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637 | Create a statistical question of your own that you could answer by surveying people in your class. Write your question here: The Utah Middle School Math Project | Your answer can be any statistical question, a question that asks for multiple data points that may vary. Example question: How old were you when you lost your first tooth? | ###Write an explanation to this problem:
Create a statistical question of your own that you could answer by surveying people in your class. Write your question here: The Utah Middle School Math Project
###Explanation:
Your answer can be any statistical question, a question that asks for multiple data points that may vary. Example question: How old were you when you lost your first tooth? |
536 | In a 90˚ rotation, what happens to the slopes of the corresponding segments of the image and pre-image? How does the coordinate rule show this? In a 180˚ rotation, what happens to the slopes of the corresponding segments of the image and pre-image? How does the coordinate rule show this? The Utah Middle School Math Project | In a 90Ë rotation, the slopes of corresponding segments are opposite reciprocals or the product of the slopes is 1. The coordinate rule for a 90Ë rotation is x, y rarr; y, -x. The x- and y-coordinates change places therefore the slope is inverted. The sign is changed on only the x-coordinates therefore changing the sign of the slope. In a 180Ë rotation, the slopes of corresponding segments are the same. The coordinate rule for a 180Ë rotation is x, y rarr; -x, -y. The signs of both the x- and y-coordinates change therefore the net result is that the slope does not change. | ###Write an explanation to this problem:
In a 90˚ rotation, what happens to the slopes of the corresponding segments of the image and pre-image? How does the coordinate rule show this? In a 180˚ rotation, what happens to the slopes of the corresponding segments of the image and pre-image? How does the coordinate rule show this? The Utah Middle School Math Project
###Explanation:
In a 90Ë rotation, the slopes of corresponding segments are opposite reciprocals or the product of the slopes is 1. The coordinate rule for a 90Ë rotation is x, y rarr; y, -x. The x- and y-coordinates change places therefore the slope is inverted. The sign is changed on only the x-coordinates therefore changing the sign of the slope. In a 180Ë rotation, the slopes of corresponding segments are the same. The coordinate rule for a 180Ë rotation is x, y rarr; -x, -y. The signs of both the x- and y-coordinates change therefore the net result is that the slope does not change. |
236 | A textbook has 428 pages numbered in order starting with 1. You flip to a random page in the book in a way that it is equally likely to stop at any of the pages. What is the sample space for this experiment? copied for free from openupresources.org | Pages 1 through 428 are the sample space for this experiment because you can pick any of those pages. | ###Write an explanation to this problem:
A textbook has 428 pages numbered in order starting with 1. You flip to a random page in the book in a way that it is equally likely to stop at any of the pages. What is the sample space for this experiment? copied for free from openupresources.org
###Explanation:
Pages 1 through 428 are the sample space for this experiment because you can pick any of those pages. |
88 | Write a pair of numbers whose LCM is 60. The Utah Middle School Math Project | There are a few different numbers that have multiples of 15. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. From this list, the only pairs of numbers that have 60 as their least common multiple are: Any of the factors and 60 4 and 30 12 and 30 20 and 30 3 and 20 6 and 20 12 and 20 15 and 20 4 and 15 12 and 15 5 and 12 10 and 12 | ###Write an explanation to this problem:
Write a pair of numbers whose LCM is 60. The Utah Middle School Math Project
###Explanation:
There are a few different numbers that have multiples of 15. The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. From this list, the only pairs of numbers that have 60 as their least common multiple are: Any of the factors and 60 4 and 30 12 and 30 20 and 30 3 and 20 6 and 20 12 and 20 15 and 20 4 and 15 12 and 15 5 and 12 10 and 12 |
758 | The sentence is stated incorrectly. Rewrite the sentence to correctly describe each situation. The temperature is -10 degrees Fahrenheit below zero. | The temperature is 10 degrees below zero on the Farenheight scale. | ###Write an explanation to this problem:
The sentence is stated incorrectly. Rewrite the sentence to correctly describe each situation. The temperature is -10 degrees Fahrenheit below zero.
###Explanation:
The temperature is 10 degrees below zero on the Farenheight scale. |
117 | During a field trip, 60 students are put into equal-sized groups. Describe two ways to interpret 60 / 5 in this context. | If there are 5 groups, how many students would be in each group? If there are 5 students in each group, how many groups would there be? | ###Write an explanation to this problem:
During a field trip, 60 students are put into equal-sized groups. Describe two ways to interpret 60 / 5 in this context.
###Explanation:
If there are 5 groups, how many students would be in each group? If there are 5 students in each group, how many groups would there be? |
125 | Describe the transformation graph shift that occurs in the following situation. Use words like translation, reflection, and rotation. The slope is increased or decreased while the y-intercept is held constant. The Utah Middle School Math Project | If the slope is increased or decreased while the y-intercept is held constant, then we know the only thing that is changing about the line is its steepness. If the steepness is changing, but the y-intercept remains the same, this transformation is described as a rotation about the y-intercept. | ###Write an explanation to this problem:
Describe the transformation graph shift that occurs in the following situation. Use words like translation, reflection, and rotation. The slope is increased or decreased while the y-intercept is held constant. The Utah Middle School Math Project
###Explanation:
If the slope is increased or decreased while the y-intercept is held constant, then we know the only thing that is changing about the line is its steepness. If the steepness is changing, but the y-intercept remains the same, this transformation is described as a rotation about the y-intercept. |
760 | A credit of 35 and a debit of 40 are applied to your bank account. What is an appropriate scale to graph a credit of 35 and a debit of 40? | Sample response: I would count by 5's because both numbers are multiples of 5 | ###Write an explanation to this problem:
A credit of 35 and a debit of 40 are applied to your bank account. What is an appropriate scale to graph a credit of 35 and a debit of 40?
###Explanation:
Sample response: I would count by 5's because both numbers are multiples of 5 |
289 | In an archery competition, scores for each round are calculated by averaging the distance of 3 arrows from the center of the target. An archer has a mean distance of 1.6 inches and a MAD distance of 1.3 inches in the first round. In the second round, the archer's arrows are farther from the center but are more consistent. What values for the mean and MAD would fit this description for the second round? Explain your reasoning. | In the second round, the archer's arrows are farther from the center but are more consistent. If the mean for the first round was 1.6 inches from the center, this number would be lower for this round. So any number less than 1.6 inches would be closer to the center. The fact that they are more consistent lets us know that the MAD is smaller than what it was. If the MAD for the first round is 1.3, then the MAD for the second round would be less than 1.3. | ###Write an explanation to this problem:
In an archery competition, scores for each round are calculated by averaging the distance of 3 arrows from the center of the target. An archer has a mean distance of 1.6 inches and a MAD distance of 1.3 inches in the first round. In the second round, the archer's arrows are farther from the center but are more consistent. What values for the mean and MAD would fit this description for the second round? Explain your reasoning.
###Explanation:
In the second round, the archer's arrows are farther from the center but are more consistent. If the mean for the first round was 1.6 inches from the center, this number would be lower for this round. So any number less than 1.6 inches would be closer to the center. The fact that they are more consistent lets us know that the MAD is smaller than what it was. If the MAD for the first round is 1.3, then the MAD for the second round would be less than 1.3. |
238 | A textbook has 428 pages numbered in order starting with 1. You flip to a random page in the book in a way that it is equally likely to stop at any of the pages. What is the sample space for this experiment? copied for free from openupresources.org | Pages 1 through 428 are the sample space for this experiment, since you can pick any of those pages. | ###Write an explanation to this problem:
A textbook has 428 pages numbered in order starting with 1. You flip to a random page in the book in a way that it is equally likely to stop at any of the pages. What is the sample space for this experiment? copied for free from openupresources.org
###Explanation:
Pages 1 through 428 are the sample space for this experiment, since you can pick any of those pages. |
0 | From 2010 to 2011, the population of Queens increased by 16,075. Brooklyn's population increased by 11,870 more than the population increase of Queens. Estimate the total combined population increase of Queens and Brooklyn from 2010 to 2011. Round the addends to estimate | The problem gives information about the increase in the population from the years 2010 to 2011. Queens increased by 16,075 Brooklyn's population increased by 11, 870 more than Queens. This means figuring out Brooklyn's population involves adding 11 870 + 16,075 = 27,945 The problem is asking you to estimate the total combined population increase of Queens and Brooklyn. 16,075 + 27, 945 You can estimate the total population by rounding up to the nearest 1000. If below 500, round down, if above 500 round up. 16,000 +30,000 = 46.000 | ###Write an explanation to this problem:
From 2010 to 2011, the population of Queens increased by 16,075. Brooklyn's population increased by 11,870 more than the population increase of Queens. Estimate the total combined population increase of Queens and Brooklyn from 2010 to 2011. Round the addends to estimate
###Explanation:
The problem gives information about the increase in the population from the years 2010 to 2011. Queens increased by 16,075 Brooklyn's population increased by 11, 870 more than Queens. This means figuring out Brooklyn's population involves adding 11 870 + 16,075 = 27,945 The problem is asking you to estimate the total combined population increase of Queens and Brooklyn. 16,075 + 27, 945 You can estimate the total population by rounding up to the nearest 1000. If below 500, round down, if above 500 round up. 16,000 +30,000 = 46.000 |
844 | Each of these is a pair of equivalent ratios. For each pair, explain why they are equivalent ratios or draw a representation that shows why they are equivalent ratios. 600 : 450 and 60:45 | To get from 600:450 to 60:45 you would divide by the same constant of 10. 600 divided by 10= 60 450 divided by 10=45 Because they are being divided by the same constant, they are equivalent ratios. | ###Write an explanation to this problem:
Each of these is a pair of equivalent ratios. For each pair, explain why they are equivalent ratios or draw a representation that shows why they are equivalent ratios. 600 : 450 and 60:45
###Explanation:
To get from 600:450 to 60:45 you would divide by the same constant of 10. 600 divided by 10= 60 450 divided by 10=45 Because they are being divided by the same constant, they are equivalent ratios. |
431 | Traveling between countries means exchanging currencies money. Look up the current exchange rate for Europe or replace Euros with the currency of a country of your choice. How many Euros would you receive for 1 dollar? The Utah Middle School Math Project | Set up the ratio according the exchange rate and find the unit rate to determine the solution. | ###Write an explanation to this problem:
Traveling between countries means exchanging currencies money. Look up the current exchange rate for Europe or replace Euros with the currency of a country of your choice. How many Euros would you receive for 1 dollar? The Utah Middle School Math Project
###Explanation:
Set up the ratio according the exchange rate and find the unit rate to determine the solution. |
743 | The sentence is stated incorrectly. Rewrite the sentence to correctly describe each situation. The temperature is -22 degrees Celsius below zero. | We don't say that a negative temperature is also below zero. In this case, we can either say the temperature is -22deg;C or we can say the temperature is 22 degrees below zero Celsius. | ###Write an explanation to this problem:
The sentence is stated incorrectly. Rewrite the sentence to correctly describe each situation. The temperature is -22 degrees Celsius below zero.
###Explanation:
We don't say that a negative temperature is also below zero. In this case, we can either say the temperature is -22deg;C or we can say the temperature is 22 degrees below zero Celsius. |
395 | Choose either the three sides condition or the two sides and included angle condition, and explain why the condition determines a unique triangle. | When we are given three sides that satisfy the triangle inequality theorem, there is only one way to draw the triangle. After drawing one length, use the other two lengths to draw circles with the lengths as the radii of each circle, centered at either end of the segment that we drew at the beginning. Regardless of which order you draw the segments in, there is only one unique triangle that can be drawn. When we draw a triangle with two side lengths with the included angle, there is only one way to draw the triangle. After drawing the angle and marking off the two side lengths on the rays of the angle, there is only one possible place to position the third side of the triangle, which also determines the two remaining angle measures of the triangle. Therefore, the two sides and included angle condition determines a unique triangle. | ###Write an explanation to this problem:
Choose either the three sides condition or the two sides and included angle condition, and explain why the condition determines a unique triangle.
###Explanation:
When we are given three sides that satisfy the triangle inequality theorem, there is only one way to draw the triangle. After drawing one length, use the other two lengths to draw circles with the lengths as the radii of each circle, centered at either end of the segment that we drew at the beginning. Regardless of which order you draw the segments in, there is only one unique triangle that can be drawn. When we draw a triangle with two side lengths with the included angle, there is only one way to draw the triangle. After drawing the angle and marking off the two side lengths on the rays of the angle, there is only one possible place to position the third side of the triangle, which also determines the two remaining angle measures of the triangle. Therefore, the two sides and included angle condition determines a unique triangle. |
512 | Solve the following problem by writing and solving a system of equations. Use any method you wish to solve. Write your answer in a complete sentence. Blake has some quarters and dimes. He has 20 coins worth a total of 2.90. How many of each type of coin does he have? The Utah Middle School Math Project | ð + ð = 20 where q is the number of quarters and d is the number of dimes 0.25ð + 0.10ð = 2.90 Blake has 14 dimes and 6 quartersEnter your explanation text here. | ###Write an explanation to this problem:
Solve the following problem by writing and solving a system of equations. Use any method you wish to solve. Write your answer in a complete sentence. Blake has some quarters and dimes. He has 20 coins worth a total of 2.90. How many of each type of coin does he have? The Utah Middle School Math Project
###Explanation:
ð + ð = 20 where q is the number of quarters and d is the number of dimes 0.25ð + 0.10ð = 2.90 Blake has 14 dimes and 6 quartersEnter your explanation text here. |
126 | Describe the transformation graph shift that occurs in the following situation. Use words like translation, reflection, and rotation. The slope is increased or decreased while the y-intercept is held constant. The Utah Middle School Math Project | The slope is the rate of change of the line and determines the steepness of the line. The y-intercept is the point where the line intersects with the y-axis. If the slope is increased or decreased while the y-intercept is held constant, then we know the only thing that is changing about the line is its steepness. If the steepness is changing, but the y-intercept remains the same, this transformation is described as a rotation about the y-intercept. | ###Write an explanation to this problem:
Describe the transformation graph shift that occurs in the following situation. Use words like translation, reflection, and rotation. The slope is increased or decreased while the y-intercept is held constant. The Utah Middle School Math Project
###Explanation:
The slope is the rate of change of the line and determines the steepness of the line. The y-intercept is the point where the line intersects with the y-axis. If the slope is increased or decreased while the y-intercept is held constant, then we know the only thing that is changing about the line is its steepness. If the steepness is changing, but the y-intercept remains the same, this transformation is described as a rotation about the y-intercept. |
278 | 1. Which one doesn't belong? Explain your reasoning. 2 * 2 * 2 * 2 16 24 4 * 2 copied for free from openupresources.org | Answers may vary here is an example: Problem D because it equals 8 while the other expressions equal 16. | ###Write an explanation to this problem:
1. Which one doesn't belong? Explain your reasoning. 2 * 2 * 2 * 2 16 24 4 * 2 copied for free from openupresources.org
###Explanation:
Answers may vary here is an example: Problem D because it equals 8 while the other expressions equal 16. |
662 | Boris is setting up an exercise schedule. For five days each week, he would like to play a sport for 30 minutes, stretch for 5 minutes, and lift weights for 10 minutes. Write two different expressions that could be used to represent the total number of minutes he will exercise in five days. The Utah Middle School Math Project | To find the total number of minutes Boris exercises over the course of five days, add the number of minutes he exercises each day and multiplies the total by 5. This can be expressed in multiple ways. 1. 530 + 5 + 10 2. 530 + 55 + 510 | ###Write an explanation to this problem:
Boris is setting up an exercise schedule. For five days each week, he would like to play a sport for 30 minutes, stretch for 5 minutes, and lift weights for 10 minutes. Write two different expressions that could be used to represent the total number of minutes he will exercise in five days. The Utah Middle School Math Project
###Explanation:
To find the total number of minutes Boris exercises over the course of five days, add the number of minutes he exercises each day and multiplies the total by 5. This can be expressed in multiple ways. 1. 530 + 5 + 10 2. 530 + 55 + 510 |
116 | Give two numbers with an absolute value greater than three. The Utah Middle School Math Project | The absolute value of a number is always positive. This question is asking: |x| > 3 We could pick a number that is already greater than 3, like 5, since taking the absolute value of a positive number is just the number itself. We could also pick a number that is negative, but whose opposite is greater than 3. One possible example is -5. |-5| = 5, and 5 is greater than 3, so -5 is another possible answer. Any number that is greater than 3 or less than -3 is a possible answer. | ###Write an explanation to this problem:
Give two numbers with an absolute value greater than three. The Utah Middle School Math Project
###Explanation:
The absolute value of a number is always positive. This question is asking: |x| > 3 We could pick a number that is already greater than 3, like 5, since taking the absolute value of a positive number is just the number itself. We could also pick a number that is negative, but whose opposite is greater than 3. One possible example is -5. |-5| = 5, and 5 is greater than 3, so -5 is another possible answer. Any number that is greater than 3 or less than -3 is a possible answer. |
228 | An experiment will produce one of ten different outcomes with equal probability for each. Why would using a standard number cube to simulate the experiment be a bad choice? | By standard number cube, the problem is referring to an average die. A standard number cube only has 6 sides and therefore only 6 possible outcomes. For this experiment, we need 10 possible outcomes. The standard number cue could not produce all 10 possibilities from the experiment. | ###Write an explanation to this problem:
An experiment will produce one of ten different outcomes with equal probability for each. Why would using a standard number cube to simulate the experiment be a bad choice?
###Explanation:
By standard number cube, the problem is referring to an average die. A standard number cube only has 6 sides and therefore only 6 possible outcomes. For this experiment, we need 10 possible outcomes. The standard number cue could not produce all 10 possibilities from the experiment. |
599 | Replace the ? in the fraction ?/12 with a number that would make the fraction greater than 2/3 but less than 1. The Utah Middle School Math Project | If you multiply both parts of 2/3 by 4, you get 8/12. If you turn 1 into a fraction over 12, you get 12/12. Thus, you need to choose a number that makes ?/12 greater than 8/12 but less than 12/12. The ? needs to be greater than 8 but less than 12. The only integers that are greater than 8 but less than 12 are 9, 10, and 11. | ###Write an explanation to this problem:
Replace the ? in the fraction ?/12 with a number that would make the fraction greater than 2/3 but less than 1. The Utah Middle School Math Project
###Explanation:
If you multiply both parts of 2/3 by 4, you get 8/12. If you turn 1 into a fraction over 12, you get 12/12. Thus, you need to choose a number that makes ?/12 greater than 8/12 but less than 12/12. The ? needs to be greater than 8 but less than 12. The only integers that are greater than 8 but less than 12 are 9, 10, and 11. |
818 | A classroom has a length of 30 ft. and a width of 20 ft. The flooring is to be replaced by tiles. If the tiles are square, 18 in. on a side. If these are to be installed, how many must be ordered? Explain your answer, and how you got it. | Sample response: An even number of tiles fit on the30 foot length of the room 20 tiles, but the width requires 13 1/3 tiles. This accounts for a 20 tile by 13 tile array. 20 * 13 = 260 260 tiles need to be ordered. The remaining area is 30 ft. * 0.5 ft. 20 * 1/3 tile Since 20 of the 1/3 tiles are needed, 7 additional tiles must be cut to form 21/3. 20 of these will be used with 1/3 of 1 tile left over. Using the same logic as above, some students may correctly say they will need 280 tiles. | ###Write an explanation to this problem:
A classroom has a length of 30 ft. and a width of 20 ft. The flooring is to be replaced by tiles. If the tiles are square, 18 in. on a side. If these are to be installed, how many must be ordered? Explain your answer, and how you got it.
###Explanation:
Sample response: An even number of tiles fit on the30 foot length of the room 20 tiles, but the width requires 13 1/3 tiles. This accounts for a 20 tile by 13 tile array. 20 * 13 = 260 260 tiles need to be ordered. The remaining area is 30 ft. * 0.5 ft. 20 * 1/3 tile Since 20 of the 1/3 tiles are needed, 7 additional tiles must be cut to form 21/3. 20 of these will be used with 1/3 of 1 tile left over. Using the same logic as above, some students may correctly say they will need 280 tiles. |
847 | List three different combinations of coins, each with a value of 30% of a dollar. | For the value to equal 30% of a dollar, you would need to have the coins equal to 0.30 Combo 1: 1 dime, 3 nickels, 5 pennies Combo 2: 1 quarter, 5 pennies Combo 3: 2 dimes, 1 nickel, 5 pennies | ###Write an explanation to this problem:
List three different combinations of coins, each with a value of 30% of a dollar.
###Explanation:
For the value to equal 30% of a dollar, you would need to have the coins equal to 0.30 Combo 1: 1 dime, 3 nickels, 5 pennies Combo 2: 1 quarter, 5 pennies Combo 3: 2 dimes, 1 nickel, 5 pennies |
274 | In your own words, what is the greatest common factor of two whole numbers? How can you find it? | The greatest common factor of two whole numbers is a factor of both numbers. It is also the largest factor that they have in common. You find it by listing the common factors of both numbers and find the greatest common factor. | ###Write an explanation to this problem:
In your own words, what is the greatest common factor of two whole numbers? How can you find it?
###Explanation:
The greatest common factor of two whole numbers is a factor of both numbers. It is also the largest factor that they have in common. You find it by listing the common factors of both numbers and find the greatest common factor. |
318 | Suppose you wanted to graph the equation y = -4x - 1. Describe the steps you would take to draw the graph. | This equation is in slope-intercept form: y = mx + b This means that we know the y-intercept: 0,b 0,-1 We also know the slope of the line: -4. This means that for every 1 unit we go to the right, the line goes down 4 units. Start with the intercept 0,-1, and use the slope of -4 to move down 1 and 4 to the right or up 1 and 4 to the left to find other points. Or, find two or more solutions to the equation by substituting in a number for x and solving for y | ###Write an explanation to this problem:
Suppose you wanted to graph the equation y = -4x - 1. Describe the steps you would take to draw the graph.
###Explanation:
This equation is in slope-intercept form: y = mx + b This means that we know the y-intercept: 0,b 0,-1 We also know the slope of the line: -4. This means that for every 1 unit we go to the right, the line goes down 4 units. Start with the intercept 0,-1, and use the slope of -4 to move down 1 and 4 to the right or up 1 and 4 to the left to find other points. Or, find two or more solutions to the equation by substituting in a number for x and solving for y |
583 | Simplify the following expression using two different methods. 710 + 3 First, show your work for each method. The Utah Middle School Math Project | One method is to first distribute the 7 to the terms within the parentheses: 710 + 3 = 710 + 73 = 70 + 21 = 91 Another method is to first add the terms in the parentheses and then distribute the 7: 710 + 3 = 713 = 91 | ###Write an explanation to this problem:
Simplify the following expression using two different methods. 710 + 3 First, show your work for each method. The Utah Middle School Math Project
###Explanation:
One method is to first distribute the 7 to the terms within the parentheses: 710 + 3 = 710 + 73 = 70 + 21 = 91 Another method is to first add the terms in the parentheses and then distribute the 7: 710 + 3 = 713 = 91 |
144 | List the sample space the available options for the chance experiment given. Flipping a coin | Sample space is the range of values for a random variable. Flipping a coin has two possibilities: Heads or tails | ###Write an explanation to this problem:
List the sample space the available options for the chance experiment given. Flipping a coin
###Explanation:
Sample space is the range of values for a random variable. Flipping a coin has two possibilities: Heads or tails |
610 | Write a situation that can be represented by the rational number. 2 The Utah Middle School Math Project | Any situation that could be represented by the rational number 2 just needs to involve words like deposit, increase, profit, and so on that would indicate a positive number. One example of a correct situation would be: 2 miles above sea level | ###Write an explanation to this problem:
Write a situation that can be represented by the rational number. 2 The Utah Middle School Math Project
###Explanation:
Any situation that could be represented by the rational number 2 just needs to involve words like deposit, increase, profit, and so on that would indicate a positive number. One example of a correct situation would be: 2 miles above sea level |
539 | Write an ordered pair that would be on the x-axis when graphed. The Utah Middle School Math Project | Any ordered pair x, y where y = 0 is on the x-axis. Examples: 7, 0 -7, 0 0, 0 | ###Write an explanation to this problem:
Write an ordered pair that would be on the x-axis when graphed. The Utah Middle School Math Project
###Explanation:
Any ordered pair x, y where y = 0 is on the x-axis. Examples: 7, 0 -7, 0 0, 0 |
784 | Name the parts of the expression. 14b | The parts of the expression are 14 and b. 14 is the coefficient, b is the variable, and 14b is a term and the product of 14*b. | ###Write an explanation to this problem:
Name the parts of the expression. 14b
###Explanation:
The parts of the expression are 14 and b. 14 is the coefficient, b is the variable, and 14b is a term and the product of 14*b. |
618 | Write a situation that can be represented by the rational number. -15,000 The Utah Middle School Math Project | Any situation that could be represented by the rational number -15,000 just needs to involve words like withdrawal, decrease, loss, and so on that would indicate a negative number. One example of a correct situation would be: 15,000 miles below sea level | ###Write an explanation to this problem:
Write a situation that can be represented by the rational number. -15,000 The Utah Middle School Math Project
###Explanation:
Any situation that could be represented by the rational number -15,000 just needs to involve words like withdrawal, decrease, loss, and so on that would indicate a negative number. One example of a correct situation would be: 15,000 miles below sea level |
369 | A company that manufactures gift boxes wants to know how many different-sized boxes having a volume of 50 cubic centimeters it can make if the dimensions must be whole centimeters. List all the possible whole number dimensions for the box. | Choice 1: 1 cm x 1 cm x 50 cm Choice 2: 1 cm x 2 cm x 25 cm Choice 3: 1 cm x 5 cm x 10 cm Choice 4: 2 cm x 5 cm x 5 cm | ###Write an explanation to this problem:
A company that manufactures gift boxes wants to know how many different-sized boxes having a volume of 50 cubic centimeters it can make if the dimensions must be whole centimeters. List all the possible whole number dimensions for the box.
###Explanation:
Choice 1: 1 cm x 1 cm x 50 cm Choice 2: 1 cm x 2 cm x 25 cm Choice 3: 1 cm x 5 cm x 10 cm Choice 4: 2 cm x 5 cm x 5 cm |
268 | A type of green paint is made by mixing 2 cups of yellow with 3.5 cups of blue. Find a mixture that will make the same shade of green but a smaller amount. | To get the same shade of green, we need the same ratio of yellow to blue. The green paint is made with 2 cups of yellow and 3.5 cups of blue, so the ratio is 2:3.5. We can divide both sides of the ratio by 2 and get the equivalent ratio of 1:1.75, meaning that for every 1 cup of yellow we use 1.75 cups of blue. Example mixture: 1 yellow, 1.75 blue | ###Write an explanation to this problem:
A type of green paint is made by mixing 2 cups of yellow with 3.5 cups of blue. Find a mixture that will make the same shade of green but a smaller amount.
###Explanation:
To get the same shade of green, we need the same ratio of yellow to blue. The green paint is made with 2 cups of yellow and 3.5 cups of blue, so the ratio is 2:3.5. We can divide both sides of the ratio by 2 and get the equivalent ratio of 1:1.75, meaning that for every 1 cup of yellow we use 1.75 cups of blue. Example mixture: 1 yellow, 1.75 blue |
797 | 7. 38% of e is 190. a. Write an equation to help you find the value of each variable. copied for free from openupresources.org | 38% of e means to multiply 38% and e. Is means equal to. In order to multiply by a percent, you will need to change the percent to a decimal first. 0.38e=190 | ###Write an explanation to this problem:
7. 38% of e is 190. a. Write an equation to help you find the value of each variable. copied for free from openupresources.org
###Explanation:
38% of e means to multiply 38% and e. Is means equal to. In order to multiply by a percent, you will need to change the percent to a decimal first. 0.38e=190 |
307 | In a video game, the chance of rain each day is always 30%. At the beginning of each day in the video game, the computer generates a random integer between 1 and 50. Explain how you could use this number to simulate the weather in the video game. | Sample Answer: If the number is between 1 and 15, the video game should create a rainy day. If the number is between 16 and 50, the video game should not create a rainy day. | ###Write an explanation to this problem:
In a video game, the chance of rain each day is always 30%. At the beginning of each day in the video game, the computer generates a random integer between 1 and 50. Explain how you could use this number to simulate the weather in the video game.
###Explanation:
Sample Answer: If the number is between 1 and 15, the video game should create a rainy day. If the number is between 16 and 50, the video game should not create a rainy day. |
423 | Tickets for a matinee are 5 for children and 8 for adults. The theater sold a total of 142 tickets for one matinee. Ticket sales were 890. How many of each type of ticket did the theater sell? First define your unknowns. The Utah Middle School Math Project | Explain what your variables will represent in your problem. For example: c = number of children tickets a = number of adult tickets | ###Write an explanation to this problem:
Tickets for a matinee are 5 for children and 8 for adults. The theater sold a total of 142 tickets for one matinee. Ticket sales were 890. How many of each type of ticket did the theater sell? First define your unknowns. The Utah Middle School Math Project
###Explanation:
Explain what your variables will represent in your problem. For example: c = number of children tickets a = number of adult tickets |
310 | Rectangle A has length 12 and width 8. Rectangle B has length 15 and width 10. Rectangle C has length 30 and width 15. Explain how you know that Rectangle C is not a scaled copy of Rectangle B. | To get a scaled copy, you have to be multiplying all of the dimensions of the original figure or shape by the same scale factor. In this case, the original figure is Rectangle B. Rectangle B measures 15 by 10. Rectangle C measures 30 by 15. We can see that Rectangle C's length is twice Rectangle B's length. BUT Rectangle C's width is not twice Rectangle B's width. This means Rectangle C cannot be a scaled copy of Rectangle B. | ###Write an explanation to this problem:
Rectangle A has length 12 and width 8. Rectangle B has length 15 and width 10. Rectangle C has length 30 and width 15. Explain how you know that Rectangle C is not a scaled copy of Rectangle B.
###Explanation:
To get a scaled copy, you have to be multiplying all of the dimensions of the original figure or shape by the same scale factor. In this case, the original figure is Rectangle B. Rectangle B measures 15 by 10. Rectangle C measures 30 by 15. We can see that Rectangle C's length is twice Rectangle B's length. BUT Rectangle C's width is not twice Rectangle B's width. This means Rectangle C cannot be a scaled copy of Rectangle B. |
567 | Give two numbers with an absolute value greater than three. The Utah Middle School Math Project | The absolute value of a number is always positive. This question is asking: |x| > 3 We could pick a number that is already greater than 3, like 5, since taking the absolute value of a positive number is just the number itself. We could also pick a number that is negative, but whose opposite is greater than 3. One possible example is -5. |-5| = 5, and 5 is greater than 3, so -5 is another possible answer. Any number that is greater than 3 or less than -3 is a possible answer. | ###Write an explanation to this problem:
Give two numbers with an absolute value greater than three. The Utah Middle School Math Project
###Explanation:
The absolute value of a number is always positive. This question is asking: |x| > 3 We could pick a number that is already greater than 3, like 5, since taking the absolute value of a positive number is just the number itself. We could also pick a number that is negative, but whose opposite is greater than 3. One possible example is -5. |-5| = 5, and 5 is greater than 3, so -5 is another possible answer. Any number that is greater than 3 or less than -3 is a possible answer. |
46 | These three points form a horizontal line: -3.5,4, 0,4, and 6.2,4. Name two additional points that fall on this line. | Since the points are in a horizontal line, the y value would always be the same. So, any point with a y value of 4 would work. 2 points that could work would be 9,4 and -100, 4. | ###Write an explanation to this problem:
These three points form a horizontal line: -3.5,4, 0,4, and 6.2,4. Name two additional points that fall on this line.
###Explanation:
Since the points are in a horizontal line, the y value would always be the same. So, any point with a y value of 4 would work. 2 points that could work would be 9,4 and -100, 4. |
349 | In the following problem, a common error has been made when simplifying the expression. Identify the mistake, explain it, and simplify the expression correctly. 5x+ 4x - x 9x Explain the mistake The Utah Middle School Math Project | Notice that when combining like terms, the coefficients were added wrong. 5x + 4x - x x5 + 4 - 1 x8 This expression simplifies to 8x, which does not equal 9x. | ###Write an explanation to this problem:
In the following problem, a common error has been made when simplifying the expression. Identify the mistake, explain it, and simplify the expression correctly. 5x+ 4x - x 9x Explain the mistake The Utah Middle School Math Project
###Explanation:
Notice that when combining like terms, the coefficients were added wrong. 5x + 4x - x x5 + 4 - 1 x8 This expression simplifies to 8x, which does not equal 9x. |
371 | Each of these is a pair of equivalent ratios. For each pair, explain why they are equivalent ratios or draw a representation that shows why they are equivalent ratios. 600 : 450 and 60:45 | If we divide both numbers in the first ratio by 10, it will equal the second ratio, showing that the two are equivalent. Ratio 1 Ratio 2 / by 1st number 2nd number 1st number 2nd number 600 450 60 45 10 60 45 | ###Write an explanation to this problem:
Each of these is a pair of equivalent ratios. For each pair, explain why they are equivalent ratios or draw a representation that shows why they are equivalent ratios. 600 : 450 and 60:45
###Explanation:
If we divide both numbers in the first ratio by 10, it will equal the second ratio, showing that the two are equivalent. Ratio 1 Ratio 2 / by 1st number 2nd number 1st number 2nd number 600 450 60 45 10 60 45 |
682 | Create a context for the numeric representation. 5001 ndash; 5001/5 = 400 The Utah Middle School Math Project | If a person has 500 apples, then loses 1/5 of the apples, this numeric expression describes how many apples they have left. | ###Write an explanation to this problem:
Create a context for the numeric representation. 5001 ndash; 5001/5 = 400 The Utah Middle School Math Project
###Explanation:
If a person has 500 apples, then loses 1/5 of the apples, this numeric expression describes how many apples they have left. |
261 | The mean weight for a sample of a certain kind of ring made from platinum is 8.21 grams. The mean weight for a sample of a certain kind of ring made from gold is 8.61 grams. Is there a meaningful difference in the weights of the two types of rings? Explain your reasoning. | The answer is unknown with this information. For example, if the MAD for each is 0.1 grams, then there would be a meaningful difference. If the MAD is greater than 0.2 grams, then there is not a meaningful difference. | ###Write an explanation to this problem:
The mean weight for a sample of a certain kind of ring made from platinum is 8.21 grams. The mean weight for a sample of a certain kind of ring made from gold is 8.61 grams. Is there a meaningful difference in the weights of the two types of rings? Explain your reasoning.
###Explanation:
The answer is unknown with this information. For example, if the MAD for each is 0.1 grams, then there would be a meaningful difference. If the MAD is greater than 0.2 grams, then there is not a meaningful difference. |
195 | The inequalities h > 42 and h < 60 represent the height requirements for an amusement park ride, where h represents a person's height in inches. Write a sentence or draw a sign that describes these rules as clearly as possible. | 60>h>42 This is saying that a person's height must be taller than 42 inches or 3 feet 6 inches or they must be shorter than 60 inches or 5 feet tall. | ###Write an explanation to this problem:
The inequalities h > 42 and h < 60 represent the height requirements for an amusement park ride, where h represents a person's height in inches. Write a sentence or draw a sign that describes these rules as clearly as possible.
###Explanation:
60>h>42 This is saying that a person's height must be taller than 42 inches or 3 feet 6 inches or they must be shorter than 60 inches or 5 feet tall. |
852 | What is the same about these two experiments? Selecting a letter at random from the word ALABAMA Selecting a letter at random from the word LAMB | If you select a letter at random from each word 'ALABAMA' or 'LAMB' you have the same opportunity to select a letter of L, A, M, B from the two words. The word 'LAMB' is in the word 'ALABAMA'. | ###Write an explanation to this problem:
What is the same about these two experiments? Selecting a letter at random from the word ALABAMA Selecting a letter at random from the word LAMB
###Explanation:
If you select a letter at random from each word 'ALABAMA' or 'LAMB' you have the same opportunity to select a letter of L, A, M, B from the two words. The word 'LAMB' is in the word 'ALABAMA'. |
714 | Create a context for the numeric representation. 1201 ndash; 1201/4 = 1203/4 = 90 The Utah Middle School Math Project | Sample Answer: If a person has 120 apples, then loses 1/4 of the apples, this numeric expression describes how many apples they have left. | ###Write an explanation to this problem:
Create a context for the numeric representation. 1201 ndash; 1201/4 = 1203/4 = 90 The Utah Middle School Math Project
###Explanation:
Sample Answer: If a person has 120 apples, then loses 1/4 of the apples, this numeric expression describes how many apples they have left. |
107 | Augusto tips 20% on his meals. If he tipped 4.80 on a meal, how much was the meal? Write an equation to solve the problem. The Utah Middle School Math Project | The percent tipped in decimal form times the original order is the amount tipped. This means 0.2*m=4.80 | ###Write an explanation to this problem:
Augusto tips 20% on his meals. If he tipped 4.80 on a meal, how much was the meal? Write an equation to solve the problem. The Utah Middle School Math Project
###Explanation:
The percent tipped in decimal form times the original order is the amount tipped. This means 0.2*m=4.80 |
59 | A water bucket is being filled with water from a water faucet at a constant rate. When will the bucket be full? What information would you need to be able to solve the problem? | The rate at which water is falling into the bucket and the volume of the bucket. | ###Write an explanation to this problem:
A water bucket is being filled with water from a water faucet at a constant rate. When will the bucket be full? What information would you need to be able to solve the problem?
###Explanation:
The rate at which water is falling into the bucket and the volume of the bucket. |
634 | The Washington Monument is 555 feet and 5 1/8 feet tall. Bob wants to create a scale model of it that is no more than 6 feet tall. What scale would you suggest Bob use for his model? The Utah Middle School Math Project | The problem gives the Washington monuments to be 560.167 feet tall. The maximum height of the scale model should be 6 feet. To find the scale factor needed, divide the desired height by the original height 6 / 560.167 = 0.011 is the ideal scale factor. | ###Write an explanation to this problem:
The Washington Monument is 555 feet and 5 1/8 feet tall. Bob wants to create a scale model of it that is no more than 6 feet tall. What scale would you suggest Bob use for his model? The Utah Middle School Math Project
###Explanation:
The problem gives the Washington monuments to be 560.167 feet tall. The maximum height of the scale model should be 6 feet. To find the scale factor needed, divide the desired height by the original height 6 / 560.167 = 0.011 is the ideal scale factor. |
830 | In triangle ABC, the measure of angle B is 50 degrees. Give possible values for the measures of angles A and C if ABC is an acute triangle. | angles A and C could both be 65 degrees since 65 is less than 90 and 50 + 65 + 65 = 180. | ###Write an explanation to this problem:
In triangle ABC, the measure of angle B is 50 degrees. Give possible values for the measures of angles A and C if ABC is an acute triangle.
###Explanation:
angles A and C could both be 65 degrees since 65 is less than 90 and 50 + 65 + 65 = 180. |
100 | The height of a parallelogram is 28.8 inches. If the area of the parallelogram is 155.52 in2, what is the length of the base of the parallelogram? Write an equation to solve the problem. The Utah Middle School Math Project | The equation for area of a parallelogram is area=base*height, or A=bh. Substitute the given values into this equation. 155.52=b*28.8. | ###Write an explanation to this problem:
The height of a parallelogram is 28.8 inches. If the area of the parallelogram is 155.52 in2, what is the length of the base of the parallelogram? Write an equation to solve the problem. The Utah Middle School Math Project
###Explanation:
The equation for area of a parallelogram is area=base*height, or A=bh. Substitute the given values into this equation. 155.52=b*28.8. |
443 | Given information: s = small angle 4s = larger angle Equation: s + 4s = 70 degrees Write a word problem to go with the information and the equation given above. The Utah Middle School Math Project | The sum of two angles is 70 degrees. The larger angle is 4 times bigger than the smaller angle. What is the measure of the two angles? | ###Write an explanation to this problem:
Given information: s = small angle 4s = larger angle Equation: s + 4s = 70 degrees Write a word problem to go with the information and the equation given above. The Utah Middle School Math Project
###Explanation:
The sum of two angles is 70 degrees. The larger angle is 4 times bigger than the smaller angle. What is the measure of the two angles? |
584 | Simplify the following expression using two different methods. 830 - 1 First, show your work for each method. The Utah Middle School Math Project | One method is to first distribute the 8 to the terms within the parentheses: 830 - 1 = 830 - 81 = 240 - 8 = 232 Another method is to first add the terms in the parentheses and then distribute the 8: 830 - 1 = 829 = 232 | ###Write an explanation to this problem:
Simplify the following expression using two different methods. 830 - 1 First, show your work for each method. The Utah Middle School Math Project
###Explanation:
One method is to first distribute the 8 to the terms within the parentheses: 830 - 1 = 830 - 81 = 240 - 8 = 232 Another method is to first add the terms in the parentheses and then distribute the 8: 830 - 1 = 829 = 232 |
765 | What is the difference between 4x and x4? | 4x means four times x; this is the same as x + x + x + x. On the other hand, x4 means x to the fourth power, or x*x*x*x. | ###Write an explanation to this problem:
What is the difference between 4x and x4?
###Explanation:
4x means four times x; this is the same as x + x + x + x. On the other hand, x4 means x to the fourth power, or x*x*x*x. |
616 | Write a situation that can be represented by the rational number. -10 The Utah Middle School Math Project | Any situation that could be represented by the rational number -10 just needs to involve words like withdrawal, decrease, loss, and so on that would indicate a negative number. One example of a correct situation would be: 10 degrees below freezing | ###Write an explanation to this problem:
Write a situation that can be represented by the rational number. -10 The Utah Middle School Math Project
###Explanation:
Any situation that could be represented by the rational number -10 just needs to involve words like withdrawal, decrease, loss, and so on that would indicate a negative number. One example of a correct situation would be: 10 degrees below freezing |
506 | Create a representation of a table of data that represents a nonlinear function. The Utah Middle School Math Project | There are many different tables that can be created that represent a nonlinear function. One possible example is: x 1 2 3 4 5 y 1 4 9 16 25 | ###Write an explanation to this problem:
Create a representation of a table of data that represents a nonlinear function. The Utah Middle School Math Project
###Explanation:
There are many different tables that can be created that represent a nonlinear function. One possible example is: x 1 2 3 4 5 y 1 4 9 16 25 |
179 | Describe a situation where the following quantities might be useful. -20 gallons per hour | One situation where the quantity -20 gallons per hour may be useful is when talking about a problem where water is being drained from a pool. | ###Write an explanation to this problem:
Describe a situation where the following quantities might be useful. -20 gallons per hour
###Explanation:
One situation where the quantity -20 gallons per hour may be useful is when talking about a problem where water is being drained from a pool. |
304 | List two different possibilities for the angle measures of an isosceles triangle. | Sample responses: 70â, 70â, 40â and 30â, 30â, 120â. | ###Write an explanation to this problem:
List two different possibilities for the angle measures of an isosceles triangle.
###Explanation:
Sample responses: 70â, 70â, 40â and 30â, 30â, 120â. |
478 | Solve the following problem by writing and solving a system of equations. Use any method you wish to solve. Write your answer in a complete sentence. Blake has some quarters and dimes. He has 20 coins worth a total of 2.90. How many of each type of coin does he have? The Utah Middle School Math Project | q + d = 20 where q is the number of quarters and d is the number of dimes 0.25q + 0.10d = 2.90 Blake has 14 dimes and 6 quarters | ###Write an explanation to this problem:
Solve the following problem by writing and solving a system of equations. Use any method you wish to solve. Write your answer in a complete sentence. Blake has some quarters and dimes. He has 20 coins worth a total of 2.90. How many of each type of coin does he have? The Utah Middle School Math Project
###Explanation:
q + d = 20 where q is the number of quarters and d is the number of dimes 0.25q + 0.10d = 2.90 Blake has 14 dimes and 6 quarters |
57 | In the cafeteria, there is one large 10-seat table and many smaller 4-seat tables. There are enough tables to fit 200 students. Write an inequality whose solution is the possible number of 4-seat tables in the cafeteria. | 10+4nge;200 | ###Write an explanation to this problem:
In the cafeteria, there is one large 10-seat table and many smaller 4-seat tables. There are enough tables to fit 200 students. Write an inequality whose solution is the possible number of 4-seat tables in the cafeteria.
###Explanation:
10+4nge;200 |
822 | 3b. Explain what a perfect cube is. copied for free from openupresources.org | A perfect cube is a number that is the result of multiplying an original number by itself 3 times. For example, 8 is a perfect cube because 2 x 2 x 2 = 8. | ###Write an explanation to this problem:
3b. Explain what a perfect cube is. copied for free from openupresources.org
###Explanation:
A perfect cube is a number that is the result of multiplying an original number by itself 3 times. For example, 8 is a perfect cube because 2 x 2 x 2 = 8. |
149 | For each equation, enter two more equations using the same numbers that express the same relationship in a different way. 3+2=5 copied for free from openupresources.org | 3 Sample Answers: 5-3=2; 5-2=3; 2+3=5 | ###Write an explanation to this problem:
For each equation, enter two more equations using the same numbers that express the same relationship in a different way. 3+2=5 copied for free from openupresources.org
###Explanation:
3 Sample Answers: 5-3=2; 5-2=3; 2+3=5 |
124 | Describe the transformation graph shift that occurs in the following situation. Use words like translation, reflection, and rotation. The y-intercept is decreased while the slope is held constant. The Utah Middle School Math Project | The slope is the rate of change of the line and determines the steepness of the line. The y-intercept is the point where the line intersects with the y-axis. If the y-intercept is decreased while the slope is held constant, then we know that the line is being moved down. This can be described as a vertical translation down. | ###Write an explanation to this problem:
Describe the transformation graph shift that occurs in the following situation. Use words like translation, reflection, and rotation. The y-intercept is decreased while the slope is held constant. The Utah Middle School Math Project
###Explanation:
The slope is the rate of change of the line and determines the steepness of the line. The y-intercept is the point where the line intersects with the y-axis. If the y-intercept is decreased while the slope is held constant, then we know that the line is being moved down. This can be described as a vertical translation down. |
735 | Give two different ratios with a description of the ratio relationship using the following information: There are 15 male teachers in the school. There are 35 female teachers in the school. | There are many possible solutions. Here are some samples: The ratio of the number of male teachers to the number of female teachers is 15:35. The ratio of the number of female teachers to the number of male teachers is 35:15. The ratio of the number of female teachers to the total number of teachers in the school is 35:50. The ratio of the number of male teachers to the total number of teachers in the school is 15:50. | ###Write an explanation to this problem:
Give two different ratios with a description of the ratio relationship using the following information: There are 15 male teachers in the school. There are 35 female teachers in the school.
###Explanation:
There are many possible solutions. Here are some samples: The ratio of the number of male teachers to the number of female teachers is 15:35. The ratio of the number of female teachers to the number of male teachers is 35:15. The ratio of the number of female teachers to the total number of teachers in the school is 35:50. The ratio of the number of male teachers to the total number of teachers in the school is 15:50. |
740 | Write a story context that would be represented by the ratio 1:4 | Sample response: Kendra's mom pays her four dollars for every load of laundry she washes and dries. | ###Write an explanation to this problem:
Write a story context that would be represented by the ratio 1:4
###Explanation:
Sample response: Kendra's mom pays her four dollars for every load of laundry she washes and dries. |
185 | A group of 100 people is divided into 5 groups with 20 people in each. One person's name is chosen, and everyone in their group wins a prize. Noah simulates this situation by writing 100 different names on papers and putting them in a bag, then drawing one out. Kiran suggests there is a way to do it with fewer paper slips. Explain a method that would simulate this situation with fewer than 100 slips of paper. | A simpler way to do this is just put 5 slips of paper in the bag with each group's number written on a slip of paper. | ###Write an explanation to this problem:
A group of 100 people is divided into 5 groups with 20 people in each. One person's name is chosen, and everyone in their group wins a prize. Noah simulates this situation by writing 100 different names on papers and putting them in a bag, then drawing one out. Kiran suggests there is a way to do it with fewer paper slips. Explain a method that would simulate this situation with fewer than 100 slips of paper.
###Explanation:
A simpler way to do this is just put 5 slips of paper in the bag with each group's number written on a slip of paper. |
809 | The volume of Box D subtracted from the volume of Box C is 23.14 cubic centimeters. Box D has a volume of 10.115 cubic centimeters. Let C be the volume of Box C in cubic centimeters. Write an equation that could be used to determine the volume of Box C. | C - 10.1115 cm3 = 23.14 cm3 | ###Write an explanation to this problem:
The volume of Box D subtracted from the volume of Box C is 23.14 cubic centimeters. Box D has a volume of 10.115 cubic centimeters. Let C be the volume of Box C in cubic centimeters. Write an equation that could be used to determine the volume of Box C.
###Explanation:
C - 10.1115 cm3 = 23.14 cm3 |
50 | If you blend 2 scoops of chocolate ice cream with 1 cup of milk, you get a milkshake with a stronger chocolate flavor than if you blended 3 scoops of chocolate ice cream with 2 cups of milk. Explain or show why. Submit your work using the tools below. | The easiest way to compare recipes is to find the ratio of ingredients for multiple batches until you find a common measurement for a specific ingredient. In this case, the following tables show both recipes up to the point that they both have 2 cups of milk. First Recipe: of batches Scoops of Choc. Ice Cream Cups of Milk 1 2 1 2 4 2 Second Recipe: of batches Scoops of Choc. Ice Cream Cups of Milk 1 3 2 Since the first recipe uses more scoops of chocolate ice cream to the same amount of milk, it means it would have a stronger chocolate flavor. | ###Write an explanation to this problem:
If you blend 2 scoops of chocolate ice cream with 1 cup of milk, you get a milkshake with a stronger chocolate flavor than if you blended 3 scoops of chocolate ice cream with 2 cups of milk. Explain or show why. Submit your work using the tools below.
###Explanation:
The easiest way to compare recipes is to find the ratio of ingredients for multiple batches until you find a common measurement for a specific ingredient. In this case, the following tables show both recipes up to the point that they both have 2 cups of milk. First Recipe: of batches Scoops of Choc. Ice Cream Cups of Milk 1 2 1 2 4 2 Second Recipe: of batches Scoops of Choc. Ice Cream Cups of Milk 1 3 2 Since the first recipe uses more scoops of chocolate ice cream to the same amount of milk, it means it would have a stronger chocolate flavor. |
341 | In a video game, the chance of rain each day is always 30%. At the beginning of each day in the video game, the computer generates a random integer between 1 and 50. Explain how you could use this number to simulate the weather in the video game. | Sample Answer: If the number is between 1 and 15, the video game should create a rainy day. If the number is between 16 and 50, the video game should not create a rainy day. This is because 15/50 = 30/100, or 30%. | ###Write an explanation to this problem:
In a video game, the chance of rain each day is always 30%. At the beginning of each day in the video game, the computer generates a random integer between 1 and 50. Explain how you could use this number to simulate the weather in the video game.
###Explanation:
Sample Answer: If the number is between 1 and 15, the video game should create a rainy day. If the number is between 16 and 50, the video game should not create a rainy day. This is because 15/50 = 30/100, or 30%. |
446 | 61% of 20 Estimate the answer. The Utah Middle School Math Project | 50% of 20 is 10, so the 61% of 20 must be greater than 10. | ###Write an explanation to this problem:
61% of 20 Estimate the answer. The Utah Middle School Math Project
###Explanation:
50% of 20 is 10, so the 61% of 20 must be greater than 10. |
842 | Diego made a graph of two quantities that he measured and said, The points all lie on a line except one, which is a little bit above the line. This means that the quantities can't be proportional. Do you agree with Diego? Explain. | I don't agree with Diego. A relationship can be considered proportional if the point fall in approximately a straight line and go through the origin. The measurements could be off just a little bit because Diego measured them himself. | ###Write an explanation to this problem:
Diego made a graph of two quantities that he measured and said, The points all lie on a line except one, which is a little bit above the line. This means that the quantities can't be proportional. Do you agree with Diego? Explain.
###Explanation:
I don't agree with Diego. A relationship can be considered proportional if the point fall in approximately a straight line and go through the origin. The measurements could be off just a little bit because Diego measured them himself. |
535 | Write an ordered pair that would be in Quadrant I when graphed. The Utah Middle School Math Project | Any ordered pair with two positive numbers is in Quadrant I. Examples: 1, 1 5, 3 1/2, 3/4 | ###Write an explanation to this problem:
Write an ordered pair that would be in Quadrant I when graphed. The Utah Middle School Math Project
###Explanation:
Any ordered pair with two positive numbers is in Quadrant I. Examples: 1, 1 5, 3 1/2, 3/4 |
321 | Draw two polygons that are not similar but could be mistaken for being similar. Submit your drawing using the tools below. | There are many correct answers to this question. To be similar, the two polygons must be the same shape, but can involve some sorts of transformations to turn one into the other. If two polygons are not similar, then their angle measures could match up but their side lengths could be not proportional. For example: A triangle with angles 45, 35, 100 and side lengths of 4, 3, and 8, and a triangle with angles 45, 35, 100 and side lengths of 5, 4, and 9. They could also have side lengths that are proportional but angles that do not match up. For example: A triangle with angles 45, 35, 100 and side lengths of 4, 3, and 8, and a triangle with angles 55, 25, 100 and side lengths of 8, 6, and 16. | ###Write an explanation to this problem:
Draw two polygons that are not similar but could be mistaken for being similar. Submit your drawing using the tools below.
###Explanation:
There are many correct answers to this question. To be similar, the two polygons must be the same shape, but can involve some sorts of transformations to turn one into the other. If two polygons are not similar, then their angle measures could match up but their side lengths could be not proportional. For example: A triangle with angles 45, 35, 100 and side lengths of 4, 3, and 8, and a triangle with angles 45, 35, 100 and side lengths of 5, 4, and 9. They could also have side lengths that are proportional but angles that do not match up. For example: A triangle with angles 45, 35, 100 and side lengths of 4, 3, and 8, and a triangle with angles 55, 25, 100 and side lengths of 8, 6, and 16. |
353 | Create a real-life example that can be modeled by the expression -2 * 4. | Maria spends 2.00 each week on coffee. She tracks her spending for 4 weeks. Write an integer to represent the change in Maria's money as a result of buying coffee for 4 weeks. ndash;24 = -8 Maria will have spent 8.00 on coffee after 4 weeks. | ###Write an explanation to this problem:
Create a real-life example that can be modeled by the expression -2 * 4.
###Explanation:
Maria spends 2.00 each week on coffee. She tracks her spending for 4 weeks. Write an integer to represent the change in Maria's money as a result of buying coffee for 4 weeks. ndash;24 = -8 Maria will have spent 8.00 on coffee after 4 weeks. |
789 | Explain why 6:4 and 18:8 are not equivalent ratios. | 6:4 and 18:8 are not equivalent because 6 x 3= 18, but 4 x 3= 12, not 8. In order for two ratios to be equivalent, both numbers have to be multiplied or divided by the same number. 6 4 12 8 18 12 | ###Write an explanation to this problem:
Explain why 6:4 and 18:8 are not equivalent ratios.
###Explanation:
6:4 and 18:8 are not equivalent because 6 x 3= 18, but 4 x 3= 12, not 8. In order for two ratios to be equivalent, both numbers have to be multiplied or divided by the same number. 6 4 12 8 18 12 |
142 | An ice cream shop offers 40 different flavors. To simulate the most commonly chosen flavor, you could write the name of each flavor on a piece of paper and put it in a bag. Draw from the bag 100 times, and see which flavor is chosen the most. Why is this simulation a bad way to figure out the most commonly chosen flavor? | This is randomly selecting a flavor and does not take into account the customers and what they choose. Some flavors may be chosen than others by customers compared to randomly choosing from a bag because flavor choice is a bias based off of the customers. | ###Write an explanation to this problem:
An ice cream shop offers 40 different flavors. To simulate the most commonly chosen flavor, you could write the name of each flavor on a piece of paper and put it in a bag. Draw from the bag 100 times, and see which flavor is chosen the most. Why is this simulation a bad way to figure out the most commonly chosen flavor?
###Explanation:
This is randomly selecting a flavor and does not take into account the customers and what they choose. Some flavors may be chosen than others by customers compared to randomly choosing from a bag because flavor choice is a bias based off of the customers. |
470 | The following situation represents a functional relationship between two quantities. Identify the independent variable and the dependent variable. A car that has more gas in its tank can drive a farther distance. The Utah Middle School Math Project | Gas is independent variable. Distance depends on the amount of gas that is left in the tank. Distance is the dependent variable. | ###Write an explanation to this problem:
The following situation represents a functional relationship between two quantities. Identify the independent variable and the dependent variable. A car that has more gas in its tank can drive a farther distance. The Utah Middle School Math Project
###Explanation:
Gas is independent variable. Distance depends on the amount of gas that is left in the tank. Distance is the dependent variable. |
370 | In a recipe for fizzy grape juice, the ratio of cups of sparkling water to cups of grape juice concentrate is 3 to 1. Find two more ratios of cups of sparkling water to cups of juice concentrate that would make a mixture that tastes the same as this recipe. | In order to make sure the mixture tastes the same as this recipe, we have to keep the ratios equivalent. We can do this by multiplying or dividing both numbers in this 3 to 1 ratio by the same number. 3 bull; 2 = 6 and 1 bull; 2 = 2 so a new ratio could be 6 to 2. We could also divide 3 / 2 = 1 1/2 and 1 / 2 = 1/2 so another ratio could be 1 1/2 to 1/2. | ###Write an explanation to this problem:
In a recipe for fizzy grape juice, the ratio of cups of sparkling water to cups of grape juice concentrate is 3 to 1. Find two more ratios of cups of sparkling water to cups of juice concentrate that would make a mixture that tastes the same as this recipe.
###Explanation:
In order to make sure the mixture tastes the same as this recipe, we have to keep the ratios equivalent. We can do this by multiplying or dividing both numbers in this 3 to 1 ratio by the same number. 3 bull; 2 = 6 and 1 bull; 2 = 2 so a new ratio could be 6 to 2. We could also divide 3 / 2 = 1 1/2 and 1 / 2 = 1/2 so another ratio could be 1 1/2 to 1/2. |
780 | Jane paid 40 for an item after she received a 20% discount. Jane's friend says this means that the original price of the item was 48. How do you think Jane's friend arrived at this amount? | Jane's friend found that 20% of 40 is 8. 0.20 * 40 = 8 Then she added 8 to the sale price: 40 + 8 = 48. Then she determined that the original amount was 48 | ###Write an explanation to this problem:
Jane paid 40 for an item after she received a 20% discount. Jane's friend says this means that the original price of the item was 48. How do you think Jane's friend arrived at this amount?
###Explanation:
Jane's friend found that 20% of 40 is 8. 0.20 * 40 = 8 Then she added 8 to the sale price: 40 + 8 = 48. Then she determined that the original amount was 48 |
141 | An experiment will produce one of ten different outcomes with equal probability for each. Why would using a standard number cube to simulate the experiment be a bad choice? | This would be a bad simulation because a standard number cube only has 6 sides options when the experiment has 10 different outcomes and would not be represented on a number cube. | ###Write an explanation to this problem:
An experiment will produce one of ten different outcomes with equal probability for each. Why would using a standard number cube to simulate the experiment be a bad choice?
###Explanation:
This would be a bad simulation because a standard number cube only has 6 sides options when the experiment has 10 different outcomes and would not be represented on a number cube. |
399 | At Shoes for Less, a pair of shoes is 15 less than a pair of boots. Cho purchased three pairs of shoes and two pairs of boots for 120. How much does a pair of boots cost? Write an equation to answer the equation. The Utah Middle School Math Project | A pair of boots can be represented by the variable b. A pair of shoes can be represented by b-15, because shoes are 15 dollars less than boots. The cost of shoes must be multiplied by 3, and the cost of boots must be multiplied by two. These total costs must be added together, and set equal to 120. The equation is 3b-15+2b=120. | ###Write an explanation to this problem:
At Shoes for Less, a pair of shoes is 15 less than a pair of boots. Cho purchased three pairs of shoes and two pairs of boots for 120. How much does a pair of boots cost? Write an equation to answer the equation. The Utah Middle School Math Project
###Explanation:
A pair of boots can be represented by the variable b. A pair of shoes can be represented by b-15, because shoes are 15 dollars less than boots. The cost of shoes must be multiplied by 3, and the cost of boots must be multiplied by two. These total costs must be added together, and set equal to 120. The equation is 3b-15+2b=120. |
263 | A steel beam can be cut to different lengths for a project. Assuming the weight of a steel beam is proportional to its length, what information would you need to know to write an equation that represents this relationship? | Sample response: Are the length of the steel beam and its weight in a proportional relationship? The weight of steel per 1 meter length of beam. The length of a steel beam that weighs 1 kilogram. The weight and length of a certain steel beam. | ###Write an explanation to this problem:
A steel beam can be cut to different lengths for a project. Assuming the weight of a steel beam is proportional to its length, what information would you need to know to write an equation that represents this relationship?
###Explanation:
Sample response: Are the length of the steel beam and its weight in a proportional relationship? The weight of steel per 1 meter length of beam. The length of a steel beam that weighs 1 kilogram. The weight and length of a certain steel beam. |
320 | Draw two polygons that are similar but could be mistaken for not being similar. Submit your drawing using the tools below. | There are many correct answers to this question. To be similar, the two polygons must be the same shape, but must involve some sorts of transformations to turn one into the other. For example, you could: start with two triangles that are exactly the same and rotate one of them by however many degrees you want. You can then multiply all of the side lengths in the rotated triangle by a certain factor. start with two triangles that are exactly the same and reflect one of them over a horizontal line. You can then multiply all of the side lengths in the reflected triangle by a certain factor. | ###Write an explanation to this problem:
Draw two polygons that are similar but could be mistaken for not being similar. Submit your drawing using the tools below.
###Explanation:
There are many correct answers to this question. To be similar, the two polygons must be the same shape, but must involve some sorts of transformations to turn one into the other. For example, you could: start with two triangles that are exactly the same and rotate one of them by however many degrees you want. You can then multiply all of the side lengths in the rotated triangle by a certain factor. start with two triangles that are exactly the same and reflect one of them over a horizontal line. You can then multiply all of the side lengths in the reflected triangle by a certain factor. |
19 | Explain what the slope and intercept mean in the situation. A graph represents the perimeter, y, in units, for an equilateral triangle with side length x units. The slope of the line is 3 and the y-intercept is 0. | The slope shows that for every 1 unit increase in side length of the equilateral triangle, the perimeter increases by 3 units. The y intercept means that an equilateral triangle with side lengths of 0 units would have a perimeter of 0 units. | ###Write an explanation to this problem:
Explain what the slope and intercept mean in the situation. A graph represents the perimeter, y, in units, for an equilateral triangle with side length x units. The slope of the line is 3 and the y-intercept is 0.
###Explanation:
The slope shows that for every 1 unit increase in side length of the equilateral triangle, the perimeter increases by 3 units. The y intercept means that an equilateral triangle with side lengths of 0 units would have a perimeter of 0 units. |
172 | Explain how to make this calculation mentally. 99 * 6 | Round the 99 to 100 so you know have: 1006 = 600. | ###Write an explanation to this problem:
Explain how to make this calculation mentally. 99 * 6
###Explanation:
Round the 99 to 100 so you know have: 1006 = 600. |
694 | A lunch table can hold 8 students. If there are 120 students who need to eat lunch at the same time, how many tables are needed? Write an equation to solve the problem. The Utah Middle School Math Project | There are 8 students at each table, and the total number of students is 120. This means that 8t=120. | ###Write an explanation to this problem:
A lunch table can hold 8 students. If there are 120 students who need to eat lunch at the same time, how many tables are needed? Write an equation to solve the problem. The Utah Middle School Math Project
###Explanation:
There are 8 students at each table, and the total number of students is 120. This means that 8t=120. |
312 | Explain what the slope and intercept mean in the situation. A graph represents the perimeter, y, in units, for an equilateral triangle with side length x units. The slope of the line is 3 and the y-intercept is 0. | The slope of 3 shows that the triangle has 3 sides. For each increase of 1 unit of the side length, the perimeter increases by 3 units. The intercept of 0 shows that the relationship is proportional--a triangle with sides of length 0 has a perimeter of length 0. | ###Write an explanation to this problem:
Explain what the slope and intercept mean in the situation. A graph represents the perimeter, y, in units, for an equilateral triangle with side length x units. The slope of the line is 3 and the y-intercept is 0.
###Explanation:
The slope of 3 shows that the triangle has 3 sides. For each increase of 1 unit of the side length, the perimeter increases by 3 units. The intercept of 0 shows that the relationship is proportional--a triangle with sides of length 0 has a perimeter of length 0. |
390 | Write a story that matches the expressions and equations. Fixing Your Car Time hours: h Cost of Mike's Mechanics dollars: 15h + 75 Cost of Bubba's Body Shop dollars: 25h 15h + 75 = 25h The Utah Middle School Math Project | The cost of Mike's Mechanics is 15h + 75, meaning there must be an initial fee of 75 and for every hour you will pay an additional 15. The cost of Bubba's Body Shop is 25h, meaning you will pay 25 dollars per hour. So, when 15h +75 = 25h, the cost of Mike's Mechanics will be equal to the cost of Bubba's Body Shop. Solve for h to find out after how many hours will both shops charge the same rate to fix your car. | ###Write an explanation to this problem:
Write a story that matches the expressions and equations. Fixing Your Car Time hours: h Cost of Mike's Mechanics dollars: 15h + 75 Cost of Bubba's Body Shop dollars: 25h 15h + 75 = 25h The Utah Middle School Math Project
###Explanation:
The cost of Mike's Mechanics is 15h + 75, meaning there must be an initial fee of 75 and for every hour you will pay an additional 15. The cost of Bubba's Body Shop is 25h, meaning you will pay 25 dollars per hour. So, when 15h +75 = 25h, the cost of Mike's Mechanics will be equal to the cost of Bubba's Body Shop. Solve for h to find out after how many hours will both shops charge the same rate to fix your car. |
796 | To make 1 batch of lavender paint, the ratio of cups of pink paint to cups of blue paint is 6 to 5. Find two more ratios of cups of pink paint to cups of blue paint that are equivalent to this ratio. | To make 2 batches, you could use the ratio of 12 to 10 which would be 12 cups of pink paint and 10 cups of blue paint. To make 1/2 of a batch you could use the ratio of 3 to 2.5 which would be 3 cups of pink paint and 2.5 cups of blue paint. | ###Write an explanation to this problem:
To make 1 batch of lavender paint, the ratio of cups of pink paint to cups of blue paint is 6 to 5. Find two more ratios of cups of pink paint to cups of blue paint that are equivalent to this ratio.
###Explanation:
To make 2 batches, you could use the ratio of 12 to 10 which would be 12 cups of pink paint and 10 cups of blue paint. To make 1/2 of a batch you could use the ratio of 3 to 2.5 which would be 3 cups of pink paint and 2.5 cups of blue paint. |
12 | Fill in the blank with numbers that correctly complete the statement. Three rational numbers between 16 and 15 < < | 15.3 < 15.6 < 15.7 | ###Write an explanation to this problem:
Fill in the blank with numbers that correctly complete the statement. Three rational numbers between 16 and 15 < <
###Explanation:
15.3 < 15.6 < 15.7 |
407 | Make a list to find the GCF of this pair of numbers. 14 and 35 First, write the list here. The Utah Middle School Math Project | Factors of 14: 1, 2, 7, 14 Factors of 35: 1, 5, 7, 35 | ###Write an explanation to this problem:
Make a list to find the GCF of this pair of numbers. 14 and 35 First, write the list here. The Utah Middle School Math Project
###Explanation:
Factors of 14: 1, 2, 7, 14 Factors of 35: 1, 5, 7, 35 |
408 | Adam is trying to solve the following riddle: The sum of three consecutive integers is -36. What are the integers? Write an equation to solve this riddle. The Utah Middle School Math Project | Consecutive integers are numbers that directly follow each other, and are whole numbers. The first integer will be set as n. The next integer follows directly after, so it can be n+1. The next integer will be n+2. These three values added together equal -36. The equation is n+n+1+n+2=-36. | ###Write an explanation to this problem:
Adam is trying to solve the following riddle: The sum of three consecutive integers is -36. What are the integers? Write an equation to solve this riddle. The Utah Middle School Math Project
###Explanation:
Consecutive integers are numbers that directly follow each other, and are whole numbers. The first integer will be set as n. The next integer follows directly after, so it can be n+1. The next integer will be n+2. These three values added together equal -36. The equation is n+n+1+n+2=-36. |
305 | In your own words, explain what a dilation is. | Sample responses: A dilation expands things. It makes scaled copies. It blows things up. It works by pushing things outward from a center. A dilation multiplies distances from the center to change the size of the figure. | ###Write an explanation to this problem:
In your own words, explain what a dilation is.
###Explanation:
Sample responses: A dilation expands things. It makes scaled copies. It blows things up. It works by pushing things outward from a center. A dilation multiplies distances from the center to change the size of the figure. |
354 | Jamie was working on his math homework with his friend, Kent. Jamie looked at the following problem. -9.5 - -8- 6.5 He told Kent that he did not know how to subtract negative numbers. Kent said that he knew how to solve the problem using only addition. What did Kent mean by that? Explain. | This is the original problem: -9.5 - -8 - 6.5 This problem has subtraction in it, so we will begin by rewriting subtraction by adding the opposite: -9.5 - -8 - 6.5 -9.5 + +8 + ndash;6.5 -9.5 + 8 + -6.5 We might begin by regrouping the terms so the negative terms are together: -9.5+ -6.5 + 8 Now we can see how this problem, which began as subtraction, can be rewritten as an addition problem. | ###Write an explanation to this problem:
Jamie was working on his math homework with his friend, Kent. Jamie looked at the following problem. -9.5 - -8- 6.5 He told Kent that he did not know how to subtract negative numbers. Kent said that he knew how to solve the problem using only addition. What did Kent mean by that? Explain.
###Explanation:
This is the original problem: -9.5 - -8 - 6.5 This problem has subtraction in it, so we will begin by rewriting subtraction by adding the opposite: -9.5 - -8 - 6.5 -9.5 + +8 + ndash;6.5 -9.5 + 8 + -6.5 We might begin by regrouping the terms so the negative terms are together: -9.5+ -6.5 + 8 Now we can see how this problem, which began as subtraction, can be rewritten as an addition problem. |
25 | Using the rule of subtraction, rewrite the following subtraction expression as an addition expression. -14 - -2 | The subtraction rule says that subtracting a number is the same as adding its opposite. To use this rule of subtraction, we start by switching the subtraction sign to an addition sign: -14 + -2 We then change the sign of -2 to its opposite and get: -14 + 2 | ###Write an explanation to this problem:
Using the rule of subtraction, rewrite the following subtraction expression as an addition expression. -14 - -2
###Explanation:
The subtraction rule says that subtracting a number is the same as adding its opposite. To use this rule of subtraction, we start by switching the subtraction sign to an addition sign: -14 + -2 We then change the sign of -2 to its opposite and get: -14 + 2 |
669 | Solve the inequality. Represent the solution on a number line. 1.2<x-9/10 The Utah Middle School Math Project | Solve for x. The number line should be greater than the solution, but not contain it. | ###Write an explanation to this problem:
Solve the inequality. Represent the solution on a number line. 1.2<x-9/10 The Utah Middle School Math Project
###Explanation:
Solve for x. The number line should be greater than the solution, but not contain it. |
169 | Name 4 points that would form a square with the origin at its center. | Here is an example, answers will vary. Check your work by graphing or ask your teacher 3, 5, -3, 5, 3, -5, and -3, -5. | ###Write an explanation to this problem:
Name 4 points that would form a square with the origin at its center.
###Explanation:
Here is an example, answers will vary. Check your work by graphing or ask your teacher 3, 5, -3, 5, 3, -5, and -3, -5. |
538 | Here, write the first problem in which the percent and the part are known, but the whole is not known. The Utah Middle School Math Project | Sample Answer: Tom bought a laptop for 680. Due to his membership, he used a 15% discount on his purchase. How much did the laptop cost before the discount? Solution: part: 680 percent: 15% = 0.15 We want to find the original price of the laptop. Note that with a 15% discount, the final price of 680 is 85% of the original price. 85% of the original price is 0.85 times the original price. With this information, we can write: 680 = original price x 0.85 Now, we can solve for the original price: 680 = original price x 0.85 680/0.85 = original price 800 = original price We get that the original price of the laptop Tom bought was 800. | ###Write an explanation to this problem:
Here, write the first problem in which the percent and the part are known, but the whole is not known. The Utah Middle School Math Project
###Explanation:
Sample Answer: Tom bought a laptop for 680. Due to his membership, he used a 15% discount on his purchase. How much did the laptop cost before the discount? Solution: part: 680 percent: 15% = 0.15 We want to find the original price of the laptop. Note that with a 15% discount, the final price of 680 is 85% of the original price. 85% of the original price is 0.85 times the original price. With this information, we can write: 680 = original price x 0.85 Now, we can solve for the original price: 680 = original price x 0.85 680/0.85 = original price 800 = original price We get that the original price of the laptop Tom bought was 800. |
38 | A green light blinks every 4 seconds and a yellow light blinks every 5 seconds. When will both lights blink at the same time? | Find the multiples of 4 and of 5. The first one they have in common will be when the lights will blink at the same time. Multiples of 4: 4,8,12,16,20 Multiples of 5: 5,10,15,20 The lights will blink at the same time after 20 seconds. | ###Write an explanation to this problem:
A green light blinks every 4 seconds and a yellow light blinks every 5 seconds. When will both lights blink at the same time?
###Explanation:
Find the multiples of 4 and of 5. The first one they have in common will be when the lights will blink at the same time. Multiples of 4: 4,8,12,16,20 Multiples of 5: 5,10,15,20 The lights will blink at the same time after 20 seconds. |
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