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Homes in a certain neighborhood sell for 117 per square foot. Can the relationship between the number of square feet in the home and the sale price of the home be modeled by a linear function? Why or why not? The Utah Middle School Math Project

This data can be modeled by a linear function, because the relationship between the number of square feet and the sale price is a constant.

###Write an explanation to this problem: Homes in a certain neighborhood sell for 117 per square foot. Can the relationship between the number of square feet in the home and the sale price of the home be modeled by a linear function? Why or why not? The Utah Middle School Math Project ###Explanation: This data can be modeled by a linear function, because the relationship between the number of square feet and the sale price is a constant.

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. In 1982, the US Mint changed the composition of pennies from all copper to zinc with copper coating. Pennies made prior to 1982 weigh 3.1 grams. Pennies made since 1982 weigh 2.5 grams. If you have a bag of 1,254 pennies, and the bag weighs 3,508.8 grams, how many pennies from each time period are there in the bag? The Utah Middle School Math Project

𝑥 + 𝑦 = 1254 where x is the number of coins made prior to 1982 and y is the number of coins made since 1982 3.1𝑥 + 2.5𝑦 = 3508.8 There are 623 pennies made prior to 1982 and 631 pennies made since 1982

###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. In 1982, the US Mint changed the composition of pennies from all copper to zinc with copper coating. Pennies made prior to 1982 weigh 3.1 grams. Pennies made since 1982 weigh 2.5 grams. If you have a bag of 1,254 pennies, and the bag weighs 3,508.8 grams, how many pennies from each time period are there in the bag? The Utah Middle School Math Project ###Explanation: 𝑥 + 𝑦 = 1254 where x is the number of coins made prior to 1982 and y is the number of coins made since 1982 3.1𝑥 + 2.5𝑦 = 3508.8 There are 623 pennies made prior to 1982 and 631 pennies made since 1982

Lin missed math class on the day they worked on expanding and factoring. Kiran is helping Lin catch up. Lin understands that expanding is using the distributive property, but she doesn't understand what factoring is or why it works. How can Kiran explain factoring to Lin? Copied for free from openupresources.org

Factoring is the distributive property in the other direction. Instead of expanding a product to a sum of terms, factoring first takes a sum of terms. It makes the sum of terms into a product by looking for common factors in the terms that can can be written outside the parentheses.

###Write an explanation to this problem: Lin missed math class on the day they worked on expanding and factoring. Kiran is helping Lin catch up. Lin understands that expanding is using the distributive property, but she doesn't understand what factoring is or why it works. How can Kiran explain factoring to Lin? Copied for free from openupresources.org ###Explanation: Factoring is the distributive property in the other direction. Instead of expanding a product to a sum of terms, factoring first takes a sum of terms. It makes the sum of terms into a product by looking for common factors in the terms that can can be written outside the parentheses.

Here, write the second problem in which the percent and the whole are known, but the part is not known, and its solution. The Utah Middle School Math Project

Sample Answer: Richard prepares 24 fluid ounces of tea for his workday. In the first hour of work, he drank 25% of the tea. How many fluid ounces of tea did he drink? Solution: whole: 24 fl.oz. percent: 25% = 0.25 To find the amount of tea he drank, we multiply the whole by the percent: whole x percent = 24 fl. oz. x 0.25 = 6 fl. oz. We get that in the first hour of work, Richard drank 6 fl. oz. of tea.

###Write an explanation to this problem: Here, write the second problem in which the percent and the whole are known, but the part is not known, and its solution. The Utah Middle School Math Project ###Explanation: Sample Answer: Richard prepares 24 fluid ounces of tea for his workday. In the first hour of work, he drank 25% of the tea. How many fluid ounces of tea did he drink? Solution: whole: 24 fl.oz. percent: 25% = 0.25 To find the amount of tea he drank, we multiply the whole by the percent: whole x percent = 24 fl. oz. x 0.25 = 6 fl. oz. We get that in the first hour of work, Richard drank 6 fl. oz. of tea.

The following situation represents a functional relationship between two quantities. Identify the independent variable and the dependent variable. Jenny has 30 coins she has collected over 6 years. The Utah Middle School Math Project

The time 6 years is the independent variable The number of coins 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. Jenny has 30 coins she has collected over 6 years. The Utah Middle School Math Project ###Explanation: The time 6 years is the independent variable The number of coins is the dependent variable.

A green light blinks every 4 seconds and a yellow light blinks every 5 seconds. When will both lights blink at the same time?

20 seconds You need to find the least common multiple of each number 4: 4, 8, 12, 16, 205: 5, 10, 15, 20

###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: 20 seconds You need to find the least common multiple of each number 4: 4, 8, 12, 16, 205: 5, 10, 15, 20

Create your own context or story that represents a relation that is not a function. Write your story.

To be a function, whatever you chose to be the independent variable x must have only one corresponding dependent variable y.

###Write an explanation to this problem: Create your own context or story that represents a relation that is not a function. Write your story. ###Explanation: To be a function, whatever you chose to be the independent variable x must have only one corresponding dependent variable y.

Use The Distributive Property to find the product. 4105 First, show your work here. The Utah Middle School Math Project

4105 = 4100 + 5 = 4100 + 45 = 400 + 20 = 420

###Write an explanation to this problem: Use The Distributive Property to find the product. 4105 First, show your work here. The Utah Middle School Math Project ###Explanation: 4105 = 4100 + 5 = 4100 + 45 = 400 + 20 = 420

As part of a competition, Diego must spin around in a circle 6 times and then run to a tree. The time he spends on each spin is represented by s and the time he spends running is r. He gets to the tree 21 seconds after he starts spinning. Write an equation showing the relationship between s and r. Copied for free from openupresources.org

Diego has to spin 6 times. If it takes him s seconds per spin, then for all of his spins together he takes 6s seconds. If it takes him 21 seconds total to get to the tree, this means the time he spent spinning 6s plus the time he spent running r equals 21 seconds. 6s + r = 21

###Write an explanation to this problem: As part of a competition, Diego must spin around in a circle 6 times and then run to a tree. The time he spends on each spin is represented by s and the time he spends running is r. He gets to the tree 21 seconds after he starts spinning. Write an equation showing the relationship between s and r. Copied for free from openupresources.org ###Explanation: Diego has to spin 6 times. If it takes him s seconds per spin, then for all of his spins together he takes 6s seconds. If it takes him 21 seconds total to get to the tree, this means the time he spent spinning 6s plus the time he spent running r equals 21 seconds. 6s + r = 21

The number e is an important irrational number. In future math classes as well as science and socialscience, you will see and use this number quite a bit. Use the approximations of e and the calculationsgiven below to answer the questions that follow. e is between 2 and 3e is between 2.7 and 2.8e is between 2.71 and 2.72e is between 2.718 and 2.719 Find a number that is between 2 and e. The Utah Middle School Math Project

If e is between 2 and 3, then any number between 2 and 3 would be a correct answer.

###Write an explanation to this problem: The number e is an important irrational number. In future math classes as well as science and socialscience, you will see and use this number quite a bit. Use the approximations of e and the calculationsgiven below to answer the questions that follow. e is between 2 and 3e is between 2.7 and 2.8e is between 2.71 and 2.72e is between 2.718 and 2.719 Find a number that is between 2 and e. The Utah Middle School Math Project ###Explanation: If e is between 2 and 3, then any number between 2 and 3 would be a correct answer.

Write three expressions that are equivalent to 2x+4x-15 The Utah Middle School Math Project

Any equation that is equivalent to 2x + 4x - 15 would be a correct answer here. Some examples include 6x - 15 6x - 17 _ 2 2x + 4x - 16 + 1

###Write an explanation to this problem: Write three expressions that are equivalent to 2x+4x-15 The Utah Middle School Math Project ###Explanation: Any equation that is equivalent to 2x + 4x - 15 would be a correct answer here. Some examples include 6x - 15 6x - 17 _ 2 2x + 4x - 16 + 1

The following situation represents a functional relationship between two quantities. Identify the independent variable and the dependent variable. The value of your car decreases with age. The Utah Middle School Math Project

Value is dependent Age is independent

###Write an explanation to this problem: The following situation represents a functional relationship between two quantities. Identify the independent variable and the dependent variable. The value of your car decreases with age. The Utah Middle School Math Project ###Explanation: Value is dependent Age is independent

Make up a situation where a quantity is changing. Explain what it means to have a negative change.

To have a negative change means that you are going down in something.

###Write an explanation to this problem: Make up a situation where a quantity is changing. Explain what it means to have a negative change. ###Explanation: To have a negative change means that you are going down in something.

Write three expressions equivalent to the expression 4c. The Utah Middle School Math Project

There are many expressions that are equivalent to 4c. One example is the expression 2c + c + c.

###Write an explanation to this problem: Write three expressions equivalent to the expression 4c. The Utah Middle School Math Project ###Explanation: There are many expressions that are equivalent to 4c. One example is the expression 2c + c + c.

Write two questions that you could ask the students in your class that would result in categorical data. For each question, explain how you know that responses to it would produce categorical data.

One question you could ask that would result in categorical data would be: What is your favorite color? This would produce categorical data because color is a category and is not a logical/measured quantity.

###Write an explanation to this problem: Write two questions that you could ask the students in your class that would result in categorical data. For each question, explain how you know that responses to it would produce categorical data. ###Explanation: One question you could ask that would result in categorical data would be: What is your favorite color? This would produce categorical data because color is a category and is not a logical/measured quantity.

Kiran drinks 6.4 oz of milk each morning. How many days does it take him to finish a 32 oz container of milk? Write and solve an equation for the situation.

The equation will have to equal 32 oz, because that is the goal. Each day x he drinks 6.4 oz of milk. This will be multiplied. This means 6.4 times x will be equal to 32. 6.4x=32 To solve, divide both sides of the equation by 6.4, because this is the inverse of multiplying x by 6.4. 32/6.4 = 5, so x=5 days to finish a 32 oz container of milk.

###Write an explanation to this problem: Kiran drinks 6.4 oz of milk each morning. How many days does it take him to finish a 32 oz container of milk? Write and solve an equation for the situation. ###Explanation: The equation will have to equal 32 oz, because that is the goal. Each day x he drinks 6.4 oz of milk. This will be multiplied. This means 6.4 times x will be equal to 32. 6.4x=32 To solve, divide both sides of the equation by 6.4, because this is the inverse of multiplying x by 6.4. 32/6.4 = 5, so x=5 days to finish a 32 oz container of milk.

Over several months, the weight of a baby measured in pounds doubles. Does its weight measure in kilograms also double? Explain

1 kg = 2.2 lbs This ratio remains the same even as the child gets heavier. A double is a number that is twice itself. For example: our baby's weight doubles from 12 lbs to 24 lbs 12 lb= 26.4 kgs Kilograms doubled that would be 52.8 kgs 24 lbs= 52.8 kgs It did double.

###Write an explanation to this problem: Over several months, the weight of a baby measured in pounds doubles. Does its weight measure in kilograms also double? Explain ###Explanation: 1 kg = 2.2 lbs This ratio remains the same even as the child gets heavier. A double is a number that is twice itself. For example: our baby's weight doubles from 12 lbs to 24 lbs 12 lb= 26.4 kgs Kilograms doubled that would be 52.8 kgs 24 lbs= 52.8 kgs It did double.

Write an equation that represents each description. The opposite of negative seven

--7 = 7

###Write an explanation to this problem: Write an equation that represents each description. The opposite of negative seven ###Explanation: --7 = 7

Write a statement relating the opposites of each of the given numbers. 7 is greater than 6

-7 is less than -6

###Write an explanation to this problem: Write a statement relating the opposites of each of the given numbers. 7 is greater than 6 ###Explanation: -7 is less than -6

Find a value for x that makes -x less than 2x. copied for free from openupresources.org

1, because -1 is less than 2*1

###Write an explanation to this problem: Find a value for x that makes -x less than 2x. copied for free from openupresources.org ###Explanation: 1, because -1 is less than 2*1

A rectangle has a height to width ratio of 3:4.5. Give two examples of dimensions for rectangles that could be scaled versions of this rectangle.

The problem gives the ratio as: height : width 3 : 4.5 To find rectangles with dimensions that fit this ratio and can be scaled versions of the rectangle given in the problem we can multiply both sides of the ratio by the same factor. If we scale both sides by a factor of 2: 3*2 = 6 and 4.5*2 = 9 We get the dimensions 6 by 9. For a second rectangle, we can scale both sides by a different factor like 4: 3*4 = 12 and 4.5*4 = 18 We get the dimensions 12 by 18. There are many other dimensions that are correct answers, but as long as they fit the ratio given in the problem, they are correct.

###Write an explanation to this problem: A rectangle has a height to width ratio of 3:4.5. Give two examples of dimensions for rectangles that could be scaled versions of this rectangle. ###Explanation: The problem gives the ratio as: height : width 3 : 4.5 To find rectangles with dimensions that fit this ratio and can be scaled versions of the rectangle given in the problem we can multiply both sides of the ratio by the same factor. If we scale both sides by a factor of 2: 3*2 = 6 and 4.5*2 = 9 We get the dimensions 6 by 9. For a second rectangle, we can scale both sides by a different factor like 4: 3*4 = 12 and 4.5*4 = 18 We get the dimensions 12 by 18. There are many other dimensions that are correct answers, but as long as they fit the ratio given in the problem, they are correct.

A piece of paper is cut into two equal sections. Each new piece is cut into two additional pieces of equal size. This pattern continues until it is no longer possible to cut the paper any more. What is the dependent variable in this relationship? The Utah Middle School Math Project

The number of pieces of paper in this relationship will be depended on the number of cuts. Number of pieces of paper will increase with each cut. The number of pieces of paper is the dependent variables in this example.

###Write an explanation to this problem: A piece of paper is cut into two equal sections. Each new piece is cut into two additional pieces of equal size. This pattern continues until it is no longer possible to cut the paper any more. What is the dependent variable in this relationship? The Utah Middle School Math Project ###Explanation: The number of pieces of paper in this relationship will be depended on the number of cuts. Number of pieces of paper will increase with each cut. The number of pieces of paper is the dependent variables in this example.

Given information: 2m = Number of red marbles 3m = Number of blue marbles Equation: 2m + 3m = 155 marbles. Write a word problem to go with the information and the equation given above. The Utah Middle School Math Project

The ratio of red marbles to blue marbles is 2:3. The total number of marbles is 155. How many red marbles and blue marbles are there?

###Write an explanation to this problem: Given information: 2m = Number of red marbles 3m = Number of blue marbles Equation: 2m + 3m = 155 marbles. Write a word problem to go with the information and the equation given above. The Utah Middle School Math Project ###Explanation: The ratio of red marbles to blue marbles is 2:3. The total number of marbles is 155. How many red marbles and blue marbles are there?

The numbers below are listed in order from least to greatest. Give two possible integer values for the ? in the problem. Use a number line to help you. ?,-10,3,8 The Utah Middle School Math Project

In this question, the only requirement is than the number that replaces the ? is an integer that is less than -10. Any number that is farther from 0 than -10 is less than -10. Possible answers include: -11, -12, -25, and so on.

###Write an explanation to this problem: The numbers below are listed in order from least to greatest. Give two possible integer values for the ? in the problem. Use a number line to help you. ?,-10,3,8 The Utah Middle School Math Project ###Explanation: In this question, the only requirement is than the number that replaces the ? is an integer that is less than -10. Any number that is farther from 0 than -10 is less than -10. Possible answers include: -11, -12, -25, and so on.

Explain a process you can use to create a number line from -5 to 5. What steps can you take? What tools can you use? The Utah Middle School Math Project

To draw a number line from -5 to 5, you can start by using a ruler on a piece of paper and drawing a line that is a certain length, like 11 centimeters long. If we count from -5 to 5 ex: -5, -4, -3, -2..., we know that we will need 11 tick marks. We can draw 11 tick marks, equally spaced, along our line. If our line is 11 cm long, then there can be a tick mark starting at 0, and one at every centimeter mark after that. We know that -5 goes with the tick mark at the left end of the line. We know that 5 goes with the tick mark at the right end of the line. We know that 0 must go on the middle tick mark, since it is halfway between -5 and 5. Finally, we can fill in the rest of the tick marks.

###Write an explanation to this problem: Explain a process you can use to create a number line from -5 to 5. What steps can you take? What tools can you use? The Utah Middle School Math Project ###Explanation: To draw a number line from -5 to 5, you can start by using a ruler on a piece of paper and drawing a line that is a certain length, like 11 centimeters long. If we count from -5 to 5 ex: -5, -4, -3, -2..., we know that we will need 11 tick marks. We can draw 11 tick marks, equally spaced, along our line. If our line is 11 cm long, then there can be a tick mark starting at 0, and one at every centimeter mark after that. We know that -5 goes with the tick mark at the left end of the line. We know that 5 goes with the tick mark at the right end of the line. We know that 0 must go on the middle tick mark, since it is halfway between -5 and 5. Finally, we can fill in the rest of the tick marks.

Write an ordered pair that would be in Quadrant II when graphed. The Utah Middle School Math Project

Any ordered pair where x is negative and y is positive is in Quadrant II.

###Write an explanation to this problem: Write an ordered pair that would be in Quadrant II when graphed. The Utah Middle School Math Project ###Explanation: Any ordered pair where x is negative and y is positive is in Quadrant II.

Make a list to find the GCF of this pair of numbers. 20 and 64 First, write the list here. The Utah Middle School Math Project

Factors of 20: 1, 2, 4, 5, 10, 20Factors of 64: 1, 2, 4, 8, 16, 32, 64 A factor is a number that you can multiply to get another larger number.

###Write an explanation to this problem: Make a list to find the GCF of this pair of numbers. 20 and 64 First, write the list here. The Utah Middle School Math Project ###Explanation: Factors of 20: 1, 2, 4, 5, 10, 20Factors of 64: 1, 2, 4, 8, 16, 32, 64 A factor is a number that you can multiply to get another larger number.

Write three numbers that are less than -1. The Utah Middle School Math Project

Possible answers: -2, -4/3, -5.75

###Write an explanation to this problem: Write three numbers that are less than -1. The Utah Middle School Math Project ###Explanation: Possible answers: -2, -4/3, -5.75

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

Here is an equation: 4x - 4 = 4x + __. What could you write in the blank so the equation would be true for: No values of x Copied for free from openupresources.org

We have 4x on both sides of this equation. We know that if we insert a constant into this blank, then there would still only be 4x on both sides of the equation. If we subtract 4x from both sides, we get: -4 = ___ If we want this to not be true, then we would need for the constant in the blank to not equal -4. The answer is any constant not equal to -4.

###Write an explanation to this problem: Here is an equation: 4x - 4 = 4x + __. What could you write in the blank so the equation would be true for: No values of x Copied for free from openupresources.org ###Explanation: We have 4x on both sides of this equation. We know that if we insert a constant into this blank, then there would still only be 4x on both sides of the equation. If we subtract 4x from both sides, we get: -4 = ___ If we want this to not be true, then we would need for the constant in the blank to not equal -4. The answer is any constant not equal to -4.

In the following problem, a common error has been made when simplifying the expression. Identify the mistake, explain it, and simplify the expression correctly. 5-x-2 5-x-2 3-x Explain the mistake The Utah Middle School Math Project

Notice that the -1 was distributed to the x in the parentheses but not distributed to the -2 within the parentheses.

###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. 5-x-2 5-x-2 3-x Explain the mistake The Utah Middle School Math Project ###Explanation: Notice that the -1 was distributed to the x in the parentheses but not distributed to the -2 within the parentheses.

How are the ordered pairs 4, 9 and 4,-9 similar?

The two ordered pairs are similar because their x-coordinates are the same.

###Write an explanation to this problem: How are the ordered pairs 4, 9 and 4,-9 similar? ###Explanation: The two ordered pairs are similar because their x-coordinates are the same.

What does the expression represent?

This is the area, because the area is length times width.

###Write an explanation to this problem: What does the expression represent? ###Explanation: This is the area, because the area is length times width.

Solve the inequality. Represent the solution on a number line. 12>x+11 The Utah Middle School Math Project

Solve for x. The number line should be less than the solution, but not contain it.

###Write an explanation to this problem: Solve the inequality. Represent the solution on a number line. 12>x+11 The Utah Middle School Math Project ###Explanation: Solve for x. The number line should be less than the solution, but not contain it.

Write an ordered pair that would be in Quadrant IV when graphed. The Utah Middle School Math Project

Any ordered pair x, y where x is positive and y is negative is Quadrant IV. Examples: 1, -1 5, -3 1/2, -3/4

###Write an explanation to this problem: Write an ordered pair that would be in Quadrant IV when graphed. The Utah Middle School Math Project ###Explanation: Any ordered pair x, y where x is positive and y is negative is Quadrant IV. Examples: 1, -1 5, -3 1/2, -3/4

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.

Kent knows that subtracting a number is the same as adding it's opposite. He then uses this idea to rewrite the problem changing all of the subtraction parts to adding the opposite: -9.5 - -8 - 6.5 can be rewritten as: -9.5 + 8 + -6.5 and now Kent can use only addition to solve this 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: Kent knows that subtracting a number is the same as adding it's opposite. He then uses this idea to rewrite the problem changing all of the subtraction parts to adding the opposite: -9.5 - -8 - 6.5 can be rewritten as: -9.5 + 8 + -6.5 and now Kent can use only addition to solve this problem.

Fill in the blank with numbers that correctly complete the statement. Three rational numbers between -1 and -2 < <

-1.9 < -1.55 < -1.02

###Write an explanation to this problem: Fill in the blank with numbers that correctly complete the statement. Three rational numbers between -1 and -2 < < ###Explanation: -1.9 < -1.55 < -1.02

Callie and Jeff each have a job delivering newspapers. Jeff gets paid 140 dollars for delivering 350 papers. Callie gets paid 100 for delivering 200 papers. Find the unit rate for Callie The Utah Middle School Math Project

The unit rate can be either the amount of dollars for every one newspaper or the amount of newspapers for every one dollar. Unlike a regular rate, unit rates have a 1 in the denominator. To turn a rate into a unit rate, divide the numerator by the denominator. The rate that Callie was delivering papers is 100 per 200 papers or 100/200 papers. To get the unit rate divide 100 by 200 to get unit rate of 0.50/paper 0.50 is the unit rate of dollars to papers The rate that Callie delivered can also be expressed in papers per dollars. The rate would be 200 papers per 100. By simplifying the rate a unit rate of 2 papers per dollar is found. 2 is the unit rate of papers to dollars.

###Write an explanation to this problem: Callie and Jeff each have a job delivering newspapers. Jeff gets paid 140 dollars for delivering 350 papers. Callie gets paid 100 for delivering 200 papers. Find the unit rate for Callie The Utah Middle School Math Project ###Explanation: The unit rate can be either the amount of dollars for every one newspaper or the amount of newspapers for every one dollar. Unlike a regular rate, unit rates have a 1 in the denominator. To turn a rate into a unit rate, divide the numerator by the denominator. The rate that Callie was delivering papers is 100 per 200 papers or 100/200 papers. To get the unit rate divide 100 by 200 to get unit rate of 0.50/paper 0.50 is the unit rate of dollars to papers The rate that Callie delivered can also be expressed in papers per dollars. The rate would be 200 papers per 100. By simplifying the rate a unit rate of 2 papers per dollar is found. 2 is the unit rate of papers to dollars.

How could you distinguish between traveling west at 5 miles per hour and traveling east at 5 miles per hour without using the words east and west?

By using positive numbers for one direction and negative numbers for the other direction.

###Write an explanation to this problem: How could you distinguish between traveling west at 5 miles per hour and traveling east at 5 miles per hour without using the words east and west? ###Explanation: By using positive numbers for one direction and negative numbers for the other direction.

Nathan is your school's star soccer player. When he takes a shot on goal, he typically scores half of the time. Suppose that he takes six shots in a game. To estimate the probability of the number of goals Nathan makes, use simulation with a number cube. One roll of a number cube represents one shot. For this problem, what represents a trial of taking six shots?

Rolling the number cube six times would represent taking six shots, since one roll represents one shot.

###Write an explanation to this problem: Nathan is your school's star soccer player. When he takes a shot on goal, he typically scores half of the time. Suppose that he takes six shots in a game. To estimate the probability of the number of goals Nathan makes, use simulation with a number cube. One roll of a number cube represents one shot. For this problem, what represents a trial of taking six shots? ###Explanation: Rolling the number cube six times would represent taking six shots, since one roll represents one shot.

Andre came up with the following puzzle. I am three years younger than my brother, and I am 2 years older than my sister. My mom's age is one less than three times my brother's age. When you add all our ages, you get 87. What are our ages? Try to solve the puzzle.

To try to solve the puzzle, we can define each person's age in terms of Andre's age. Andre's age is x. He is 3 years younger than his brother, so his brother's age would be 3 more years than his age. His brother's age is x + 3. He is 2 years older than his sister. Her age would be his age minus 2 years. His sister's age is x - 2 His mom's age is 1 less than 3 times his brother's age. His brother's age is x + 3. 3 times his brother's age is 3x + 3 1 less than this is 3x + 3 - 1. If you add all of their ages, you get 87. x + x + 3 + x - 2 + 3x + 3 - 1 = 87 First, we should distribute the 3 to the terms in parentheses. x + x + 3 + x - 2 + 3x + 9 - 1 = 87 Next, we should combine like terms. 6x + 9 = 87 Now, we can solve for x by subtracting 9 from both sides. 6x = 78 x = 78/6 = 13 Andre is 13. His brother is 16. His sister is 11. His mother is 47.

###Write an explanation to this problem: Andre came up with the following puzzle. I am three years younger than my brother, and I am 2 years older than my sister. My mom's age is one less than three times my brother's age. When you add all our ages, you get 87. What are our ages? Try to solve the puzzle. ###Explanation: To try to solve the puzzle, we can define each person's age in terms of Andre's age. Andre's age is x. He is 3 years younger than his brother, so his brother's age would be 3 more years than his age. His brother's age is x + 3. He is 2 years older than his sister. Her age would be his age minus 2 years. His sister's age is x - 2 His mom's age is 1 less than 3 times his brother's age. His brother's age is x + 3. 3 times his brother's age is 3x + 3 1 less than this is 3x + 3 - 1. If you add all of their ages, you get 87. x + x + 3 + x - 2 + 3x + 3 - 1 = 87 First, we should distribute the 3 to the terms in parentheses. x + x + 3 + x - 2 + 3x + 9 - 1 = 87 Next, we should combine like terms. 6x + 9 = 87 Now, we can solve for x by subtracting 9 from both sides. 6x = 78 x = 78/6 = 13 Andre is 13. His brother is 16. His sister is 11. His mother is 47.

List the sample space for selecting a letter a random from the word PINEAPPLE.

P, I, N, E, A, L

###Write an explanation to this problem: List the sample space for selecting a letter a random from the word PINEAPPLE. ###Explanation: P, I, N, E, A, L

Write a story problem using sea level that includes both integers -110 and 120.

Answers may vary. On the beach, a man's kite flew at 120 feet above sea level, which is indicated by the water's surface. In the ocean, a white shark swims at 110 feet below the water's surface.

###Write an explanation to this problem: Write a story problem using sea level that includes both integers -110 and 120. ###Explanation: Answers may vary. On the beach, a man's kite flew at 120 feet above sea level, which is indicated by the water's surface. In the ocean, a white shark swims at 110 feet below the water's surface.

Determine if the relation or situation defines a function. Justify your answer. {-1, 0, 1, 2, 1, 4, 5, 2} The Utah Middle School Math Project

No, the data does not describe a function For the x value of 1 there are two y values, 2 and 4 This means each input does not have exactly one output, and is therefore not a function

###Write an explanation to this problem: Determine if the relation or situation defines a function. Justify your answer. {-1, 0, 1, 2, 1, 4, 5, 2} The Utah Middle School Math Project ###Explanation: No, the data does not describe a function For the x value of 1 there are two y values, 2 and 4 This means each input does not have exactly one output, and is therefore not a function

In the problem below, a starting equation and transformed equation have been given. State the action that produced the equivalent equation. Original Equation: x = 9 Transformed Equation: 6x = 54 Action: ______ The Utah Middle School Math Project

On the left side of the equation, x has been multiplied by 6.

###Write an explanation to this problem: In the problem below, a starting equation and transformed equation have been given. State the action that produced the equivalent equation. Original Equation: x = 9 Transformed Equation: 6x = 54 Action: ______ The Utah Middle School Math Project ###Explanation: On the left side of the equation, x has been multiplied by 6.

Create a context for the numeric representation. 801 + 80.40 = 801.40 = 112 The Utah Middle School Math Project

Sample Answer: If a person has 80 apples, then gains 40% more apples than the original 80, this numeric expression describes the total number of apples.

###Write an explanation to this problem: Create a context for the numeric representation. 801 + 80.40 = 801.40 = 112 The Utah Middle School Math Project ###Explanation: Sample Answer: If a person has 80 apples, then gains 40% more apples than the original 80, this numeric expression describes the total number of apples.

Draw a square with side length 7 cm. Predict the perimeter.

Use the information given in the problem to predict the perimeter of a square with a side length of 7 cm. Keep in mind that: A square has 4 side lengths of the same measurement The perimeter is the distance around the shape The problem says that the side length is 7 cm long To predict the perimeter, we can multiply the length of 1 side by 4 to get the sum of all of the lengths of the sides of the square. 7 cm * 4 sides = 28 cm A good prediction for the perimeter of the square would be 28 cm.

###Write an explanation to this problem: Draw a square with side length 7 cm. Predict the perimeter. ###Explanation: Use the information given in the problem to predict the perimeter of a square with a side length of 7 cm. Keep in mind that: A square has 4 side lengths of the same measurement The perimeter is the distance around the shape The problem says that the side length is 7 cm long To predict the perimeter, we can multiply the length of 1 side by 4 to get the sum of all of the lengths of the sides of the square. 7 cm * 4 sides = 28 cm A good prediction for the perimeter of the square would be 28 cm.

Describe the relative position of the numbers on a horizontal number line. -0.2 < -0.1

-0.2 is to the left of -0.1, or -0.1 is to the right of -0.2.

###Write an explanation to this problem: Describe the relative position of the numbers on a horizontal number line. -0.2 < -0.1 ###Explanation: -0.2 is to the left of -0.1, or -0.1 is to the right of -0.2.

Determine whether the three given side lengths can form a triangle. If you chose no, explain why it doesn't work. 5.1 inches, 3.24 inches, 8.25 inches The Utah Middle School Math Project

By the Triangle Inequality Theorem, the sum of the lengths of two sides is greater than the length of the third side. 5.1 in + 3.24 in = 8.34 in > 8.25 in 5.1 in + 8.25 in = 13.35 in > 3.24 in 3.24 in + 8.25 in = 11.49 in > 5.1 in Therefore, by Triangle Inequality, the three side lengths can form a triangle.

###Write an explanation to this problem: Determine whether the three given side lengths can form a triangle. If you chose no, explain why it doesn't work. 5.1 inches, 3.24 inches, 8.25 inches The Utah Middle School Math Project ###Explanation: By the Triangle Inequality Theorem, the sum of the lengths of two sides is greater than the length of the third side. 5.1 in + 3.24 in = 8.34 in > 8.25 in 5.1 in + 8.25 in = 13.35 in > 3.24 in 3.24 in + 8.25 in = 11.49 in > 5.1 in Therefore, by Triangle Inequality, the three side lengths can form a triangle.

Six teams are out on the field playing soccer. The teams all have the same number of players. The head coach asks for 2 players from each team to come help him move some equipment. NOW there are 78 players on the field. Write an equation whose solution is the number of players on each team. copied for free from openupresources.org

Sample Answer: An equation can be 6x-2 = 78.

###Write an explanation to this problem: Six teams are out on the field playing soccer. The teams all have the same number of players. The head coach asks for 2 players from each team to come help him move some equipment. NOW there are 78 players on the field. Write an equation whose solution is the number of players on each team. copied for free from openupresources.org ###Explanation: Sample Answer: An equation can be 6x-2 = 78.

Simplify the following expressions using two different methods. 9100 + 10 + 1 First, show your work for each method. The Utah Middle School Math Project

One method is to first distribute the 9 to the terms within the parentheses: 9100 + 10 + 1 = 9100 + 910 + 91 = 900 + 90 + 9 = 999 Another method is to first add the terms in the parentheses and then distribute the 9: 9100 + 10 + 1 = 9111 = 999

###Write an explanation to this problem: Simplify the following expressions using two different methods. 9100 + 10 + 1 First, show your work for each method. The Utah Middle School Math Project ###Explanation: One method is to first distribute the 9 to the terms within the parentheses: 9100 + 10 + 1 = 9100 + 910 + 91 = 900 + 90 + 9 = 999 Another method is to first add the terms in the parentheses and then distribute the 9: 9100 + 10 + 1 = 9111 = 999

Gavin is out shopping. After he buys 2 pairs of pants for 27 each, he has 60. How much money did he have before he bought the 2 pairs of pants? Write an equation to solve the problem. The Utah Middle School Math Project

Gavin has already spent 27 twice, and can spend 60 total. This means that m-227=60

###Write an explanation to this problem: Gavin is out shopping. After he buys 2 pairs of pants for 27 each, he has 60. How much money did he have before he bought the 2 pairs of pants? Write an equation to solve the problem. The Utah Middle School Math Project ###Explanation: Gavin has already spent 27 twice, and can spend 60 total. This means that m-227=60

Write two ratios that are equivalent to 3:11.

Answers will vary. You might write the following ratios: 6:22, 9:33, etc.

###Write an explanation to this problem: Write two ratios that are equivalent to 3:11. ###Explanation: Answers will vary. You might write the following ratios: 6:22, 9:33, etc.

9. Noah raised 54 to support the animal shelter, which is 60% of his fundraising goal. a. Write an equation to represent the situation. copied for free from openupresources.org

0.60x = 54 This is saying that 60% times some amount x is equal to 54.

###Write an explanation to this problem: 9. Noah raised 54 to support the animal shelter, which is 60% of his fundraising goal. a. Write an equation to represent the situation. copied for free from openupresources.org ###Explanation: 0.60x = 54 This is saying that 60% times some amount x is equal to 54.

Write three expressions that are equivalent to the expression + + 4 + 3. The Utah Middle School Math Project

Example: 2x+4x+12; 6x+12; x+x+4x+43

###Write an explanation to this problem: Write three expressions that are equivalent to the expression + + 4 + 3. The Utah Middle School Math Project ###Explanation: Example: 2x+4x+12; 6x+12; x+x+4x+43

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 can 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 can 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 a context to match the numeric expression given. 0.232 + 0.228 The Utah Middle School Math Project

Dom is hungry at the store. He wants to buy fancy ham that is 32 a pound, and magic beans that are 28 a pound. He is going to buy 0.2 pounds of each. How much is this going to cost?

###Write an explanation to this problem: Write a context to match the numeric expression given. 0.232 + 0.228 The Utah Middle School Math Project ###Explanation: Dom is hungry at the store. He wants to buy fancy ham that is 32 a pound, and magic beans that are 28 a pound. He is going to buy 0.2 pounds of each. How much is this going to cost?

Write an inequality that relates the rational numbers. In December, the total snowfall was 13.2 inches, which is more than the total snowfall in October and November, which was 3.7 inches and 6.15 inches, respectively.

13.2 > 6.15 > 3.7

###Write an explanation to this problem: Write an inequality that relates the rational numbers. In December, the total snowfall was 13.2 inches, which is more than the total snowfall in October and November, which was 3.7 inches and 6.15 inches, respectively. ###Explanation: 13.2 > 6.15 > 3.7

Create a real-life example that can be modeled by the expression -2 * 4.

Your answer might vary. One example of a real life situation is: Tobi spends 2 each week on a candy bar for four weeks. Write an integer to represent Tobi's change in money after four weeks. Tobi's money changes by -2 X 4 after four weeks.

###Write an explanation to this problem: Create a real-life example that can be modeled by the expression -2 * 4. ###Explanation: Your answer might vary. One example of a real life situation is: Tobi spends 2 each week on a candy bar for four weeks. Write an integer to represent Tobi's change in money after four weeks. Tobi's money changes by -2 X 4 after four weeks.

Using circles and squares, create pictures that represent the ratio given. The ratio of squares to circles is 3:5. Draw your pictures on paper, take a picture, and upload it using the image upload icon If you do not have the ability to upload an image of your work type Picture is on paper. The Utah Middle School Math Project

DO NOT DO THIS ON SEPARATE PAPER. TYPE C for CIRCLE and S for SQUARE into the box making the correct ratio.

###Write an explanation to this problem: Using circles and squares, create pictures that represent the ratio given. The ratio of squares to circles is 3:5. Draw your pictures on paper, take a picture, and upload it using the image upload icon If you do not have the ability to upload an image of your work type Picture is on paper. The Utah Middle School Math Project ###Explanation: DO NOT DO THIS ON SEPARATE PAPER. TYPE C for CIRCLE and S for SQUARE into the box making the correct ratio.

Are you ready for more? The effects of each move can be undone by using another move. For example, to undo the effect of translating 3 units to the right, we could translate 3 units to the left. What move undoes each of the following moves? 1. Translate 3 units up. Copied for free from openupresources.org

Translate 3 units down.

###Write an explanation to this problem: Are you ready for more? The effects of each move can be undone by using another move. For example, to undo the effect of translating 3 units to the right, we could translate 3 units to the left. What move undoes each of the following moves? 1. Translate 3 units up. Copied for free from openupresources.org ###Explanation: Translate 3 units down.

Round 1 of a tennis tournament starts with 64 players. After each round, half the players have lost and are eliminated from the tournament. Therefore, in round 2 there are 32 players, in round 3 there are 16 players and so on. What is the dependent variable in this relationship? The Utah Middle School Math Project

The number of players left is determined by the round number of the tournament. This means that players is the dependent variable in the equation.

###Write an explanation to this problem: Round 1 of a tennis tournament starts with 64 players. After each round, half the players have lost and are eliminated from the tournament. Therefore, in round 2 there are 32 players, in round 3 there are 16 players and so on. What is the dependent variable in this relationship? The Utah Middle School Math Project ###Explanation: The number of players left is determined by the round number of the tournament. This means that players is the dependent variable in the equation.

Write a phrase for the following algebraic expression 7 - x The Utah Middle School Math Project

7 - x is a number x less than seven.

###Write an explanation to this problem: Write a phrase for the following algebraic expression 7 - x The Utah Middle School Math Project ###Explanation: 7 - x is a number x less than seven.

What is a net? Describe it in your own words.

Answers will vary but should capture the essence of the definition used in this lesson. A net is a two-dimensional figure which can be folded to create a three-dimensional solid with no gaps or overlaps.

###Write an explanation to this problem: What is a net? Describe it in your own words. ###Explanation: Answers will vary but should capture the essence of the definition used in this lesson. A net is a two-dimensional figure which can be folded to create a three-dimensional solid with no gaps or overlaps.

A green light blinks every 4 seconds and a yellow light blinks every 5 seconds. When will both lights blink at the same time?

20 seconds You need to find the least common multiply of each number 4: 4, 8, 12, 16, 205: 5, 10, 15, 20

###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: 20 seconds You need to find the least common multiply of each number 4: 4, 8, 12, 16, 205: 5, 10, 15, 20

Describe some things you could notice in two polygons that would help you decide that they were not scaled copies.

I would check the angles in each polygon first. If any of the corresponding angle measures are not congruent they would not be scaled copies. After checking the angles, I would check the ratios of the corresponding sides to see if they are equivalent. If they are scaled copies, all of the ratios of the corresponding sides will be the same.

###Write an explanation to this problem: Describe some things you could notice in two polygons that would help you decide that they were not scaled copies. ###Explanation: I would check the angles in each polygon first. If any of the corresponding angle measures are not congruent they would not be scaled copies. After checking the angles, I would check the ratios of the corresponding sides to see if they are equivalent. If they are scaled copies, all of the ratios of the corresponding sides will be the same.

Round 1 of a tennis tournament starts with 64 players. After each round, half the players have lost and are eliminated from the tournament. Therefore, in round 2 there are 32 players, in round 3 there are 16 players and so on. What is the dependent variable in this relationship? The Utah Middle School Math Project

The number of players left is determined by the round of the tournament. This means that players is the dependent variable in the equation.

###Write an explanation to this problem: Round 1 of a tennis tournament starts with 64 players. After each round, half the players have lost and are eliminated from the tournament. Therefore, in round 2 there are 32 players, in round 3 there are 16 players and so on. What is the dependent variable in this relationship? The Utah Middle School Math Project ###Explanation: The number of players left is determined by the round of the tournament. This means that players is the dependent variable in the equation.

Norman purchased 3.5 lb. of his favorite mixture of dried fruits to use in a trail mix. The total cost was 16.87. How much does the fruit cost per pound? Estimate the quotient.

Estimate: 16 / 4 = 4

###Write an explanation to this problem: Norman purchased 3.5 lb. of his favorite mixture of dried fruits to use in a trail mix. The total cost was 16.87. How much does the fruit cost per pound? Estimate the quotient. ###Explanation: Estimate: 16 / 4 = 4

The owner of a new restaurant is ordering tables and chairs. He wants to have only tables for 2 and tables for 4. The total number of people that can be seated in the restaurant is 120. Describe some possible combinations of 2-seat tables and 4-seat tables that will seat 120 customers. Explain how you found them.

There are several combinations of tables that could be used to seat 120 customers. There could be no 2-seat and 30 4-seat, 10 2-seat and 25 4-seat, 40 2-seat and 10 4-seat. There are several ways to find the combinations, the number of 2-seat tables could be determined and then multiplied by 2 to find the number of people that could sit at those tables. Then the remaining number of people could be divided by 4 to get the number of 4-seat tables needed.

###Write an explanation to this problem: The owner of a new restaurant is ordering tables and chairs. He wants to have only tables for 2 and tables for 4. The total number of people that can be seated in the restaurant is 120. Describe some possible combinations of 2-seat tables and 4-seat tables that will seat 120 customers. Explain how you found them. ###Explanation: There are several combinations of tables that could be used to seat 120 customers. There could be no 2-seat and 30 4-seat, 10 2-seat and 25 4-seat, 40 2-seat and 10 4-seat. There are several ways to find the combinations, the number of 2-seat tables could be determined and then multiplied by 2 to find the number of people that could sit at those tables. Then the remaining number of people could be divided by 4 to get the number of 4-seat tables needed.

Are all rectangles scaled versions of each other? Justify your answer. The Utah Middle School Math Project

The answer is no. This is because the ratios can differ. Rectangles all have the same angles, but consecutive sides are not always the same ratio.

###Write an explanation to this problem: Are all rectangles scaled versions of each other? Justify your answer. The Utah Middle School Math Project ###Explanation: The answer is no. This is because the ratios can differ. Rectangles all have the same angles, but consecutive sides are not always the same ratio.

Write a statement relating the opposites of each of the given numbers. 7 is greater than 6

-7 is less than -6

###Write an explanation to this problem: Write a statement relating the opposites of each of the given numbers. 7 is greater than 6 ###Explanation: -7 is less than -6

A team has a 75% chance to win each of the 3 games they will play this week. Clare simulates the week of games by putting 4 pieces of paper in a bag, 3 labeled win and 1 labeled lose. She draws a paper, writes down the result, then replaces the paper and repeats the process two more times. Clare gets the result: win, win, lose. What can Clare do to estimate the probability the team will win at least 2 games?

Sample Answer: She needs to repeat the process many more times to get a good estimate of the probability. She has only done it once right now. After she has repeated the simulation of the week many times, she could count the fraction of simulated weeks that included at least 2 wins and use that as an estimate for the probability.

###Write an explanation to this problem: A team has a 75% chance to win each of the 3 games they will play this week. Clare simulates the week of games by putting 4 pieces of paper in a bag, 3 labeled win and 1 labeled lose. She draws a paper, writes down the result, then replaces the paper and repeats the process two more times. Clare gets the result: win, win, lose. What can Clare do to estimate the probability the team will win at least 2 games? ###Explanation: Sample Answer: She needs to repeat the process many more times to get a good estimate of the probability. She has only done it once right now. After she has repeated the simulation of the week many times, she could count the fraction of simulated weeks that included at least 2 wins and use that as an estimate for the probability.

In your own words, explain what a dilation is.

A dilation is a transformation that changes the size but not the shape of a figure.

###Write an explanation to this problem: In your own words, explain what a dilation is. ###Explanation: A dilation is a transformation that changes the size but not the shape of a figure.

Give an example of a rational number, and explain how you know it is rational. Copied for free from openupresources.org

A rational number is a fraction or its opposite or any number equivalent to a fraction or its opposite. Sample response: 2/3 is a rational number because rational numbers are fractions and their opposites and 2/3 is a fraction.

###Write an explanation to this problem: Give an example of a rational number, and explain how you know it is rational. Copied for free from openupresources.org ###Explanation: A rational number is a fraction or its opposite or any number equivalent to a fraction or its opposite. Sample response: 2/3 is a rational number because rational numbers are fractions and their opposites and 2/3 is a fraction.

Your teacher will show you two rods. Does it take more green rods or blue rods lined up end to end to measure the width of a piece of printer paper? copied for free from openupresources.org

More green rods are used because they are shorter than the blue rods.

###Write an explanation to this problem: Your teacher will show you two rods. Does it take more green rods or blue rods lined up end to end to measure the width of a piece of printer paper? copied for free from openupresources.org ###Explanation: More green rods are used because they are shorter than the blue rods.

Simplify the following expression using two different methods. Four multiplied by the sum of thirty and six First, show your work for each method. The Utah Middle School Math Project

The sum of thirty and six is 30 + 6, so four multiplied by the sum of thirty and six is 430 + 6 One method is to first distribute the 4 to the terms within the parentheses: 430 + 6 = 430 + 46 = 120 + 24 = 144 Another method is to first add the terms in the parentheses and then distribute the 4: 430 + 6 = 436 = 144

###Write an explanation to this problem: Simplify the following expression using two different methods. Four multiplied by the sum of thirty and six First, show your work for each method. The Utah Middle School Math Project ###Explanation: The sum of thirty and six is 30 + 6, so four multiplied by the sum of thirty and six is 430 + 6 One method is to first distribute the 4 to the terms within the parentheses: 430 + 6 = 430 + 46 = 120 + 24 = 144 Another method is to first add the terms in the parentheses and then distribute the 4: 430 + 6 = 436 = 144

Describe a situation that could be modeled with the ratio 4:1.

Answers may vary. You might write something like For every four teaspoons of cream in a cup of tea, there is one teaspoon of honey.

###Write an explanation to this problem: Describe a situation that could be modeled with the ratio 4:1. ###Explanation: Answers may vary. You might write something like For every four teaspoons of cream in a cup of tea, there is one teaspoon of honey.

Divide: 994.14 / 18.9 Estimate the quotient.

Estimate: 100 / 2 = 50

###Write an explanation to this problem: Divide: 994.14 / 18.9 Estimate the quotient. ###Explanation: Estimate: 100 / 2 = 50

Name the parts of the expression. 30jk

The parts of the expression are 30, j, k. 30 is the coefficient, and j and k are the variables, and 30jk is a term and the product of 30*j*k.

###Write an explanation to this problem: Name the parts of the expression. 30jk ###Explanation: The parts of the expression are 30, j, k. 30 is the coefficient, and j and k are the variables, and 30jk is a term and the product of 30*j*k.

Here is an inequality: -2x > 10. List some values for x that would make this inequality true.

Any number less than -5 would make this inequality true. Some examples: If x = -6, then -2-6 = 12 > 10. If x = -7.5 then -2 -7.5 = 15 > 10. If x = -10 frac12; then -2-10 frac12; = 21 > 10. If x = -5, then -2-5 = 10 which is not greater than 10.

###Write an explanation to this problem: Here is an inequality: -2x > 10. List some values for x that would make this inequality true. ###Explanation: Any number less than -5 would make this inequality true. Some examples: If x = -6, then -2-6 = 12 > 10. If x = -7.5 then -2 -7.5 = 15 > 10. If x = -10 frac12; then -2-10 frac12; = 21 > 10. If x = -5, then -2-5 = 10 which is not greater than 10.

Solve the inequality. Represent the solution on a number line. 9<d/3 The Utah Middle School Math Project

Solve for d. 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. 9<d/3 The Utah Middle School Math Project ###Explanation: Solve for d. The number line should be greater than the solution, but not contain it.

There are 88 seats in a theater. The seating in the theater is split into 4 identical sections. Each section has 14 red seats and some blue seats. Draw a tape diagram to represent the situation. Submit your tape diagram using the tools below.

All together the tape diagram would equal the 88 seats. The tape diagram should be split into 4 equal pieces to represent the seating sections. Inside each equal piece you should see 14 + x. The 14 represents the red seats and the x represents how many blue seats are there.

###Write an explanation to this problem: There are 88 seats in a theater. The seating in the theater is split into 4 identical sections. Each section has 14 red seats and some blue seats. Draw a tape diagram to represent the situation. Submit your tape diagram using the tools below. ###Explanation: All together the tape diagram would equal the 88 seats. The tape diagram should be split into 4 equal pieces to represent the seating sections. Inside each equal piece you should see 14 + x. The 14 represents the red seats and the x represents how many blue seats are there.

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.

Make a list to find the GCF of this pair of numbers. 20 and 64 First, write the list here. The Utah Middle School Math Project

Factors of 20: 1, 2, 4, 5, 10, 20Factors of 64: 1, 2, 4, 8, 16, 32, 64 Factors are numbers that you can multiply together to get another number.

###Write an explanation to this problem: Make a list to find the GCF of this pair of numbers. 20 and 64 First, write the list here. The Utah Middle School Math Project ###Explanation: Factors of 20: 1, 2, 4, 5, 10, 20Factors of 64: 1, 2, 4, 8, 16, 32, 64 Factors are numbers that you can multiply together to get another number.

Given information: a = Measure of angle A a + 20 = Measure of angle B a ndash; 10 = Measure of angle C Equation: a + a + 20 + a ndash; 10 = 180deg;. Write a word problem to go with the information and the equation given above. The Utah Middle School Math Project

In a triangle, the largest angle is 20 degrees more than the medium angle. The smallest angle is 10 degrees less than the medium angles. What is the measure of the three angles?

###Write an explanation to this problem: Given information: a = Measure of angle A a + 20 = Measure of angle B a ndash; 10 = Measure of angle C Equation: a + a + 20 + a ndash; 10 = 180deg;. Write a word problem to go with the information and the equation given above. The Utah Middle School Math Project ###Explanation: In a triangle, the largest angle is 20 degrees more than the medium angle. The smallest angle is 10 degrees less than the medium angles. What is the measure of the three angles?

Make a list to find the GCF of this pair of numbers. 20 and 64 First, write the list here. The Utah Middle School Math Project

Factors of 20: 1, 2, 4, 5, 10, 20Factors of 64: 1, 2, 4, 8, 16, 32, 64 The GCF is 4.

###Write an explanation to this problem: Make a list to find the GCF of this pair of numbers. 20 and 64 First, write the list here. The Utah Middle School Math Project ###Explanation: Factors of 20: 1, 2, 4, 5, 10, 20Factors of 64: 1, 2, 4, 8, 16, 32, 64 The GCF is 4.

Solve the inequality. Represent the solution on a number line. 2ge;0.4b The Utah Middle School Math Project

Solve for b. The number line should be less than and equal to the solution.

###Write an explanation to this problem: Solve the inequality. Represent the solution on a number line. 2ge;0.4b The Utah Middle School Math Project ###Explanation: Solve for b. The number line should be less than and equal to the solution.

Below is a statistical question. What data would you collect and analyze to answer each question? For numerical data, include the unit of measurement that you would use. What is a typical height of female athletes on a team in the most recent international sporting event?

This question will provide numerical data and should use a unit of measurement like inches or centimeters.

###Write an explanation to this problem: Below is a statistical question. What data would you collect and analyze to answer each question? For numerical data, include the unit of measurement that you would use. What is a typical height of female athletes on a team in the most recent international sporting event? ###Explanation: This question will provide numerical data and should use a unit of measurement like inches or centimeters.

The meat department manager at a grocery store is worried some of the packages of ground beef labeled as having one pound of meat may be under-filled. He decides to take a sample of 5 packages from a shipment containing 100 packages of ground beef. The packages were numbered as they were put in the box, so each one has a different number between 1 and 100. Describe how the manager can select a fair sample of 5 packages.

The manager could use a random number generator to pick a random number from 1 to 100 five times.

###Write an explanation to this problem: The meat department manager at a grocery store is worried some of the packages of ground beef labeled as having one pound of meat may be under-filled. He decides to take a sample of 5 packages from a shipment containing 100 packages of ground beef. The packages were numbered as they were put in the box, so each one has a different number between 1 and 100. Describe how the manager can select a fair sample of 5 packages. ###Explanation: The manager could use a random number generator to pick a random number from 1 to 100 five times.

Solve the inequality. Represent the solution on a number line. 8+n<11 The Utah Middle School Math Project

Solve for n. The number line should be less than the solution, but not contain it.

###Write an explanation to this problem: Solve the inequality. Represent the solution on a number line. 8+n<11 The Utah Middle School Math Project ###Explanation: Solve for n. The number line should be less than the solution, but not contain it.

Imagine cutting a square based right pyramid parallel to the base. What shape is the plane section? The Utah Middle School Math Project

Cutting a square based right pyramid parallel to the base will result in a square cross-section

###Write an explanation to this problem: Imagine cutting a square based right pyramid parallel to the base. What shape is the plane section? The Utah Middle School Math Project ###Explanation: Cutting a square based right pyramid parallel to the base will result in a square cross-section

Each statement contains a quantity. Rewrite each quantity using a power of 10. There are about 37 trillion cells in an average human body. copied for free from openupresources.org

3.7*1013

###Write an explanation to this problem: Each statement contains a quantity. Rewrite each quantity using a power of 10. There are about 37 trillion cells in an average human body. copied for free from openupresources.org ###Explanation: 3.7*1013

Let and ' be two lines with the same slope passing through the same point . Show that there is only one line with a slope , where < 0, passing through the given point . Draw a diagram if needed.

First, assume that there are two different lines l and l' with the same negative slope passing through P. From point P, mark a point Q one unit to the right. Then, draw a line parallel to the y-axis through point Q. The intersection of this line and line l and l' are noted with points R and R' respectively. By definition of slope, the lengths |QR| and |QR'| represent the slopes of lines l and l', respectively. It has been given that the lines have the same slope, which means that lengths |QR| and |QR'| are equal. Since that is true, then points R and R' coincide and so do lines l and l'. therefore, our assumption that they are different lines is false; l and l' must be the same line. Therefore, there is only one line with slope m passing through the given point P.

###Write an explanation to this problem: Let and ' be two lines with the same slope passing through the same point . Show that there is only one line with a slope , where < 0, passing through the given point . Draw a diagram if needed. ###Explanation: First, assume that there are two different lines l and l' with the same negative slope passing through P. From point P, mark a point Q one unit to the right. Then, draw a line parallel to the y-axis through point Q. The intersection of this line and line l and l' are noted with points R and R' respectively. By definition of slope, the lengths |QR| and |QR'| represent the slopes of lines l and l', respectively. It has been given that the lines have the same slope, which means that lengths |QR| and |QR'| are equal. Since that is true, then points R and R' coincide and so do lines l and l'. therefore, our assumption that they are different lines is false; l and l' must be the same line. Therefore, there is only one line with slope m passing through the given point P.

Write a pair of numbers whose GCF is 10. The Utah Middle School Math Project

Factors of 10: 1, 2, 5, 10 Factors of 20: 1, 2, 4, 5, 10, 20 GCF is 10

###Write an explanation to this problem: Write a pair of numbers whose GCF is 10. The Utah Middle School Math Project ###Explanation: Factors of 10: 1, 2, 5, 10 Factors of 20: 1, 2, 4, 5, 10, 20 GCF is 10

In the following problem, a common error has been made when simplifying the expression. Identify the mistake, explain it, and simplify the expression correctly. 2x+3+4x9x Explain the mistake The Utah Middle School Math Project

Notice that 2x + 3 + 4x = 6x + 3, since the 3 in this expression is not being multiplied by x.

###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. 2x+3+4x9x Explain the mistake The Utah Middle School Math Project ###Explanation: Notice that 2x + 3 + 4x = 6x + 3, since the 3 in this expression is not being multiplied by x.

Below is a statistical question. What data would you collect and analyze to answer each question? For numerical data, include the unit of measurement that you would use. What is a typical height of female athletes on a team in the most recent international sporting event?

Collect data on the heights of female athletes in a sport and use inches as the units.

###Write an explanation to this problem: Below is a statistical question. What data would you collect and analyze to answer each question? For numerical data, include the unit of measurement that you would use. What is a typical height of female athletes on a team in the most recent international sporting event? ###Explanation: Collect data on the heights of female athletes in a sport and use inches as the units.

How can you find 50% of a number quickly in your head?

50% is the same as half. The easy way to find 50% of a number is to divide it by 2.

###Write an explanation to this problem: How can you find 50% of a number quickly in your head? ###Explanation: 50% is the same as half. The easy way to find 50% of a number is to divide it by 2.

Imagine cutting a sphere with diameter 10 cm parallel to the table through the center. What shape will any plane section be? The Utah Middle School Math Project

The shape of any place section taken through the center of a sphere will be a circle.

###Write an explanation to this problem: Imagine cutting a sphere with diameter 10 cm parallel to the table through the center. What shape will any plane section be? The Utah Middle School Math Project ###Explanation: The shape of any place section taken through the center of a sphere will be a circle.

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

The numbers 1 through 428

###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: The numbers 1 through 428

7. Write three expressions containing exponents so that each expression equals 81. copied for free from openupresources.org

Answers will vary Here are some examples. 92 = 81because 9 x 9 = 81 34 = 81because 3 x 3 x 3 x = 81 811because 81 x 1 = 81

###Write an explanation to this problem: 7. Write three expressions containing exponents so that each expression equals 81. copied for free from openupresources.org ###Explanation: Answers will vary Here are some examples. 92 = 81because 9 x 9 = 81 34 = 81because 3 x 3 x 3 x = 81 811because 81 x 1 = 81