Provides a great vehicle for explorations and class discussions that students and professors can pursue together.
NEW--Updated problems and examples. Many of the problems and examples have been updated and improved based on reviewer/user feedback.
Provides updated examples and additional exercises.
Carefully chosen topics--Meet the following criteria:
Accessibility--the material does not require a heavy mathematical infrastructure.
Applicability--the presentation connects the mathematics presented and the real-life problems that motivate it.
Currency--much of the material dates within the last 100 years, and some--fractals for instance--within the last 15.
Aesthetics--develops an appreciation for mathematics by combining its elegance with its simplicity.
Numerous exercises--Over 1500 in total.
Exercises are divided into 3 levels of difficulty:
Walking: straight forward applications of the concepts discussed in the chapter.
Jogging: exercises that require extra effort and/or insight on the part of the student.
Running: exercises that really challenge the students' ability and understanding | 677.169 | 1 |
This concise, accessible text provides a thorough introduction to quantum computing - an exciting emergent field at the interface of the computer, engineering, mathematical and physical sciences. Aimed at advanced undergraduate and beginning graduate students in these disciplines, the text is technically detailed and is clearly illustrated throughout with diagrams and exercises. Some prior knowledge of linear algebra is assumed, including vector spaces and inner products. However, prior familiarity with topics such as quantum mechanics and computational complexity is not required.
Readership:
Undergraduate and beginning graduate students in mathematics, computer science, physics, and engineering | 677.169 | 1 |
More About
This Textbook
Overview
REA's Plane and Solid (Space) Geometry Problem Solver
Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. Answers to all of your questions can be found in one convenient source from one of the most trusted names in reference solution guides. More useful, more practical, and more informative, these study aids are the best review books and textbook companions available. They're perfect for undergraduate and graduate studies.
Related Subjects
Meet the Author
Founded in 1959, Research & Education Association is dedicated to producing the finest and most effective digital and print materials in educational publishing, including some of America's most popular test preps and study aids. REA's wide-ranging catalog is a leading resource for teachers, students, and professionals.
Read an Excerpt
HOW TO USE THIS BOOK
This book can be an invaluable aid to students in geometry as a supplement to their textbooks. The book is divided into 52 chapters, each dealing with a separate topic. The subject matter is developed beginning with lines and angles and extending through analytic (coordinate) and solid geometry. Sections on constructions, coordinate conversions, polygons, surface areas, and volumes have also been included.
Each chapter in the book starts with a section titled Basic Attacks and Strategies for Solving Problems in this Chapter. This section explains the principles that are applicable to the topics in the chapter. By reviewing these principles, students can acquire a good grasp of the underlying techniques and strategies through which problems related to the chapter may be solved.
HOW TO LEARN AND UNDERSTAND
A TOPIC THOROUGHLY
1. Refer to your class text and read the section pertaining to the topic. You should become acquainted with the principles discussed there. These principles, however, may not be clear to you at the time.
2. Then locate the topic you are looking for by referring to the Table of Contents in the front of this book. After turning to the beginning of the appropriate chapter, read the section titled Basic Attacks and Strategies for Solving Problems in this Chapter. This section is a review of the important principles related to the chapter, and it will help you to understand further how and why problems in the chapter are solved in the manner shown.
3. Turn to the page where the topic begins and review the problems under each topic, in the order given. For each topic, the problems are arranged in order of complexity, from the simplest to the more difficult. Some problems may appear similar to others, but each problem has been selected to illustrate a different point or solution method.
To learn and understand a topic thoroughly and retain its contents, it will generally be necessary for students to review the problems several times. Repeated review is essential in order to gain experience in recognizing the principles that should be applied and to select the best solution technique.
HOW TO FIND A PARTICULAR PROBLEM
To locate one or more problems related to particular subject matter, refer to the index. In using the index, be certain to note that the numbers given there refer to problem numbers, not to page numbers. This arrangement of the index is intended to facilitate finding a problem more rapidly, since two or more problems may appear on a page.
If a particular type of problem cannot be found readily, it is recommended that the student refer to the Table of Contents and then turn to the chapter which is applicable to the problem being sought. By scanning or glancing at the material that is boxed, it will generally be possible to find problems related to the one being sought, without consuming considerable time. After the problems have been located, the solutions can be reviewed and studied in detail.
For the purpose of locating problems rapidly, students should
acquaint themselves with the organization of the book as found in the Table of Contents.
In preparing for an exam, it is useful to find the topics to be covered in the exam from the Table of Contents, and then review the problems under those topics several times. This should equip the student with what might be needed for the | 677.169 | 1 |
...
More About
This Book
From basic arithmetic, to pre-algebra, geometry, ratio and proportions, algebra, measurements and graphs, statistics, and some trigonometry, this learning tool provides more than 800 mathematical terms and their definitions. Enjoy!
Product Details
ISBN-13: 9781475126389
Publisher: CreateSpace Independent Publishing Platform
Publication date: 3/31/2012
Pages: 110
Product dimensions: 6.00 (w) x 9.00 (h) x 0.23 (d)
Meet the Author
Mark J. Curry earned a Bachelor's Degree in Elementary Education with a mathematics concentration from East Stroudsburg University, located in East Stroudsburg, Pennsylvania. He also earned a Master of Education Degree in Elementary Education, also from East Stroudsburg University. Mark has taught basic mathematics through beginning algebra at a New Jersey state prison for nearly six years. Mark is married to Kristie and they have two daughters; Alexa, who is seven-years-old and Abigail, who went home to the Lord shortly before birth in 2007. They currently have an unborn son on | 677.169 | 1 |
Barron's SAT Subject Test 2009: Math Level newly updated edition of this manual presents a diagnostic test, a review of all Math Level 2 test topics, and six full length model tests with answer keys and answer explanations. Test topics are reviewed in five separate chapters, each containing many practice exercises and answers. Major topics covered include functions and algebra; trigonometry; coordinate geometry; three-dimensional geometry; data analysis, statistics, and probability; and number and operations. Also included is detailed instruction on the use of graphing calculators. ... MOREThis version of the manual comes with an enclosed CD-ROM that presents two additional practice tests with answers, explanations, and automatic scoring. | 677.169 | 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.