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MATHEMATICS WORKBOOK
Higher physics demands fluency and ability in physical and
mathematical analysis that are not given prominence in school physics
syllabii. During the SPC we shall be developing both problem solving
skills and experimental techniques. In order to provide a common
assumed background we have prepared a set of questions based (roughly)
on the core AS-level syllabus. You may have already mastered all the
material, in which case this workbook will provide a useful set of
revision problems. However, if some of the material is new to you or
if you are uncertain about it or get stuck on any of the questions,
then we suggest that you refer to an appropriate A-level textbook.
These questions are selected and adapted from the University of
Cambridge Natural Sciences Tripos mathematics workbook:
Any comments or queries about this
workbook should be sent to [email protected] | 677.169 | 1 |
Mathematics Extension 2 tutoring - year 12
HSC Mathematics Extension 2 is often said to be the most challenging HSC course, but it scales incredibly high. For almost a decade, maths extension 2's scaled mean has hovered around the low to mid 40s (source: Table A3, UAC Scaling Report), making it the highest scaled subject commonly offered by schools. The majority of students who attain 99+ ATAR have done this course. Therefore, we strongly urge all students with at least moderate mathematical capability to consider doing this course. Dux College prides itself in the high standards of our Extension 2 classes.
Course information
HSC Mathematics Extension 2 will be taught by topic throughout year 12. Students who take this course should have a strong foundation in the Preliminary 2 unit and Extension 1 content. In this course, we focus on maximising our students' exposure to the largest variety of exam-style questions possible. Practice of all variations of the core topics will prepare students to get almost full marks for Questions 1 to 6. We aim higher than this. Our students are also given a variety of the more difficult, creative questions which are considered Question 7-8 standard. Theory and harder questions will be discussed in class through worked examples, paying particular attention to technique, manipulation and pattern recognisation.
A common concern with this subject is the difficulty of Questions 7 and 8. Often these questions are left substantially unattempted. We've even seen school teachers advise their students to "not worry about it too much". We believe this complacent approach towards the subject is ill-advised. We disagree with the popular notion that some students are born with exceptional lateral thinking skills needed to get near full marks in this subject. It is simply a matter of experience. Students will see patterns emerge after they have done a large quantity and variety of questions. It is this skill gained from doing many Question 8 standard questions that allows students to comfortably tackle the more challenging Questions 7 and 8.
Like all the other courses, we provide HSC exam style homework booklets for students to practice each week. Exposure to exam-style questions and exam conditions is the best preparation for the real thing. Questions will range in difficulty, with the bulk of questions being slightly harder than average (question 4-6 standard). There will also be challenging questions which are compulsory for all students to attempt. These will be discussed fully in class.
All our homework questions will be marked and provided with teacher's comments and worked solutions where appropriate. Students are expected to use the discussion forums for homework assistance when they face any difficulty.
Term 1 starts 2nd of February - for all new students interested in joining us for Term 1 2013, contact us to book a free trial lesson first.
Harshala Bharath - Leumeah High School, 2010
My class teacher was very helpful and to the point when I got confused. The teaching style was methodic and simple to understand compared to school. The study notes were very helpful and easy to understand. My marks went up progressively ... | 677.169 | 1 |
Advances in technology, together with an increased
interest in dynamical systems, are influencing the
nature of many first courses in ordinary differential
equations (ODEs). In addition, there is an increased
emphasis on nonlinear differential equations, systems
of differential equations, and mathematical modeling,
as well as on qualitative and numerical approaches
that shed light on the behavior of solutions. Analytic
techniques are still important, but they no longer
tend to be the sole focus. As these new directions
come into classrooms, research is beginning to
illuminate aspects of learning and teaching ODEs that
can inform ongoing curricular innovations.
In a French study aimed at exploring the teaching of
qualitative solutions, Michèle Artigue conducted a
three-year project with first-year students at the
University of Lille I. Approximately 100 students per
year received nearly 35 hours of instruction on
first-order differential equations. Students attended
common lecture sessions with smaller exercise sessions
using computers. As evidenced by their lab reports and
examinations, early on students were able to
successfully complete tasks where information was given
simultaneously in two settings and the problem to be
solved required interpretation between the two settings.
For example, one interpretation task asked students to
find and justify the correct match between seven
different differential equations and corresponding
graphs of solution curves. With little or no intervention
from the instructor, these students were successful
because they were able to employ a variety of familiar
criteria for determining and checking their answers.
These criteria included connections between the sign of
f (where dy/dx = f(x,
y)) and properties of monotonicity for solution
curves, zeros of f and horizontal slope, infinite
limit of f and vertical slope, the value of
f at a particular point and the slope of a
solution curve at that point, and recognizing particular
solutions associated with straight lines in the graphic
setting and checking them in the algebraic setting. We
would hope all students are familiar with these criteria
from calculus, and thus, that such criteria can serve as
a basis for further study of differential equations.
[see Artigue (1992)]
Michelle Zandieh and Michael McDonald also studied
students' underlying understanding of solutions
and equilibrium solutions. They interviewed a
total of 23 students from two separate
reform-oriented differential equations classes, one
at a large state university in the southwest and one
at a small liberal arts college on the west coast.
In addition to asking students open-ended questions
such as, "What is a differential equation?" and "What
is a solution to a differential equation?", they
posed several tasks for students to solve. One of
the tasks was the same matching task used by
Rasmussen and another asked students to draw
representative solutions on a given slope field for
dy/dt = y + 1. Much like
previous research findings, 7 of the 23 students
overgeneralized the notion of equilibrium solution
to include all values for which dy/dt
is zero. When asked to draw representative solution
functions, 3 of the 23 students failed to sketch the
equilibrium solution y(t) = 1.
Mathematically, we would expect students' notion of
equilibrium solution to be a subset of their notion
of solution, but for these students this did not
appear to be the case. Consistent with Rasmussen's
findings, these results underscore an important
conceptual difficulty that may lie beneath many
correct answers. [see Zandieh &
McDonald (1999)]
In another task Rasmussen provided students with
the autonomous differential equation
dN/dt = 4N(1N/3)
(1N/6) and the corresponding graph of
dN/dt vs. N. He asked the
following three questions: (a) What are the
equilibrium solutions? (b) Which of the
equilibrium solutions are stable and which are
unstable? (c) What is the limiting population for
N(0) = 2, N(0) = 3, N(0) = 4, and
N(0) = 7? All six interview subjects figured
out the correct answers to parts (a) and (b) but
four of the six students were unable to address
part (c). This was particularly surprising because
the typical student approach to this problem was to
figure out the first two parts by creating a sketch
like that in Figure 1.
Figure 1. Typical student sketch
Why would students be able to do parts (a) and
(b), but fail to "see" the connection between their
sketches and the long-term behavior for various
initial populations? This question is especially
intriguing because these students had just created
for themselves what is from our perspective a
sketch like that in Figure 1 of various solution
functions. During the interviews, the answer to
this question became quite clear. Students did not
view the sketch they had just created as a plot of
the functions that solve the differential
equation. In the words of one student, his sketch
was "just a test for stability." These students had
learned a graphical approach for determining
stability where the graphs they created did not
carry the intended conceptual meaning. [see Rasmussen (2001)]
Research findings focusing on student understanding
in courses taking new directions in ODEs indicate
that graphical and qualitative approaches do not
automatically translate into conceptual
understanding. In a traditional course, a typical
complaint is that students often learn a series of
analytic techniques without understanding important
connections and conceptual meanings. Care must be
taken or else students are likely to supplement
mindless symbolic manipulation with mindless
graphical manipulation. Of course, how a student
thinks and reasons is as much a reflection of his
or her individual cognitive development as it is a
reflection of the mathematics classroom. For
example, if students are not routinely expected to
explain and mathematically defend their
conclusions, it is more likely that they will
learn to proceduralize various graphical and
qualitative approaches in ways that are
disconnected from other aspects of the problem.
Since graphical predictions are playing an
increasingly prominent role in reform-oriented
approaches to ODEs (see for example, Blanchard, Devaney, and Hall, 2002;
Borrelli and Coleman, 1998; Diacu, 2000; Kostelich
and Armbruster, 1997), it makes sense to explore
the extent to which students are able to create
geometric proofs. Artigue's work specifically
examined this issue; she reported on students' work
on three types of tasks — prove that a solution
intersects a given curve; prove that it cannot
intersect a given curve; and prove that it has an
asymptote or rule out the possibility of such an
infinite branch. She found that students had great
difficulties generating these proofs. She
attributes this to two causes. First, students had
not been exposed to the delicate tools that they
needed to use in qualitative proofs in the graphical
setting. For example, the helpful ideas of
fence, funnel, and area had not
been introduced to students because, as Artigue
suggests, mathematics professors have been slow to
accept the graphical setting as a place for proof.
Second, many students have strong monotonic
conceptions that interfere with their proof efforts.
For example, students had an intuitive belief in the
following false statement: If f(x) has
a finite limit when x tends towards infinity,
its derivative f '(x) tends toward
zero. Yet another reason for students' difficulty
was that moving from predictions about how a
solution might look to actually proving these
statements requires the use of elementary analysis.
[see Artigue (1992)]
A different approach to proofs involves emphasizing
argumentation as a routine part of everyday classroom
discussions. In a multi-year project at a mid-sized
university in the Midwest, researchers2 are studying
student learning in a first course in ODEs as it
occurs in classrooms over the course of an entire
semester. An interesting example from this research
related to proof involves the arguments students
developed to justify that two solutions to a logistic
growth differential equation with different initial
conditions would never touch. Although these
students had not yet studied the uniqueness theorem,
they argued that since graphs of solutions to
autonomous different equations were shifts of each other
along the t-axis, there would never be a point
in time when the solutions intersected each other. For
another example of arguments involving short chains of
deductive reasoning, consider the following question
that students in this project asked and answered: Is it
possible for a graph of a solution to an autonomous
differential equation to oscillate? The typical
argument these students developed to reject this
possibility was to argue that since the slopes in a
slope field for an autonomous differential equation
would have to be the same "all the way across" the
slope field, a graph of a solution would not oscillate
because if it did, there would be a value for y
where the slope would be both positive and negative.
For students like those in this class who have little
to no experience in developing mathematical arguments
to support or refute claims, significant progress in
their ability to create and defend short deductive
chains of reasoning was observed. This progress was
due in large part to the explicit attention paid to
classroom norms pertaining to explanation and
justification. These social aspects of the mathematics
classroom are reviewed in the final section. [see
Stephan & Rasmussen (2002)]
In Rasmussen's study, students discussed a
previously completed Mathematica
assignment where they had generated and
interpreted graphs of the angular position (in
radians) versus time for the linear and
non-linear differential equations similar to
those shown in Figure 3. Each plot in Figure
3 shows a different set of initial conditions
for the solutions to the undamped linear model,
'' + = 0
and to the undamped nonlinear model,
'' + sin = 0.
Graphs of solutions to the nonlinear model are
indicated with NL.
Figure 3. Solution graphs for a linear and
nonlinear pendulum
As might be expected, students experienced the most
difficulty interpreting the graphs in Plots C and D.
Students tended to interpret the graph as a literal
picture of the situation. For example, one student
said that the graph of the solution to the
nonlinear model in Plot C indicates that "it starts
increasing and remains at a constant distance from,
whatever, and then it would start increasing again
spontaneously, plateau again and then start
increasing." He also acknowledged that he had
never seen a pendulum do something like that, but
was unable to interpret the plot otherwise. In
Plot D, this same student explained that the graph
of the solution to the nonlinear model shows the
pendulum "increasing and increasing and this thing
wouldn't be able to hold it and it would just fly
off." [see Rasmussen (2001)]
The studies by Trigueros and Rasmussen suggest that
developers of both curriculum and instruction need
to be cautious about what is assumed will be
obvious to students when dealing with rich and
complex graphical representations. Perhaps
further and deeper classroom conversations
surrounding the interpretation of such
representations might help minimize the types of
student difficulties highlighted in these studies.
In the study conducted by Rasmussen at the large
mid-Atlantic university, students worked on CAS
labs outside of class time and only rarely did
class discussion focus on interpretations or
analysis of their labwork. As documented by
interviews and surveys, students viewed these
CAS labs as unrelated to what they saw as the
main ideas of the course and they did not think
that the work they put into the labs furthered
their understandings of important ideas or
methods, which was contrary to the instructor's
goals of the course. However, when technology
is integrated into the course, there is
some evidence that this does help promote
better understandings of various graphical
representations. For example, in Habre's study
students used computer modules designed for
specific course goals of the course and
intended to introduce students to specific
concepts. Although this was not the focus of
his research, some of Habre's interview data
suggests that these modules might have been
helpful to students in their development of
mathematics in the graphical setting. For
example, when given a vector field for the
system of ODEs x'(t)
= x + 4y,
y'(t) = 3xy, all the
students he interviewed were able to draw an
appropriate trajectory in the xy-plane
and to draw reasonable x(t) and
y(t) graphs corresponding to this
trajectory. Some students, however, faced
difficulties in drawing the 3D-parametric curve.
Habre suggests that the role of such computer
modules in student learning warrants further study.
[see Rasmussen (1997); Habre (2000)]
What this limited research does indicate is that
we need to be deliberate in how and why we decide
to implement technology in the classroom. It
shows that students' visual understanding of phase
portraits, slope fields, and solutions of
differential equations is an area where we might
consider integrating technology into students'
experiences in the classroom. Using a computer
algebra system as a separate lab component or
only as a demonstration tool seems less likely
to achieve the intended learning goals.
Findings include the significant role of
explanation and justification as a normal part of
classroom discussion. The following classroom
features, critical to the success of the project
in terms of student learning, were initiated by
the instructor and sustained throughout the
semester: Students routinely explained their
thinking and reasoning (versus just providing
answers), listened to and tried to make sense of
other students' thinking, indicated agreement or
disagreement with other students' thinking, and
responded to other students' challenges and
questions. Such aspects of classroom social
interactions involving explanation and
justification that become routine are referred to
as social norms. The initiation and
maintenance of such norms was a challenge because
students in the project classes were used to and
expected traditional patterns of interaction
where the instructor talked and the students
listened.
Given that many undergraduate students are not
used to explaining their reasoning and making
sense of other students' thinking, a pervasive
and important question is: How can instructors (1)
initiate a shift in social norms, and (2) sustain
these norms over time? The studies conducted by
Rasmussen and colleagues offer useful responses.
For example, in the semester-long classroom studies
described, the instructor devoted explicit attention
to initiating the social norms described above.
During an approximately twenty-minute whole class
discussion on the second day of class, the
instructor led a whole discussion where he offered
no mathematical explanation himself. Rather,
he strove to initiate new social norms by inviting
students to discuss their thinking and reasoning
through remarks and questions such as:
Tell us how you thought about it. That's what
we're interested in.
What do some of the rest of you think about
what Jason just said?
Did anyone think about that in a different
way?
That's a good question. Let's put that
question out to the rest of the class. What do
the rest of you think about it?
Tell us why you are thinking that.
I'm not sure that everyone heard what you
were saying. Say it again please.
Say a bit more about that.
Social norms are not rules set out in
advance on a syllabus. Although being explicit
about expectations can be useful, such explicit
statements are insufficient. Norms are
regularities in the ways individuals interact.
As such, an instructor alone cannot establish them.
They are constituted and sustained through
participation and interaction over time. As
students and the instructor act in ways that are
consistent with new expectations regarding
explanation and justification, they contribute to
their ongoing constitution.
Another point, which is illustrated in two case
studies at two different U.S. universities, is that
every class, from the most traditional to the most
reform-oriented, has social norms that are
operative for that particular class. It is not the
presence or absence of social norms that
differentiates one class from one another. Rather,
it is the nature of the norms that differ from
class to class. Of course the social norms
pertaining to explanation and justification might
apply to a history class or an English literature
class, as well as a mathematics class. The term
sociomathematical norm refers to the fact
that the subject being learned is mathematics. The
expectation that one is to give an
explanation is a social norm, but what is
considered to be an elegant solution, a different
solution, an efficient solution, or an acceptable
mathematical explanation are sociomathematical
norms. For example, when students develop
predictions and explanations about the future of
say, the population of fish in a lake, it is
imperative that these explanations move beyond
conclusions based solely on contextual reasons
(e.g., the fish are going to run out of food,
so their numbers are going to decrease) to
include reasons that rely on an
interpretation of the mathematical idea of rate
grounded in the differential equation. Fostering
a classroom learning environment that promotes
the types of explanations valued by the mathematics
community is a process that evolves over time as
students and instructor interact in the classroom
setting. If instructors are interested in
promoting a classroom environment where students
routinely give and evaluate mathematical arguments,
explicit attention to the processes by which norms
are constituted is a first step.
Finally, this research team documented how these
evolving norms fostered a shift in student beliefs
about their role as learners, about their
instructor's role, and about the general nature of
mathematical activity. These beliefs shifted from
seeing their role as passive absorbers of
information to active participants in knowledge
creation. When the classroom is viewed as a
dynamic system that includes the way in which
students participate in mathematical learning,
we can account not only for how student beliefs
evolve and develop, we can also promote student
beliefs about mathematics more compatible with the
discipline itself. [see Yackel &
Rasmussen (in press)] | 677.169 | 1 |
Materials to be ordered via the DLD
Description
Algebra II is a comprehensive course that builds on the algebraic concepts covered in Algebra I and prepares students for advanced-level courses. Through a "Discovery-Confirmation-Practice" based exploration of intermediate algebra concepts, students are challenged to work toward a mastery of computational skills, to deepen their conceptual understanding of key ideas and solution strategies, and to
Within each Algebra II lesson, students are supplied with a post-study "Checkup" activity, providing them the opportunity to hone their computational skills in a low-stakes, 10-question problem set before moving on to a formal assessment. Additionally, many Algebra II lessons include interactive-tool-based exercises and/or math explorations to further connect lesson concepts to a variety of real-world contexts.
The content is based on the National Council of Teachers of Mathematics (NCTM | 677.169 | 1 |
Week 1 (USC classes begin on Monday; Friday is the last day to drop without a grade of W)
Jan 12 (Mon)
Read page xv from the preface as well as sections 1.1 and 1.2. In 1.1 do #2, 4, 7, 9, 14, 17. In 1.2 do #3, 7, 11, 12, 14, 15, 16. Get a graphing calculator by Wednesday. If you don't already own one, I suggest any of the TI-83 or TI-84 calculators. If you need to take or retake the Algebra Placement Test, go to and select the third link Take Me To The Tests. You could also look at the first two links which include practice tests.
Jan 14 (Wed)
Finish the assigned homework from 1.1-1.2 and read section 1.3. Bring your calculator to class from now on.
Jan 16 (Fri)
Read section 1.5. In 1.3 do #5, 9, 10, 11, 12, 20, 25, 26, 31. In 1.5 do #11, 12, 17, 18. There will be a quiz Wednesday on sections 1.1-1.3. Today we spent a lot of time on calculator usage. We discussed how to use the Y=, 2nd-TBLSET, and 2nd-TABLE features of your calculator in order to get a table of values for a function. We also discussed how to use the GRAPH and 2nd-CALC features of your calculator in order to find the intersection of two graphs. The quiz will include testing your ability to use your calculator effectively. We did not spend much class time on section 1.3 today but it will still be included on the quiz so be sure to read the book carefully and do all of the homework.
Read section 1.6 and do #2, 5, 10, 21, 22, 23, 24, 31, 36 from that section. For many of the problems in sections 1.5 and 1.6, I recommend that you try solving them 3 ways — (1) with a table of values on your calculator, (2) with a graph along with the intersect or trace features of your calculator, (3) by hand with rules of algebra including logarithm rules.
Week 3
Jan 26 (Mon)
Read section 1.7. For now do #1, 5, 8. If you want to get ahead then do #10, 11, 15, 16, 18, 19. There will be a quiz Friday on sections 1.5-1.6.
Jan 28 (Wed)
Finish the problems from 1.7. There will be a quiz Friday on sections 1.5-1.6.
Jan 30 (Fri)
If you missed today's quiz, download a copy and allow yourself around 20 minutes to complete it. For homework read section 2.1 and do #1, 3, 5, 6, 12, 13, 15 from that section.
Week 4
Feb 2 (Mon)
Read section 2.2 and do #13, 17, 20, 24, 26, 27 from that section. There will be a quiz Friday on sections 1.7, 2.1, 2.2.
Monday's test will cover sections 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 2.1, 2.2, and 2.3. You should also be able to use your calculator effectively. In particular you should be able to easily generate graphs and tables of values. You should also know how to use the 2nd-CALC features of your calculator to find intersections. For this test, in section 1.4 you may skip the part on "marginal cost", "marginal revenue", and "marginal profit" at the bottom of page 25. You may also skip the part on "supply and demand curves" from the middle of page 26 to the end of page 28. You should be able to solve any of this semester's assigned homework problems or quiz problems. I recommend that you look at past quizzes and tests with solutions from every semester I have taught this course. Remember to bring a student ID and a graphing calculator with fresh batteries.
Read section 3.3 and do the odd problems from #1–33 for practice with the chain rule. Also do #35, 36, 37, 39, 42, 43, 44, 45. There will be a quiz Wednesday on sections 3.1-3.2 and another quiz Friday on section 3.3. Now is a great time to use the free services of the Math Tutoring Center as well as Supplemental Instruction.
Feb 25 (Wed)
If you missed today's quiz, download a copy and allow yourself around 15 minutes to complete it. Read section 3.4 and do the odd problems #1–27 for practice with the product rule and quotient rule. Also do #34, 35, 36, 39, 41. There will be a quiz Friday on section 3.3 and a quiz Monday on section 3.4.
Feb 27 (Fri)
If you missed today's quiz, download a copy and allow yourself around 15 minutes to complete it. There will be a quiz Monday on section 3.4. There is no new homework - just get caught up this weekend.
Week 8
Mar 2 (Mon)
If you missed today's quiz, download a copy and allow yourself around 15 minutes to complete it. Read sections 4.1-4.2. Without using a calculator, graph the functions found in #8, 9, 10, 11, 12 in section 4.1. Don't be too concerned with the terminology just yet – we'll get to that next time.
Mar 4 (Wed)
Read section 4.3. Without using a calculator, graph the functions found in section 4.2 (#11, 12, 13, 14, 15, 16, 17, 18, 19, 20) and section 4.3 (#24, 25, 26). Now compare your results to graphs obtained with your calculator. Be able to state the intervals upon which the function is increasing, decreasing, concave up, or concave down. Be able to find any local max/min, global max/min, or inflection points. There will be a quiz Friday on this material.
There will be a quiz Wednesday based on the homework from sections 4.3 and 4.4 assigned the Friday before Spring Break.
Mar 18 (Wed)
If you missed today's quiz, download a copy and allow yourself around 15 minutes to complete it.
Mar 20 (Fri)
Monday's test will cover sections 3.1, 3.2, 3.3, 3.4, 4.1, 4.2, 4.3, 4.4. No calculators are allowed for this test. You should be able to solve any assigned homework problem from these sections as well as any problem from quiz 4, quiz 5, quiz 6, quiz 7 or quiz 8. I also recommend that you look at past quizzes and tests with solutions from every semester I have taught this course. Remember to bring a student ID.
Week 11
Mar 23 (Mon)
Test 2
Mar 25 (Wed)
Read sections 5.1-5.2. In section 5.1 do #6, 8, 9, 11, 14, 15, 17. In class we worked on the following problem.
A population changes at a rate of 5e0.05t people per year where t is the number of years since 1980. If the population is 4000 in 1980 then estimate the population in 2000. First we approximate the total change in population between 1980 and 2000.
Estimate 3 (average of the two estimates above): 172.7 people (do you know if this is an overestimate or underestimate?)
This tells us that the population in 2000 is somewhere between 4151.2 and 4194.2 people. An estimate of 4172.7 is probably closer to the actual population but it is more difficult to say whether this is an underestimate or an overestimate. For homework get more refined estimates by choosing Δ t = 1. Be sure to make all 3 estimates and state when you know that your estimate is an overestimate or an underestimate.
Mar 27 (Fri)
No new homework.
Week 12
Mar 30 (Mon)
In section 5.2 do #3, 6, 10, 11, 12, 17.
Apr 1 (Wed)
Read section 5.3. No new homework.
Apr 3 (Fri)
Do #1-11 on the handout Old test and quiz problems — Chapter 5. There will be a quiz Monday on sections 5.1-5.2 and #1-6, 11 on the handout. There will be a quiz Wednesday on section 5.3 and #7-10 on the handout.
Week 13
Apr 6 (Mon)
If you missed today's quiz, download a copy and allow yourself around 10 minutes to complete it. Do #3, 4, 5, 6, 7, 8, 11, 19 from section 5.3. There will be a quiz Wednesday on section 5.3 and problems #7-10 on the recent handout Old test and quiz problems — Chapter 5.
Monday's test will cover sections 5.1, 5.2, 5.3, 5.4, and 5.5. It will include a non-calculator part. Bring your student ID. Be sure to look carefully over quiz 9 (blank copy, solutions), quiz 10 (blank copy, solutions), the handout on old tests and quiz problems (blank copy, solutions), and the handout on the Fundamental Theorem of Calculus (blank copy, solutions). On the Fundamental Theorem of Calculus handout, problems #1 and #2 will definitely be on the test. I also strongly suggest looking at my tests and quizzes from past semesters. | 677.169 | 1 |
5 2011Product Description
Review
. (Cut the Knot )
[. (Blake Mellor Journal of Mathematics and the Arts )
. (Choice )
From the Inside Flap
"The book's emphasis on a workshop approach is good and the authors offer rich insights and teaching tips. The inclusion of work by contemporary artists--and the discussion of the mathematics related to their work--is excellent. This will be a useful addition to the sparse literature on mathematics and art that is currently available for classroom use."--Doris Schattschneider, author of M. C. Escher: Visions of Symmetry
"Concentrating on perspective and fractal geometry's relationship to art, this well-organized book is distinct from others on the market. The mathematics is not sold to art students as an academic exercise, but as a practical solution to problems they encounter in their own artistic projects. I have no doubt there will be strong interest in this book."--Richard Taylor, University of Oregon
5.0 out of 5 starsExcellent textbook on the mathematics of PerspectiveSep 5 2011
By Ed Pegg - Published on Amazon.com
Format:Hardcover
At the start of the book, students are looking at normal hallways, rooms, and buildings through sheets of plexiglas, and tracing the outlines of what they see with drafting tape. From there, it's easy to see the concept of vanishing points.
A few pages later, an image from Jurassic Park with a velociraptor walking towards Sam Neill is shown. As an exercise, the students must compare the position of a clawtip to the bottom of a doorframe. I've messed up this issue of image placement many times, so this simple exercise brought home a lesson for me.
The core part of the book is 1, 2, and 3 point perspective, but with the idea that you'll be using a modern program of some sort. Then they introduce fractal geometry in a way I didn't expect, by taking a picture of a patch of grass and a small rock, and photopasting in a toy gnu and a climber. Small rocks look like big rocks look like mountains. I knew that, but I hadn't been tricked by it before, so I got the lesson better this time.
Recommended.
4.0 out of 5 starschallengingFeb 13 2013
By Proteus - Published on Amazon.com
Format:Kindle Edition|Amazon Verified Purchase
I have to admit, this book is pretty challenging. There is a ton of geometry math that is used to describe the mathematical aspects of perspective. But it is a face, it increased my understanding of perspective. It is not an easy book to get through, and frankly I probably only understood about 20% of it, but that 20% was useful, and some day I will probably go back and actually try to do the exercises. | 677.169 | 1 |
Student Solution Manual for Foundation Mathematics for the Physical Sciences
Choose a format:
eBook - PDF
Overview
Book Details
Student Solution Manual for Foundation Mathematics for the Physical Sciences
English
ISBN:
0511911181
EAN:
9780511911187
Category:
Mathematics / General
Publisher:
Cambridge University Press
Release Date:
03/28/2011
Synopsis:
This Student Solution Manual provides complete solutions to all the odd-numbered problems in Foundation Mathematics for the Physical Sciences. It takes students through each problem step-by-step, so they can clearly see how the solution is reached, and understand any mistakes in their own working. Students will learn by example how to arrive at the correct answer and improve their problem-solving skills.
Student Solution Manual for Foundation Mathematics for the Physical Sciences | 677.169 | 1 |
Three Uses for Khan Academy
Three Uses for Khan Academy
If you don't know, Khan Academy is a free online educational website that consists of an amazing set of videos (over 3000) made by the amazing, Salman Khan. Khan uses a Wacom type of tablet and a headset to make videos to teach a variety of subjects. Khan Academy is mostly known for Khan's mathematics and science videos but he also covers economics, art, history and more. Khan has made hundreds of videos for math including ones covering about every subject imaginable in algebra. The best thing about Khan Academy is that not only are there thousands of videos on multiple topics, but they are taught extremely well; he is definitely the teacher you dream of having in college. Khan explains more than just how to do problems, but actually gives you the intuition behind them. Khan Academy can be very a handy reource, and here are three good reasons to use it.
Review:
If you finished your high school math requirements during your junior year of high school and then you decided to not take math your freshman year of college, then the chances are you have forgotten everything about math. So, before you go back into calculus after not thinking about math for three years it might be wise to review a bit. Once you watch a couple of videos you'll probably get a sense of how behind you are and can then go from there.
Study:
Khan Academy is also a good resource when you are currently enrolled in a class. Books in general can be annoying and difficult to learn from, especially with subjects like math and science. Khan has most likely made a video on the subject you are struggling with and can help you with that upcoming quiz.
Test out of Classes:
If you were not an amazing high school student and did not take calculus, but have decided you want to in college, you have a long and slow ladder to climb. You can only take one math class at a time, and if you only took algebra in high school, it's going to take you a while to catch up. However, there is a solution. Most colleges will allow you to "test out of" or actually test into a certain class instead of taking all the prerequisites. So, if you think you could test out of trigonometry or whatever class you're missing, you can use Khan Academy to do so. During the summer, you could watch all of the Khan Academy videos on that subject, and try to test out | 677.169 | 1 |
Tuesday, May 15, 2012
Period 2 We revisited the quiz we took on Friday. We discussed what the problems meant and how we could better solve them. Students were then able to correct errors. Assignment: no homework tonight.
Period 3 We began some work on evaluating expressions and powers. These are skills that we have used in previous units and concepts that will be needed in Algebra 1. We reviewed vocabulary terms and related them to things we already know. We will continue this discussion tomorrow. Assignment: wksheet 1.1A, #1 - 22
Period 4 We went over the concepts that were covered in Friday's quiz and then students were able to correct errors. We also discussed how we can solve quadratic equations that are in factored form. This is something that we will continue tomorrow. Assignment: no homework tonight.
Period 5 We continued our work with the quadratic formula. This is just one way to solve quadratic equations but the method that works for all of them. Tomorrow we will discuss how to use the discriminant to determine the number of solutions. This builds directly on what we have been learning when using the quadratic formula. Assignment: p. 674, #7 - 12, 19 -26 | 677.169 | 1 |
MATHEMATICAL IDEAS
MATHEMATICAL IDEAS
2012 Fall Term
3 Units
Mathematics 140
Designed to give students a broad understanding and appreciation of mathematics. Includes topics not usually covered in a traditional algebra course. Topics encompass some algebra, problem solving, counting principles, probability, statistics, and consumer mathematics. This course is designed to meet the University Proficiency Requirement in mathematics for those students who do not wish to take any course which has 760-141 as a prerequisite. ACT Math subscore 19-23 (SAT 460-550)
Other Requirements: PREREQ: MATH 041 WITH A GRADE OF C OR BETTER, OR DEMONSTRATION OF EQUIVALENT CAPABILITY
Class Schedule
Disclaimer
This schedule is informational and does not guarantee availability for registration.
Sections may be full or not open for registration. Please use WINS if you wish to register for a course. | 677.169 | 1 |
The CLEP Exam
The CLEP Exam in Algebra is similar to a one semester course taught at many
colleges.
NOTE: There are three online sites that appear to cover much of this exam, and many
additional resources are available on the Web. Featured faculty and their home pages can be found
at the end of this page.
The topics in bold face are those The College Board indicates will be found on the
exams. The percentages given after the main topic headings are only approximate. Always contact
The College Board for the latest information.
(Click on description.)
Basic algebraic operations -- 25%
Combining algebraic expressions
Factoring
Simplifying algebraic functions
Operating with powers and roots
Equations, inequalities and their graphs -- 20%
Algebraic, exponential and logarithmic functions and their graphs -- 25%
Getting Started
Using the Free University Project Study Guide
A) Read the Introductory Material suggested in the Study Guide.
B) Read the material in the first two or three topics in the Study Guide. In order to stay
focussedRepeat the cycle. Periodically take time to review; do suggested exercises; take a practice CLEP
exam and review areas of weakness.
Remember to keep your journal up to date.
PLEASE NOTE: Free Online video series -- A complete listing of a 26 part Annenberg/CPB series is provided on a separate page | 677.169 | 1 |
MATHEMATICS
Mathematics Department Promotional Guidelines:
·
A student must have an overall average of 60% or more in order to be
promoted from one level to the next in Regular Math.
·
Promotion from Regular Math to Enriched Math is NOT a frequent occurrence.
Such promotions will be done if and only if there is a recommendation from
the Math Department. This recommendation will be done based primarily on the
work ethic and motivation of the student and secondly on the student's
results.
·Students
in Enriched Math must maintain a strong average (specified by the teacher).
·Students
who do not wish to stay in Enriched Math or whose results are less than
specified (but more than 60%) will be promoted to regular Math in June.
·Students
in regular Math in Secondary III will be enrolled in Cultural Secondary IV
Math.
·Students
in Enriched Math in Secondary III have the choice among Cultural Math,
Technical Math and Scientific Math in Secondary IV.
·Students
with an overall average of 60% or more in Cultural Secondary IV Math will be
promoted to Cultural Secondary V Math.
·Students
with an overall average of 60% or more in Technical Secondary IV Math will
be promoted to Technical OR Cultural Secondary V Math.
·Students
with an overall average of 60% or more in Scientific Secondary IV Math will
be promoted to Scientific OR Technical OR Cultural Secondary V Math.
Mathematics-Regular (MAT100-563100)
Natural Numbers: order of
operations, patterns, exponents, estimating
Integers: four
operations, order of operations
Rational Numbers: understanding,
four operations, order of operations, convert fractions to and from decimals
and percents, word problems using rational numbers
This is an enriched program in which students cover the regular
MAT100 program and additional topics. Number sense skills, fractions,
pre-algebra, and introductions to MAT212 material are strongly emphasized.
Students are chosen for this program based on very strong Grade 6 Math
results and specific teacher recommendation.
Mathematics – Regular (MAT212-563212)
Algebra: representations of a situation, sequences, the Cartesian
plane, variables, tables of values and the rule, graphs, ratios and
proportions, percentage, equations
Geometry: similarity, regular polygons, the circle, and solids
Probability: events and outcomes
Mathematics – Enriched (MEN212–563212)
This is an enriched program in which students cover the regular
MAT212 program and additional topics. The main emphasis is on increasing
algebra skills and number sense skills (specifically fractions) to better
prepare these students for success in the enriched Math 306 program.
Mathematics – Individualized Program (MAT310-568310)
Math 310 is intended for students who have had trouble completing
Secondary II. The goal is for students to successfully complete Secondary
III at a slower pace. The class covers the objectives of Secondary III but
does not go as in-depth as a regular Secondary III class. The class size is
smaller which allows for more individual interactions with the teacher.
Students may be eligible to write the regular Secondary III exam in June.
This is an enriched program in which students cover the regular
MAT306 program and additional topics. All topics are substantially enriched,
with particular emphasis on Algebra (includes Factoring, Absolute Value,
Quadratics, Fractional Equations and more).
Mathematics – Cultural (MATCUL-563404)
Arithmetic and Algebra
·First
inequality in two variables, real function: polynomial of degree less
than 3, exponential, periodic, step, piecewise, system of first-degree
equations in two variables
·Distance
between two points, coordinates of a point of division, straight line:
equation, slope, parallel and perpendicular lines, perpendicular
bisectors, metric and trigonometric relations in right triangles | 677.169 | 1 |
Explore how the parameters in a quadratic equaiton in standard form affect the graph of the equation. Dynamically change the parameters a, b, and c and immediately see the effect on the graph. Try t... More: lessons, discussions, ratings, reviews,...
Explore how the parameters in a rational equation affect the graph of the equation. Dynamically change the parameters and immediately see how the graph changes. Undefined points are clearly visible.... More: lessons, discussions, ratings, reviews,...
Use this savings calculator to see how a consistent approach to investing can make your money grow. Whether saving for a house, a car, or other special purchase, the savings calculator will help you d... More: lessons, discussions, ratings, reviews,...
This is a full lesson that gives students experience with exponential functions in an application format. There is a movie, an applet for gathering information, and a graphing applet. There is a les... More: lessons, discussions, ratings, reviews,...
Play this customizable game by entering functions that "hit" certain coordinates while avoiding others. Players (or teachers) can add as many of the coordinates to target or avoid, as well as set colo... More: lessons, discussions, ratings, reviews,...
The user can change the values of the initial population size, the yearly restocking amount, and the growth factor of the trout population in a pond, and then view the graph of the population size. More: lessons, discussions, ratings, reviews,...
Explore how the parameters in a quadratic equation in vertex form affect the graph of the equation. Dynamically change the parameters and immediately see how the graph changes. Try to change the par... More: lessons, discussions, ratings, reviews,...
This is a Java graphing applet that can be used online or downloaded. The purpose it to construct dynamic graphs with parameters controlled by user defined sliders that can be saved as web pages or em... More: lessons, discussions, ratings, reviews,...
This page contains a graphing applet that can be used online or downloaded. The applet can create dynamic graphs with sliders that can be saved as web pages. Users can plot points, functions, parametr...Students can choose from six types of factoring problems, or mixtures of the six types. The student enters the correct factoring and is given immediate feedback which includes steps needed in order | 677.169 | 1 |
If learners in the classroom are to be excited by mathematics, teachers need to be both well informed about current initiatives and able to see how what is expected of them can be translated into rich and stimulating classroom strategies.
The book examines current initiatives that affect teaching mathematics and identifies pointers for action inGraduate textbooks often have a rather daunting heft. So it's pleasant for a text intended for first-year graduate students to be concise, and brief enough that at the end of a course nearly the entire text will have been covered. This book manages that feat, entirely without sacrificing any materia more...
The leading reference on probabilistic methods in combinatorics-now expanded and updated When it was first published in 1991, The Probabilistic Method became instantly the standard reference on one of the most powerful and widely used tools in combinatorics. Still without competition nearly a decade later, this new edition brings you up to speed on... more...
In 2003 the British Combinatorial Conference conference was held at the University of Wales, Bangor. The papers contained here are high quality surveys contributed by the invited speakers, covering topics of significant interest. Ideal for established researchers and graduate students who will find much here to inspire future workA unique approach illustrating discrete distribution theory through combinatorial methods This book provides a unique approach by presenting combinatorial methods in tandem with discrete distribution theory. This method, particular to discreteness, allows readers to gain a deeper understanding of theory by using applications to solve problems. The... more... | 677.169 | 1 |
Mathematics Algebra I This course allows students to develop a solid foundation in basic
algebra skills and concepts. Topics include algebraic vocabulary,
properties and their operations, linear sentences, lines and distance,
slopes and lines, exponents and powers, polynomials, and systems of
equations. Further, Algebra I allows students to develop mathematical
power through problem solving strategies, reasoning activities and
cooperative learning projects using appropriate tools.
Geometry This course reviews basic algebraic concepts and then introduces the
elements of inductive and deductive reasoning as they relate to the
study of geometry. Geometry topics include perpendicular lines,
parallel lines and planes, congruent triangles, similar polygons,
circles, arcs, triangles, geometric constructions and loci coordinate
geometry, and areas and volumes of various types of figures. Through
the use of geometry, students become problem solvers who are able to
meet the demands of tomorrow's world. Prerequisite: Algebra I.
Geometry Honors This is an accelerated course of study with an in-depth development of the topics listed in Geometry.
Algebra II This course consists of a review of Algebra I topics and further
develops the concepts of polynomials, factoring, relations, functions,
solutions of linear equations, rational, irrational and complex
numbers. The course then introduces the study of quadratic equations,
logarithms, and elementary trigonometry. Algebra II allows students to
enhance their creative thinking by interpreting the application of
algebraic principles to related technology and scientific use.
Algebra II Honors This is an accelerated course of study with an in-depth development
of the topics listed in Algebra II. The course is also technology rich
with project-based learning including a summer technology component and
a collaboration of student, CTE instructor and academic instructor in
the creation of a video showing the relationship of mathematical
concepts to a student's specific program of study.
Pre-Calculus
This course consists of a review of the concepts taught in Algebra
II and geometry as they relate to the principles of trigonometry.
Development of the relationship between functions and their graphs is
explored with extensive use of the graphing calculator incorporated
throughout the course. Systems of linear equations and inequalities,
including matrices are covered with application to technology where
possible. After completing Pre-Calculus, students have a strong
foundation for work in calculus and problem-solving applications
necessary in a technical field. Prerequisite: Algebra II and Geometry.
Pre-Calculus Honors
This is an accelerated course of study with an in-depth development of the topics listed in Pre-Calculus.
Calculus
This course introduces elementary topics of Calculus including
limits, continuity and curve sketching. It also applies differentiation
to minimum and maximum and related rate problems and integration to
surface areas and volumes. It is recommended for students who will need
to take calculus in college. Prerequisite: Pre-Calculus.
Probability and Statistics
Probability emphasizes simulations of real world problems that
involve students in experimenting, collecting, organizing and using
data. Statistics emphasizes "making sense of data" by exploring and
organizing relevant data in a variety of ways. Students learn about
modeling trends and predicting the behavior of systems over time.
Extensive use of the statistical calculator and applications that are
relevant to the student's occupational program are incorporated.
Prerequisite: Algebra II and Geometry.
Algebra I Math Lab 9
This math class provides an individual approach to Algebra I along
with reinforcement of basic math skills. IEP goals are targeted with
progress being closely monitored so that instruction can be adjusted as
needed. It is a progressively challenging class. Algebra I topics
taught are: simplifying expressions, plotting coordinate plan, solving
slope, order of operations, linear equations and data analysis.
Geometry Math Lab 10
This math class provides an individual approach to Geometry along
with reinforcement of basic math skills. IEP goals are targeted with
progress being closely monitored so that instruction can be adjusted as
needed. It is a progressively challenging class. Geometry topics taught
are: basic geometry concepts (points, lines, distance, midpoints)
angles, perpendicular and parallel lines and triangles.
Algebra II Math Lab 11
This math class provides an individual approach to Algebra II along
with reinforcement of basic math skills. IEP goals are targeted with
progress being closely monitored so that instruction can be adjusted as
needed. It is a progressively challenging class. Algebra II topics
taught are: exponents, scientific notation, probability, quadratic
functions and equations, polynomials and factoring, and rational
expressions and equations.
Consumer Math Lab 12
This math class provides an individual approach to consumer math
along with reinforcement of basic math skills. IEP goals are targeted
with progress being closely monitored so that instruction can be
adjusted as needed. It is a progressively challenging class. Consumer
math topics taught are: expenses (wants and needs), making a budget,
salary (gross, net, deductions) retirement, stock market, using a
checking account, finding suitable housing and transportation.
Integrated Math
This course provides a structured approach to a variety of topics
such as ratios, percents, equations, inequalities, geometry, graphing
and probability and statistics. A solid foundation in these topics with
real-world applications to the more abstract algebraic concepts can be
found throughout the text. Various activity labs in each chapter
ensures students receive the visual and special instruction necessary
to conceptualize these abstract concepts, better preparing them for
life in the work force. | 677.169 | 1 |
Hey, This morning I started solving my math assignment on the topic Pre Algebra. I am currently not able to finish the same because I am not familiar with the fundamentals of angle suplements, angle suplements and cramer's rule. Would it be possible for anyone to assist me with this?
I have no clue why God made math, but you will be happy to know that a group of people also came up with Algebra Buster! Yes, Algebra Buster is a software that can help you solve math problems which you never thought you would be able to. Not only does it provide a solution the problem, but it also explains the steps involved in getting to that solution. All the Best!
Actually even I like this software and I can't explain a variety of distinct problems, really helped me. | 677.169 | 1 |
MATH 1503
CONTEMPORARY MATHEMATICAL CONCEPTS
This course is designed to give students a measure of insight into
modern mathematics, especially those looking forward to a career in
elementary education. Topics will include propositional logic, number
systems, calculus of sets, solution of equations and inequalities,
and geometry. Emphasis is placed on the understanding and use of the
various concepts that are introduced. Science students, business students,
and mathematics and statistics majors may not receive credit for this course. | 677.169 | 1 |
A) Communicate Effectively
1)Learning Outcome: Student will be able to communicate effectively about
mathematics through symbolic language and translate English expressions into
algebraic expressions. Course Objective : Understand and communicate the meaning of mathematical
symbols used in algebra. Assessment is based on tests, quizzes, and daily work in math portfolio
B) Think critically Assessment is based on tests, quizzes, and daily work in math portfolio. 1. Learning Outcome- Students will perform basic algebraic operations. Course Objective:Solve problems using algebraic formulas 2. Learning Outcome- Students will perform basic algebraic operations. Course Objective : Solve algebraic equations with 2 or more
operations. 3. Learning Outcome- Students will perform basic algebraic operations. Course Objective : To solve inequalities 4. Learning Outcome : Students will be able to construct graphs. Course Objective : Construct graphs of linear equations in 2 variables 5. Learning Outcome : Students will be able to simplify expressions. Course Objective : Apply rules of exponents to simplify expressions 6. Learning Outcome : Students will be able to perform operations on
Polynomials Course Objective : Students will factor polynomials and
perform 4 basic operations on polynomials.
C. Learn Independently Learning Outcome : The student will apply their learning. Course Objective : The student will understand applications that lead to
the
solution of problems.
Method of Instruction: Lectures on topics will be presented by the
instructor working
and explaining problems on the board. These problems will copied by the students
in
their course notebook. There will be time and opportunity for any student to ask
questions. Students will be individually encouraged to work problems on the
board and
explain them to the class. There may be up to three motivational films viewed in
class
where the students do brief reports on the content.
*All students must complete the course to receive credit. If a student stops
attending class toward the end of the semester and does not take the final
examination, a grade of "F" will be given.
Course Requirements:
You are expected to attend all classes and to be on time. Any combination of
three
tardies or early dismissals count as an absence. A very detailed record of your
attendance
will be kept by your instructor. This record may be shared with the Dean of
Enrollment
through the Early Alert System. Your attendance record may be given to the
Registrar for Financial Aid purposes. If you are in an athletic program, your
coach will probably be interested in your attendance history.
You are given the opportunity to prepare in class a
mathematics portfolio during the semester according to the following
guidelines:
Your math notes should be kept in a 3-ring binder with 2 pockets. It can be
organized in the following way: 1) Daily notes clipped in the middle.
2) Assignments in one pocket of notebook
3) All graded tests and quizzes in another pocket
4) Other written assignments from a math film or computer retrieved topic
5) Course Syllabus
If your mathematics portfolio is neat, complete, and meets all of the above
guidelines you can present this work at the end of the semester to receive an
additional grade of 100 points to be computed into your average.
* If you do not adhere to the above guidelines, do not present this work for a
grade.
Approximate dates of tests: Test 1: 1/17/08 (Order of Operations, simplification of numeric and alg.
expressions )
Test 2: 2/12/08 ( Solutions of Equations and Inequalities)
Test 3: 3/13/08 (Graphing linear equations in 2 variables by using different
forms of linear equations)
Test 4: 4/17/08 ( Laws of Exponents & Polynomials)
Test 5 :TBA (Final Exam.) SC 101 Date: 4/30/08 Time: 8:00a-10:00a The lowest test score will be dropped with the exception of the final
examination if the student has perfect attendance. *Remember, any combination of 3 tardies or early dismissals are
counted as an absence.
Quizzes are brief, simple, and unannounced. About 15 to 20 quizzes will be given
during the course and will be averaged for a semester quiz grade of 100. Two
quiz scores can be dropped. If you are absent on the day of a quiz, it cannot
be made up. The quizzes are a participatory grade given on what was covered
in class on that particular day. Your own personal notes may used when taking a
quiz.
Student's responsibility in case of an absence:
Absences and withdrawal from course: Refer to the Attendance information
provided in the Saint Catharine College Catalog. If you miss a class for any
reason, you are responsible for all material covered and assigned. The
Mathematics- Developmental Education Notebook (M-DEN) for your class will be
kept in the Resource Center which contains the daily notes and the assignment.
If you miss a class, report to the Resource Center and refer to this course
notebook with a tutor. If you miss an hour of class, spend an hour of time in
the Resource Center to catch up. Your time in the Resource Center will be kept
in a log book for both a reference and record. Documents that explain absences
may be given to the instructor to be filed as a record. (Parental notes will not
be accepted.) Exceptions will be made for student athletes when they are absent
due to school- related sporting events. Also exceptions will be made for any
student who is representing our college in a school function. ( All students regardless of the reasons for being absent should make up
their time at the Resource Center to learn the material covered on the days of
absence.) * If a student is absent on a test day, arrangements should be made with the
instructor and the test made up within one week. If these conditions are not
met, the student will receive a grade of 0 on that test. Academic Integrity: Academic dishonesty, whether intentional or not, is a
serious offense. Read the Saint Catharine College Catalog for the basic
statement of principle, definitions, responsibilities, policies, and penalties
of academic dishonesty. Classroom Behavior Statement:
You are expected to behave in a Christian manner toward students and
instructors. No disruptive or disrespectful behavior will be tolerated. Any
student who is guilty of any of these behaviors will be dismissed from class and
counted absent. Any student who is found sleeping during class will be dismissed
from class for that day and be counted absent. Cell phones must be turned off
and kept in book bags. Students are not allowed to share calculators during
tests or quizzes. There may also be some tests or quizzes where you will not be
allowed to use a calculator. Classroom learning accommodations: Inform instructor in writing by the end of the 2nd class of needed
accommodation as certified by the college.
Important dates:
January 7 First day of class for 16 week courses
January 11 Last day to withdraw from a 16 week course(s) without record
January 21 Martin Luther King Jr. Holiday
March 3 – 7 Spring Vacation
March 4 Mid term Grades Due
March 7 Midterm Grades Mailed to Students
March 20-23 Holy Thursday, Good Friday, Easter Holiday
March 25 Last Day to Withdraw from a 16 week course(s) with grade of W or change
to audit
April 14-18 Spring Convocation Activities (Classes Continue)
April 25 Last Day of Class
April 28-May 2 Final Examinations
Special Note to Student
Mathematics is a course that needs to be studied daily. Your class may only meet
2 or 3 times per week. However you need to focus on it Monday through Friday.
Sometimes working out of class at home may not prove to be too productive for
you . Even though I recommend that you work at every opportunity, the following
special accommodations will be made for you.
My office hours will be posted and given to you. My office is downstairs in
Bertrand Hall in Room #9. Feel free to drop by and see me for any help in this
course. The Resource Center will be opened daily with set hours where you can
receive tutoring. An appointment is not necessary when you report to Resource
Center.
Math – DEN stands for Mathematical Developmental
Educational Notebook. This is a special notebook left in the
Resource Center that contains the notes for your specific math class. The M-DEN
system will be explained in detail when we begin our classes.
The MathZone is an Internet Web based program in which you will register.
This program is an excellent learning tool for mathematics. | 677.169 | 1 |
focuses on the fundamental concepts of arithmetic, algebra, geometry and trigonometry needed by learners in technical trade programs. A wealth of exercises and applications, coded by trade area, include such trades as machine tool, plumbing, carpentry, electrician, auto mechanic, construction, electronics, metal-working, landscaping, drafting, manufacturing, HVAC, police science, food service, and many other occupational and vocational programs. The authors interviewed trades workers, apprentices, teachers, and training program directors to ensure realistic problems and applications and added over 100 new exercises to this edition. Chapter content includes arithmetic of whole numbers, fractions, decimal numbers, measurement, basic algebra, practical plane geometry, triangle trigonometry, and advanced algebra. For individuals who will need technical math skills to succeed in a wide variety of trades.
Table of Contents
Arithmetic of Whole Numbers
Fractions
Decimal Numbers
Ration, Proportion, and Percent
Measurement
Pre-Algebra
Basic Algebra
Pratical Plane Geometry
Solid Figures
Triangle Trigonometry
Advanced Algebra
Statistics Answers to Previews Answers
Index
Index of Applications
Table of Contents provided by Publisher. All Rights Reserved.
Excerpts
This book provides the practical mathematics skills needed in a wide variety of trade and technical areas, including electronics, auto mechanics, construction trades, air conditioning, machine technology, welding, drafting, and many other occupations. It is especially intended for students who have a poor math background and for adults who have been out of school for a time. Most of these students have had little success in mathematics, some openly fear it, and all need a direct, practical approach that emphasizes careful, complete explanations and actual on-the-job applications. This book is intended to provide practical help with real math, beginning at each student's own individual level of ability. Features Those who have difficulty with mathematics will find in this book several special features designed to make it most effective for them: Careful attention has been given toreadability.Reading specialists have helped plan both the written text and the visual organization. Adiagnostic pretestand performanceobjectiveskeyed to the text are included at the beginning of each unit. These clearly indicate the content of each unit and provide the student with a sense of direction. Each unit ends with aproblem setcovering the work of the unit. Theformatis clear and easy to follow. It respects the individual needs of each reader, providing immediate feedback at each step to ensure understanding and continued attention. The emphasis is onexplainingconcepts rather than simplypresentingthem. This is a practical presentation rather than a theoretical one. Special attention has been given toon-the-job math skills,using a wide variety of real problems and situations. Many problems parallel those that appear on professional and apprenticeship exams. The answers to all problems are given in the back of the book. A light, livelyconversational styleof writing and a pleasant, easy- to understand visual approach are used. The use of humor is designed to appeal to students who have in the past found mathematics to be dry and uninteresting. Seven editions and over two decades of experience with a wide variety of students indicate that this approach is successful--the book works and students learn, many of them experiencing success in mathematics for the first time. Flexibility of use was a major criterion in the design of the book. Field testing and extensive experience with the first five editions indicate that the book can be used successfully in a variety of course formats. It can be used as a textbook in traditional lecture-oriented courses. It is very effective in situations where an instructor wishes to modify a traditional course by devoting a portion of class time to independent study. The book is especially useful in programs of individualized or self-paced instruction, whether in a learning lab situation, with tutors, with audio tapes, or in totally independent study. Calculators Calculators are a necessary tool for workers in trade and technical areas, and we have recognized this by using calculators extensively in the text, both in fording numerical solutions to problems, including specific keystroke sequences, and in determining the values of transcendental functions. We have taken care to first explain all concepts and problem solving without the use of the calculator and to estimate and check answers. Many realistic problems included in the exercise sets involve large numbers, repeated calculations, and large quantities of information and are ideally suited to calculator use. They are representative of actual trades situations where a calculator is needed. Detailed instruction on the use of calculators is included in special sections at the end of appropriate chapters or is integrated into the text. Supplements An extensive package of supplementary | 677.169 | 1 |
SMART Notebook software is included free with all SMART Board interactive whiteboards and SMART Sympodium interactive lecterns. If you would like to view SMART Notebook files, you can download a tr... More: lessons, discussions, ratings, reviews,...
The SimCalc Project aims to democratize access to the Mathematics of Change for mainstream students by combining advanced simulation technology with innovative curriculum that begins in the early g... More: lessons, discussions, ratings, reviews,...
Simplesim is suited for modelling of non-analytic relations in systems which are causal in the sense that different courses of events interact in a way that is difficult to see and understand. Exam... More: lessons, discussions, ratings, reviews,...
TeaCat is a dynamic mathematical application that aims to enable high school students to experiment with and to exercise a variety of mathematical topics. Even engineers may benefit of TeaCat for rath... More: lessons, discussions, ratings, reviews,...
WCMGrapher is a free graphing application. It is designed to enable teachers to create graphs of functions, format the graphs, then copy and paste the graph into other applications. For example, yo... More: lessons, discussions, ratings, reviews,...
Web Components for Mathematics (webcompmath or WCM) is a library used to create interactive graphing web applets for teaching mathematics. It is written in the Java programming language and is baseWe present two versions of a 3D function grapher--one on a white background, one on a black background. The user enters a formula for f(x,y) in terms of x and y and the applet draws its graph in 3D. TFlash introduction to finding the equation of an ellipse centered on (0,0) and with its major axis on the x-axis. Students can use this Tab Tutor program to learn about the equation of this ellipse an... More: lessons, discussions, ratings, reviews,...
Flash introduction to finding the equation of an hyperbola centered on (0,0) and with its major axis on the x-axis. With step-by-step instructions and an illustrated glossary, students can learn how | 677.169 | 1 |
Book DescriptionProduct Description
From the Inside Flap
Algebra is used by virtually all mathematicians, be they analysts, combinatorists, computer scientists, geometers, logicians, number theorists, or topologists. Nowadays, everyone agrees that some knowledge of linear algebra, groups, and commutative rings is necessary, and these topics are introduced in undergraduate courses. We continue their study.
This book can be used as a text for the first year of graduate algebra, but it is much more than that. It can also serve more advanced graduate students wishing to learn topics on their own; while not reaching the frontiers, the book does provide a sense of the successes and methods arising in an area. Finally, this is a reference containing many of the standard theorems and definitions that users of algebra need to know. Thus, the book is not only an appetizer, but a hearty meal as well.
Let me now address readers and instructors who use the book as a text for a beginning graduate course. If I could assume that everyone had already read my book, A First Course in Abstract Algebra, then the prerequisites for this book would be plain. But this is not a realistic assumption; different undergraduate courses introducing abstract algebra abound, as do texts for these courses. For many, linear algebra concentrates on matrices and vector spaces over the real numbers, with an emphasis on computing solutions of linear systems of equations; other courses may treat vector spaces over arbitrary fields, as well as Jordan and rational canonical forms. Some courses discuss the Sylow theorems; some do not; some courses classify finite fields; some do not.
To accommodate readers having different backgrounds, the first three chapters contain many familiar results, with many proofs merely sketched. The first chapter contains the fundamental theorem of arithmetic, congruences, De Moivre's theorem, roots of unity, cyclotomic polynomials, and some standard notions of set theory, such as equivalence relations and verification of the group axioms for symmetric groups. The next two chapters contain both familiar and unfamiliar material. "New" results, that is, results rarely taught in a first course, have complete proofs, while proofs of "old" results are usually sketched. In more detail, Chapter 2 is an introduction to group theory, reviewing permutations, Lagrange's theorem, quotient groups, the isomorphism theorems, and groups acting on sets. Chapter 3 is an introduction to commutative rings, reviewing domains, fraction fields, polynomial rings in one variable, quotient rings, isomorphism theorems, irreducible polynomials, finite fields, and some linear algebra over arbitrary fields. Readers may use "older" portions of these chapters to refresh their memory of this material (and also to see my notational choices); on the other hand, these chapters can also serve as a guide for learning what may have been omitted from an earlier course (complete proofs can be found in A First Course in Abstract Algebra). This format gives more freedom to an instructor, for there is a variety of choices for the starting point of a course of lectures, depending on what best fits the backgrounds of the students in a class. I expect that most instructors would begin a course somewhere in the middle of Chapter 2 and, afterwards, would continue from some point in the middle of Chapter 3. Finally, this format is convenient for the author, because it allows me to refer back to these earlier results in the midst of a discussion or a proof. Proofs in subsequent chapters are complete and are not sketched.
I have tried to write clear and complete proofs, omitting only those parts that are truly routine; thus, it is not necessary for an instructor to expound every detail in lectures, for students should be able to read the text.
When I was a student, Birkhoff and Mac Lane's A Survey of Modern Algebra was the text for my first algebra course, and van der Waerden's Modern Algebra was the text for my second course. Both are excellent books (I have called this book Advanced Modern Algebra in homage to them), but times have changed since their first appearance: Birkhoff and Mac Lane's book first appeared in 1941, and van der Waerden's book first appeared in 1930. There are today major directions that either did not exist over 60 years ago, or that were not then recognized to be so important. These new directions involve algebraic geometry, computers; homology, and representations (A Survey of Modern Algebra has been rewritten as Mac Lane-Birkhoff, Algebra, Macmillan, New York, 1967, and this version introduces categorical methods; category theory emerged from algebraic topology, but was then used by Grothendieck to revolutionize algebraic geometry).
Here is a more detailed account of the later chapters of this book.
Chapter 4 discusses fields, beginning with an introduction to Galois theory, the interrelationship between rings and groups. We prove the insolvability of the general polynomial of degree 5, the fundamental theorem of Galois theory, and applications, such as a proof of the fundamental theorem of algebra, and Galois's theorem that a polynomial over a field of characteristic 0 is solvable by radicals if and only if its Galois group is a solvable group.
Chapter 6 introduces prime and maximal ideals in commutative rings; Gauss's theorem that R x is a UFD when R is a UFD; Hilbert's basis theorem, applications of Zorn's lemma to commutative algebra (a proof of the equivalence of Zorn's lemma and the axiom of choice is in the appendix), inseparability, transcendence bases, Lüroth's theorem, affine varieties, including a proof of the Nullstellensatz for uncountable algebraically closed fields (the full Nullstellensatz, for varieties over arbitrary algebraically closed fields, is proved in Chapter 11); primary decomposition; Gröbner bases. Chapters 5 and 6 overlap two chapters of A First Course in Abstract Algebra, but these chapters are not covered in most undergraduate courses.
Chapter 8 introduces noncommutative rings, proving Wedderburn's theorem that finite division rings are commutative, as well as the Wedderburn-Artin theorem classifying semisimple rings. Modules over noncommutative rings are discussed, along with tensor products, flat modules, and bilinear forms. We also introduce character theory, using it to prove Burnside's theorem that finite groups of order pmqn are solvable. We then introduce multiply transitive groups and Frobenius groups, and we prove that Frobenius kernels are normal subgroups of Frobenius groups.
Chapter 9 considers finitely generated modules over PIDs (generalizing earlier theorems about finite abelian groups), and then goes on to apply these results to rational, Jordan, and Smith canonical forms for matrices over a field (the Smith normal form enables one to compute elementary divisors of a matrix). We also classify projective, injective, and flat modules over PIDs. A discussion of graded k-algebras, for k a commutative ring, leads to tensor algebras, central simple algebras and the Brauer group, exterior algebra (including Grassman algebras and the binomial theorem), determinants, differential forms, and an introduction to Lie algebra.
Chapter 10 introduces homological methods,beginning with semidirect products and the extension problem for groups. We then present Schreier's solution of the extension problem using factor sets, culminating in the Schur-Zassenhaus lemma. This is followed by axioms characterizing Tor and Ext (existence of these functors is proved with derived functors), some cohomology of groups, a bit of crossed product algebras, and an introduction to spectral sequences.
Chapter 11 returns to commutative rings, discussing localization, integral extensions, the general Nullstellensatz (using Jacobson rings), Dedekind rings, homological dimensions, the theorem of Serre characterizing regular local rings as those noetherian local rings of finite global dimension, the theorem of Auslander and Buchsbaum that regular local rings are UFDs.
Each generation should survey algebra to make it serve the present time.
From the Back Cover is a tough book to review, because it is not clear who the real audience is supposed to be. The author says that it is aimed at first-year graduate students, with a bunch of extra material that can be referred back to during the second year and beyond. The earlier chapters also include efficient reviews (with sketched proofs) of material that should be familiar to those who have taken undergraduate algebra.
This characterization is debatable. Based on my experience reading most of the first six chapters (the first 400 out of about 1000 pages), I would say that the level of sophistication is roughly that of Dummit and Foote's "Abstract Algebra", which is usually considered an undergraduate book. D&F can sometimes be harder to read, and that is in part because Rotman's exposition is better (in my opinion), but also because D&F introduce more difficult material earlier. Whether D&F's approach is better is questionable; I find Rotman to be a much smoother read, but the organization is quite different -- for example, one does not encounter noncommutative rings until deep into the book, whereas Dummit and Foote introduce them immediately upon defining rings. On the other hand, early in the coverage of D&F's chapter on rings, one has to digest Zorn's Lemma and its applications almost from the beginning, whereas Rotman (I think wisely) pushes this back into a later section. In general, D&F introduce a lot of hairy examples that by themselves require a lot of effort to digest (thereby impeding the reader's progress through the core material), whereas Rotman's examples tend to be straightforward, at least as new concepts are being presented.
So, overall, the exposition flows more smoothly in Rotman's book, and the reader can cover the basics more quickly with less time spent on tangential examples and early generalizations. Also, Rotman's proofs are usually much cleaner and the overall style is very nice. It's more pleasant to read than Dummit and Foote. But this comes at a cost: Dummit and Foote do cover more material, and generalize at an earlier stage, than Rotman does.
But my biggest gripe concerns the exercises. Put simply, Rotman's are far too easy for what is being pitched as a graduate course. In fact, they are in general far easier than the homework problems I sweated through when I took honors undergraduate algebra. They're barely adequate to convince the reader that he has a basic grasp on the material, and there are almost no hard ones, let alone really tough, thought-provoking open-ended problems like one encounters in Herstein's "Topics in Algebra" (an undergraduate book). There are certainly no exercises in Rotman's book that would be of any use for a graduate student preparing for qualifying exams. They're not even much of a workout for a decent (honors student) undergraduate.
So, what is this book good for? I think it's great for reading material that is usually harder to understand elsewhere. Rotman has a real knack for clear mathematical exposition, and some of the chapters are a real joy to read. (Side note: there are also a lot of typos, at least in the first printing. The author maintains an errata list at his web site, and a second printing is coming soon. There are still many errata that he didn't catch, but they're fairly minor and do not detract significantly from the reading.) But this is simply not suitable for a primary graduate text or reference. Most good schools are going to demand more of their graduate students, and one is inevitably going to have to read Lang or Hungerford (and work through their exercises) to achieve competence at the graduate level. Rotman's book is a kinder, gentler book upon which to fall back when those books are inscrutable, as is all too common. I do recommend it highly for that purpose -- I think it's a very good secondary book.
To begin with, don't let the title scare you. After having read through Rotman's book I am suprised that this text had not crossed my path earlier. It is a wonderful book and must have for any inspiring Algebraist. Moreover, I am quite shocked that the larger universities have not adopted this book.
(a) This book could quite easily be used as the standard third/fourth year undergraduate introduction to Abstract Algebra. In particular, the first four chapters provide a solid foundation for a moderate paced one semester course at which point the instructor has many different options for additional topics based on the performance of his/her class.
(b) Those students that move on to the graduate level, and obviously to a university using this book, would both be familiar with the temperment and flow of the author as well as devoid of the requirement of having to purchase another expensive Mathematics text. For example, my undergraduate Algebra text was Hungerford's and post completion the logical step, being familiar with his style, was to purchase Hungerford's graduate text. For those not familiar, let me tell you there is a night and day difference with repsect to how the material is presented.
(c) The remaining 7 chapters take the willing student on a pleasant tour of ring/module theory, some advanced group theory (for the inspiring group theorist I highly recommend the authors graduate text "Group Theory"), algebras(linear included), Homology(some cohomology) and finally some algebraic number theoretic concept under the heading of Commutative Rings III.
(d) Lastly, Rotamn does not get needlessly bogged down in any one section of the book. The flow is smooth, to the point with precise definitions, examples, and ample exercises.
I have only two negative remarks: one, the failure to include more aspects of field/Galois theory. This may be due to the author already having published a book entitled "Galois Theory". Two, the failure to devote an entire section to Finite Fileds and possibly some its applications. But this failure is minimal since, at present, the majority of Algebra texts, fail to adequately introduce and motivate Finite Fields.
I previously purchased Rotman's First Course in Abstract Algebra, and fell in love with it. So when I saw he a Second Abstract Algebra book, I had to have it. I am currently taking a Graduate Level Modern Algebra course, and I find this book to be a great help in my Studies. I wouldn't be as interested in Modern Algebra as I am now if it weren't for this book. I love this book and I would reccomend it to anyone who is interested in Modern Algebra, or taking a course in Modern or Abstract Algebra. | 677.169 | 1 |
One of the most intimidating tasks facing a homeschooling parent, next to teaching your early elementary student how to read, is the whole issue of tackling high school math. Precalculus and calculus are particularly challenging to most average high school parents, and yet they are both subjects that many homeschooling high schoolers will find themselves wanting or needing to complete.
I am one of those "average" homeschooling parents, and my oldest son is an "above-average" math student. This means at age fifteen, he has already surpassed my competency level in mathematics and we have had to find creative solutions to teaching the subjects he is ready to learn. Online classes have been one way we have addressed this issue, and yet there are still times that he needs additional help working through a solution to a problem or times that it would benefit him to be able to work problems in addition to those provided by his curriculum. These software programs address that need.
Basically a student can enter a problem into these programs and receive not only an answer, but also the step-by-step solution. In addition, you can have the program generate example problems, generate interactive texts and even track your progress.
While these programs wouldn't necessarily take the place of a standard math curriculum, they are an easy to use supplement that my son has enjoyed using. They are flexible but comprehensive. Priced at $49.99, they should be accessible to most homeschooling families, particularly considering the programs are an investment that can be used for more than one student.
In addition to these titles, Bagatrix also offer several others including Trigonometry
Solved!, Geometry Solved! and College Algebra Solved! As more and more families
are exploring more non-traditional methods of addressing not only high school
educational needs but also college, I can see that these software programs could
be used even beyond high school. My son and I would recommend these programs
to advanced students, or even to those who need remedial help in mathematics. | 677.169 | 1 |
The National Council of Teachers of Mathematics
National Council of Teachers of Mathematics
1906 Association Drive
Reston, VA 22091
The NCTM is a seventy-five year old professional organization for mathematics teachers of
grades K-14. It contains approximately 106,000 members, of whom 79,000 are individual memberships.
The primary purpose of NCTM is to provide leadership in the
improvement of the teaching and learning of mathematics. To stimulate
students' interest and accomplishments in mathematics and to promote a
comprehensive education for every child, the Council has established
three goals:
To foster excellence in school mathematics curricula and
instructional programs, including assessment and evaluation
To promote professional excellence in mathematics teaching
To strengthen NCTM's leadership in mathematics education
The NCTM Statement on Algebra
The following text is based on the Presidential Address at the 72nd Annual Meeting of the NCTM in
Indianapolis, IN 14, April 1994. Mary M. Lindquist was President
First-year algebra in its present form is not the algebra for everyone. In fact, it is not the algebra for most high school graduates today.
Weaknesses of First Year Algebra in its present form:
They advance only a narrow range of by-hand skills for transforming, simplifying, and solving symbolic expressions, most often divorced from any natural context.
As a separate course, they effectively isolate the concepts and methods of algebra from the other major strands of school mathematics: statistics, geometry, and discrete mathematics.
They neither acknowledge nor encourage the development of informal understanding of algebraic ideas in grades K-8.
The first step toward algebra for everyone is a reconceptualization of the algebra strand within the fabric of school mathematics.
The reconceptualization of the algebra strand of the high school curriculum should be guided by the following two general principles about goals and teaching approaches to the subject:
The primary role of algebra at the school level is to develop confidence and facility in using variables and functions to model numerical and quantitative relations -- both within pure mathematics and in a broad range of settings in which numerical data are important.
The use of graphing calculators and computers makes the focus on modeling and functions attractive and accessible for students across a broad range of interests, aptitudes, and prior achievement. The use of these calculating tools will offer students a variety of powerful new learning and problem-solving strategies. | 677.169 | 1 |
The theory is all there, but it's placed nicely in a context appropriate for a mixed bag of undergrad students by a large number of interesting-but-doable exercises and informative historical notes. Modern applications to computer science, cryptography, etc are all there and can be emphasized (or not) as you see fit.
This is what I'd read if I were you. Last time I checked, the book was annoyingly expensive - but this is the only criticism of it I have. Most students give this book very favorable reviews, too. | 677.169 | 1 |
Sharp Math
Building Better Math Skills
A 10-question diagnostic quiz in every chapter to show readers where they need the most help.
Math from basic arithmetic to Algebra 2, broken down by subject and then building up from chapter to chapter so readers can group concepts together for easier learning.
A variety of practice exercises with detailed answer explanations for every topic.
A 15-20 question recognition and recall practice set that includes material from the entire chapter (and a few questions that cover material from the previous chapters), to once again reinforce what the reader has learned on a larger scale. Detailed answer explanations follow the practice set. | 677.169 | 1 |
Algebra And Trigonometry - 01 edition
ISBN13:978-0534434120 ISBN10: 0534434126 This edition has also been released as: ISBN13: 978-0534380298 ISBN10: 0534380298
Summary: Algebra and Trigonometry was designed specifically to help readers learn to think mathematically an...show mored to develop true problem-solving skills. Patient, clear, and accurate, the text consistently illustrates how useful and applicable mathematics is to real life. The new book follows the successful approach taken in the authors' previous books, College Algebra, Third Edition, and Precalculus, Third Edition. ...show less
The text has light marking, the cover has a small "slit" on the upper back edge and several tiny soil marks on the back cover, otherwise in nice condition. Quantity Available: 1. ISBN: 0534434126. IS...show moreBN/EAN: 9780534434120. Inventory No: 1560779853. ...show less
0534434126 | 677.169 | 1 |
Calculus is an instrument of cruel and unusual punishment created by those who derive immense pleasure from the agony of high school students.
Like most of mathematics, it is strict, precise, and insufferable. Blood, sweat, and tears were shed by the millions before you who have trodden the same beat-up path that now beckons before you. By merit of natural selection, only the fittest will survive.
But don't worry, the future isn't all too bleak: by the end of this class, you will have developed a swelling appreciation for the intricacies of the great outdoors.
Just kidding!
Calculus isn't that painful. In fact, it was the first class where math made sense to me. No longer will you be burdened by tedious formulas or meaningless variables. Instead, to learn calculus is to understand a few basic concepts forwards, backwards, and upside-down until you are able to apply your understanding of key ideas to any problem. In short:
Understanding > Rote Memorization
Like variations on a theme in music, calculus revolves around a few key concepts that are articulated in different ways: derivatives are inversion of integrals, second derivatives are simply the derivative taken twice, etc.
As long as you do (do, not B.S.) your homework and understand the concepts, you'll do great—even if you are left-brained, mathematically challenged, or numerically illiterate. Just look at me.
--
Ke Zhao is a senior at Amador Valley High School. Mathematics, a subject that never came easily, was made several degrees more understandable when she took Advanced Placement Calculus AB with Mrs. James last year.
Though she has conquered Calculus AB, Ke faces a new challenge this school year: Calculus BC | 677.169 | 1 |
Andhra Pradesh Board Sample Papers for Math
The Andhra Pradesh educational board is one of the old and prestigious educational board of India. Maths is one of the difficult subjects which require practice from Andhra Pradesh Board Maths Sample Papers by the students. For the same, our website edurite provides the sample appears which are made by the team of ex – teachers of reputed schools of Andhra Pradesh educational board and our experts. The Math AP Sample Paper follows the updated syllabus of the board. AP Maths Sample Paper compiled by our team covers the topics like geometry, progression, statements and sets, polynomials over integers, statistics, functions, linear programming, real numbers, analytical geometry, trigonometry, matrix and determinants and computing. As the students always remain tenses during Maths exam, so the Andhra Pradesh educational board whose AP Maths Sample Paper we provide will help the students to remove their fear as by solving this sample papers, they can have an idea about how much they are prepared as the questions in the sample paper comes with marks and they will also know the time taken by them in solving the papers. Solutions to some of the Andhra Pradesh Board Maths Sample Papers are also there on our sites to help the students with the correct answers and their preparation.
Apart from Math AP Sample Paper we also do provide course books, question paper, study materials of different subjects, teacher's suggestion are also posted by us to help the students with their exams. As we focuses on spreading our helping hands towards every students of India, so we provide all possible things required by the students for exam preparation of all regional and national board. We never wanted to limit our focus only to a certain area that is why we decided to focus on the regional boards too as students of regional boards also need a site where they can find every material required for studies.
Andhra Pradesh Board Sample Papers for Math
Andhra Pradesh Board Sample Papers by Years for Math | 677.169 | 1 |
The mathematics department at Tolland High School will strive to have each student understand and use mathematical concepts and fundamental processes, i.e., experimentation, logical reasoning, computational skills, and analysis of both theory and applications at a level which is consistent with their ability, maturity, and needs. A variety of challenging courses are offered to students of all ability levels. Technology is incorporated appropriately within the lessons. Graphing Calculators have been integrated into the college preparatory and honor courses. | 677.169 | 1 |
Introduces the process of abstraction, studies two elementary structures on sets, and covers the necessary generalities concerning algebraic structures. Presents powerful abstract mathematical concepts from algebra and combinatorics, supported by concrete applications. All background material is provided, including elements of logic, set theory, abstract algebra, linear algebra, and graph theory. Each chapter develops a new mathematical concept, then shows how to apply it. Includes numerous end-of-chapter problems and exercises. | 677.169 | 1 |
Introductory Algebra for Collegesemester undergraduate introductory algebra course. The goal of this text is to provide students with a strong foundation in Basic Algebra skills; to develop students' critical thinking and problem-solving capabilities and prepare students for Intermediate Algebra and some service math courses. Topics are presented in an interesting and inviting format incorporating real world sourced data modeling. A 4-color hardback book w/complete text-specific instructor and student print/enhanced media supplement pac... MOREkage. AMATYC/NCTM Standards of Content and Pedagogy integrated in current, researched, real-world Applications, Technology Boxes, Discover For Yourself Boxes and extensively revised Exercise Sets. Early introduction and heavy emphasis on modeling demonstrates and utilizes the concepts of introductory algebra to help students solve problems as well as develop critical thinking skills. One-page Chapter Projects (which may be assigned as collaborative projects or extended applications) conclude each chapter and include references to related Web sites for further student exploration. The influence of mathematics in fine art and their relationships are explored in applications and chapter openers to help students visualize mathematical concepts and recognize the beauty in math. | 677.169 | 1 |
A comprehensive source of mathematical definitions. With over 2000 terms defined, this dictionary is ideal for supporting students who are studying mathematics or related subjects. All terms in our dictionary are cross-referenced and linked for ease of use, making finding information quick and easy. The definitions of terms and concepts included within our dictionary include the majority of words that students will come across when studying mathematics at secondary school.
Free on-line Mathemeatics Dictionary for students studying mathematics subjects and courses. The definitions of terms and concepts included within our dictionary include the majority of words that students will come across when studying mathematics at secondary school. | 677.169 | 1 |
Horizons Algebra 1 Grade 8
The Horizons Math series from Alpha Omega Publications is a very highly rated
Christian based math curriculum. The Horizons Algebra 1 course is now available
and recommended for students in grades 8 or 9. In this course your student will
learn about exponents and powers, absolute value, radical expressions,
multiplying and dividing monomials and polynomials, the Foil Method, and
factoring trinomials, as well as solving, writing, and graphing linear
equations.
Horizons Algebra 1 Student Book
Publisher: Alpha Omega Publications
The Algebra 1 Student Book has 160 daily lessons, 16 sports and real-life
applications, and 15 college test-prep problem sets. Your student will learn basic
operations with monomials, polynomials, and rational expressions, as well as
linear equations and graphing, quadratic equations and functions, conjunctions,
and disjunctions. In addition there are | 677.169 | 1 |
Algebra II
Course Description: This course is for the highly competitive college-bound student. It is open to those students who excelled in Algebra I grade 8, Plane Geometry grade 8, or excelled in both Algebra I 12 and Plane Geometry 22. Students expand their knowledge of concepts and skills developed in Algebra I using them to solve problems from many areas of study. Quadratic equations and functions are studied along with their graphic representations. Students also study the complex number system, logarithms, and exponential and radical equations.
Algebra II 32
Full Year (two semesters) 1 Credit
Prerequisites: Successful completion of Algebra I
Course Description: This course reviews the concepts developed in Algebra I and pursues them to completion. The student will use these skills to solve verbal problems from many different areas of study. Quadratic equations and functions are studied along with their graphic representations. Graphic representation of other functions, along with base b logarithms and complex numbers will be stressed. Conic sections also are studied using graphing calculator.
Students successfully completing this course may qualify to receive college credit for Intermediate Algebra from Naugatuck Valley Community College through the College Career Pathways Program beginning in the 2012-2013 school year. See News for details.
Algebra II33
1 Credit Full Year (two semesters)
Prerequisites: Successful completion of Algebra I 13
Course Description: Algebra II 33 is intended for students needing a slower, more concrete, and/or visual approach to develop concepts and methods to solve algebraic problems. The concepts developed in Algebra I are reviewed and extended to solve verbal problems from many different areas of study. Graphic representation of families of functions, particularly quadratic functions, will be a major area of study along with logarithms and the complex number system | 677.169 | 1 |
Intermediate Algebra - 3rd edition
Summary: KEY BENEFIT:Intermediate Algebra, Third Edition, by Tom Carson, addresses two fundamental issues-individual learning styles and student comprehension of key mathematical concepts-to meet the needs of today's students and instructors.Carson's Study System, presented in the ldquo;To the Studentrdquo; section at the front of the text, adapts to the way each student learns to ensure their success in this and future courses. The consistent emphasis on thebig picture of algebra, with pedag...show moreogy and support that helps students put each new concept into proper context, encourages conceptual understanding. KEY TOPICS: Real Numbers and Expressions; Linear Equations and Inequalities in One Variable; Equations and Inequalities in Two Variables and Functions; Systems of Linear Equations and Inequalities; Exponents, Polynomials, and Polynomial Functions; Factoring; Rational Expressions and Equations; Rational Exponents, Radicals, and Complex Numbers; Quadratic Equations and Functions; Exponential and Logarithmic Functions; Conic Sections MARKET: For | 677.169 | 1 |
Best Math and verbal books - advice pleasethanks
I saw a book published by EZ methods. I am planning to get one. Remeber the key to solve math is understand the trick and use concepts rather than breath of problems! | 677.169 | 1 |
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This website introduces the TI-83 Graphics Calculator, an advanced piece of technology that is actually more like a hand held computer. You will learn here some operations, keystrokes, and hints. The graphics calculator will make the learning process more exciting and enjoyable. The TI-83 will be used to verify answer/solutions and to discover more about mathematics. This calculator is a great aid for teachers and students, but it will never be a substitute for learning. With this in mind, most of these instructional mathematics videos also contain the analytic or hand methods. These videos are designed for all age groups/skill levels and no prior calculator experience is needed. A major benefit of these instructional videos are that they are short and easy to understand and straight to the point. Click on the testimonials link and view the comments of others.
Youtube
Testimonials
gene06825:
"That was an awesome video. Seriously. You really showed how to do it step by step, and you made it very clear. I really do appreciate it. Keep up the good work. Thanks again."
misspopular333
"Thank you for your help.? This was perfect. No unneccesary info. Just the facts! Perfect. Thank you!!!!
."
ashleypina1:
Wow! I did not know you could put these in a calculator. Thanks.
nterry23:
THANK YOU SOOOO MUCH! I? have been looking every where for a tutorial on how to graph these. THANKS AGAIN. GREAT VID!!!
OlympicClassDandy:
Thanks, this was a very helpful lesson! The calculator trick was? especially useful. | 677.169 | 1 |
Unit specification
Aims
The programme unit aims to introduce students to theoretical and practical aspects of the numerical solution of
linear and nonlinear equations, the approximation of functions by polynomials and the approximation of integrals
via quadrature schemes.
Brief description
Numerical analysis is concerned with finding numerical solutions to
problems for which analytical solutions either do not exist or are not
readily or cheaply obtainable. This course provides an introduction to
the subject, focusing on the three core topics of iteration,
interpolation and quadrature.
The module starts with 'interpolation schemes', methods for
approximating functions by polynomials, and 'quadrature schemes',
numerical methods for approximating integrals, will then be explored
in turn. The second half of the module looks at solving systems of
linear and nonlinear equations via iterative techniques. In the case
of linear systems, examples will be drawn from the numerical solution
of differential equations.
Students will learn about practical and theoretical aspects of all
the algorithms. Insight into the algorithms will be given through
MATLAB illustrations, but the course does not require any
programming.
Intended learning outcomes
On completion of this unit successful students will be able to:
practical knowledge of a range of iterative techniques for solving
linear and nonlinear systems of equations, theoretical knowledge of
their convergence properties, an appreciation of how small changes in
the data affect the solutions and experience with key examples arising
in the solution of differential equations; | 677.169 | 1 |
An Introduction to Mathematics for Mathematics for Economics introduces quantitative methods to students of economics and finance in a succinct and accessible style. The introductory nature of this textbook means a background in economics is not essential, as it aims to help students appreciate that learning mathematics is relevant to their overall understanding of the subject. Economic and financial applications are explained in detail before students learn how mathematics can be used, enabling students to learn how to put mathematics into practice. Starting ... MOREwith a revision of basic mathematical principles the second half of the book introduces calculus, emphasising economic applications throughout. Appendices on matrix algebra and difference/differential equations are included for the benefit of more advanced students. Other features, including worked examples and exercises, help to underpin the readers' knowledge and learning. Akihito Asano has drawn upon his own extensive teaching experience to create an unintimidating yet rigorous textbook. A concise, accessible introduction to quantitative methods for economics and finance for students who are new to the subject. This textbook contains lots of practical applications to show why maths is necessary and relevant to economics, as well as worked examples and exercises to help students learn and revise. | 677.169 | 1 |
Short Description: This book is primarily about complex numbers. I can't remember it that well, but most of the book doesn't require calculus (though a fair part definitely does). Most of it is about the algebra and geometry of complex numbers and is thus accessible to the nonmathematician. The last chapter has a very fun introduction to complex analysis. | 677.169 | 1 |
Haverford TrigonometryGraph theory deals with the study of graphs and networks and involves terms such as edges and vertices. This is often considered a very specific branch of combinatorics. Lastly, probability in discrete math deals with events that occur in discrete sample spaces | 677.169 | 1 |
N-RN.1
Explain how the definition of the meaning of rational
exponents follows from extending the properties of integer
exponents to those values, allowing for a notation for
radicals in terms of rational exponents. For example, we
define to be the cube root of 5 because we want to hold,
so must equal 5.
N-Q.1 Use
units as a way to understand problems and to guide the
solution of multi-step problems; choose and interpret units
consistently in formulas; choose and interpret the scale and
the origin in graphs and data displays.
A-SSE
2. Use the structure of an expression to identify ways to
rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus
recognizing it as a difference of squares that can be
factored as (x2 – y2)(x2 + y2).
Interpret parts of an
expression, such as terms, factors, and coefficients.
Interpret complicated
expressions by viewing one or more of their parts as a
single entity. For example, interpret P(1+r)n as the
product
A-APR.1
Understand that polynomials form a system analogous to the
integers, namely, they are closed under the operations of
addition, subtraction, and multiplication; add, subtract,
and multiply polynomials.
A-CED
3. Represent constraints by equations or inequalities, and
by systems of equations and/or inequalities, and interpret
solutions as viable or nonviable options in a modeling
context. For example, represent inequalities describing
nutritional and cost constraints on combinations of
different foods.
STRAND:
Understanding solving
equations as a process of reasoning and explain the
reasoning
A-REI 1
Explain each step in solving a simple equation as following
from the equality of numbers asserted at the previous step,
starting from the assumption that the original equation has
a solution. Construct a viable argument to justify a
solution method.
A-REI
11. Explain why the x-coordinates of the points where the
graphs of the equations y = f(x) and y = g(x) intersect are
the solutions of the equation f(x) = g(x); find the
solutions approximately, e.g., using technology to graph the
functions, make tables of values, or find successive
approximations. Include cases where f(x) and/or g(x) are
linear, polynomial, rational, absolute value, exponential,
and logarithmic functions.★
A-REI
12. Graph the solutions to a linear inequality in two
variables as a halfplane (excluding the boundary in the case
of a strict inequality), and graph the solution set to a
system of linear inequalities in two variables as the
intersection of the corresponding half-planes.
STRAND:
Understand the
Concept of a Function and use Function Notation.
F-IF 1.
Understand that a function from one set (called the domain)
to another set (called the range) assigns to each element of
the domain exactly one element of the range. If f is a
function and x is an element of its domain, then f(x)
denotes the output of f corresponding to the input x. The
graph of f is the graph of the equation y = f(x).
STRAND:
Interpret functions
that arise in applications in terms of the context.
F-IF 4.
For a function that models a relationship between two
quantities, interpret key features of graphs and tables in
terms of the quantities, and sketch graphs showing key
features given a verbal description of the relationship. Key
features include: intercepts; intervals where the function
is increasing, decreasing, positive, or negative; relative
maximums and minimums; symmetries; end behavior; and
periodicity.★
STRAND:
Interpret functions
that arise in applications in terms of the context.
F-IF 5.
Relate the domain of a function to its graph and, where
applicable, to the quantitative relationship it describes.
For example, if the function h(n) gives the number of
person-hours it takes to assemble n engines in a factory,
then the positive integers would be an appropriate domain
for the function.★
Grade 07-08:
The Slope of a Line: Writing and Comparing Unit Rates and
Graphs from Word Problems
MATH :
COURSE ONE
Interpreting Functions
STRAND:
Analyze functions
using different representations.
F-IF 7.
Graph functions expressed symbolically and show key features
of the graph, by hand in simple cases and using technology
for more complicated cases.★
a.
Graph linear and quadratic functions and show intercepts,
maxima, and minima.
b.
Graph exponential and logarithmic functions, showing
intercepts and end behavior, and trigonometric functions,
showing period, midline, and amplitude. At this level, for
part e, focus on exponential functions only
F-IF 9.
Compare properties of two functions each represented in a
different way (algebraically, graphically, numerically in
tables, or by verbal descriptions). For example, given a
graph of one quadratic function and an algebraic expression
for another, say which has the larger maximum.
STRAND:
Build a Function that
Models a Relationship Between two Quantities.
BF 1.
Write a function that describes a relationship between two
quantities.★
a.
Determine an explicit expression, a recursive process, or
steps for calculation from a context.
b.
Combine standard function types using arithmetic operations.
For example, build a function that models the temperature of
a cooling body by adding a constant function to a decaying
exponential, and relate these functions to the model.
c. (+)
Compose functions. For example, if T(y) is the temperature
in the atmosphere as a function of height, and h(t) is the
height of a weather balloon as a function of time, then
T(h(t)) is the temperature at the location of the weather
balloon as a function of time.
F-LE
2. Construct linear and exponential functions, including
arithmetic and geometric sequences, given a graph, a
description of a relationship, or two input-output pairs
(include reading these from a table).
G-CO 1.
Know precise definitions of angle, circle, perpendicular
line, parallel line, and line segment, based on the
undefined notions of point, line, distance along a line, and
distance around a circular arc.
G-GPE
4. Use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a figure
defined by four given points in coordinate plane is a
rectangle; prove or disprove that the point (1, √3) lies on
the circle centered at the origin and containing the point
(0, 2).
G-GPE
5. Prove the slope criteria for parallel and perpendicular
lines and use them to solve geometric problems (e.g., find
the equation of a line parallel or perpendicular to a given
line that passes through a given point).
G-MD 1.
Give an informal argument for the formulas for the
circumference of a circle, area of a circle, volume of a
cylinder, pyramid, and cone. Use dissection arguments,
Cavalieri's principle, and informal limit arguments.
STRAND:
Summarize, Represent,
and Interpret Data on Two Categorical and Quantitative
Variables.
S-ID 6.
Represent data on two quantitative variables on a scatter
plot, and describe how the variables are related.
a. Fit
a function to the data; use functions fitted to data to
solve problems in the context of the data. Use given
functions or choose a function suggested by the context.
Emphasize linear, quadratic, and exponential models.
b.
Informally assess the fit of a function by plotting and
analyzing residuals.
c. Fit
a linear function for a scatter plot that suggests a linear
association. | 677.169 | 1 |
Reviewing for the AP Statistics Exam with Fathom
Description
Are you looking for some review activities for the AP® Statistics Exam that will enhance students' understanding and refresh their memories? This webinar will focus on short and informative lessons using Fathom that will strengthen your students' skills in preparation for the three-hour exam. The lessons focus on concepts from throughout the AP® Statistics curriculum, from descriptive statistics through inference. The activities can be used in an AP® or general statistics class.
Presenter
Beth Benzing is a moderator of the Teaching Statistics using Fathom online course. She has been teaching AP® Statistics for 12 years and is a reader for the AP® Statistics exam. She has taught a statistics institute for the Math and Science Partnership Program at Arcadia University and will be teaching a statistics institute at West Chester University this summer. Beth sits on the board of a regional affiliate of NCTM in the Philadelphia area. She is a regular presenter at local, state, and national math conferences. She teaches at Strath Haven High School in Wallingford, PA, a southwest suburb of Philadelphia where she lives with her husband and three children. | 677.169 | 1 |
This lesson is designed
to provide you with an understanding of algebraic structure
and to introduce you to complex situations. Most of the work
for this unit deals with polynomials, with an emphasis on
symbolic manipulations.
To be able
to describe and use the concepts of zeroes and end behavior
of different functions.
To be able to use polynomial
and rational functions to model physical observations.
To be able to describe and illustrate
relationships between graphs of polynomials or rational
functions with its symbolic representation.
To be able to develop facility
with manipulation and reasoning about polynomials and rational
symbolic representations.
To be able to determine all complex
number roots of polynomials and be able to perform mathematicaloperations on them. | 677.169 | 1 |
Costs
Course Cost:
$300.00
Materials Cost:
None
Total Cost:
$300
Special Notes
State Course Code
02003 and in SpanishMath Foundations I offers a structured remediation solution based on the NCTM Curricular Focal Points and is designed to expedite student progress through 3rd- to 5th-grade skills. The course is appropriate for use as remediation for students in grades 6 to 12. When used in combination, Math Foundations I and Math Foundations II (covering grades 6 to 8) effectively remediate computational skills and conceptual understanding needed to undertake high school–level math courses with confidence.
Math Foundations I empowers students to progress at their optimum pace through over 80 semester hours of interactive instruction and assessment spanning 3rd- to 5th-grade math skills. Carefully paced, guided instruction is accompanied by interactive practice that is engaging and accessible. Formative assessments help students to understand areas of weakness and improve performance, while summative assessments chart progress and skill development. Early in the course, students develop general strategies to hone their problem-solving skills. Subsequent units provide a problem-solving strand that asks students to practice applying specific math skills to a variety of real-world contexts.
The content is based on the National Council of Teachers of Math (NCTM) April 2006 publication, Curricular Focal Points for Prekindergarten through Grade 8 Mathematics: A Quest for Coherence and is aligned to state standards. WA State standards correlations available upon request.
Foundations courses meet the needs of both high school students and transitioning middle school students who are not prepared for grade-level academic challenges. Foundations courses develop skills and strategies in math, reading, and writing with the goal of raising achievement to a high school level. Courses feature structured remediation designed to accelerate mastery of required skills appropriate to grades 3–8. Foundations courses have been designed to be age-appropriate with respect to content, illustrations, and examples for students ages 13 and older. | 677.169 | 1 |
ALGEBRA II A, the first in a two-semester course, begins with a review of algebraic properties. Students will study properties and applications of linear and quadratic functions, radical functions, and rational functions. Students will identify how the major topics in algebra relate to real-world applications. Students will also explore exponential and logarithmic functions and trigonometric functions. | 677.169 | 1 |
EGT on the Web
Related Links
Exploratory Galois Theory adopts an exploration-based approach, using a variety of problems with hints and proof sketches to help students participate in the development of the theory.
The notion of algebraic number is developed from first principles, with explicit examples of algebraic numbers and finite extensions of the rationals (number fields) providing the primary context for a discussion of field extensions. In this setting of subfields of the complex numbers, some proofs and sketches may be successfully left for students to complete in exercises. Later on, in an optional section, the text outlines the changes necessary to expand the treatment to cover the Galois theory of finite extensions of arbitrary fields.
EGT encourages experimentation, with broadly functional Maple and Mathematica packages allowing students to examine extensions of the rationals generated by one or more algebraic numbers. These packages, called AlgFields, permit students to use the powerful symbolic computation systems in a user-friendly way.
Extensions may be constructed using particular roots of irreducible polynomials, and Galois groups may be explicitly calculated. The functions employ the same procedures described in the text, such as factoring polynomials over extensions of Q, determining the irreducible factor corresponding to a particular root, determining whether a particular isomorphism may extend to a simple extension by determining what roots of a related polynomial lie in the target field. Free for educational distribution, the source code for the packages is available for interested students and faculty.
EGT assumes only a first course in abstract algebra (using any popular text, such as Gallian or Hungerford), with a review of necessary material in the first chapter.
EGT introduces concepts, theorems, and proofs at an extremely accessible level. The course may then be followed in one of several forms: a traditional lecture format, a seminar-style format with students presenting sections from the text, or a self-paced independent study. | 677.169 | 1 |
Algebra I Workbook For Dummies
Synopsis
From signed numbers to story problems — calculate equations with ease
Practice is the key to improving your algebra skills, and that's what this workbook is all about. This hands-on guide focuses on helping you solve the many types of algebra problems you'll encounter in a focused, step-by-step manner. With just enough refresher explanations before each set of problems, this workbook shows you how to work with fractions, exponents, factoring, linear and quadratic equations, inequalities, graphs | 677.169 | 1 |
Build Algebra Understanding Using Three-Dimensional Manipulatives and the Interactive Whiteboard!
Help students master algebra concepts with a hands-on system that includes powerful instructional resources, three-dimensional manipulatives, and interactive whiteboard technology. By providing both hands-on and visual representations, Algeblocks' takes advantage of the natural link to geometry. The use of Algeblocks spatial models that represent both X and Y variables up to the third power puts the solutions to algebra at students' fingertips! | 677.169 | 1 |
The essential guide to MATLAB as a problem solving tool This text presents MATLAB both as a mathematical tool and a programming language, giving a concise and easy to master introduction to its potential and power. The fundamentals of MATLAB are illustrated throughout with many examples from a wide range of familiar scientific and engineering areas, as well as from everyday life. The new edition has been updated to include coverage of Symbolic Math and SIMULINK. It also adds new examples and applications, and uses the most recent release of Matlab.
Audience First time users of Matlab. Undergraduates in engineering and science courses that use Matlab. Any engineer or scientist needing an introduction to MATLAB. | 677.169 | 1 |
This course covers plane and solid geometric topics. The curriculum includes formal proofs through deductive and inductive reasoning, congruency, perpendicularly, parallelism, similarity, inequalities, and area of polygons, volumes of three-dimensional solids, angle measures and extensive work with cubes.
This is a rigorous course in plane, solid, and coordinate geometry designed for the outstanding math student. There is an emphasis on proofs, using deductive and inductive reasoning. The course develops concepts in depth and deals with extensive applications of modern geometry.
The purpose of this class is to provide an Honors level alternative for seniors. Statistics Honors will introduce students to the major concepts and tools for collecting, analyzing, and drawing conclusions from data. There are four broad conceptual themes to this course:
1) Exploring Data: Describing patterns and departures from patterns
2) Sampling and Experimentation: Planning and conducting a study
3) Anticipating Patterns: Exploring random phenomenon using probability and simulation
4) Statistical Inference: Estimating population parameters and testing hypotheses | 677.169 | 1 |
An excellent, very comprehensive study guide containing all the maths you should be able to do before moving on to your final year in school. A MUST-HAVE FOR ALL GRD. 11 & 12 LEARNERS. The book includes lots of questions and answers on Number Patterns; Increase & decrease (Growth and Decay) Graphs of functions; Mathematical modelling; Length, area & volume; Co-ordinate Geometry; Transformation Geometry; Data handling & Probability. Optional work will be published in a separate book. | 677.169 | 1 |
Fr...
For courses in secondary or middle school math. This text focuses on all the complex aspects of teaching mathematics in today's classroom and the most current NCTM standards. It demonstrates how to creatively incorporate the standards into teaching along with inquiry-based instructional strateg... | 677.169 | 1 |
Non-HP RPN Scientific Calculators?
Non-HP RPN Scientific Calculators?
I'm a freshman now in college and I'm looking to buy a scientific calculator, since graphing calculators aren't allowed in my exams. I've been running through high school with my TI-89 in class, though in senior year I became interested in RPN and have been using an HP-48 emulator for Android in RPN mode just for kicks | 677.169 | 1 |
Calculus : Easy Way - 4th edition
Summary: This ingenious, user-friendly introduction to calculus recounts adventures that take place in the mythical land of Carmorra. As the story's narrator meets Carmorra's citizens, they confront a series of practical problems, and their method of working out solutions employs calculus. As readers follow their adventures, they are introduced to calculating derivatives; finding maximum and minimum points with derivatives; determining derivatives of trigonometric functions; ...show morediscovering and using integrals; working with logarithms, exponential functions, vectors, and Taylor series; using differential equations; and much more. This introduction to calculus presents exercises at the end of each chapter and gives their answers at the back of the book. Step-by-step worksheets with answers are included in the chapters. Computers are used for numerical integration and other tasks. The book also includes graphs, charts, and whimsical line illustrations. Barron's Easy Way books focus on both practical and academic topics, presenting fundamental subject matter in clear, understandable language. Equally popular as self-teaching manuals and supplementary texts for classroom use, they are written to help students improve their grades and review subject matter before tests. They are also useful for introducing general readers to a new career-related skill. Easy Way titles cover virtually all subjects that are taught on advanced high school and college-101 levels. New subjects are periodically added, and existing titles are frequently updated to keep them timely and relevant to students' needs. Subject heads and key phrases are set in a second color5.16 +$3.99 s/h
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2006-01-01 Paperback | 677.169 | 1 |
Course Syllabus for Math 241:
Honors Calculus III (Fall 2012)
COURSE OVERVIEW:
Calculus is the mathematics of quantities that vary in time,
with a wealth of applications to many different branches of
science. It's also a marvelous synthesis of algebra and
geometry, tying together pretty much everything you learned
in high school math, and serving as a bridge to higher
mathematics.
In Math 241, we'll take a more in-depth study of calculus
than is offered in Math 231,
with more of a focus on the
rigorous underpinnings of the subject: not just what's
true, but why it's true. Also, since the class
is small (fewer than 20 students), class discussions can
play a big role in helping you learn to view the material
from many different angles, and you'll get more direct
contact with a faculty member.
You'll also learn the answer to the questions:
What do you get when you mate a mountain-climber and a mosquito?
And: Why do pirates like polar coordinates?
I want all of you to succeed in this class; below you will find
some tips for how you can help make this happen.
USEFUL LINKS:
Course
overview (the lecture notes for the first day;
these notes will probably answer most of your questions
about the structure of the course)
CONTACTING THE INSTRUCTOR:
Professor James Propp
Email: jpropp at cs dot uml dot edu
(note: I also have a James_Propp account but I don't read it very often).
Phone: (978) 934-2438. I'll leave a message on my voice mail if the
university is open but I'm unable to attend class. To check whether the
university has been closed because of weather, call (978) 934-2121.
Fax: (978) 934-3053 ("Attn: James Propp").
Office: Olney 428C.
Consultation Hours: TBA.
Meetings at times other than my office hours can be arranged by
appointment; see me after class, call me on the phone, or send me
an email message.
Suggestions about how the course is being run are welcome at any time.
If something isn't working for you, please don't wait until the end of the
semester to tell me!
GENERAL COURSE INFORMATION:
Expectations: You're expected to attend classes, do the reading
in advance, ask questions, and make serious attempts to answer
questions raised by me or by other students during class.
If you miss a class, it's your responsibility to make
sure you obtain all information (course material, assignments,
changes in exam dates, etc.) presented that day.
TEXT:
James Stewart, Essential Calculus: Early Transcendentals (2nd edition),
2012. A copy of this book will be placed on reserve at Lydon Library.
(On the other hand, the Stewart book Calculus: Early Transcendentals ---
note the absence of the word ``Essential'' --- is structured differently
and cannot be used as a textbook for this class.)
During the Fall semester, we'll cover Chapters 10 through 13.
GRADING POLICY:
Course grades
Course grades will be based on three numbers: your Homework score,
your score on the in-class Midterm, and your score on the Final.
Your average score for the course will be computed as a weighted average
of your Homework, Midterm, and Final scores in which the highest
of the three scores is assigned weight 40% and the other two scores
are assigned weight 30%. (For instance, if your highest score was
on the Midterm, your average score for the course would be 30% of
your Homework score plus 40% of your Midterm score plus 30% of your
Final score.) Since this is an Honors class with challenging
problems, the scheme for computing letter-grades is on the lenient
side, and is determined from your weighted average score according
to the following table:
Average
[85, 100)
[82, 85)
[80, 82)
[75, 80)
[72, 75)
[70, 72)
Grade
A
A-
B+
B
B-
C+
Average
[65, 70)
[62, 65)
[60, 62)
[55, 60)
[0, 55)
Grade
C
C-
D+
D
F
(I may raise your grade above what's shown in the table
if your class participation is strong: one more reason
to come to class. Also, coming to office hours counts
as a form of class participation.)
Exam dates: Midterm TBA; final exam TBA.
Exam Policy
It's important that everyone take the same exams under the same
conditions for maximum fairness and reliability of testing. I therefore
don't give makeup exams unless you have a valid reason for missing
the scheduled exam (for example, illness or a religious holiday), and I
don't allow extra time on exams unless you have a note from Disability
Services (see below).
If you have to miss a scheduled exam, please let me know
ahead of time if at all possible; I'm much more likely to be
sympathetic if you call me the morning of the exam and say "I have the
flu and can't take the exam" than if you come in two days after the
exam and say "I missed the exam. When can I take a makeup?"
You may not use a cell phone in any way during an exam.
Use of calculators is prohibited during exams.
You can always reschedule an exam
that falls on a day that is a religious holiday for you, but you must
make these arrangements ahead of time.
Tips on Preparing for Exams
Start studying for an exam at least one week ahead of time.
Begin by reviewing the homework problems for the sections that will
be covered on the exam. Make sure you know how to solve each problem.
If you can't solve a particular problem, make a note of the problem
number and move on to the next problem; you can go back to the problem
later with a fresh head (yours or someone else's!).
You can test your knowledge by trying odd-numbered problems
for which the answer is given at the back of the book.
Try the review problems that appear at the end of each chapter.
Ask me or someone else for help on any homework problem that gave
you trouble, then try to solve a similar problem from the textbook.
Get a good night's sleep the night before the exam. You'll
perform better if you are fresh and able to think clearly.
Tips on Taking Exams
Read every question on the exam before you start working. This will
give you a feel for how long the exam is and how you should pace
yourself. It'll also give your subconscious mind a chance to start
working on the questions.
If you're not sure what a question means, please ask me. I'm
trying to see how well you know the material, not to trick you with
ambiguous wording.
Show as much of your work as possible, in as clear a way as possible.
Even if you get the wrong answer, I'll try to award you as much partial
credit as I feel I can conscientiously give you, but it's hard for me to
do this if you don't show your thought-processes.
Look at the point value of each question. Obviously, it's more
important to do well on the questions that count the most than the ones
that count the least.
It's generally best to do the easiest problem first, then the next
easiest, and so on. You don't have to do the problems in the order
they appear on the exam.
If you get stuck on one question, move on to the next. Come back
later to the question that is giving you trouble.
Be aware of how much time you have left. Don't spend too much time
on a single question. It's generally better to get partial credit on
every question than full credit on a small number of questions.
If you have extra time, use it to check your work! Better still,
if there's more than one natural approach to the problem, try to
solve the problem with a different method; this can be a better way
to catch mistakes than just re-reading your calculations. If you get
the wrong answer with one approach but the right answer with the
other approach, I'll give you nearly full credit (especially if
you speculate intelligently on where you might have made an error).
If you get an answer that doesn't make sense but don't have time to
trace where your error came from, don't just cross out your answer;
explain why you think the answer you got looks wrong,
and you may get some extra points for having good instincts.
Never be afraid to ask for extra paper. (If you want to write
on the reverse side of a page, please write "see other side".)
Homework
Typically there'll be one homework assignment per week,
due one week after it is assigned. (We may deviate from this
schedule at the beginning of the term and around the time of
the midterm.)
In order for you to understand the material in this course, it's
extremely important that you do the assigned homework problems.
Working with your classmates can be a great help, and I strongly
encourage it, subject to certain provisos (see below).
I also urge you to ask questions about any problems that give you
trouble.
Homework will usually be due each week on Friday
(except during the week of an exam).
Your grade will be based on clarity as well as correctness, so
neatness, grammar, and punctuation should not be neglected.
Harder problems will in general be worth more points.
You are required to include an estimate of how much time you spent
on each and every assigned problem; this will help me assess
which of the problems are the harder ones. (I reserve the
right to throw out a problem entirely if it turns out to be
too hard.)
Barring unusual circumstances, late homeworks will not be accepted.
Each student will be allowed to skip two assignments without penalty;
additional skipped homeworks will only be permitted if a valid
excuse is presented, preferably ahead of time rather than afterwards.
Don't use up your "free skips" too early in the semester!
If you skip just one assignment, your lowest homework score gets dropped.
If you don't skip any assignments, your two lowest homework scores get dropped.
While you can discuss the exercises with classmates, the work you hand in
should be your own write-up and not copied from someone else. When leaving a
joint homework-solving session, don't carry away anything that doesn't fit
in your own brain. Also, you must acknowledge who you worked with.
(If you didn't work with anyone, please write "I worked alone on this
assignment".)
Academic honesty in homeworks is expected. (E.g., if you use web-resources
or tutors or collaborators of any kind, the role of their contribution must
be acknowledged; you won't receive a lower grade for using such resources,
but if the grader and I feel you're relying on them too heavily, we may
require you to change your way of doing homework.) My expectations for
appropriate ways of doing the homework will be discussed in class; in
case you are in any doubt about what is expected, it is your responsibility
to contact me for clarification. See
the UMass Lowell catalogue
for a definitive statement of UMass Lowell's academic honesty policy.
It is not required that you submit your solutions in
LaTeX,
but if you are planning to be a mathematician,
scientist, or engineer, it's never too early to learn!
LaTeX is free software that lets you typeset formulas
about as fast as you can write them (with some practice).
Composing your homework in LaTeX will help you pay attention
to your communication of mathematics, and make it much easier
to edit your work as you go along. There will be an initial
hump of getting started, but after a couple of problem sets,
using LaTeX will become quite natural. You'll probably still
want to draw your diagrams and figures free-hand, but knowing
how to write equations in LaTeX is a life-skill that will
serve you well in later courses in which homeworks involve
fewer pictures and more formulas.
Also, if you want to use Mathematica as an aid to your learning, check out
Effective Fall 2011, students will be able to download Mathematica
as part of the campus license, so using it for classes will be more
convenient than in the past.
You shouldn't use Mathematica as a substitute for being able
to do the work yourself the old-fashioned way,
but it's a great way to check your work.
Also, Mathematica features many demonstrations
(see
that can bring course material to life in a vivid way.
You will be expected to fill out a time sheet
that tells me how much time you spent on each problem
(rounded to the nearest minute, or the nearest five minutes;
there's no need to be super-precise).
This helps me improve the course from year to year
by spreading out the work-load more evenly from week to week.
Points may be deducted from students who repeatedly
fail to submit legible time sheets.
Many students find it profitable to read the solutions to all the
problems in the current assignment (posted on the web each week).
Even if you got a problem correct, you may learn something from
reading the posted solution, such as an alternate approach to
the problem or a good clear way to express the main ideas.
Attendance
Regular attendance is expected. It is not part of the grading scheme,
but it may be used to adjust grades upward in the event of a borderline
grade. Class participation that shows that you have read the assigned
material may also be helpful in borderline cases.
SPECIAL NEEDS:
If you have any special needs, e.g., you need more time on exams
because of a disability, I'll do my best to accomodate you.
Please notify me at least two weeks in advance. | 677.169 | 1 |
Mathematics
Mathematics is a useful, exciting, creative and vital area of study that is taught and explored by faculty who foster an appreciation and sense of enjoyment for all students.
The rigorous and varied curriculum provides opportunities for students to develop their abilities to solve problems and reason logically.
The curriculum offers courses of study, which will enable students to explore and make sense of their world. The study of mathematics is an ongoing process that will extend to and enhance every facet of the students' lives.
As students reach the junior high years, their course placement will be determined by a variety of assessment tools. In 7th grade, students take Pre-Algebra at either the college placement or honors level. Occasionally 7th grade students are placed in Algebra I Honors. Most eighth grade students follow the progression to Algebra I or Algebra I Honors, while those students who successfully complete Algebra I Honors in 7th grade move to Geometry Honors. | 677.169 | 1 |
Peer review is a process of self-regulation by a profession or a process of evaluation involving qualified individuals within the relevant field. Peer review methods are employed to maintain standards, improve performance and provide credibility...
The Mathematical Association of America is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists;...
. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a journal of mathematics rather than pedagogy. Rather than articles in the terse "theorem-proof" style of research journals, it seeks articles which provide a context for the mathematics they deliver, with examples, applications, illustrations, and historical background.
Paid circulation in 2008 was 9,500 and total circulation was 10,000. | 677.169 | 1 |
Search Course Communities:
Course Communities
Lesson 21: Variation
Course Topic(s):
Developmental Math | Variation
The lesson begins with a comparison of data tables and graphs of two functions, one directly proportional (cost of gas) and the other exponential (population), before a definition for direct variation is introduced. Direct variation is then linked to linear function (f(x)= kx) and the scaling property of direct variation is examined (i.e. a multiple of the independent variable will always correspond to that same multiple of the dependent variable). Direct variation with a power of (x) follows with a test for direct variation before indirect variation and indirect variation with a power of (x) are introduced. | 677.169 | 1 |
Whether teaching remedial, mainstream, or honors classes,
in a segregated or integrated program, these 179 dynamic animations bring to
life algebraic topics from pre-algebra through pre-calculus. Packaged on a
CD-ROM (with a basic license for 4 on-site computers to use at a variety of
levels), Algebra In Motion™
animations perform equally well on either the Windows or Macintosh platform.
They must be run by The Geometer's Sketchpad v4 or v5 (no prior versions),
owned and sold by Key Curriculum (
on either Windows or Macintosh platforms.
Although a detailed instruction manual is included on the CD-ROM (PDF
format), most of the animations can be run successfully using only the
on-screen information.
THE BASICS Visually explore or review fractions (meanings,
comparisons, improper, LCD, adding, decimals, percents), signed number
operations, absolute value, and introduce the concept of an equation as a
balance of values and use that balance to solve preset equations or create
your own.
INTRODUCING THE COORDINATE PLANE
& GRAPHING Introduce students to the basic vocabulary and
characteristics of the coordinate plane (+ history). Dynamically display the
definition of slope. Practice graphing lines from y = mx+b, Ax+By = C, and
y-y1 = m(x-x1) forms.
Explore the relationship of parallel and perpendicular lines to slope.
Develop the formulas for midpoint and distance. Test relations using an
animated vertical line test
Present 4 different graphing grids on the same
screen.
axes~quadrants~coordinates
vertical line test, domain/range
exploring domain/range
with any function
visualizing slope
graph y =
mx + b
graph Ax +
By = C
graph y-y1
= m(x-x1)
parallel
and perpendicular
developing
the midpoint formula
developing
the distance formula
4 grids on
1 screen
evolution
of a polynomial
parabola evolution
parabola 3 graphing forms
transform
f(x) to f(x) + a
transform f(x) to
af(x)
transform f(x) to f(x-a)
transform f(x) to f(ax)
transformation practice
(line)
transformation practice
(absolute value)
transformation example
(parabola)
transformation practice
(sine)
transformation practice
(exponential)
MULTIPLICATION & FACTORING The distributive property is geometrically demonstrated
for products of all combinations of monomials, binomials, and trinomials.
The factored form of a2 – b2 is developed and proved.
Special emphasis is given to (x+h)2 ≠ x2+h2
and (x+h)3 ≠ x3+h3. "Completing
the square" is modeled physically.
FOIL (a+a)(x+b)
(a+b)(c+d) = ac+ad+bc+bd
(a+b+c)(d+e+f)
a2 - b2 = (a+b)(a-b)
(x+h)2 = x2 + 2xh + h2
(x+h)3 = x3 + 3x2h + 3xh2 +
h3
completing
the square
CONNECTING SOLUTIONS TO GRAPHS Reinforce meaningful understanding of solutions to
sentences with absolute value, systems of 2 linear equalities or
inequalities, and finding the roots of a quadratic equation (including
complex roots) using its related parabola.
|ax+b| ≥ c
system of
linear equalities
system of
linear inequalities
complex
roots of quadratic equations
polynomial
root dragging
(set of 7 animations)
ADVANCED GRAPHING Dynamically graph points in 3D space or on the complex
number plane (history included). Control coefficients of a polynomial to
"morph" it from a constant function up through a 5th degree polynomial.
Similarly "morph" graphs of logarithmic & exponential functions, parametric
& polar graphs, greatest integer functions, and inverses. Explore how
composites of functions are created, and create linear programming examples.
points in
xyz-space
points in
complex plane
"morphing
polynomials"
"morph"
exponential functions
"morph"
logarithmic functions
parametric
graphs
polar
graphs
your
choice
greatest
integer functions
more
greatest integer
inverses
creating
composites
(adapts to any example)
composite
ex. 1
composite
ex. 2
composite
ex. 3
linear
programming
(create your own example)
linear
programming
(preset example 1)
linear
programming
(preset example 2)
CONICS &
THEIR APPLICATIONS Dynamically create each conic section from its
definition. Graph and "morph" all features of each. Alter coefficients of
the general equation or eccentricity to "morph" one conic into another.
Explore applications to satellite dishes, elliptical pool tables, whisper
chambers, & falling objects.
overview
construct
circle by def. & graph
construct parabola by def.
construct
ellipse by definition
construct
hyperbola by definition
graphing
parabolas
graphing
ellipses
graphing
hyperbolas
"morphing"
from general equation
parabaloid,
ellipsoid, hyperboloid
family of
hyperbolas & ellipses
mutually
orthogonal
reflections & collections
falling
projectile
altering
eccentricity
TRIGONOMETRY In an environment where rotation is real, not merely
imagined, thoroughly investigate the unit circle's angles, coordinates, and
ratios. Literally unwrap the unit circle to form sine and cosine waves.
Dilate and translate trigonometric graphs to explore amplitude, period, and
shift. Explore and prove Pythagorean identities, the Law of Sines, and the
Law of Cosines. Convincingly demonstrate that sin (a+b) can't be (sin a +
sin b).
unit
circle angles
sine,
cosine, tangent, definitions
sin, cos, tan, sec, cec, cot on the unit circle
special angles of
the unit circle
unwrapping the unit circle
"morphing" trig graphs
Pythagorean Identities
Law of
Sines
Law of Sines- ambiguous case
Law of
Coines
sin (a+b),
Q1
sin (a+b),
Q2 both acute
sin (a+b), Q2 acute+obtuse
cos (a+b),
Q1
sin
(a+b), cos (a+b)
geometric approach
THEOREMS Dynamically explore conventional theorems such as the
Pythagorean Theorem (+ history) along with 7 different visual proofs of it.
In addition, discover a large selection of unusual and unexpected theorems
concerning tangents to parabolas, cubics, and quartics that will amaze and
fascinate your students while laying an excellent foundation for more
advanced mathematical study.
Pythagorean Theorem
More
Pythagorean Theorem
any 3
tangents to any parabola
3 tangent
proof (part 1)
3 tangent
proof (part 2)
parallel
tangent at midpoint
inflection
point at midpoint
use 2
roots to find 3rd (+ extend to imag. rts)
ratio of
areas in cubic
more
ratios of areas in cubic
ratios of
areas in quartic
PT –
shearing proof
PT –
Chinese proof
PT –
Pythagoras proof
PT –
Bhaskara proof
PT –
DaVinci proof
PT –
Garfield proof
PT –
Generalized
any 2
tangents to any parabola
2 tangent
proof (part 1)
2 tangent
proof (part 2)
MORE APPLICATIONS & VERTICAL
TEAMING Build basic graphing sense using intriguing questions
about real-world situations that animate at the click of a button.
Thoroughly explore classic problems such as the "open box" (vary the size of
squares removed from the corners of the original rectangle), the sliding
ladder, flying kite, etc. As the level of the class increases, more and more
features can be explored at the click of a button. Finally, even calculus
students will benefit by experiencing the varying rates of change in
familiar favorites – perfect for vertical teaming! | 677.169 | 1 |
Linear Algebra, 2nd Edition
This popular textbook was thoughtfully and specifically tailored to introducing undergraduate students to linear algebra. The second edition has been ...Show synopsisThis popular textbook was thoughtfully and specifically tailored to introducing undergraduate students to linear algebra. The second edition has been carefully revised to improve upon its already successful format and approach. In particular, the author added a chapter on quadratic forms, making this one of the most comprehensive introductory texts on linear algebra.Hide synopsis288 pages) this popular textbook was thoughtfully and specifically tailored to introducing undergraduate students to linear algebra. the second edition has been carefully revised to improve upon its already successful format and approach. in particular, the author added a chapter on quadratic forms, making this one of the most comprehensive introductory texts on linear algebra. a system of vectors. matrices. elementary row operations. an introduction to determinants. vector spaces. linear mappings. matrices from linear mappings. eigenvalues, eigenvectors, and diagonalization. euclidean spaces. quadratic forms. appendix: mappings. index. (Paperback)
Description:New. 0751401595 ***BRAND-NEW*** FAST Fedex shipping, so you'll...New. 0751401595 | 677.169 | 1 |
Mathematics Course Descriptions
The following course descriptions are from the Clarke University 2012-2013 Academic Catalog.
COURSE DESCRIPTIONS: MATHEMATICS
MATH 005 ELEMENTARY ALGEBRA 3 hours Students learn the numeric and algebraic skills necessary for future mathematics work. The focus is on the development of the real number system with emphasis on relationships and applications. Credit for this course does not count toward the 124 credits required for graduation. Students who are enrolled in this course may use it to fulfill athletic and financial aid eligibility.
MATH 090 INTERMEDIATE ALGEBRA 3 hours Explores linear, quadratic, exponential and logarithmic functions. This course may include polynomial functions of higher order, rational functions and systems of equations. Students develop their skills by approaching functions numerically, algebraically, graphically and verbally. Credit for this course does not count toward the 124 credits required for graduation. Students who are required to enroll in this course may use it to fulfill athletic and financial aid eligibility.
MATH 105 FOUNDATIONS OF MATHEMATICS I 3 hours Designed for strengthening the mathematical backgrounds of elementary teachers, this course may cover topics such as numeration systems, number systems, problem solving, and topics in number theory, statistics and geometry.
MATH 110 MATH AS A LIBERAL ART 3 hours Enables students to appreciate mathematics in the world around them. The emphasis is on reading, writing and conceptual understanding as opposed to rote skills. Topics may include networks, voting, games, statistics, coding, tiling, symmetry and patterns, infinity, personal finance, and the fourth dimension. This course is designated as a mathematics and natural science division general education course, and fulfills the university mathematics requirement.
MATH 117 PRECALCULUS WITH ALGEBRA 3 hours Oriented to preparation for calculus, this course continues the exploration of functions, including algebraic, exponential, and trigonometric functions. Students learn about functions through symbolic, numerical, graphical and verbal techniques. This course may not be taken for credit if a grade of C or above was achieved in a higher mathematics course (with the single exception of MATH 220 Statistics). This course is designated as a mathematics and natural science division general education course, and fulfills the university mathematics proficiency requirement. Prerequisites: Three years of high school mathematics or equivalent and appropriate placement.
MATH 180 TOPICS IN MATHEMATICS 1-3 hours A study of basic concepts in various areas of mathematics.
MATH 220 STATISTICS 3 hours Using technology and real-world data, this course explores descriptive and inferential statistics in preparation for research in various fields of study. A TI-83 graphing calculator or equivalent may be required. This course is designated as a mathematics and natural science division general education course, and fulfills the university mathematics proficiency requirement. Prerequisite: MATH 090 or equivalent or appropriate placement.
MATH 225 CALCULUS I 4 hours Includes the study of functions via rates of change. The main tool is the derivative, and it is approached from algebraic, numerical and graphical points of view. There are applications of differentiation and an introduction to integration. Meets five days per week, which may include time in the computer lab. A TI-83 graphing calculator or equivalent is required. This course is designated as a mathematics and natural science division general education course, and fulfills the university mathematics proficiency requirement. Prerequisite: Four years of high school mathematics or equivalent or MATH 117.
MATH 226 CALCULUS II 4 hours Sequel to MATH 225 Calculus I, in which functions are studied via integration. Topics include applications of the definite integral and an introduction to infinite series and differential equations. Meets five days per week, which may include time in the computer lab. A TI-83 graphing calculator or equivalent is required. Prerequisite: MATH 225 or consent.
MATH 230 STATISTICS FOR MAJORS 3 hours This course explores the statistical concepts in the MATH 220 course at a depth more appropriate for mathematics majors and minors. Prerequisite: permission of department.
MATH 280 TOPICS IN MATHEMATICS 1-3 hours Students will study basic concepts in various areas of mathematics.
MATH 336 GEOMETRY SEMINAR 3 hours Topics include Euclidean and non-Euclidean geometries. Emphasis is on student exploration, communication and research skills. This course is offered every other year. Prerequisite: MATH 226 or consent.
MATH 395 INTERNSHIP, UPPER DIVISION CV A professional experience in mathematics as arranged with department or off-campus supervisors.
MATH 443 ABSTRACT ALGEBRA 3 hours Includes the study of abstract algebraic structures, including groups, rings and fields. This course is offered every other year. Prerequisites: MATH 226 and MATH 333 or consent.
MATH 487 RESEARCH 1-4 hours Students read and conduct research or do creative work on a problem in mathematics and/or computer science.
MATH 499 CAPSTONE: MATHEMATICS SEMINAR 3 hours This course focuses on discipline-specific topics and expands to include breadth of knowledge and synthesis. Interdisciplinary integration of knowledge and research is emphasized. General education and major outcomes are integral to course assessment. Prerequisites: Ordinarily, a student must have senior standing with a minimum of 42 credit hours in general education completed, MATH 333 Linear Algebra, MATH 336 Geometry Seminar, and consent. | 677.169 | 1 |
Science Books
Intermediate Algebra
The Lial series has helped thousands of students succeed in developmental mathematics through its friendly writing style, numerous realistic examples, extensive problem sets, and complete supplements packageTurning The Tables In Chemistry(June 8, 2007) — What do glowing veggies have to do with a career in science? It just so happens that electrified pickles swimming in metal ions are one example of the type of undergraduate chemistry class ... > read more
Test After Test Turns Students Off Math(October 17, 2007) — The ever-growing strain of examinations, cramming and top-down teaching is turning students off studying maths at university - according to new research. Researchers in the UK says the pressures ... > read more | 677.169 | 1 |
Algebraic Formalism within the Works of Servois and Its Influence on the Development of Linear Operator Theory
Introduction
Before the nineteenth century, algebra usually referred to the theory of solving equations. However, the field of algebra experienced an extensive transformation during the nineteenth century, a time period referred to by many historians as the Golden Age of mathematics. Consequently, by 1900 algebra encompassed the study of algebraic structures. One contributor to the advancement of algebra was François-Joseph Servois (1767-1847). Servois was a priest, artillery officer, professor of mathematics, and museum curator. Not only did he battle to defend Paris in 1814, but he also fought for an algebraic foundation for calculus. As we will see, Servois was an advocate of "algebraic formalism," and the majority of his contributions to the field of mathematics fall under this category. In an "Essai" written in 1814, Servois attempted to provide a rigorous foundation for the calculus by introducing several algebraic properties, such as "commutativity" and "distributivity." Essentially, he presented the notion of a field, an idea far ahead of his time. Although Servois was not successful in providing calculus with a proper foundation, his work did have an impact on the field of algebra, and influenced several mathematicians, including the English mathematicians Duncan Gregory and Robert Murphy. Many English mathematicians of this period used the works of French mathematicians to aid in their development of linear operator theory and abstract algebra [Koppleman 1971].
This article further illustrates that while the disciplines of algebra and analysis are studied separately today, mathematicians of the eighteenth century (and before) made little, if any, distinction between them. Finally, instructors can use the original sources found in this article to demonstrate to students the connection between classical and modern day mathematics.
A Concise History of Abstract Algebra
According to Piccolino [1984], the nineteenth century is considered by many historians to be a Golden Age in the development of mathematics. Advancements in several branches of mathematics, such as geometry and analysis, occurred during this revolutionary time period. Another area that experienced change was algebra [Piccolino 1984]. Prior to the nineteenth century, algebra usually referred to the theory of solving equations; however, by 1900 it involved the study of mathematical structures, such as groups, rings, and fields [Katz 2009]. Mathematicians found that these structures often did not share properties found in the real and complex number systems, such as commutativity. This fundamental change in algebraic thought is characteristic of modern or abstract algebra.
One of the earliest instances where we see a mathematician's work on the solvability of algebraic equations is in the writings of Mohammed ibn Mûsâ al-Khowârizmî (ca. 790 CE - ca. 850 CE). Al- Khowârizmî provided solutions to linear (first-degree) equations and quadratic (second-degree) equations, but his results were presented verbally, without the use of algebraic symbols [Dunham 1990]. Al-Khowârizmî did not recognize either negative coefficients or negative solutions in his general solution to the quadratic equation \(ax^2 + bx + c = 0\), which he broke up into six cases. According to David Eugene Smith [1958], the first significant treatment of negative numbers was by Girolamo Cardano (1501-1576) in his 1545 book on algebra, Ars Magna. The first consideration of imaginary solutions occurred a few years later when Rafael Bombelli (1526-1572) used imaginary numbers as a "tool" for solving cubic equations [Dunham 1990, pp. 150-151]. According to Victor Katz, Bombelli's work "provided mathematicians with the first hint that there was some sense to dealing with" imaginary numbers in their algebraic work [2009, p. 407].
Figure 2. Joseph-Louis Lagrange (public domain).
Initial developments in abstract algebra occurred in Continental Europe during the late eighteenth and early nineteenth centuries. These changes were driven by problems in classical algebra, such as the solvability of third, fourth, and higher degree equations [Piccolino 1984]. The works of Joseph-Louis Lagrange (1736-1813), Augustin-Louis Cauchy (1789-1857), Paolo Ruffini (1765-1822), Niels Henrik Abel (1802-1829), and Evariste Galois (1811-1832) were of central importance during this time period and contained several concepts associated with modern group theory.
Lagrange explored the solvability of equations via the theory of permutations. Cauchy also made contributions to the theory of permutations by introducing concepts such as the identity permutation, a permutation that does not change a given arrangement of objects. Furthermore, Ruffini made several attempts at proving that the general equation of degree five is unsolvable in terms of radicals. Although Ruffini's efforts were not successful, his work provided a foundation for Abel's proof that such a solution cannot exist [Katz 2009]. Finally, Galois made significant contributions to the theory of solvability of algebraic equations by studying the structure of algebraic equations, particularly what he called "the group of the equation" [Katz 2009, p. 726]. We also owe to Galois the first known use of the term "group" in mathematics, which appeared in 1830 [Boyer 1989].
Although the work on algebraic solvability was carried out on the Continent, it was the British school of algebra that was primarily responsible for the shift in algebraic thinking towards abstract structural properties. As we shall see, this was not necessarily done by building on continental work on algebraic solvability, but rather by extending properties of ordinary arithmetic and of what we would call functions from analysis. Important figures in this movement included George Peacock (1791-1858), Duncan Farquharson Gregory (1813-1844), and William Rowan Hamilton (1805-1865). Peacock introduced the notions of arithmetical algebra and symbolical algebra. He defined arithmetical algebra as a universal arithmetic (using letters instead of numbers) of positive numbers [Katz 2009]. In this system, the term \(a - b\) had meaning only if \(a\) was greater than or equal to \(b\). On the other hand, symbolic algebra referred to the study of operations that were defined through arbitrary laws. In Peacock's symbolic algebra, \(a - b\) was valid regardless of the relationship between the symbols \(a\) and \(b\) [Piccolino 1984]. However, his laws in symbolic algebra were derived using principles found in his arithmetical algebra [Katz 2009]. Peacock was on the cusp of formulating an internally consistent algebra and his efforts in that direction were extended by Gregory.
Gregory, founder of the Cambridge Mathematical Journal, focused on algebraic structure. In his works, he often referred to the ideas of commutativity, distributivity, index operations (a sort of law of exponents for operators), and inverses, which he described as "circulating operations." He also mentioned the principle of the separation of symbols of operation, crediting the French mathematician François-Joseph Servois (1767-1847) as the first to "correctly give" the procedure [Allaire and Bradley 2002, p. 410]. Gregory appears to have been one of the first mathematicians to establish a connection between differentiation in calculus and the ordinary symbols of algebra, noting that the commutative and distributive laws hold true for what he referred to as the symbols of differentiation. Despite his contributions to the development of abstract algebra, Gregory, like Peacock, maintained the stance that results in symbolic algebra had to suggest results in arithmetical algebra [Piccolino 1984].
Figure 4. Duncan Farquharson Gregory (public domain).
A new, internally consistent algebraic system was finally introduced by Hamilton with his discovery of quaternions on October 16, 1843. Hamilton extended the algebra of number pairs to ordered quadruples of numbers, \((a, b, c, d)\), and defined the quaternions as ordered quadruples of numbers that followed several rules [Katz 2009]. Most notably, Hamilton's quaternions did not satisfy the commutative postulate for multiplication [Boyer 1989]. His system was the first algebra that did not follow all of the laws established by Peacock [Katz 2009]. The freedom and structure present in Hamilton's system was unprecedented and, as a result, many historians consider his discovery of the quaternions as the beginning of abstract algebra [Piccolino 1984].
Fundamental structures in abstract algebra, such as groups and fields, were formally defined in the later part of the nineteenth century. Heinrich Weber (1842-1913) was the first mathematician to present detailed, axiomatic definitions of groups and fields [Katz 2009]. Weber's definition of a finite group was slightly different than the one that most mathematicians are familiar with today. His definition included three axioms analogous to the modern day ideas of closure, associativity, and left- and right-hand cancellation laws. The terminology of closure under an operation is first found in Saul Epsteen and J. H. Maclagan-Wedderburn's "On the Structure of Hypercomplex Number Systems," which appeared in Transactions of the American Mathematical Society, Vol. 6, No. 2. in April of 1905 (see [Epsteen and Maclagan-Wedderburn 1905] and [Miller 2010]). Weber then showed that his three laws imply the existence of a unique identity element, and for each element, the existence of a unique inverse. He also incorporated his notion of group in the definition of a field. He defined a field as a set with two operators, addition and multiplication. In Weber's field, the entire set forms a commutative group under addition and the nonzero elements form a commutative group under multiplication as well. Weber also noted several properties of fields including the distributive law, which states that \(a\cdot (b + c) = a\cdot b + a\cdot c\) for all elements in the field [Katz 2009].
It is clear that mathematicians of the nineteenth century were concerned with foundational issues that spanned across several different areas of mathematics. Servois was no different and concentrated mainly on the foundational issues of the calculus. His attempts to settle the foundational issues of calculus were not successful [Bradley and Petrilli 2010]; however, as we will see, his work had a direct influence on the development of abstract algebra, and in particular, linear operator theory.
Francois-Joseph Servois
François-Joseph Servois was born on July 19, 1767, in the village of Mont-de-Laval, located in the Department of Doubs close to the Swiss border. Throughout his youth, Servois attended several religious schools in Mont-de-Laval and Besançon, the capital of Doubs, aspiring to become a priest. He was ordained a priest at Besançon shortly before the start of the French Revolution. He then left the priesthood in 1793 and became an officer in the Foot Artillery (sometimes referred to as the Heavy Artillery) with the outbreak of the revolutionary wars. In his leisure time Servois studied mathematics and his mathematical talents were apparent when he made improvements to one of the cannons, increasing its firing range significantly [Boyer 1895]. He suffered from poor health during his military career and, as a result, requested a non-active military position in the field of academia. He was assigned his first academic position on July 7, 1801, as a professor at the artillery school in Besançon, by virtue of a recommendation from the great mathematician Adrien-Marie Legendre (1752-1833). Throughout his academic career, Servois was on faculty at numerous artillery schools, including Besançon (1801), Châlons (March 1802 - December 1802), Metz (December 1802 - February 1808, 1815-1816), and La Fère (February 1808-1814, 1814-1815). His research spanned several areas, including mechanics, geometry, and calculus; however, he is best known for first introducing the words "distributive" and "commutative" to mathematics. On May 2, 1817, Servois was assigned to what would be his final position, as Curator of the Artillery Museum, which is currently part of the Museum of the Army in Paris. Servois retired to his hometown of Mont-de-Laval in 1827 and lived for another twenty years with his sister and his two nieces. He died on April 17, 1847. Readers interested in a more extensive biography and a review of Servois' other mathematical works can refer to Petrilli [2010].
It would be customary to include a painting or photograph of Servois in this biographical section, but there are no known images of him. However, due to Anne-Marie Aebischer and Hombeline Languereau [2010], there is now a photograph of his signature available to the public.
Servois' Belief in Algebraic Formalism
During the years 1811 to 1817, the majority of Servois' works were published in Joseph Diaz Gergonne's (1771-1859) Annales des mathématiques pures et appliquées. Much of his work focused on what Taton [1972a and 1972b] called "algebraic formalism." In 1814, we witness Servois' first defense of algebraic formalism, when he began a heated debate with Jean Robert Argand (1768-1822) and Jacques Français (1775-1833). In 1813, Français published a paper based on the work of Argand, in which he viewed complex numbers geometrically. In modern day mathematics, the view of complex numbers in the plane is known as the Argand Plane. Servois highly criticized the work of these two mathematicians, saying: "I had long thought of calling the ideas of Messrs. Argand and Français on complex numbers by the odious qualifications of useless and erroneous ...." [Servois 1814b, p. 228]. For instance, Servois argued against Français' geometric "demonstration," given in the preceding issue of Gergonne's journal [1813], that the quantity \(a\sqrt{-1}\) can be seen as the geometric mean of \(-a\) and \(a\). (In the field of complex numbers, we define the geometric mean of two real numbers \(a\) and \(b\) as \(\sqrt{ab}\).) Furthermore, he went on to state that it was in the best interest of the science to express his personal view, because in this work he saw nothing but "a geometric mask applied to analytic forms ...." [Servois 1814b, pp. 228-230].
Servois' fight for "algebraic formalism" continued in 1814 with the publication of his "Essai sur un nouveau mode d'exposition des principes du calcul différential" [Servois 1814a] ("Essay on a New Method of Exposition of the Principles of Differential Calculus"). This work was an extension of Lagrange's research on the foundations of the differential calculus. In his "Essai," Servois stated his belief that the differential calculus could be unified through algebraic generality:
In the preceding article, we have sketched the set of laws that brings together and unites all the differential functions, that is, the most general theory of the differential calculus. The practice of this calculus, which is nothing other than the execution of the operations given in the definitions .... [Servois 1814a, p. 122].
The notion of algebraic generality is apparent in the opening sections of his "Essai," where Servois essentially defined a field for his set of functions under the operations of addition and composition [Bradley and Petrilli 2010]. However, it was not Servois' intention to create formal structures within algebra, but rather he "was concerned above all else to preserve the rigor and purity of algebra" [Taton 1972a].
Algebraic Structure within Servois' 'Essai'
In his "Essai," Servois attempted to provide a rigorous foundation for the calculus through algebra. In light of what we know today, Servois did not fully succeed in putting calculus on a mathematically correct foundation. It was Cauchy [1821] who, through his approach to calculus by means of limits and inequalities, moved the subject into the modern age. Although Servois' efforts were ultimately unsuccessful, we find several ideas associated with abstract algebra in his "Essai."
Figure 6. Title page of Servois' "Essai" (public domain).
In Sections 1-4 of the "Essai," Servois presented several definitions that would be crucial to his work. He began by introducing his notation for a function as \(f\,z\), where a modern reader would understand this as \(f(z)\). The formal definition of a function that we use today would not be introduced until 1837 by Lejeune Dirichlet (1805-1859). A reader will notice that Servois used the term function not only to describe ordinary functions of an independent variable, but also to describe operators, such as the difference and differential operators. After he presented his preliminary definitions, Servois introduced two functions or operators with special properties, namely the identity \(f^0\) and inverse \(f^{-1}\). Undergraduate students will notice that these operators and properties are analogous to the modern day notions of an identity element and inverse element. For instance, Servois explained that when the identity function or operator is applied to \(z\), "\(z\) does not undergo any modification" [Servois 1814a, p. 96]. Additionally, Servois provided many examples of inverse functions. For instance, he considered the inverse of the sine function, noting that \[z = \mbox{sin}(\mbox{sin}^{-1} (z)) = \mbox{sin}(\mbox{arcsin}(z)).\] For the difference operator \(\Delta\), he noted that \[\Delta^n (\Delta^{-n} (z)) = \Delta^{-n} (\Delta^n (z)) = z.\]
In Section 3, Servois defined a function or operator \(\varphi\) to be distributive if it satisfied \[\varphi (x + y + ...) = \varphi(x) + \varphi(y) + \cdots.\] He presented several examples of functions and operators that are distributive and others that are not. One such distributive function was \(f(x) = ax\), because \[a(x + y + \cdots) = ax + ay + \cdots.\] Later, Cauchy [1821] showed that this is the only continuous function that satisfies this property. Conversely, Servois provided as a function that does not satisfy the distributive property \(f(x) = \mbox{ln}\,x\). He also demonstrated that the differential and integral operators are distributive.
Actually, undergraduates are exposed to specific examples of Servois' distributive property in a first-year calculus course, namely differentiation and integration under the operation of addition. Additionally, students see a generalized version of Servois' distributive property in a beginning linear algebra course. One of the properties of a linear transformation is that it preserves the addition operation.
Finally, in Section 4, Servois stated that two functions or operators \(f\) and \(\varphi\) are commutative between themselves if \[f(\varphi (z)) = \varphi (f (z)).\] For example, he stated that \(z\) commutes with any constants \(a\) and \(b\) because \[abz = baz;\] however, the sine function is not commutative with any constant \(a, a\not= -1,0,1\), because \[\mbox{sin}(az) \neq a \mbox{sin}(z).\]
If we consider the commutative operators \(f(z)\) and \(\varphi (z) = kz\), where \(k\) is any scalar, \[f(kz) = kf(z),\] then we have the familiar scalar multiplication property that undergraduates would see in a first-year calculus course for vectors or beginning linear algebra course for linear transformations.
In modern day mathematics, we speak of an algebraic structure (ring, field, etc.) as being commutative when \(a \cdot b = b \cdot a\) and distributive when \(a \cdot (b + c) = a \cdot b + a \cdot c\), for all elements \(a\), \(b\), and \(c\) in the structure. The hallmark of Servois' calculus was his examination of the set of all functions and operators that satisfy these properties. Interestingly, the words "commutative" and "distributive" were medieval legal terms [Bradley 2002] and Servois was the first to use them in a modern mathematical sense.
In Sections 5-9 of his "Essai," Servois examined the "closure" properties of distributive and commutative functions or operators. Servois demonstrated that distributivity is closed under composition and addition, and if \(f\) and \({\rm f}\) are commutative functions or operators, then each commutes with the inverse of the other. An examination of Servois' proof of the latter theorem reveals a law from abstract algebra. By the definition of the inverse function, we have \[f( \mbox{f(f}^{-1}(z))) = \mbox{f(f}^{-1}(f(z))),\] and by virtue of the commutativity of \(f\) and \({\rm f}\), we get \[f( \mbox{f(f}^{-1}(z))) = \mbox{f}(f(\mbox{f}^{-1}(z))).\] Now, substitute \(\mbox{f}(f(\mbox{f}^{-1}(z)))\) for \(f( \mbox{f(f}^{-1}(z)))\) in equation (1), and we get \[\mbox{f}(f(\mbox{f}^{-1}(z))) = \mbox{f(f}^{-1}(f(z))).\] Finally, apply \(\mbox{f}^{-1}\) to both sides of equation (2) and we arrive at the desired result that: \[f(\mbox{f}^{-1}(z)) = \mbox{f}^{-1}(f(z)).\] Essentially, when Servois applied \(\mbox{f}^{-1}\) to both sides of equation (2), he invoked a familiar theorem from group theory, the left-hand cancellation law.
After he considered the properties of commutativity and distributivity separately, Servois examined the collection of functions or operators that satisfy both of these properties. He used this as a launching point to introduce his theory for the differential calculus.
From a modern standpoint, the first twelve sections of Servois' "Essai" constitute the creation of an algebraic structure. In them, he showed that the set of invertible, distributive, and pairwise commutative functions or operators forms a field with respect to the operations of addition and composition. Servois never discussed the associative property with respect to these two operations. However, the importance of associativity was being uncovered during the nineteenth century. For instance, Carl Friedrich Gauss (1777-1855) did prove an associative law in 1801 [Gauss 1801, Section 240] and Hamilton stated the importance of associativity in 1843 after his discovery of the quaternions [Crilly 2006, p. 102]. Hamilton's statement was actually the first appearance of the term [Miller 2010]. Additionally, Servois assumed the existence of inverses for all of his functions. Again, a reader must keep in mind that Servois worked only with functions that were well-behaved and he did not examine the issue of domain.
Servois' Influence on the Development of Linear Operator Theory
Servois' influence on the development of Linear Operator theory can be traced through the works of several well-known mathematicians, including Murphy and Gregory. Although the formalization of symbolic algebra is generally credited to these two mathematicians, we see several ideas associated with Linear Operator Theory in by a lesser-known academic, Thomas Jarrett (1805-1882). Jarrett was an English cleric, Professor of Arabic at the University of Cambridge, and a linguist. According to the biography by E. J. Rapson [1892], he knew at least twenty different languages and would translate Chinese characters into Roman characters using a system that he devised himself. There is no biographical information that indicates that he had any formal mathematical training.
In the Preface to his book, Jarrett stated that part of his work was taken from the following mathematicians: Servois, Louis François Antoine Arbogast (1759-1803), John Frederick William Hershel (1792-1871), Carl Friedrich Hindenburg (1741-1808), Sylvestre François Lacroix (1765-1843), Pierre-Simon Laplace (1749-1827), Ferdinand Franz Schwiens (1780-1856), and Josef-Maria Hoëné-Wronski (1776-1853). Interestingly, Wronski's calculus was based on infinitesimals and Jarrett used no such foundations. Jarrett went on to say that some of the material was partly original; however, according to his biographers [Rapson 1892], the original contributions could simply have been new notation. Their contention is supported by Jarrett himself: "In the present Work [Algebraic Notation] is applied to the demonstration of the most important series in pure Analysis" [Jarrett 1831, p. III].
Jarrett's work is similar to Servois' "Essai" in that they both used algebra as a foundation for calculus; however, Jarrett presented Servois' material in a more structured format. Interestingly, when Jarrett discussed the summation operator he distinguished between operators (which he called operations) and functions, but when he presented his theory of the calculus he made no such distinction and classified both as functions. He derived many of the same results as Servois, only using different notation. Jarrett's calculus was based on the concept of the separation of symbols, which he credited to Servois in his Preface: "The demonstration of the legitimacy of the separation of the symbols of operation and quantity, with certain limitation, belongs to Servois ..." [Jarrett 1831, p. III]. At the heart of Jarrett's theory were Servois' distributive and commutative properties:
If \(\varphi (u)\) is such a function of \(u\) that \(\varphi (u + v) = \varphi (u) + \varphi (v)\), then \(\varphi (u)\) is called a distributive function of \(u\)
If \(\varphi (u)\) and \(\psi (u)\) are such functions of \(u\) that \(\varphi \psi (u) = \psi \varphi (u)\), then the functions \(\varphi (u)\) and \(\psi (u)\) are said to be commutative with each other.
From a modern standpoint, Jarrett defined a field in a fashion similar to Servois' by introducing the notions of an identity, inverses, and closure, in addition to these two properties. Using properties of this field he derived his theory of the differential and integral calculus.
We also see Servois' ideas in the work of another mathematician, Robert Murphy (1806-1843). Murphy's [1837] "First Memoir on the Theory of Analytic Operations" is a detailed exposition on the theory of operators. Murphy clearly distinguished between functions and operations, and called the objects on which operations are performed subjects [Allaire and Bradley 2002]. With respect to his notation, if Murphy wanted to discuss the operator \(\psi\) applied to the function \(f(x)\), then he denoted it as \([f(x)] \psi\), where the subject is contained within brackets.
Murphy [1837] began his paper by examining special types of operators. He considered the operators \(p\) and \(q\) as fixed or free, where "in the first case a change in the order in which they are to be performed would affect the result, in the second case it would not do so" [Murphy 1837, p. 181]. To relate this to Servois' work, a free operator would be one that satisfied Servois' commutative property. Now, let \(a\) and \(b\) be subjects and \(p\) be the operation of multiplying by the quantity \(p\). Then \[\left[a \pm b\right]p = \left[a\right]p \pm \left[b\right]p,\] which makes \(p\) (or multiplication by \(p\)) a linear operator according to Murphy's definition. Thus, a linear operator is one that satisfies Servois' distributive property. In modern day mathematics, in order for \(p\) to be a linear operator it would also have to be free with respect to a constant \(k\). Additionally, Murphy was the first mathematician to use the term linear to describe a special class of operators [Allaire and Bradley 2002].
Bradley and Allaire [2002] state that Murphy derived many of the same results as Servois, only with greater clarity and brevity. However, Murphy also expanded on the theory of linear operators. For example, he defined the appendage of a linear operator as "the result of its action on zero" [Murphy 1837, p. 188]. Here, "action" refers to the inverse image of the operator. In modern day mathematics, the appendage would be referred to as the kernel of a linear transformation, and this was the first time that the kernel of an operator had been considered [Allaire and Bradley 2002]. The kernel is an important concept in understanding the behavior of linear transformations.
Unlike Jarrett, Murphy did not acknowledge a debt to Servois nor is there solid evidence that he actually read his work. Murphy's opening sections are devoted to the important properties of operators – that is, Servois' commutativity and distributivity – but Murphy gave these properties different names. However, Servois began the study of linear operators and his work was read in England, as demonstrated by Jarrett's use of his research in 1831. As we will see, Duncan Gregory gave Servois the credit he was due.
According to Piccolino [1984], Gregory played a vital role in the development of symbolic algebra and was a key contributor to the overall advancement of mathematics in England during the late 1830s and early 1840s. In addition to his mathematical demonstrations, Gregory often provided philosophical insights to reinforce his views on algebra. For example, Gregory considered symbolic algebra as "the science which treats of the combinations of operations defined not by their nature, that is, by what they are or what they do, but by the laws of combination to which they are subject" [Gregory 1865, p. 2]. Essentially, Gregory believed that the general principles of algebra must fit a certain structure, which he called a class.
Throughout his Mathematical Writings [1865], Gregory provided several examples illustrating the "laws of combination" to which operations are subject. For example, he considered two classes of operations \(F\) and \(f\), which are connected by the following laws:
\(FF(a) = F(a)\)
\(ff(a) = F(a)\)
\(Ff(a) = f(a)\)
\(fF(a) = f(a)\)
His most general interpretation of these laws was multiplication for positive and negative numbers [Allaire and Bradley 2002]. For instance, (2.) shows that a negative number multiplied by a negative number yields a positive number. In this case, \(F\) and \(f\) should not be interpreted as functions, but rather as operations.
Next, he considered a general class of operations, which satisfy the following laws:
\(f(a) + f(b) = f(a + b)\)
\(f_1 f(a) = f f_1 (a)\)
Gregory credited Servois with the classification of these laws, writing, "Servois, in a paper which does not seem to have received the attention it deserves, has called them, in respect of the first law of combination, distributive functions, and in respect to the second law, commutative functions" [Gregory 1865, pp. 6-7]. Whereas Murphy [1837] considered the special class of functions satisfying (1.) to be linear operators, Gregory noticed that these two laws together constitute a special class of operations, which are called linear transformations or linear operators in modern mathematics.
Gregory then provided an example to demonstrate his first law, where \(f\) is taken to be the operation of multiplying by a constant \(a\): \[a(x) + a(y) = a(x + y).\] In his Mathematical Writings, Gregory stated that Cauchy sometimes utilized the "laws of combination" [p. 7], so Gregory may have been familiar with the fact that this is the only continuous function that satisfies the distributive law.
Finally, Gregory defined a class of operations by the law \[f(x) + f(y) = f(xy).\] This is the first time we see a general law for a class in which two different operations are considered. Gregory related this definition to a familiar law of logarithms, that \(\ln (x) + \ln (y) = \ln (xy)\), saying "when \(x\) and \(y\) are numbers, the operation is identical with the arithmetical logarithm" [Gregory 1865, p. 11].
Figure 7. Augustus De Morgan (public domain).
Petrova [1978] stated that linear operator theory began with Servois and was continued by Murphy. According to Allaire and Bradley [2002], Gregory was the next key figure in the development of this theory. We wondered, however, if any other mathematicians were influenced by the work of Servois. The authors examined numerous works written by mathematicians during the Golden Age of mathematics, including Peacock, Augustus De Morgan (1806-1871), and George Boole (1815-1864). These mathematicians made great advances in algebra; however, they appear to have made no significant contributions to the theory of linear operators. After examining several of their works, we now make some observations regarding the influence Servois may have had on these mathematicians. It should be noted that this analysis is highly subjective, because even though some of these mathematicians used methods similar to Servois', none gave him direct credit. Additionally, a majority of these works began to appear in the 1840s, and the works of Murphy and Gregory were already available by this time.
According to O'Connor and Robertson [1996], Peacock was interested in making reforms to Cambridge mathematics and he aided in the creation of the Analytical Society in 1815 as a result. The society was intended to bring the continental methods of the calculus to Cambridge. The reform began when Peacock translated Lacroix's calculus text, Traité élémentaire de calcul differéntiel et du calcul intégral [1802]. Using Lacroix's ideas, Peacock published his Collection of Examples of the Applications of the Differential and Integral Calculus [1820]. Lacroix's work was based on the calculus of Lagrange. Consequently, Peacock adopted many of the methods presented by Lacroix. Peacock did not explicitly use Servois' methods in his work and made no claim about the algebraic properties of operators. However, because Peacock was interested in the continental calculus, it is possible that he was familiar with the works of Servois.
Now, De Morgan, who was a student of Peacock's, presented the algebraic definitions for distributivity and commutativity in his work, Trigonometry and Double Algebra. These definitions are very similar to the ones that students would learn in a high school algebra course today. For instance, De Morgan stated, "A symbol is said to be distributive over terms or factors when it is the same thing whether we combine that symbol with each of the terms or factors, or whether we make it apply to the compound term or factor" [De Morgan 1849, pp. 102-103]. Being a student of Peacock's, De Morgan could have been familiar with the works of the continental mathematicians. This is further supported by the fact that he used the terms distributive and commutative in a fashion similar to Servois'.
Figure 8. George Boole (public domain).
Finally, in his Treatise on the Calculus of Finite Differences [1860], Boole gave the laws for the symbols \(\Delta\) and \(\frac{d}{dx}\). For instance, he stated:
Since Boole was a student of Gregory's, it is reasonable to conjecture that he was introduced to the works of Servois via Gregory's teachings. Servois' influence can be seen in Boole's own statements about \(\Delta\) and \(\frac{d}{dx}\) being distributive and commutative operators.
Conclusions and Recommendations
Although Servois was not successful in providing calculus with a proper foundation, his work did have an influence on the field of algebra, where he was a pioneer ahead of his time. He knew that, when performing algebraic manipulations on quantities, he needed to have a structure consisting of a set that obeyed certain axioms. Additionally, his work on analysis spread to England and significantly influenced mathematicians such as Duncan Gregory and Robert Murphy in their efforts to establish the foundations of linear operator theory. Servois' main contribution to this development was recognizing the "distributive" and "commutative" properties of operators, terms that he coined, and the method of separating symbols and their operations. Koppleman [1971] stated that the English mathematicians found the tools for the calculus of operations in the works of French mathematicians, such as Servois'. She went on further to say that "it was the English who developed this work in the calculus of operations both in extending the scope of its applications and in relating it to the theory of abstract algebra" [Koppleman 1971, p. 175].
The material discussed in this paper can aid teachers and students of abstract algebra, linear algebra, and the history of mathematics. By presenting the history of mathematics, instructors can illustrate the idea that mathematics is a constantly evolving field. Besides providing a readable account of the history of algebraic structures and the beginnings of linear operator theory, this paper contains many explanations of and references to original sources. Additionally, this article highlights the fact that while the disciplines of algebra and analysis are studied separately today, mathematicians of the eighteenth century (and before) made little, if any, distinction between them. When analyzing the original sources, the student co-author of this article, Anthony, was initially somewhat shocked to see so many calculus-like and algebraic ideas presented together. As an undergraduate, he had rarely seen ideas from both analysis and algebra meshed so closely together.
In the spirit of Victor Katz, we would encourage instructors to incorporate original sources within their classrooms. This paper provides references to original sources that can easily be found on the internet. With a little consideration on the part of the instructor, it is easy to create historical activities that can fit in any mathematics course. For instance, pages 1-13 of Gregory's The Mathematical Writings contain many examples of symbolical algebra that can be incorporated into any course, such as a first-year calculus course. An instructor could provide a copy of Gregory's discussion of operators that satisfy the property \(f(x) + f(y) = f(xy)\) and ask students to write a list of functions that satisfy this property.
Furthermore, these sources provide students with an opportunity to conduct research on the history of mathematics. Open questions include, for instance:
Who was Thomas Jarrett? Did he receive any formal mathematical training? If so, from whom did he receive training? Are there any other mathematical works published by him?
Jarrett's work is taken from a few mathematicians [Jarrett 1831, pp. III-V]. Many are well-known, but who was Ferdinand Franz Schwiens? Other than the analysis textbook mentioned by Jarrett, what did Schwiens write?
Many unanswered questions still remain regarding Servois' mathematical career. For instance, was there correspondence between Servois and any English mathematicians? If so, it would be interesting to use it to explore the extent of Servois' influence on these mathematicians.
Acknowledgments / About the Authors
Acknowledgments
The authors are extremely grateful to referees for their many helpful suggestions and corrections.
About the Authors
Anthony J. Del Latto is an undergraduate mathematics major at Adelphi University. He is currently a senior and serves as a tutor for the Department of Mathematics and Computer Science. After completing his bachelor's degree, he wishes to continue his studies in mathematics at the graduate level. His research interests include abstract algebra, history of mathematics, and applied statistics.
Salvatore J. Petrilli, Jr. is an assistant professor at Adelphi University. He has a B.S. in mathematics from Adelphi University and an M.A. in mathematics from Hofstra University. He received an Ed.D. in mathematics education from Teachers College, Columbia University, where his advisor was J. Philip Smith. His research interests include history of mathematics and mathematics education.
[Boyer 1989] Boyer, C. B. (1989). A History of Mathematics. New York: John Wiley and Sons.
[Bradley 2002] Bradley, R. E. (2002). "The Origins of Linear Operator Theory in the Work of François-Joseph Servois," Proceedings of Canadian Society for History and Philosophy of Mathematics14, 1 - 21.
[Jarrett 1831] Jarrett, T. (1831).. London: Cambridge University.
[Petrova 1978] Petrova, S. S. (1978). "The Origin of Linear Operator Theory in the Works of Servois and Murphy." History and Methodology of the Natural Sciences20, 122-128. (Unpublished translation by Valery Krupkin.)
[Piccolino 1984] Piccolino, A. V. (1984). A Study of the Contributions of Early Nineteenth Century British Mathematicians to the Development of Abstract Algebra and Their Influence on Later Algebraists and Modern Secondary Curricula. Doctoral thesis: Columbia University Teachers College. | 677.169 | 1 |
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Publisher:
Wiley & Sons, Incorporated, John
Release Date:
08/11/2008
Synopsis:
The mere thought of having to take a required calculus course is enough to make legions of students break out in a cold sweat. Others who have no intention of ever studying the subject have the notion that calculus is impossibly difficult unless you happen to be a direct descendant of Einstein.Well, the good news is that you can master calculus. Its not nearly as tough as its mystique would lead you to think. Much of calculus is really just very advanced algebra, geometry, and trigonometry. It builds upon and is a logical extension of those subjects. If you can do algebra, geometry, and trig, you can do calculus.Calculus For Dummies is intended for three groups of readers: Students taking their first calculus course If youre enrolled in a calculus course and you find your textbook less than crystal clear, this is the book for you. It covers the most important topics in the first year of calculus: differentiation, integration, and infinite series. Students who need to brush up on their calculus to prepare for other studies If youve had elementary calculus, but its been a couple of years and you want to review the concepts to prepare for, say, some graduate program, Calculus For Dummies will give you a thorough, no-nonsense refresher course. Adults of all ages whod like a good introduction to the subject Non-student readers will find the books exposition clear and accessible. Calculus For Dummies takes calculus out of the ivory tower and brings it down to earth.This is a user-friendly math book. Whenever possible, the author explains the calculus concepts by showing you connections between the calculus ideas and easier ideas from algebra and geometry. Then, youll see how the calculus concepts work in concrete examples. All explanations are in plain English, not math-speak. Calculus For Dummies covers the following topics and more: Real-world examples of calculus The two big ideas of calculus: differentiation and integration Why calculus works Pre-algebra and algebra review Common functions and their graphs Limits and continuity Integration and approximating area Sequences and seriesDont buy the misconception. Sure, calculus is difficult but its manageable and doable. You made it through algebra, geometry, and trigonometry. Well, calculus just picks up where they left off its simply the next step in a logical progression. Add Calculus Workbook For Dummies into the equation, and youre sure to be understanding calculus quicker and sooner than you ever thought possible. Help is here again with 275 pages of equations and answers, with ample room for you to work out the problems. Not sure where you went wrong (or right)? The answer section explains everything.AUTHOR BIO: Mark Ryan has taught algebra through calculus since 1989. He is a member of the National Council of Teachers of Mathematics. | 677.169 | 1 |
The unifying theme of this text is the development of the skills necessary for solving equations and inequalities, followed by the application of those skills to solving applied problems. An earlier introduction to the coordinate system and graphing is a focus of this fifth edition. Tables, graphs, and other visuals have been added to give students practice interpreting different forms of data display. | 677.169 | 1 |
PIMS Bar Model Workshop: Melania Alvarez
Date: 03/08/2012
Speaker(s):
Melania Alvarez
Location:
Topic:
The Bar Model a visual representation for problem solving
Description:
The main purpose of this workshop is to show how the Bar Model method can be used not only as a problem solving technique, but also to develop in students a deeper understanding of fundamental concepts in mathematics. | 677.169 | 1 |
Engaging in mathematical investigations yourself will help you be a better mathematics teacher!
National, state, and local curriculum requirements and recommendations increasingly emphasize active student involvement in exploratory investigations. Teachers now face new demands when they field unexpected student discoveries, determine when some open-ended activity has played out its usefulness or is just on the verge of paying off in a major way, and judge which student-initiated directions are likely to lead to development of important mathematical ideas and which are dead ends. These pedagogical judgments depend heavily on deep mathematical knowledge.
Ways to Think About Mathematics uses immersion experiences in algebra, geometry, and statistics to help mathematics teachers improve their knowledge and understanding of mathematical concepts. By experiencing open-ended problems, making and checking conjectures, and evaluating problem solving strategies, every math teacher can become better prepared to deal with day-to-day classroom decisions.
Funded by the National Science Foundation and successfully field-tested in a wide variety of professional development and preservice settings, the materials in this book integrate mathematical thinking, effective teaching practices, and explicit connections to exemplary curricula. Because it is aligned with the principles and standards of the National Council of Teachers of Mathematics but is not fixed to any single curriculum, the materials in this book can be aligned with any state or district standards.
Ways to Think About Mathematics gives teachers the opportunity to grow as learners and teachers of mathematics by including problems for teachers to explore as well as selected "Problems for the Classroom" for the classes they teach. | 677.169 | 1 |
Intermediate Algebra
9780495108405
ISBN:
0495108405
Edition: 8 Pub Date: 2007 Publisher: Thomson Learning
Summary: Algebra is accessible and engaging with this popular text from Charles "Pat" McKeague! INTERMEDIATE ALGEBRA is infused with McKeague's passion for teaching mathematics. With years of classroom experience, he knows how to write in a way that you will understand and appreciate. McKeague's attention to detail and exceptionally clear writing style help you to move through each new concept with ease. Real-world applicatio...ns in every chapter of this user-friendly book highlight the relevance of what you are learning. And studying is easier than ever with the book's multimedia learning resources, including ThomsonNOW for INTERMEDIATE ALGEBRA, a personalized online learning companion.[read more] | 677.169 | 1 |
SMTE
1351:
Fundamentals of Math II
This course is the second
in a sequence of three mathematics content courses for students
seeking
certification for EC-6 Generalist, Special Education, Bilingual Education,
and grade 4-8 disciplines. This
course provides the conceptual framework for application of rational numbers, probability and statistics in
a problem solving setting.
"These courses should
be designed to ensure that the material is understood by the students
at a deeper level than would be the case if they took a more traditional
mathematics course....the material should be presented as much as possible
in a form that connects to the ways in which the subject comes up in
the elementary classroom....the courses should be such that they motivate
and engage students who have come to fear mathematics and mistrust their
own abilities to understand it al all. Finally, the course should involve
opportunities and requirements for communicating understanding of mathematics"
(Jonker, PRIMUS, 2008). | 677.169 | 1 |
All Matters of Math by Canaa
Thanks for Visiting!
My name is Canaa Lee, and I am proud to announce that my dream of becoming an author has finally come true! My first book, Algebra for the Urban Student is the first algebra book
that was written for the typical math student. Math books are written for people who love and understand the jargon. From my experience as a high school teacher, most students dislike
mathematics because it has always been difficult for them and they have never been good at it.
Algebra for the Urban Student is a book that is written in the language that I use to personalize my classroom. I have interjected my personality in my book to guide my students
through their assignments in class. Now, when students leave my classroom, it is as if they have taken me home with them to help complete their assignments.
The chapters in Algebra for the Urban Student illustrate a significant algebra concept, such as solving linear equations and inequalities and finding the slope of a line. Then, the
chapter includes homework assignment that provides students with the opportunity to "demonstrate your understanding." In addition, there are real life projects for both algebra and geometry,
grading rubrics for whole and small class discussions. Furthermore, there are algebra 2 lessons that utilize the graphing calculator and takes the pain out of learning upper lever algebra.
This is just the first. I am writing the sequel to Algebra for the Urban Student. I anticipate its release in August 2012! You can purchase Algebra for the Urban
Student today! | 677.169 | 1 |
MATH 070: Basic Mathematical Skills
All course material below are in PDF format. In order to view and print then You will need Adobe Acrobat reader installed on your computer which you can download for free form Adobe at
These calendars are intended as a guide to pace you through the semester. Check your instructor's D2L for specific course assignments.
Practice exams are a sample of the type of questions you could see on the unit exams. Use these as if you were taking the exam. This means no book or notes. PRACTICE PROBLEMS ARE UNAVAILABLE AT THIS TIME. | 677.169 | 1 |
Guys and Gals! Ok, we're tackling printable high school algebra math worksheets and I was absent in my last algebra class so I have no notes and my teacher explains lessons way bad that's why I didn't get to understand it very well when I went to our algebra class a while ago. To make matters worse, we will have our quiz on our next meeting so I can't afford not to study printable high school algebra math worksheets. Can somebody please help me try to understand how to answer couple of questions regarding printable high school algebra math worksheets so that I can prepare for the test. I'm hoping that someone would assist me as soon as possible.
Hello Friend How are you?. Well I've been reading your post and believe me : I had the same problem. Some time ago I was in the same problem, but before you get a professor, I will like to recommend you one software that's very good: Algebrator. I really tried a lot of other programs but that one it's definitely the the real deal! The best luck with that! Let me know what you think!.
I verified each one of them myself and that was when I came across Algebrator. I found it really appropriate for graphing function, rational expressions and rational expressions. It was actually also easy to operate this. Once you feed in the problem, the program carries you all the way to the solution clearing up every step on its way. That's what makes it outstanding. By the time you arrive at the result, you by now know how to crack the problems. I took great pleasure in learning to crack the problems with Basic Math, Algebra 1 and Remedial Algebra in algebra. I am also certain that you too will love this program just as I did. Wouldn't you want to test this out?
A truly piece of math software is Algebrator. Even I faced similar problems while solving fractional exponents, algebra formulas and converting decimals. Just by typing in the problem workbookand clicking on Solve – and step by step solution to my math homework would be ready. I have used it through several algebra classes - College Algebra, Pre Algebra and Intermediate algebra. I highly recommend the program. | 677.169 | 1 |
Students must receive a grade of at least a "C-" in previous courses in order to have the opportunity to take the next level math course. If a student receives a grade below a "C-" in a math course, he/she may opt to retake that class for no additional credit, but to raise the grade in order to advance to the next level math course.
Introduction to Algebra - Year Course Grades 9 – 12
(Value: 1 credit for the entire year – Mathematics)
This course is designed for the student who needs additional preparation prior to taking Algebra I. It is also the basic grade (9) course for students who do not take Algebra I. The course topics are similar to one semester of Algebra I with less degree of difficulty.
Algebra I - Year course - Recommended for colleges and technical programs
Prerequisite: Prealgebra (grade 8) or Introduction to Algebra
(Value: 1 credit for the entire year – Mathematics)
This is a beginning course in algebra. It covers operations on positive and negative numbers, fractions, solving equations with one and two variables, factoring polynomials, graphing, the study of squares and square roots and the use of the quadratic equation.
UNITS:
Introduction to Algebra
Working with real numbers
Solving equations and problems
Polynomials
Factoring polynomials
Fractions
Applying fractions
Introductions to functions
Systems of linear equations
Rational and irrational numbers
Algebra II - Year course *Intensive Homework*
Prerequisite: Algebra I and Geometry
Grades 10-12 (Value: 1 credit for the entire year – Mathematics)
The basic facts and rules studied the first year of Algebra I are briefly reviewed. Taking these facts as a background, new and more advanced use of them is studied in detail. A few of these new concepts are: imaginaries, logarithms, slide rule, progressions, determinants, statistics, calculators, etc. Students who are interested in math or those who plan to study additional math or enter the physical science area of study will find that Algebra II is an important mathematics course to take.
UNITS:
Sets of numbers: Axioms
Open sentences in one variable
Systems of linear open sentences
Relations and functions
Rational numbers and expressions
Relations and functions
Irrational numbers and quadratics equations
Quadratic relations and systems
Exponential functions and logarithms
Progressions and binomial expressions
Matrices and determinants
Geometry - Year course - Grades 9-12*Intensive Homework*
Prerequisites: Algebra I Required by most colleges.
(Value: 1 credit for the entire year – Mathematics)
Geometry is the study of plane and solid figures and their relationships to each other and to other mathematical principles and concepts. Attention will also be given to logic and formal proof.
In this class students use a discovery approach to develop an awareness of basic geometry concepts and their applications in the real world. Using geometry tools, pencil and paper, manipulatives, and the computer, students discover geometric properties by experimentation and observation. The development of reasoning skills is stressed throughout the course but the emphasis is on inductive reasoning skills with limited time spent on deductive reasoning and proof as in all academic classes. Good study skills are necessary for students to be successful in this class. (Notetaking, organization and completing homework)
UNITS:
Inductive Reasoning Introducing Geometry
Using Tools of Geometry
Line and Angle Properties
Triangle Properties
Polygon Properties
Circles
Geometry in Arts & Nature
Transformation & Tessallations
Area
Pythagorean Theorem
Volume
Similarity & Trignometry
Deductive Reasoning
Geometric Proof
Probability and Statistics – Semester Course – Grades 10 – 12
Prerequisite: Algebra I and Geometry
(Value: 1/2 credit - Mathematics)
This course is meant to introduce students to the mathematical concepts of both probability and statistics. In the statistics sessions, students will learn about how quantative data is collected, displayed, analyzed, and interpreted. This part of the class will also focus on the effects of how data can be misrepresented. In the probability session, students will learn what probabilities actually mean, what things affect certain probabilities, and how probabilities can help us make decisions in life. Each session will share equal time, which is approximately one quarter. This is a project-intensive course where students may have to present their findings to the class.
The word trigonometry is a derivation of three Greek words, which mean "three angles measurement". By means of trig, it is possible to determine the remaining sides and angles of a triangle when some of them are known. Algebra I and Algebra II are required prior to taking this course.
This is a complete course in precalculus topics. It is intended for use by students who have completed two years of high school algebra and one year of high school geometry. It is written for average and above average students who would like to prepare for college mathematics, review for college entrance examinations, or simply study more mathematics.
Chapters 1 through 5 are devoted to topics from advanced algebra. Chapter 1 offers a thorough review of quadratic equations and coordinate geometry. Chapter 2 includes a review of polynomial algebra and theory of equations as well as some material that will be new to most students. Chapter 3 is devoted to inequalities, and Chapter 4 to functions. Chapter 5 presents exponents and logarithms: it includes exponential growth and decay and natural logarithms. Other chapters include sequences and series, statistics, probability, and introductory calculus.
Senior Math - Year Course Grades 11 & 12
(Value: 1 credit for the entire year – Mathematics)
The purpose of this course is to review basic mathematical fundamentals while investigating topics of math as they apply to living in today's world. This class is designed to meet the needs of grade 12 students not enrolled in college prep curriculum. Grade 11 students may take this course with prior approval of the instructor. This course is offered alternate years.
Calculus is a branch of mathematics that makes use of plane geometry and algebra to which we add the idea of limit and of the limiting process. From the idea of limit, we study the two principal concepts of calculus, the derivative and the integral. We study various applications of the derivative and integral, including area, functions, sequences and series.Students have the opportunity to take the A.P. Calculus test for college credit at the conclusion of the course at their own expense. | 677.169 | 1 |
MATH 547: Introduction to Group Theory
Course ID
Mathematics 547
Course Title
MATH 547: Introduction to Group Theory
Credits
3
Course Description
A group is an algebraic system described by a set equipped with one associative operation. Groups contain an identity element and every element has an inverse. Group theory has applications in diverse areas, such as art, biology, geometry, linguistics, music and physics. The kinds of groups covered in this class include permutation, symmetric, alternating and dihedral groups. Some of the important theorems covered are Cayley's Theorem, Fermat's Little Theorem, Lagrange's Theorem and the Fundamental Theorem of Finite Abelian Groups. 342/542 | 677.169 | 1 |
Adobe PageMaker Pro for print, press, and electronic distributionThe element calculates determinants, linear equation systems and generates matrices. It provides additional basic functionality like faculty, subdeterminant and matrix reduction calculations. Further more an event is implemented to support a progress bar for time intensive operations | 677.169 | 1 |
Fort Mcdowell StatisticsWhen students can grasp the concept first, they are more likely to grasp more complex concepts later (math builds). At the least, they develop critical thinking skills grow more comfortable visualizing abstract relationships. I took college calculus and might only need to briefly brush-up on so... of learning. | 677.169 | 1 |
Mathematics
Why study Mathematics?
An essential element of mathematical learning is the development of mathematical knowledge in a way which encourages confidence and provides satisfaction and enjoyment. It is expected that students will gain an appreciation of the use of mathematical skills within other subjects as well as an understanding of problem solving in the real world.
Which specification is followed?
Girls in Lower School follow the National Framework for Mathematics as laid down by the National Curriculum and the National Numeracy Strategy. From Lower 4 onwards pupils are taught in sets according to attainment.
In 2009, Bradford Girls' Grammar School embarked on the iGCSE course following the Edexcel specification. Candidates are encouraged to develop a feel for numbers; to recognise patterns and relationships; to generalise results; and to use the language of mathematics to communicate their ideas effectively and efficiently. All candidates will be studying a course leading to the higher tier examinations, but individual students may be entered at Foundation Level if appropriate. The iGCSE does not have a coursework element. Marks are obtained from sitting two examination papers at the end of Upper 5. The iGCSE is an excellent preparation for students intending to study mathematics at A-level.
Mathematics and Further Mathematics are offered at both AS and A level following the Edexcel specifications. In recent years the role of Further Mathematics has changed. It is seen as enriching and deepening the curriculum: truly further maths rather than just harder maths – it provides able students with a course which stimulates them mathematically and prepares them for a wide range of higher education options.
At A level students study three main areas of mathematics:
Pure (or Core) Mathematics which develops and extends topics already met at GCSE including algebra, trigonometry and graphs, it also introduce new topics such as calculus.
Statistics which includes the presentation, analysis and interpretation of data and the study of probability.
Mechanics which involves the study of the motion of objects and how they respond to forces acting on them.
Further Mathematics will cover Decision Mathematics in addition to the three areas mentioned above.
Workshops, conferences and visits
Mathematics workshops are offered during lunchtimes, providing help and support to students in all years.
Girls throughout Bradford Girls' Grammar School participate in the Mathematical challenges, including the Junior and Senior Team Challenges, run by the UK Mathematics Trust.
Degree and career choices
There are many opportunities to study Mathematics in the Sixth Form at Bradford Girls' Grammar and it is an excellent support subject for any combination of 'A' levels. It is not an easy subject to study at this level and its academic rigour means that it is highly valued by universities for entry into most degree programme,s particularly the sciences, geography, economics, psychology, medicine and engineering. | 677.169 | 1 |
MATH40237 Fundamental Mathematics for University
Course details
Fundamental Mathematics for University is designed to provide students with foundation concepts, rules and methods of elementary mathematics. The main aim of this course is to provide the fundamentals of mathematics, which are necessary to develop a unified body of knowledge. Topics covered in the course include operations, percentages, introductory algebra, simple equation solving, exponents, linear equations, introductory statistics, and units and conversions | 677.169 | 1 |
COURSE SYLLABUS
Course Number: MATH1000
Course Title: COLLEGE ALGEBRA
Credit Hours: 3
Prerequisites: High school geometry and second year high school algebra.
Corequisite:
Objectives:
To develop student mathematical and analytic skill with particular emphasis on establishing student skills at algebraic manipulation. To give students the background and preparation necessary for core mathematics. Does not satisfy the core requirement in mathematics.
Schedule and Outline of Course Content:
Fundamental concepts of algebra ( 3 weeks).
Equations and inequalities ( 3 weeks).
Functions and graphs ( 5 weeks).
Polynomial and rational functions ( 4 weeks).
Possible textbooks:
Algebra and Trigonometry, 9th edition by Swokowski and Cole, 1997, Brooks/Cole.
Sample Grading and Evaluation Procedures:
Students will be expected to have prepared the daily homework assignments. Homework will occasionally be collected. This will be part of the participation grade.
Grade Calculation
Participation grade
(includes: blackboard presentation and classwork, attendance, homework): 10%
Quizzes (quizzes are approximately 10-minutes long and may be announced or unannounced): 10%
Hour Tests (four tests at 12 % each): 48%
Final Exam: 32%
Tentative Test Schedule
Hour tests are given approximately every three weeks and will be announced a week ahead of time. Quizzes may or may not be announced; at least four quizzes will be given in the course of the semester. Friday is typically a good day for quizzes
The course will be used in conjunction with the mathematics placement test and is designed to bring students to the level of preparation which will enable them to take core mathematics courses. | 677.169 | 1 |
Helping Undergraduates Learn to Read Mathematics
Although most students "learn to read" during their first
year of primary school, or even before, reading is a skill which continues
to develop through primary, secondary and post-secondary school, as the
reading material becomes more sophisticated and as the expectations for
level of understanding increase. However, most of the time spent deliberately
helping students learn to read focuses on literary and historical texts.
Mathematical reading (and for that matter, mathematical writing) is rarely
expected, much less considered to be an important skill, or one which can
be increased by practice and training.
Even as an undergraduate mathematics major, I viewed mathematical reading
as a supplementary way of learning--inferior to learning by lecture or
discussion, but necessary as a way of "filling in the gaps."
Not until graduate school was I responsible for reading new material at
a high level of comprehension. And, as I began to study primarily written
mathematics (texts and articles) rather than spoken mathematics (lectures),
I discovered that the activities and habits needed to learn from written
mathematics are quite different from those involved in learning from a
mathematics lecture or from those used in reading other types of text.
As I consciously considered how to read mathematics more effectively and
to develop good reading habits, I observed in my undergraduate students
an uneasiness and lack of proficiency in reading mathematics.
In response to this situation, I wrote for my students (mostly math
majors in Introductory Abstract Algebra at the University of Chicago) two
handouts, one on reading theorems and the other
on reading definitions. These describe some
of the mental activities which help me to read mathematics more effectively.
I also gave a more specific written assignment, applying some of these
questions to a particular section of assigned reading. My hope was that,
as they were forced to actively engage in reading, they would discover
that reading mathematics could be a profitable pursuit, and that that they
would develop habits which they would continue to use.
More than one such written exercise is needed to significantly affect
the way that students view reading. While the students seemed to understand
the types of questions that are helpful, they needed some practice in carrying
these out, and even more practice using these activities in the absence
of a written assignment.
A Few Mathematical Study Skills... Reading Theorems
In almost any advanced mathematics text, theorems, their proofs, and
motivation for them make up a significant portion of the text. The question
then arises, how does one read and understand a theorem properly? What
is important to know and remember about a theorem?
A few questions to consider are:
What kind of theorem is this? Some possibilties are:
A classification of some type of object (e.g., the classification of
finitely generated abelian groups)
An equivalence of definitions (e.g., a subgroup is normal if, equivalently,
it is the kernel of a group homomorphism or its left and right cosets coincide)
An implication between definitions (e.g., any PID is a UFD)
A proof of when a technique is justified (e.g., the Euclidean algorithm
may be used when we are in a Euclidean domain)
Can you think of others?
What's the content of this theorem? E.g., are there some cases in which
it is trivial, or in which we've already proven it?
Why are each of the hypotheses needed? Can you find a counterexample
to the theorem in the absence of each of the hypotheses? Are any of the
hypotheses unneccesary? Is there a simpler proof if we add extra hypotheses?
How does this theorem relate to other theorems? Does it strengthen
a theorem we've already proven? Is it an important step in the proof of
some other theorem? Is it surprising?
What's the motivation for this theorem? What question does it answer?
We might ask more questions about the proof of theorem. Note that, in
some ways, the easiest way to read a proof is to check that each step follows
from the previous ones. This is a bit like following a game of chess by
checking to see that each move was legal, or like running the spell checker
on an essay. It's important, and necessary, but it's not really the point.
It's tempting to read only in this step-by-step manner, and never put together
what actually happened. The problem with this is that you are unlikely
to remember anything about how to prove the theorem. Once you've read a
theorem and its proof, you can go back and ask some questions to help synthesize
your understanding. For example:
Can you write a brief outline (maybe 1/10 as long as the theorem) giving
the logic of the argument -- proof by contradiction, induction on n, etc.?
(This is KEY.)
What mathematical raw materials are used in the proof? (Do we need
a lemma? Do we need a new definition? A powerful theorem? and do you recall
how to prove it? Is the full generality of that theorem needed, or just
a weak version?)
A Few Mathematical Study Skills... Reading Definitions
Nearly everyone knows (or think they know) how to read a novel, but
reading a mathematics book is quite a different thing. To begin with, there
are all these definitions! And it's not always clear why one would care
to know about these things being defined. So what should you do when you
read a definition?
Ask yourself (or the book) a few questions:
What kind of creature does the definition apply to? integers? matrices?
sets? functions? some pair of these together?
How do we check to see if it's satisfied? (How would we prove that
something satisfied it?)
Are there necessary or sufficient conditions for it? That is, is there
some set of objects which I already understand which is a subset or a superset
of this set?
Does anything satisfy this definition? Is there a whole class
of things which I know satisfy this definition?
Does anything not satisfy this definition? For example?
What special properties do these objects have, that would motivate
us to make this definition?
Is there a nice classification of these things?
Let's apply this to an example, abelian groups:
What kind of creature does it apply to? Well, to groups... in particular,
to a set together with a binary operation.
How do we check to see if it's satisfied? The startling thing is that
we have to compare every single pair of elements! This would be a big job,
so:
Are there necessary or sufficient conditions for it? Well, it's sufficient
that the group be cyclic, as we saw in the homework. Do you know of any
neccesary conditions?
Does anything satisfy this definition? Well, yes... the group of rational
numbers under addition, for example. We have a whole class of things which
satisfy the definition, too -- cyclic groups.
Does anything not satisfy this definition? Yes, matrix groups come
to mind first. There are finite non-abelian groups, but this is
harder to see... do you know of one yet?
What special properties do these objects have, that would motivate
us to make this definition? Some of these properties are obvious, others
are things which we had to prove. One example: If H and K are subgroups
of an abelian group, then HK is also a subgroup.
Is there a nice classification of these things? Why, yes, at least
for a large subcategory of them. We'll get to it later... it says, basically,
that a finite abelian group is always built in a simple way from cyclic
groups (Zn's). | 677.169 | 1 |
The credit crisis that started in 2007, with the collapse of well-established financial institutions and the bankruptcy of many public corporations, has clearly shown the importance for any company entering the derivative business of modelling, pricing, and hedging its counterparty credit exposure. Building an accurate representation of firm-wide credit... more...
This book covers the necessary topics for learning the basic properties of complex manifolds, offering an easy, friendly, and accessible introduction into the subject while aptly guiding the reader to topics of current research and to more advanced publications. The first half of the book provides an introduction to complex differential geometry and... more...
Algorithmic composition - composing by means of formalizable methods - has a century old tradition not only in occidental music history. This book provides an overview of prominent procedures of algorithmic composition in a pragmatic way. It is suitable for the musicians and the researchers. more...
Accessible Mathematics is Steven Leinwand?s latest important book for math teachers. He focuses on the crucial issue of classroom instruction. He scours the research and visits highly effective classrooms for practical examples of small adjustments to teaching that lead to deeper student learning in math. Some of his 10 classroom-tested teaching shiftsThis book is devoted to Quisped, Roberts, and Thompson (QRT) maps, considered as automorphisms of rational elliptic surfaces. The theory of QRT maps arose from problems in mathematical physics, involving difference equations. The application of QRT maps to these and other problems in the literature, including Poncelet mapping and the elliptic billiard,... more... | 677.169 | 1 |
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11th Grade: Logarithmic Equations Word Problems
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How Others Use Our Site
I hope it will teach me the fundamentals for 11th grade and 12th grade Math. I am helping a student with her 11th grade math and will be using this website as a refresher. Because i am in 9th grade n im supposed 2 be in the 11th grade and i was told this could catch me up. I teach resource math 6th grade-11th grade and am always looking for sources. 11th grade math is hard, need examples of solved problems. As a math coach I am always trying to improve upon my own skills before implementing ideas in staff development. I also have an 11th grader who needs my help. | 677.169 | 1 |
SUPPLIES: Texas Instruments TI-83 Graphing
Calculator (note: If you are purchasing a calculator for this class, you are
required to purchase the TI-83. If you already have a graphing calculator,
consult your instructor about its acceptability)
EXPECTED STUDENT COMPETENCIES TO BE
ACQUIRED: The
successful student, at the end of the course, will be able to produce
well-written correct solutions for problems similar to those assigned for
homework in this course.
COURSE OBJECTIVE: To solve, both graphically and by
calculation, mathematical problems that involve:equations and inequalities, graphs,
functions, and inverse functions, polynomial, logarithmic, exponential, and
trigonometric expressions.
ASSIGNMENTS: Homework will be assigned daily and may
occasionally be collected as a check on how you are keeping up. Although most
of the homework assignments will not be collected, that doesn't mean you don't
have to do it! A major part of learning mathematics involves DOING mathematics! Also, homework is useful
in preparing for the type of questions, which may appear on quizzes or
exams.Many homework problems will be
given on quizzes and some on tests.
Evaluations:There will be given three tests and one final exam during the
semester.There will also be given
quizzes once a week approximately.It is
important that you work all of the assigned homework problems to practice for
quizzes, tests and the final exam. Also rework at home examples done in class.Some of these examples or homework problems
might given on a quiz, a test or the final exam.
Here are tentative dates for
the tests:
Test 1
Week of September
25-29
Test 2
Week ofOctober 23-27
Test 3
Week of Nov.
27-Dec. 1
Final Exam: The
final exam will be given on Wednesday December
13 at 11:00 AM.The final exam is comprehensive.
GRADING
The weights
of the various components of your grade in determining your final course grade
are shown below, along with the grade scale for the course.
WEIGHTS:
GRADE SCALE
1. Three Tests : 300 points) (100 each)
90-100
A
70-75
C
2. Quizzes, homework 150 pts
86-89
B+
66-69
D+
3. Cumulative Final Exam 150 pts
80-85
B
60-65
D
76-79
C+
0-59
F
NOTES:
Two quiz/homework
grades will be dropped to determine your final quiz/homework average.There will be no makeup quizzes.There will be no makeup tests, except under
special (documented) circumstances.In
the case you cannot take an exam at the scheduled time, contact the instructor
as soon as possible after (or before) the test, to arrange a make up.Exams not made up within one week of the
scheduled exam date will be recorded 0.
SPECIAL NOTES: If you have a physical, psychological,
and/or learning disability which might affect your performance in this class,
please contact the Office of Disability Services, 126A B&E, (803) 641-3609, and/or see
me, as soon as possible. The Disability Services Office will determine
appropriate accommodations based on medical documentation.
ATTENDANCE POLICY: I may occasionally take attendance. It
is highly recommended that the student not miss any class.However, the Attendance Policy established by
the Department
of
Mathematical Sciences states that the maximum number of unexcused absences
allowed in this class before a penalty is imposed is four for a regular
semester.
ACADEMIC CODE OF HONESTY: Please read and review the Academic
Code of Conduct relating to Academic Honesty located in the Student Handbook.
If you are found to be in violation of this Code of Honesty, a grade of F(0)
will be given for the work. Additionally, a grade of F may be assigned for the
course and/or further sanctions may be pursued. | 677.169 | 1 |
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Mathematics
Apply the central concepts of Mathematics and Logic to any career path.
If you like solving puzzles and figuring things out, then our mathematics major may interest you. Applications of mathematics are everywhere and a strong background in mathematics can help you in many different careers.
Ashland's Mathematics program is taught in small classes that allow you to explore and learn the language of numbers in-depth.
You will have many opportunities to participate actively in class and to get to know your professors well. Your professors have earned three awards for excellence in teaching, a testimony to their exceptional classroom skills and the strong mentoring relationships they develop with students.
What You'll Love About the Mathematics Program:
Unlike Mathematics programs in large universities where you're no more than a face in a crowd, at Ashland you get to know each of your professors well and benefit from their knowledge of Mathematics.
All classes are taught exclusively by our Mathematics professors and not by graduate students or teaching assistants.
You have many opportunities to participate in mathematics challenges and contests such as programming competitions and activities made available through the Ohio conference of the Mathematical Association of America (MAA).
The department actively participates in the national Mathematical Association of America (MAA) as well as in the Ohio section of the organization, attending conferences and presenting papers and workshops.
As a student you will have opportunities to make presentations at the Ohio section of MAA and at national meetings.
You have full access to the extensive computer resources within the department's technology center including a variety of hardware running Linux, Solaris, MacOS and Windows operating systems. The resources also include computer algebra systems, statistical and geometric software and other applications that will facilitate your learning.
Reach Your Career Goals
In addition to pursuing graduate school, our graduates are well prepared to begin careers as:
Actuaries in insurance companies
Operations and research analysts
Quality control engineers
Mathematics consultants
In addition, the analytical and logical abilities developed through the program equip you to pursue further study in areas such as business, law or medicine.
Outstanding Educators
Faculty members are very active in the field of mathematics often attending regional and national meetings with student groups.
Professors are excellent classroom educators who mentor students both in and out of the classroom through problem solving challenges and in-depth discussions of math topics.
Organizations for Mathematics Majors
Academic Department Info
Explore the language of numbers in small class settings.
Mathematics is the doorway to science and technology. Computer Science is the study of algorithmic processes to ultimately...read moreMeet our Faculty...
Also Solved by Paul S. Bruckman: An Interview by Dr. Thomas P. Dence, Professor of Mathematics Read More
Career Outlook for Mathematics Majors
Mathematics training often leads to careers in actuarial science, statistics, engineering and physics. Those with a bachelor's degree and proper licensure often become teachers of mathematics. Many mathematicians work for the federal and state governments including the Department of Defense. Private sector employers include research and development companies, technical consulting firms and insurance companies. Experts anticipate average job growth of about 10 percent between 2006 and 2016.1 Learn More about careers for mathematics majors!
What Students Say About Ashland
"Ashland University has played a large role in developing my character, providing me with the opportunities, resources, and experiences I needed to become a scholar, a leader and servant to others." -- Rachel Cordy from Elyria, Ohio, involved in Alpha Phi sorority, Judicial Board, Math Club, Orientation Team and Kappa Delta Pi. | 677.169 | 1 |
Set theory permeates much of contemporary mathematical thought. This text for undergraduates offers a natural introduction, developing the subject through observations of the physical world. Its progressive development leads from concrete finite sets to cardinal numbers, infinite cardinals, and ordin... read more
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Axiomatic Set Theory by Paul Bernays A historical introduction by A. A. Fraenkel to the original Zermelo-Fraenkel form of set-theoretic axiomatics, plus Paul Bernays' independent presentation of a formal system of axiomatic set theory.
An Introduction to Algebraic Structures by Joseph Landin This self-contained text covers sets and numbers, elements of set theory, real numbers, the theory of groups, group isomorphism and homomorphism, theory of rings, and polynomial rings. 1969Sets, Sequences and Mappings: The Basic Concepts of Analysis by Kenneth Anderson, Dick Wick Hall This text bridges the gap between beginning and advanced calculus. It offers a systematic development of the real number system and careful treatment of mappings, sequences, limits, continuity, and metric spaces. 1963 edition.
Lattice Theory: First Concepts and Distributive Lattices by George Grätzer This outstanding text is written in clear language and enhanced with many exercises, diagrams, and proofs. It discusses historical developments and future directions and provides an extensive bibliography and references. 1971 edition.
Product Description:
Set theory permeates much of contemporary mathematical thought. This text for undergraduates offers a natural introduction, developing the subject through observations of the physical world. Its progressive development leads from concrete finite sets to cardinal numbers, infinite cardinals, and ordinals. Although set theory begins in the intuitive and the concrete, it ascends to a very high degree of abstraction. All that is necessary to its grasp, declares author Joseph Breuer, is patience. Breuer illustrates the grounding of finite sets in arithmetic, permutations, and combinations, which provides the terminology and symbolism for further study. Discussions of general theory lead to a study of ordered sets, concluding with a look at the paradoxes of set theory and the nature of formalism and intuitionalism. Answers to exercises incorporated throughout the text appear at the end, along with an appendix featuring glossaries and other helpful information | 677.169 | 1 |
The Mathematics Department offers courses from the intervention level through Advanced Placement levels. All courses cover the state standards relevant to that course and many cover additional enrichment topics, especially academy and pathway sections. Students also have opportunities to participate in contests throughout the school year and the Math-Science Club meets weekly during lunch.
Math Department Course Offerings
Algebra 1-2
This course is required for graduation and for admission to the University of California or California State University systems. College eligibility requires a grade of C or better.
Students complete one year of college preparatory Algebra in this two semester sequence. Incoming ninth graders are placed in this class based on scores on the placement exam and record from middle school.
This course is required for graduation and for admission to the University of California or California State University systems. College eligibility requires a grade of C or better.
Students take this class when they have successfully completed Algebra 1 and 2. The course covers all of the California State Standards for Geometry. Topics include proofs, congruence, similarity, properties of parallel lines cut by a transversal, polygons, circles, perimeter, area, volume, three dimensional figures, special triangles and polygons such as squares, Pythagorean theorem, basic constructions, coordinate geometry, basic right triangle trigonometry, and transformations. ALHS uses the Michael Serra textbook Discovering Geometry.
Advanced Algebra 1 and 2
This course is required for graduation for the class of 2014 and beyond. It is required for admission to the University of California or California State University systems. College eligibility requires a grade of C or better.
Topics included in this course cover all of the state standards. The topics include solving equations and inequalities using absolute value, solving systems of equations in a variety of methods, performing operations on polynomials, complex numbers,, rational expressions, graphs of quadratic functions, logarithms, exponential functions, conic sections, combinations and permutations, probability involving combinations and permutations, binomial theorem, arithmetic and geometric series, inverse functions, and composition of functions.
Statistics and Probability 1 and 2
This is an introductory high school level course to statistics and probability. Students taking this course should have completed at least through Geometry and Advanced Algebra is highly recommended since topics covered in that course are often used in this course.
This course covers all of the standards listed in the California Framework, but covers many other topics in addition to those listed in the Framework. The course tends to have a hands-on approach so that students can generate their own data in many situations. Topics include linear regression, analysis of one variable data, introductory inference, and probability. Teachers often use a variety of sources for the class, including websites for additional enrichment.
Pre-Calculus 1 and 2
Prerequisites for this course are Algebra, Geometry, and Advanced Algebra. Success in this course is very dependent on students having a thorough knowledge of second year algebra and geometry since both are used and built on throughout the course. Students must have teacher recommendation to take this course and scores on CST math subjects are considered. A diagnostic test is given to determine if placement in this course is appropriate.
This course covers all of the traditional trigonometry topics as well as many analytic geometry topics. The standards listed under Trigonometry and Mathematical Analysis in the California Mathematics Framework are addressed in this course as well as almost all the topics in the Linear Algebra listing in the Framework.
Advanced Placement Statistics 1 and 2
This course is the College Board Advanced Placement Statistics course and includes all of the topics listed on the College Board website. The course has four main components: exploratory analysis, probability, experimental design, and inference. Students enrolling in this course must have fairly high reading and writing ability and must have successfully completed through Advanced Algebra (An A or B in that course is recommended). Teacher recommendation and parent approval is required to be eligible to take this course.
Advanced Placement Calculus 1 and 2
Students must complete Precalculus successfully to enroll in this course. Students must also pass a placement test and have teacher recommendation to enroll. This course is very similar to the UC Berkeley course and assumes students have had some background in Analytic Geometry in addition to Trigonometry before entering this course.
The course covers all of the topics in the College Board syllabus for this course as well as a few additional topics. Students taking this course must take the Advanced Placement exam in May of the year they take the course. Students who pass the Advanced Placement exam are then eligible for college credit at many universities.
How to Contact Faculty & Staff
To the left of Faculty/Staff names click on the icon to send a message to that person through Loop Mail. | 677.169 | 1 |
books.google.com - The... graph theory | 677.169 | 1 |
Brand New!
Who doesn't need help with math? Practically everybody… And we have the answer! Basic Shop Math is designed to assist students who need extra help or simply need to refresh their basic math skills. A practical, crystal-clear, Auto Shop centered math course that makes math understandable!
Most instructors start the school year with their first priority, shop safety. Right after that, math is often the next topic.
Basic Shop Math is designed to assist students with the basic math skills most often used in the Auto shop. Whether they need a little extra help or just need to refresh their math skills, this course covers what they need to know to work successfully in the Auto Shop. All math examples are expressed in terms associated with typical auto shop tasks or jobs.
Students who possess a solid understanding of fundamental math skills are more likely to succeed in the auto shop. | 677.169 | 1 |
Once you've mastered the algebra, geometry, and coordinate geometry topics covered in Pre-GED Mathematics, you'll be effectively prepared for this comprehensive review. If you're math-phobic, or just need a quick refresher, put your mind at ease. This unique program helps you prepare for even the most challenging problems on the GED Mathematics Test. Work along as your friendly math professor guides you through many practice examples on the chalkboard, offering step-by-step solutions and clear explanations along the way. This easy-to-follow review course includes sample problems in all of the following areas: number operations; probability; statistics; data analysis; algebra; geometry; and coordinate geometry. You'll also learn about the structure of the exam and the rules for filling in the grids; plus, you'll learn how to judge when to use the calculator – and when not to! | 677.169 | 1 |
Easy Solutions and Visualizations: Exploring a System of Three Equations in Three Variables Using Mathcad
In scientific calculator was considered high tech. Last year I taught Grade 8 algebra at The Park School in Brookline, MA. Students used graphing calculators, laptops, iPhone Apps, Java Applets, etc. Despite the incredible change in available technology, many people presume that the key skills required in an Algebra 1 course remain the same as they were when I took the course.
In recent years, as I have learned more about the use of mathematics in the workplace, I have grown increasingly disenchanted with the traditional Algebra 1 course that I received in 1979 (and I still see in existence in many schools). Topics like factoring, which I was quite good at in Grade 8, seem unconnected to any real vocation. Tools like matrices, which were left out of the course, seem to be ubiquitous in many modern, high paying STEM fields. I can think of two defensible reasons for this fact: (a) matrices can be time consuming to do by hand and (b) even with a graphing calculator they are difficult to visualize in more than the two variables.
Recently I was exploring some problems involving systems of three equations in three variables. I used Mathcad to solve the problems using matrices and then to visualize the solution on a 3-D plot. A portrait of my experiences follows. I propose that in 2010, matrices are a much more accessible topic for school algebra.
Solving a System and Visualizing a Solution
The first problem that I explored had a unique solution. Below I show how I defined matrices F and G to represent the system of three equations. Of course, I also checked the determinant of F to see if the system would yield a solution. Since |F| = 70, I solved the system by multiplication of F-1*G. In pursuing this strategy I asked Mathcad to solve the system using both numeric and symbolic evaluation. Note how the results are equivalent, but the symbolic result is a fraction instead of a decimal. Next, I used the lsolve function to see if Mathcad would replicate the same results and wrote an explanation of my findings. (Ref Fig. 1)
Some may consider the above work to be a black box solution. Mathcad has completed the calculations without showing any work. Yet, I have determined the results efficiently and explained them in writing using Mathcad's math and text capabilities. In addition, below I use Mathcad's graphing capabilities to further illustrate the problem. It is easy to create a 3D quick plot in Mathcad. The graph below clearly illustrates the intersection of the three planes at a single point. (Ref Fig. 2)
For contrast, I also explored a system that did not have a solution. In this case, Mathcad's determinant function helped me to identify that the system was unsolvable. However, the more enlightening exercise was graphing the system in order to observe how the planes intersect in three-space.
When Mathcad evaluated the determinant at zero, I knew that there would not be a solution. The symbolic results confirmed this, identifying the solution as undefined. The lsolve command, however, produced an error (shown in red text beneath my explanation). Thus, working with Mathcad to solve systems of equations offers a number of opportunities for mathematical explanation. First, the user needs to interpret the results intelligently. Do we need to debug the lsolve error or interpret its meaning? Second, there is a rich opportunity for thinking about the system by interpreting the graph. How is this graph different from the previous graph that depicted a solution? Comparing and describing the pair of graphs above can help students ground their understanding of the nature of the solution to a system of three equations in three variables.
Give Matrices their due in the algebra curriculum
Matrices are challenging, but they are really important in applied mathematics – they are a critical STEM topic. Engineers and scientists use matrices to solve challenging problems in many, many dimensions. Mathcad's matrix and graphing tools offer capabilities that can help students' explore matrices early in their school experience so that they are both prepared to use and aware of the importance of matrices. With currently available technologies matrices can be used, explored, and visualized effectively in Algebra 1 class. Systems of equations in three variables need not be avoided any longer. Matrices can be an efficient and powerful way to solve systems, with increased clarity now that we have tools to graph 3D plots. (Ref Fig. 4).... (Show more)(Show less) | 677.169 | 1 |
Obama to students: You will need algebra
Through education, you can also better yourselves in other ways. You learn how to learn – how to think critically and find solutions to unexpected challenges. I remember we used to ask our teachers, "When am I going to need algebra?" Well, you may not have to solve for x to get a good job or be a good parent. That's true. But you will need to think through tough problems. You will need to think on your feet. So, math teachers, you can tell your students that the President says they need algebra.
I really don't get how solving for x in 3x + 17 = 5x + 2 will "help students think through tough problems [in life]" and "think on their feet." I see algebra skills as useful, but not out of context of the types of problems they help us solve. Instead of students learning an algorithm for which almost none of our students will ever get to see a real application; what if we taught students areas of mathematics which had direct application in their lives, and which actually helped them think?
I think that we do need some people who learn algebra in a really deep way, but the type of algebra that people use in their day to day lives is fairly simple, and doesn't take very long to teach, especially if students see the value in what they are learning. Too long people have learned math because someone said they should.
Well Mr. President, I don't think that telling my students that just because you think it is useful will mean they will want to learn it. I'm going to keep focusing on presenting the mathematics I teach in the context of the lives of my students instead, thank you.
Comments
Although Obama's statement begs a lot of question, I don't think, however, that he meant that you actually need algebra to solve problems in life, but I think what he meant was that the level of focus needed to solve tough problems is present in algebra.
It's amazing the type of critical thinking skills you gain if you keep an open mind about math, and that you dedicate just a little bit of your time to at least solve just one challenging problem, such as 3x + 17 = 5x + 2.
I'll give an example that is sort of off topic: at SFU (and maybe in other Universities as well), it's mandatory for software engineering students to take an introductory course to circuit design. Will software engineers ever need to build circuit boards themselves? No, of course not. That's not what they do. Software engineers work through a series of abstraction, and there absolutely no point in trying to dabble with hardware. Even the professor that taught the circuit design course said that you will never need to know all the nitty gritty details of circuit design. But it sure helps to think critically about how data and instructions are passed from the memory system to the CPU. That way, someone trying to come up with an efficient algorithm to solve a problem will be two steps ahead. Not only will the engineer come up with a solution that is algorithmically efficient (that is, having a very low big-order O), but one that is also physically efficient when it comes to CPU usage.
I'm not saying that you should coerce kids to learn algebra. I'm just saying that the level of thinking required to solve an algebraic problem to find x is at the same level of thinking in the real world. Sure by the time that kids grow up, they will forget a lot of the math stuff, but the one thing that they will retain forever is the ability to stay focused to solve problems, and I think that's what counts, and I think that's what Obama meant.
It probably is what he meant, but my question is, can we find something more engaging for kids to work on and end up with the same result, an adult who is willing to struggle through a problem in order to find a solution?
Kids solve problems every day playing videogames, with a laser-like focus. They strategize, plot courses, engage in basic physics. Not much of that skill transfers to real life though. It takes a huge mental jump and a lot of initiative to bridge one domain to another, and most students won't do it on their own.
The same is more or less true with algebra.
Teaching algebra to teach basic problem solving skill works on some level, but it's about as efficient as trying air condition your back yard. Almost none of the effort that students spend learning algebra will go to anything else in life, even indirectly. And because it goes unused, even direct domain specific knowledge will fade away within a couple years.
Meanwhile, Alan Simpson, who the president appointed to reform Social Security, said recently that it was nonsense to say a phrase like "The average life expectancy of someone that reaches 65 is 78." He literally did not understand why the average life expectancy at 65 would be different than at 12, or at birth.
I bet Alan could solve for x just fine. But, as the cognitive science literature will tell you, skills aren't easily transferable/generalizable, and we never bothered to teach him stats, and because of that a lot of old people may go very cold and hungry. So it really does matter, and matter quite a bit what we choose to teach people -- it really is a matter of life and death, and is that quite frequently.
I agree completely. I didn't think of transfer being a problem, but I can definitely see that it is. Students are expected to learn problem solving skills from algebra and then transfer those skills to the ability to fix a flat in their car, or whatever other problems life throws their way. Not going to happen.
Hey David-- another great post. I'd like to see you flesh out your ideas starting from scratch. If you could design a year-long curriculum that accomplished what Algebra advocates claim Algebra accomplishes, but has that missing ingredient of tangible application I'd be interested for sure.
Mike makes a huge point about domain knowledge. Anyone who has ever worked with smart but "non tech savvy" people know how vital specific knowledge is to problem solving. I can't debug Java because I have no domain knowledge. I can't correct spelling on a French test for the same reason. Deep knowledge is almost the whole game when it comes to problem solving. Without broadening the schema of our students, they will be cut-off from the ability to learn deeply and solve problems in most areas. Some educators claim to want to teach problem solving skills at the expense of a broad schema. Never understood why those beliefs still circulate. Maybe it's because they never read a comment like yours! Valuable piece of the puzzle for sure.
Your analogies are good, and I think they point to an important issue in mathematics education. Many math teachers teach nothing but domain knowledge, and produce kids that can't solve real problems. So domain knowledge isn't everything. Similarly, as you point out, student's can't solve any problems if they don't have the tools. So we need to teach both domain knowledge and the problem solving tools students need to be able to move on in their lives. Since we are teaching hardly any of the problem solving skills in mathematics, and ending up with a largely innumerate public, I think we should teach more of the problem solving and less of the domain knowledge.
I'm not sure exactly what that looks like yet. I'm still thinking about it, and think I am not qualified to make that determination alone. I am working with a small team right now of other progressive mathematics educators interested in improving mathematics instruction in the same way.
re: "Many math teachers teach nothing but domain knowledge, and produce kids that can't solve real problems." I think that is essentially what Mike was talking about. There is no real thing as "problem solving skills" there is just a certain amount of domain knowledge needed to solve a specific problem.
Understanding how my lawn mower works is all I really need to be able to know how to solve any problem with it. Knowing how my lawn mower works doesn't help me figure out why my soccer team can't breakdown a defense at midfield. In order to solve my soccer problem, I need to know a lot about soccer. Knowing Algebra doesn't really help me fix that lawn mower either. Understanding Algebra might help me understand more about the relationships between numbers though, which might help me in other areas of math. Understanding how my lawn mower works might help me fix a go-cart or a motorcycle because there are similar patterns and vocabulary and materials involved, meaning, I already have *some* of that domain knowledge or I can more easily add new info to my schema because I have a solid foundation in "motors" or "combustion" or whatever.
"Problem solving skills" is a phrase that doesn't have much real meaning. Asking people to use strategies like "take a break and look at it fresh tomorrow" or "examine your assumptions and see if your assuming something to be true that isn't true" are certainly helpful reminders, but they are a miniscule part of problem solving. "Problem solving ability" essentially = "level of domain knowledge."
I might add that confidence/freedom to act is an important part of the equation, but that *usually* comes with experience as well. Sometimes students are capable of solving a problem, they have the know-how, but they are afraid for some reason. In those cases, domain knowledge is insufficient. I am not saying domain knowledge is everything, but it is an overwhelmingly large piece of the puzzle.
To put it in numbers, maybe 95% of problem solving is simply domain knowledge coupled with freedom to act on that knowledge.
Boy, I can't say I agree with that. The problem with domain level knowledge is that it comes from someone else, and is tied to their understanding of the situation, not yours. The more domain knowledge you learn, the less able you are to think about that domain knowledge outside of the language constructed around it. In other words, being deeply entrenched in domain knowledge is to be entrenched in the ideas of others, which limits your ability to think creatively about the problem.
If we carefully instructed all of our students in the ideologies of environmentalism, for example, whatever lense we chose to share those ideologies would become the lens through which they would examine all new instances of environmental problems. Ask a typical politician about the environment for example, and because they've been so deeply rooted in the economic model of environmentalism, they can't see solutions to problems outside of that domain.
I think your example proves the opposite point though-- asking a politician how to solve problems with our global ecosystem is a bad idea because they have virtually zero domain knowledge. I think climate scientists have a much better chance of finding root causes to environmental problems because they understand the subject better. Plus, hey have a much more developed schema to weave new knowledge into.
The solution they come up with might be something like "we need some way to build taller buildings, or invent new drainage systems or something. They would be poor choices to pick to actually build those things though because they have no domain knowledge. Engineers, on the other hand, have that domain knowledge so they would be better to solve those problems. People can't solve any problem without specific domain knowledge of the problem area. You don't necessarily need to be a master to begin solving problems, but the more knowledge you have the better chance you have to solve the problem. Not only that, but the more knowledge you have, the better chance you have to learn more about the problem on your own in the future.
If someone actually was 95% steeped in the domain knowledge of others, how would they look at the problem from a fresh perspective? Further, achieving such a high level of domain knowledge in students is only possible through a standardization of the curriculum. Do we really want all kids to have the same understanding of the issues relevant to our world? How could we possibly find solutions?
Some of the freshest and best approaches to solving problems have come from amateurs and people outside of the field. Cross-pollination of fields has produced some of the most amazing ideas. The entire field of chaos theory, essentially invented by Lorenz, happened because a mathematician (and a nonstandard one at that) explored an area outside of his original expertise, meteorology.
Not sure what you mean by "95% steeped in the domain knowledge of others."
Are you saying that 7x7=49 to you because you are being influenced by people who knew this before you were born? If someone with no knowledge of multiplication looked at 7x7, they might find a different, better answer?
Totally confused.
How about this-- can you give me at least one example of a problem in math that people can solve without domain knowledge? If I couldn't add, subtract multiply or divide and I didn't know what the number values were or what any formulas were or anything in the domain, what problem could I solve with problem solving skills alone?
Also- can you give an example of a "problem solving skill?" What would you teach someone instead of building their schema? You are advocating sacrificing domain knowledge and increasing problem solving skills, but I don't understand what you mean by that. Can you give me a concrete example?
Now you are being ridiculous, and not reading my post correctly. When did I say that domain knowledge was completely pointless? Obviously I can't solve any problems without some domain knowledge, but to think that you need absolutely every piece of the puzzle in order to solve problems is also completely crazy in my mind. When in life do we actually have every piece of the puzzle?
As for your second question, there are a few things kids need to be able to do to solve problems.
For simple problems, they need to know how to find the answer when they don't know it.
For more complex problems, they need to know how to build a team of people capable of working on the problem.
Students need to be able to communicate what they know in a variety of representations. This means to me, being able to write about the problem, graph it, map it out, create multimedia representations for sharing with an audience, and a whole lot of other representations.
They need to look at "the problem" from different perspectives. What are the cultural implications of this problem? What are the environmental consequences of this solution?
They need to be able to look at a problem and even recognize what they know, what their social circle knows, and what other people know about the problem. This is also called researching the problem.
They need to be able to deconstruct problems into smaller "chewable" pieces. Take a big problem, find the smaller sub-problems, and then distribute those among their community to solve. If there are only a few pieces of problem to chew on, then obviously they can solve it themselves.
David, not being ridiculous. Trying to understand the comment about "domain knowledge of others". You seemed to imply that domain knowledge was something that hurts problem solving. I am guessing what you meant to imply is that people can teach other people misleading things and give them a false sense of having acquired knowledge. I agree that if someone has the wrong answers to questions, those answers will not help them solve problems. We agree on that, if that was what you meant.
If you were saying all knowledge we learn from books, teachers, TV, radio, friends, is somehow wrong . . . I don't know how to respond to that.
re: "to think that you need absolutely every piece of the puzzle in order to solve problems is also completely crazy in my mind." I agree. I don't even think it is possible to know everything there is to know about a subject. The inescapable fact though is that the more we know about something, the better chance we have to solve problems that arise in that domain. If we know a little, we can solve some problems. If we know a lot we can solve many problems. I don't think anyone would debate that. There is a direct relationship between domain knowledge and problem solving ability.
The Lorenz example is a perfect illustration of how important domain knowledge is to problem solving. Lorenz was a mathematician. If he was a janitor or a sportcaster or a basket weaver, the fact that he solved a math/weather problem would be unusual. The fact that he took his domain knowledge of math and then added to it an extensive set of really knowledge from the domain of meteorology, underscores the fact that he was possibly the *most likely* person on earth to uncover chaos theory. He was able to see what others couldn't because he had more knowledge of the domains than others did. Mathematicians don't know weather and meteorologists don't know math well enough. He knew both. His domain knowledge separated him from others.
His story is a great example of why building not just a deep schema, but a *broad* schema is so valuable. When we know a lot about a few different subjects, we can make connections and see things others who lack our knowledge can't see.
I think everything on your list is just another way to emphasize the value of domain knowledge. I think we are actually saying the same thing just using different words.
re: finding answers/building teams-- I am saying we can't solve problems without domain knowledge. You are saying we can, just find people that have that knowledge or find a resource where you can learn it. Same difference to me. Either way, the problem doesn't get solved without domain knowledge. It is the only piece of the problem solving puzzle that is mandatory. I think we both agree on that, no?
re: variety of representations-- not sure what sharing via multimedia has to do with solving a problem. If the specific problem they are solving is "how do I communicate X to others?" then having knowledge of multimedia tools will help them find a solution, but for most problems, multi-media presentation tools are not applicable. Fixing that lawnmower takes knowledge of lawnmowers, not multi-media presentation tools.
My understanding is that the goal is not to teach kids how to get other people to solve their problems, but to help build within them the ability to solve their own problems. teaching kids how to ask for help when they can't solve a problem on their own is great. I lean on people all the time. It doesn't help me learn to be a better problem solver though- it helps the person who solves my problem for me. What helps me become better is learning why the solution they proposed worked/didn't work. That knowledge adds to my understanding in that domain and may help me solve my own problem in the future.
re: Cultural implications and consequences-- these skills assume that the problem solver has or will need to have domain knowledge to come up with answers. Can't know the repercussions of something you don't understand. If I don't understand gravity, the repercussions of space travel on my bones will be a terrible shock to me. If I know about gravity and bones, I will have a better understanding of the consequences.
re: research- Researching is defined as a way to increase your domain knowledge when you don't have enough to solve the problem. I agree this is vital. The more knowledge we have, the more problems we can solve.
I agree with all of this information, excepting that multimedia representations is pretty broad and would include pictures and diagrams, so a lawnmower-fixing-person might absolutely have to be able to explain their problem using a picture, or a diagram to explain exactly where the problem is if they are unable to solve the problem themselves.
Where I think we differ is when this gathering of domain knowledge occurs. Should it occur during school, in an out of context setting, or should it occur when the person actually needs the knowledge? I'd prefer to see people who knew a lot about finding accurate, reliable, and useful information about an area of need, and perhaps a bit less than we do now of people expecting to know everything about a topic when they leave school. In other words, in our current system, I think we spend way too much time front-loading kids with information, in the expectation that they will find it useful, and not nearly enough time spending time actually finding useful information for themselves.
A consequence of this, is that sites like Answers.com and similar junky-user-entered-info-with-lots-of-ads profilerate through the web, and people have very little idea on how to actually find useful information. I was talking about this problem with our school librarian, and she was complaining that kids just do random Google searches to find information, and lack any kind of critical awareness of the reliability of that information. She said that the student at the very least should check to see if the article they are reading has references. So I opened up a textbook, and pointed out that traditional textbooks lack references as well.
I know what your counterargument will be, which is that people who have vast exposure to domain level knowledge can spot the BS more easily than people with little exposure to domain knowledge. However, it's always a trade-off. We can't possible train the kids in everything they could be exposed to, so we might be better off giving them a moderate amount of domain knowledge in a broader range of topics, and focusing on techniques they can use to establish the reliability of new knowledge.
However, there is also value when people make leaps and try to fill in gaps in their existing domain knowledge. Most of the time, they get it wrong, but once in a while they stumble across a different perspective in the gap, and it ends up re-arranging our existing knowledge in an area. It feels to me like your perspective on domain knowledge is that of a static representation of the world, when in fact I see our knowledge over time as having been dynamic. We know the world differently than we used to, our perspective on it has changed. How has that happened? At least occasionally, there are people muddling around and messing around and making assumptions about the world, and at least some of these assumptions change the nature of what we know. While domain knowledge is absolutely important for expanding areas of knowledge, too much of it is a hindrance in terms of finding flaws with existing knowledge.
I agree, it is a trade-off. The tug of war between depth and breadth is tricky.
The deeper the knowledge, the more higher-order thinking skills begin to emerge. Critical thinking, problem solving, etc., all emerge only after a certain tipping point. "Covering" a lot of different material (in a surface way) prevents these higher-order thinking skills from emerging. I think this is the real issue with increasing the importance of fact-based testing in many schools-- standardized test designers value breadth over depth.
The broader the knowledge, the greater the student's ability to learn new things in the future. We can only learn things that connect to our current schema, so leaving things out of the curriculum basically eliminates a student's ability to learn those things later in life. It's not impossible, but it makes it highly unlikely that a student will learn that thing from scratch later in life. This is the reason why millions of Mexican children speak Spanish, but millions of American (or Canadian) adults who want to learn Spanish never do.
In some ways it is a balancing act between higher-order thinking ability and the ability to be a life-long learner. Keeping both of those plates spinning is the difficult part.
Yes, we completely agree on this. Finding the right balance point is the tricky part. Right now, I think we are too far on the content area of the spectrum, but as you point out, we have to be careful not to go too far in the other direction. | 677.169 | 1 |
Formula Editor
Mathematics uses symbols that most text editors can't produce, so we've created the formula editor. This gives teachers the ability to create mathematics formulas directly in the material they are working on. With the formula editor, teachers can embed example formulas and problems directly in assignments, tests or classroom exercises – and learners can also create formulas when showing their answers.
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If you're used to working with a specific formula editor, you may be able to install it as a plug-in directly into the itslearning text editor. If you can't see it the list of available plug-ins, ask the vendor to install it. | 677.169 | 1 |
Jacumba Precalculus have worked with university and industry scientists analyzing data from simple and complex experiments on people and animals. I have published papers. I have written grant proposals and IRB submissionsImprove your chances at succeeding in school and being hired for a job by improving your grammar. Excel is a very useful program for a variety of reasons. A few examples are making graphs, performing multiple math computations at once, and manipulating data. | 677.169 | 1 |
for College Students
A 4-color hardback book w/complete text-specific instructor and student print/"enhanced" media supplement package. AMATYC/NCTM Standards of Content ...Show synopsisA 4-color hardback book w/complete text-specific instructor and student print/"enhanced" media supplement package. AMATYC/NCTM Standards of Content and Pedagogy integrated in current, researched, real-world Applications, Technology Boxes, Discover For Yourself Boxes and extensively revised Exercise Sets. Early introduction and heavy "emphasis on modeling" demonstrates and utilizes the concepts of introductory algebra to help students solve problems as well as develop critical thinking skills. One-page Chapter Projects (which may be assigned as collaborative projects or extended applications) conclude each chapter and include references to related Web sites for further student exploration. The influence of mathematics in fine art "and their relationships" are explored in applications and chapter openers to help students visualize mathematical concepts and recognize the beauty in mathUnfortunately, this text was required for my class. I got to use the Lial series for PreAlgebra, and I will get to for Intermediate as well. They are much better for those who need more examples, description, worked problems, etc. This one assumes you know | 677.169 | 1 |
Applied mathematics (minor subject)
An applied mathematician works with solving problems which lie beyond the realm of regular mathematics, and therefore Applied mathematics is the programme for you if you want to work with mathematics and still keep in mind how mathematics can be of use in your future career.
Applied mathematics is not a particular branch of mathematics, but rather a way of working with mathematics. In the Applied mathematics programme, you will learn to create models of and solve problems from the practical world by utilising advanced mathematical tools.
The Applied mathematics programme gives you the opportunity to solve complex problems and to create new insight and recognition. Applied mathematics is for you if you want to learn to utilise advanced mathematical tools and computers to model, analyse and solve complex problems in the business sector or in research. An applied mathematician masters and is able to further develop the mathematical tools which have contributed to the development of the modern society of information. | 677.169 | 1 |
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