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Lin and Noah are solving the equation 7x+2=91 . Lin starts by using the distributive property. Noah starts by dividing each side by 7. Show what Lin's and Noah's full solution methods might look like. copied for free from openupresources.org

7x + 2 = 91 Lin used the distributive property to solve the equation. To solve she would have to multiply 7 times x and then 7 times 2. 7x + 14. Then subtract 14 from both sides. 7x = 77 Divide both sides by 7 x = 11 For Noah's method he took the same equation but divided each side by 7 first. 7x + 2 = 91 x + 2 = 91/7 x + 2 = 13 -2 - 2 x = 11 Both Lin and Noah solved the equations differently but still were able to come up with the same solution which was 11.

###Write an explanation to this problem: Lin and Noah are solving the equation 7x+2=91 . Lin starts by using the distributive property. Noah starts by dividing each side by 7. Show what Lin's and Noah's full solution methods might look like. copied for free from openupresources.org ###Explanation: 7x + 2 = 91 Lin used the distributive property to solve the equation. To solve she would have to multiply 7 times x and then 7 times 2. 7x + 14. Then subtract 14 from both sides. 7x = 77 Divide both sides by 7 x = 11 For Noah's method he took the same equation but divided each side by 7 first. 7x + 2 = 91 x + 2 = 91/7 x + 2 = 13 -2 - 2 x = 11 Both Lin and Noah solved the equations differently but still were able to come up with the same solution which was 11.

The equation d=3t represents the relationship between the distance d in inches that a snail is from a certain rock and the time t in minutes. What does the number 3 represent?

The number 3 is a coefficient. It is what we are multiplying our time by to get distance. Since we are getting units of distance from units of time, we must be multiplying by a rate of distance per unit time. This means that 3 is the constant of proportionality or the speed of the snail, 3 inches per minute.

###Write an explanation to this problem: The equation d=3t represents the relationship between the distance d in inches that a snail is from a certain rock and the time t in minutes. What does the number 3 represent? ###Explanation: The number 3 is a coefficient. It is what we are multiplying our time by to get distance. Since we are getting units of distance from units of time, we must be multiplying by a rate of distance per unit time. This means that 3 is the constant of proportionality or the speed of the snail, 3 inches per minute.

Give a number between -1 and -2. The Utah Middle School Math Project

Your answer can be any negative number that is less than -1 and greater than -2. Possible answers: -1 1/4, -1.01, -1.99, -3/2

###Write an explanation to this problem: Give a number between -1 and -2. The Utah Middle School Math Project ###Explanation: Your answer can be any negative number that is less than -1 and greater than -2. Possible answers: -1 1/4, -1.01, -1.99, -3/2

Write a word problem whose solution is |20| = 20.

Sample response: Kelli flew a kite 20 feet above the ground. Determine the distance between the kite and the ground.

###Write an explanation to this problem: Write a word problem whose solution is |20| = 20. ###Explanation: Sample response: Kelli flew a kite 20 feet above the ground. Determine the distance between the kite and the ground.

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?

Write three expressions that are equivalent to 3x + 15 The Utah Middle School Math Project

Any equation that is equivalent to 3x + 15 would be a correct answer here. Some examples include: 3x + 5 3x + 17 - 2 x + 2x + 15

###Write an explanation to this problem: Write three expressions that are equivalent to 3x + 15 The Utah Middle School Math Project ###Explanation: Any equation that is equivalent to 3x + 15 would be a correct answer here. Some examples include: 3x + 5 3x + 17 - 2 x + 2x + 15

22% of 54 Estimate the answer. The Utah Middle School Math Project

20% of 50 is 10, so 22% of 54 is close to 10.

###Write an explanation to this problem: 22% of 54 Estimate the answer. The Utah Middle School Math Project ###Explanation: 20% of 50 is 10, so 22% of 54 is close to 10.

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.

Han and Priya want to know the mean height of the 30 students in their dance class. They each select a random sample of 5 students. The mean height for Han's sample is 59 inches. The mean height for Priya's sample is 61 inches. Does it surprise you that the two sample means are different?

No, because the students would be a variety of heights. I would expect the sample means to be close but not necessarily exactly the same.

###Write an explanation to this problem: Han and Priya want to know the mean height of the 30 students in their dance class. They each select a random sample of 5 students. The mean height for Han's sample is 59 inches. The mean height for Priya's sample is 61 inches. Does it surprise you that the two sample means are different? ###Explanation: No, because the students would be a variety of heights. I would expect the sample means to be close but not necessarily exactly the same.

Draw a diagram to represent each situation: The number of miles driven this month was a 30% decrease of the number of miles driven last month. Submit your diagram using the tools below.

Sample Answer: A tape diagram showing 10 equal pieces labeled number of miles driven last month on the top with one below it that is just 7 pieces long and is labeled, number of miles driven this month.

###Write an explanation to this problem: Draw a diagram to represent each situation: The number of miles driven this month was a 30% decrease of the number of miles driven last month. Submit your diagram using the tools below. ###Explanation: Sample Answer: A tape diagram showing 10 equal pieces labeled number of miles driven last month on the top with one below it that is just 7 pieces long and is labeled, number of miles driven this month.

Why do whole numbers raised to an exponent get greater, while fractions raised to an exponent get smaller?

As whole numbers are multiplied by themselves, products are larger because there are more groups. As fractions of fractions are taken, the product is smaller. A part of a part is less than how much we started with.

###Write an explanation to this problem: Why do whole numbers raised to an exponent get greater, while fractions raised to an exponent get smaller? ###Explanation: As whole numbers are multiplied by themselves, products are larger because there are more groups. As fractions of fractions are taken, the product is smaller. A part of a part is less than how much we started with.

Write your own absolute value statement for which there is not a number that makes the statement true. The Utah Middle School Math Project

One example of an absolute value statement for which there is not a number that makes the statement true is: |x| = -6 The absolute value of x, our number, must always be positive, so -6 makes the statement impossible

###Write an explanation to this problem: Write your own absolute value statement for which there is not a number that makes the statement true. The Utah Middle School Math Project ###Explanation: One example of an absolute value statement for which there is not a number that makes the statement true is: |x| = -6 The absolute value of x, our number, must always be positive, so -6 makes the statement impossible

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

Sample Space is the letters that make up the word: A,E,I,L,N,P

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

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?

What you would collect is heights of female athletes and you would gather numerical data. The unit of measurement would be both feet and inches.

###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: What you would collect is heights of female athletes and you would gather numerical data. The unit of measurement would be both feet and inches.

For each expression, write an equivalent expression that uses only addition. 20 - 9 + 8 - 7 copied for free from openupresources.org 20-9+8-7

We know that subtracting a number is the same as adding its opposite. That means we can transform the expression: 20 - 9 + 8 - 7 20 + -9 + 8 + -7

###Write an explanation to this problem: For each expression, write an equivalent expression that uses only addition. 20 - 9 + 8 - 7 copied for free from openupresources.org 20-9+8-7 ###Explanation: We know that subtracting a number is the same as adding its opposite. That means we can transform the expression: 20 - 9 + 8 - 7 20 + -9 + 8 + -7

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

Sample Answer: Richard prepares tea for his workday. In the first hour of work, he drank 6 fl. oz. of the tea, which was 25% of the tea he prepared. How many fluid ounces of tea did he prepare? Solution: part: 6 fl.oz. percent: 25% = 0.25 We want to find the amount of tea Richard prepared started with. Note that 25% of the original amount is 6 fl. oz. 25% of the original amount is 0.25 times the original amount. With this information, we can write: 6 fl. oz. = original amount x 0.25 Now, we can solve for the original price: 6 fl. oz. = original amount x 0.25 6 fl. oz./0.25 = original amount 24 fl. oz. = original amount We get that Richard prepared 24 fl. oz. of tea.

###Write an explanation to this problem: Here, write the second 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: Richard prepares tea for his workday. In the first hour of work, he drank 6 fl. oz. of the tea, which was 25% of the tea he prepared. How many fluid ounces of tea did he prepare? Solution: part: 6 fl.oz. percent: 25% = 0.25 We want to find the amount of tea Richard prepared started with. Note that 25% of the original amount is 6 fl. oz. 25% of the original amount is 0.25 times the original amount. With this information, we can write: 6 fl. oz. = original amount x 0.25 Now, we can solve for the original price: 6 fl. oz. = original amount x 0.25 6 fl. oz./0.25 = original amount 24 fl. oz. = original amount We get that Richard prepared 24 fl. oz. of tea.

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

Note that we can simplify the inequality: -2x > 10 We can divide both sides by -2 to get one side equal to x. Since we are dividing both sides by a negative number, change the inequality sign from > to <. -2x/-2 < 10/-2 x < -5 Therefore, any number less than -5 is a solution.

###Write an explanation to this problem: Here is an inequality: -2x > 10. List some values for x that would make this inequality true. ###Explanation: Note that we can simplify the inequality: -2x > 10 We can divide both sides by -2 to get one side equal to x. Since we are dividing both sides by a negative number, change the inequality sign from > to <. -2x/-2 < 10/-2 x < -5 Therefore, any number less than -5 is a solution.

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.

There are multiple answers to this question, this is just an example: Are you on Instagram? Answers: Yes or No It will produce categorical data because they will be given answer choices to choose from. It will not be an open-ended question.

###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: There are multiple answers to this question, this is just an example: Are you on Instagram? Answers: Yes or No It will produce categorical data because they will be given answer choices to choose from. It will not be an open-ended question.

At Discovery Preschool, parents who have two students enrolled get a discount on the second child. The second child's tuition is 10 dollars less per day than the first child's. If Tess has her two children enrolled for 5 days and her total bill for both children is 200, how much does she pay each day for her second child to attend daycare? Write an equation to solve the question. The Utah Middle School Math Project

The tuition for one day for the first child is t and the tuition for one day for the second child is t-10. Both of these prices must be multiplied by 5, for the 5 days the children are in attendance. Then these values must be added together, and set equal to the total bill. This equation will be 5t+5t-10=200.

###Write an explanation to this problem: At Discovery Preschool, parents who have two students enrolled get a discount on the second child. The second child's tuition is 10 dollars less per day than the first child's. If Tess has her two children enrolled for 5 days and her total bill for both children is 200, how much does she pay each day for her second child to attend daycare? Write an equation to solve the question. The Utah Middle School Math Project ###Explanation: The tuition for one day for the first child is t and the tuition for one day for the second child is t-10. Both of these prices must be multiplied by 5, for the 5 days the children are in attendance. Then these values must be added together, and set equal to the total bill. This equation will be 5t+5t-10=200.

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.

Sample answer: I don't agree with Diego, since the quantities could be proportional if the line goes through the origin. Measurements are not perfect and the relationship could be proportional.

###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: Sample answer: I don't agree with Diego, since the quantities could be proportional if the line goes through the origin. Measurements are not perfect and the relationship could be proportional.

Kiran wrote the expression x - 10 for this number puzzle: Pick a number, add -2, and multiply by 5. Lin thinks Kiran made a mistake. How can she convince Kiran he made a mistake?

We can choose a value for x and then solve for the answer using Kiran's expression and using the number puzzle's phrasing. If x = 1, then the number puzzle should result in -5: Add -2: 1 + -2 = -1 Multiply by 5: -1 * 5 = -5 The number puzzle gives an answer of -5. If x = 1, then Kiran's expression would give: x - 10 1 - 10 = -9 Kiran's expression gives an answer of -9. The two answers do not match because Kiran's expression is wrong.

###Write an explanation to this problem: Kiran wrote the expression x - 10 for this number puzzle: Pick a number, add -2, and multiply by 5. Lin thinks Kiran made a mistake. How can she convince Kiran he made a mistake? ###Explanation: We can choose a value for x and then solve for the answer using Kiran's expression and using the number puzzle's phrasing. If x = 1, then the number puzzle should result in -5: Add -2: 1 + -2 = -1 Multiply by 5: -1 * 5 = -5 The number puzzle gives an answer of -5. If x = 1, then Kiran's expression would give: x - 10 1 - 10 = -9 Kiran's expression gives an answer of -9. The two answers do not match because Kiran's expression is wrong.

Give the dimensions of a right triangle and a parallelogram with the same area. Explain how you know.

Answers will vary.

###Write an explanation to this problem: Give the dimensions of a right triangle and a parallelogram with the same area. Explain how you know. ###Explanation: Answers will vary.

The following equation is given: x/2 ndash; 5 = 10 Provide a context/word problem for the given information. Write contexts that are more real life. The Utah Middle School Math Project

I spent half my money on clothes and then 5 on lunch. If I have 10 left over, how much did I start with?

###Write an explanation to this problem: The following equation is given: x/2 ndash; 5 = 10 Provide a context/word problem for the given information. Write contexts that are more real life. The Utah Middle School Math Project ###Explanation: I spent half my money on clothes and then 5 on lunch. If I have 10 left over, how much did I start with?

A hockey team has a 75% chance of winning against the opposing team in each game of a playoff series. To win the series, the team must be the first to win 4 games. Design a simulation for this event.

There are several ways that the event can be simulated. One way to simulate the event is to make a spinner with 4 equal sections labeled 1, 2, 3, and 4. Spin the spinner, and record the outcomes as wins if the spinner lands on 1, 2, or 3 or losses when the spinner lands on 4 until one team wins the series. This process can be repeated several times. Other simulations can be used as long as the odds of winning and losing are the same.

###Write an explanation to this problem: A hockey team has a 75% chance of winning against the opposing team in each game of a playoff series. To win the series, the team must be the first to win 4 games. Design a simulation for this event. ###Explanation: There are several ways that the event can be simulated. One way to simulate the event is to make a spinner with 4 equal sections labeled 1, 2, 3, and 4. Spin the spinner, and record the outcomes as wins if the spinner lands on 1, 2, or 3 or losses when the spinner lands on 4 until one team wins the series. This process can be repeated several times. Other simulations can be used as long as the odds of winning and losing are the same.

Think about this... In this lesson, you were given the slope and a point on the line and used this information to write the equation of the line in slope-intercept form. In the next lesson, you will be given 2 points and asked to write the equation in slope-intercept form. Write down your thoughts on how you might do this. The Utah Middle School Math Project

First find the slope and then plug in the slope and one of the points to find b. Now you have the slope and y-intercept needed to create the equation in slope-intercept form.

###Write an explanation to this problem: Think about this... In this lesson, you were given the slope and a point on the line and used this information to write the equation of the line in slope-intercept form. In the next lesson, you will be given 2 points and asked to write the equation in slope-intercept form. Write down your thoughts on how you might do this. The Utah Middle School Math Project ###Explanation: First find the slope and then plug in the slope and one of the points to find b. Now you have the slope and y-intercept needed to create the equation in slope-intercept form.

Write an inequality that relates the rational numbers. Sixteen degrees Celsius is warmer than zero degrees Celsius.

Since sixteen degrees Celsius is warmer than zero degrees Celsius the inequality is: 16>0

###Write an explanation to this problem: Write an inequality that relates the rational numbers. Sixteen degrees Celsius is warmer than zero degrees Celsius. ###Explanation: Since sixteen degrees Celsius is warmer than zero degrees Celsius the inequality is: 16>0

Describe the transformation graph shift that occurs in the following situation. Use words like translation, reflection, and rotation. The y-intercept is increased 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 increased while the slope is held constant, then we know that the line is being moved up. This can be described as a vertical translation up.

###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 increased 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 increased while the slope is held constant, then we know that the line is being moved up. This can be described as a vertical translation up.

At Discovery Preschool, parents who have two students enrolled get a discount on the second child. The second child's tuition is 10 dollars less per day than the first child's. If Tess has her two children enrolled for 5 days and her total bill for both children is 200, how much does she pay each day for her second child to attend daycare? Write an equation to solve the question. The Utah Middle School Math Project

The tuition for one day for the first child is t and the tuition for one day for the second child is t-10. Both prices must be multiplied by 5, for the 5 days the children are in attendance. Then these values must be added together, and set equal to the total bill. This equation will be 5t+5t-10=200.

###Write an explanation to this problem: At Discovery Preschool, parents who have two students enrolled get a discount on the second child. The second child's tuition is 10 dollars less per day than the first child's. If Tess has her two children enrolled for 5 days and her total bill for both children is 200, how much does she pay each day for her second child to attend daycare? Write an equation to solve the question. The Utah Middle School Math Project ###Explanation: The tuition for one day for the first child is t and the tuition for one day for the second child is t-10. Both prices must be multiplied by 5, for the 5 days the children are in attendance. Then these values must be added together, and set equal to the total bill. This equation will be 5t+5t-10=200.

Solve the inequality. Represent the solution on a number line. 7/8tge;56 The Utah Middle School Math Project

Solve for t. The number line should be greater than and equal to the solution.

###Write an explanation to this problem: Solve the inequality. Represent the solution on a number line. 7/8tge;56 The Utah Middle School Math Project ###Explanation: Solve for t. The number line should be greater than and equal to the solution.

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?

The weight and length of at least one 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: The weight and length of at least one beam.

What does the expression + + + represent?

This is the perimeter, because the perimeter is the length twice plus the width twice.

###Write an explanation to this problem: What does the expression + + + represent? ###Explanation: This is the perimeter, because the perimeter is the length twice plus the width twice.

Lin wants to know how many games teenagers in the United States have on their phones. What is the population for Lin's question?

All 13 to 19 year olds in the United States who have a phone.

###Write an explanation to this problem: Lin wants to know how many games teenagers in the United States have on their phones. What is the population for Lin's question? ###Explanation: All 13 to 19 year olds in the United States who have a phone.

If you flipped the coin 100 times, how many heads would you expect to get? Explain your answer. The Utah Middle School Math Project

Sample Answer: We would expect to get a head about 1/2 of the time. So, if we flip the coin 100 times, we would expect to get 1001/2 = 50 heads.

###Write an explanation to this problem: If you flipped the coin 100 times, how many heads would you expect to get? Explain your answer. The Utah Middle School Math Project ###Explanation: Sample Answer: We would expect to get a head about 1/2 of the time. So, if we flip the coin 100 times, we would expect to get 1001/2 = 50 heads.

Sean was at batting practice. He swung 25 times but only hit the ball 15 times. Describe and write more than one ratio related to this situation.

Ratio of the number of hits to the total number of swings is 15:25. Ratio of the number of hits to the number of misses is 15:10. Ratio of the number of misses to the number of hits is 10:15. Ratio of the number of misses to the total number of swings is 10:25.

###Write an explanation to this problem: Sean was at batting practice. He swung 25 times but only hit the ball 15 times. Describe and write more than one ratio related to this situation. ###Explanation: Ratio of the number of hits to the total number of swings is 15:25. Ratio of the number of hits to the number of misses is 15:10. Ratio of the number of misses to the number of hits is 10:15. Ratio of the number of misses to the total number of swings is 10:25.

The sentence is stated incorrectly. Rewrite the sentence to correctly describe each situation. The temperature is -10 degrees Fahrenheit below zero.

We don't say that a negative temperature is also below zero. In this case, we can either say the temperature is -10deg;F or we can say the temperature is 10 degrees below zero Fahrenheit.

###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: We don't say that a negative temperature is also below zero. In this case, we can either say the temperature is -10deg;F or we can say the temperature is 10 degrees below zero Fahrenheit.

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

The x-coordinates are the same.

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

Give a number between -6.1 and -6.2. The Utah Middle School Math Project

Your answer can be any negative number that is less than -6.1 and greater than -6.2. Possible answers: -6.11, -6.19, -6 1/2

###Write an explanation to this problem: Give a number between -6.1 and -6.2. The Utah Middle School Math Project ###Explanation: Your answer can be any negative number that is less than -6.1 and greater than -6.2. Possible answers: -6.11, -6.19, -6 1/2

Imagine cutting a cube, parallel to the base. What shape is the plane section? The Utah Middle School Math Project

If a cube was cut parallel to the base the resulting cross-section would have to be a square

###Write an explanation to this problem: Imagine cutting a cube, parallel to the base. What shape is the plane section? The Utah Middle School Math Project ###Explanation: If a cube was cut parallel to the base the resulting cross-section would have to be a square

3 x -6 = _____ Write the meaning of the multiplication problem. The Utah Middle School Math Project

This multiplication problem asks that you take the integer 3 and multiply it -6 times, so 3 x -6 can be thought of as -3 + -3 + -3 + -3 + -3 + -3

###Write an explanation to this problem: 3 x -6 = _____ Write the meaning of the multiplication problem. The Utah Middle School Math Project ###Explanation: This multiplication problem asks that you take the integer 3 and multiply it -6 times, so 3 x -6 can be thought of as -3 + -3 + -3 + -3 + -3 + -3

Based on your current preparation, summarize the four steps you are expected to complete as part of presenting a statistical study.

Step 1: State my statistical question. My question is based on collecting data that will vary. Step 2: Devise a plan to collect data. I prepared a question to ask the students in my class. Step 3: Summarize my data. I prepared a dot plot of the responses to the question. My dot plot indicated that the responses to my question were skewed to the left; therefore, I used the median of the data distribution to describe my center and the IQR to describe the variability. Step 4: Based on my graphs and numerical summaries, I answered my question.

###Write an explanation to this problem: Based on your current preparation, summarize the four steps you are expected to complete as part of presenting a statistical study. ###Explanation: Step 1: State my statistical question. My question is based on collecting data that will vary. Step 2: Devise a plan to collect data. I prepared a question to ask the students in my class. Step 3: Summarize my data. I prepared a dot plot of the responses to the question. My dot plot indicated that the responses to my question were skewed to the left; therefore, I used the median of the data distribution to describe my center and the IQR to describe the variability. Step 4: Based on my graphs and numerical summaries, I answered my question.

List two different possibilities for the angle measures of an isosceles triangle.

One possibility could be 40deg;, 40deg;, 100deg; because 40 + 40 + 100 = 180 and two of the angle measurements are the same. A second possibility could be 20deg;, 20deg;, 140deg; because 20 + 20 + 140 = 180 and two of the angle measurements are the same.

###Write an explanation to this problem: List two different possibilities for the angle measures of an isosceles triangle. ###Explanation: One possibility could be 40deg;, 40deg;, 100deg; because 40 + 40 + 100 = 180 and two of the angle measurements are the same. A second possibility could be 20deg;, 20deg;, 140deg; because 20 + 20 + 140 = 180 and two of the angle measurements are the same.

Write a situation that can be represented by the rational number. 0 The Utah Middle School Math Project

Any situation that involves the base level of something, like having no net profit or being at boiling temperature in degrees Fahrenheit, could be represented by 0. One example of a correct situation would be: At sea level.

###Write an explanation to this problem: Write a situation that can be represented by the rational number. 0 The Utah Middle School Math Project ###Explanation: Any situation that involves the base level of something, like having no net profit or being at boiling temperature in degrees Fahrenheit, could be represented by 0. One example of a correct situation would be: At sea level.

Five students came for after-school tutoring. Priya's teacher assigned each of them the same number of problems to complete. Then she assigned 2 more problems to one of the students. 27 problems were assigned in all. Write an equation for the story.

Let x be the number of problems the teacher assigned to all of the five students. An additional two problems are assigned to one of the students. So, an expression for the number of problems assigned would be 5x + 2. Also, we are given the total number of problems assigned, which is 27 problems. Now, we can set the expression equal to 27. Therefore, the equation is 5x + 2 = 27

###Write an explanation to this problem: Five students came for after-school tutoring. Priya's teacher assigned each of them the same number of problems to complete. Then she assigned 2 more problems to one of the students. 27 problems were assigned in all. Write an equation for the story. ###Explanation: Let x be the number of problems the teacher assigned to all of the five students. An additional two problems are assigned to one of the students. So, an expression for the number of problems assigned would be 5x + 2. Also, we are given the total number of problems assigned, which is 27 problems. Now, we can set the expression equal to 27. Therefore, the equation is 5x + 2 = 27

Show more than one way you could use the structure of the table below to find the unknown value. Number of Weeks Amount of Money in Account 2 350 4 700 6 1,050 8 A 10 B

I can add two to the weeks each time to get the next number. I can add 350 to the money to get the next values. In the rows, we have 2:350, which is equal to 1:175. So the money is 175 times larger than the week. I can just multiply the week by 175 to get the amount of money in the account. The ratio used to create the table was 1:175.

###Write an explanation to this problem: Show more than one way you could use the structure of the table below to find the unknown value. Number of Weeks Amount of Money in Account 2 350 4 700 6 1,050 8 A 10 B ###Explanation: I can add two to the weeks each time to get the next number. I can add 350 to the money to get the next values. In the rows, we have 2:350, which is equal to 1:175. So the money is 175 times larger than the week. I can just multiply the week by 175 to get the amount of money in the account. The ratio used to create the table was 1:175.

Given information: e = Elisabeth's age now e ndash; 7 = Zack's age now e + 4 = Gail's age now 2e = Bob's age now Equation: e + e - 7 + e + 4 + 2e = 82 Write a word problem to go with the information and the equation given above. The Utah Middle School Math Project

Zack is 7 years younger than Elizabeth, Gail is 4 years older than Elizabeth, and Bob is 2 times the age of Elizabeth. The sum of their ages is 82. How old are Elizabeth, Zack, Gail and Bob?

###Write an explanation to this problem: Given information: e = Elisabeth's age now e ndash; 7 = Zack's age now e + 4 = Gail's age now 2e = Bob's age now Equation: e + e - 7 + e + 4 + 2e = 82 Write a word problem to go with the information and the equation given above. The Utah Middle School Math Project ###Explanation: Zack is 7 years younger than Elizabeth, Gail is 4 years older than Elizabeth, and Bob is 2 times the age of Elizabeth. The sum of their ages is 82. How old are Elizabeth, Zack, Gail and Bob?

Create a representation of a context story that can be modeled by a nonlinear function. The Utah Middle School Math Project

There are many different stories that can be created that represent a nonlinear function. One possible example is: Kelly hears a rumor about the class receiving extra recess time. She tells two of her friends, who each tell two more of their friends. Each person that hears the rumor tells two more people. How many people have heard the rumor in 4 hours if it is shared every 30 minutes?

###Write an explanation to this problem: Create a representation of a context story that can be modeled by a nonlinear function. The Utah Middle School Math Project ###Explanation: There are many different stories that can be created that represent a nonlinear function. One possible example is: Kelly hears a rumor about the class receiving extra recess time. She tells two of her friends, who each tell two more of their friends. Each person that hears the rumor tells two more people. How many people have heard the rumor in 4 hours if it is shared every 30 minutes?

The numbers below are listed in order from least to greatest. Give two possible integer values for the ? in each problem. Use a number line to help you -2,-1,0,? The Utah Middle School Math Project

In this question, the only requirement is than the number that replaces the ? is an integer that is greater than 0. Any number that is not 0 in the positive direction is greater than 0, so any number like 1, 2, 5, 10, and so on is a correct answer.

###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 each problem. Use a number line to help you -2,-1,0,? 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 greater than 0. Any number that is not 0 in the positive direction is greater than 0, so any number like 1, 2, 5, 10, and so on is a correct answer.

There is a bag of 50 marbles. Andre takes out a marble, records its color, and puts it back in. In 4 trials, he gets a green marble 1 time. Jada takes out a marble, records its color, and puts it back in. In 12 trials, she gets a green marble 5 times. Noah takes out a marble, records its color, and puts it back in. In 9 trials, he gets a green marble 3 times. Estimate the probability of getting a green marble from this bag.

Sample Answer: A good estimate of the probability of getting a green marble comes from combining Andre, Jada, and Noah's trials. They took a marble out of the bag a total of 25 times and got a green marble 9 of those times. So, the probability of getting a green marble appears to be close to 9/25=0.36. Since there are 50 marbles in the bag, it is a reasonable estimate that 18 of the 50 marbles are green, though this is not guaranteed.

###Write an explanation to this problem: There is a bag of 50 marbles. Andre takes out a marble, records its color, and puts it back in. In 4 trials, he gets a green marble 1 time. Jada takes out a marble, records its color, and puts it back in. In 12 trials, she gets a green marble 5 times. Noah takes out a marble, records its color, and puts it back in. In 9 trials, he gets a green marble 3 times. Estimate the probability of getting a green marble from this bag. ###Explanation: Sample Answer: A good estimate of the probability of getting a green marble comes from combining Andre, Jada, and Noah's trials. They took a marble out of the bag a total of 25 times and got a green marble 9 of those times. So, the probability of getting a green marble appears to be close to 9/25=0.36. Since there are 50 marbles in the bag, it is a reasonable estimate that 18 of the 50 marbles are green, though this is not guaranteed.

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.

Since 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, the inequality is: 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: Since 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, the inequality is: 13.2 > 6.15 > 3.7

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.

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 find more ratios of cups of pink paint to cups of blue paint, you will have to apply a scale factor. For example, 6 to 5 is equivalent to 12 to 10 6*2 = 12 and 5 *2=10 when we apply a scale factor of 2. Here is another example: 6 to 5 is equivalent to 18 to 15 6*3=18 and 5*3=15 Here is one more example: 6 to 5 is equivalent to 3 to 2.5 6 * 1/2 = 3 and 5 * 1/2 = 2.5

###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 find more ratios of cups of pink paint to cups of blue paint, you will have to apply a scale factor. For example, 6 to 5 is equivalent to 12 to 10 6*2 = 12 and 5 *2=10 when we apply a scale factor of 2. Here is another example: 6 to 5 is equivalent to 18 to 15 6*3=18 and 5*3=15 Here is one more example: 6 to 5 is equivalent to 3 to 2.5 6 * 1/2 = 3 and 5 * 1/2 = 2.5

Explain why zero is considered its own opposite.

Numbers have an opposite because the number line has a line of symmetry on 0. Since 0 is the mid-point, it's also its own opposite. The distance from 0 to 0 is 0.

###Write an explanation to this problem: Explain why zero is considered its own opposite. ###Explanation: Numbers have an opposite because the number line has a line of symmetry on 0. Since 0 is the mid-point, it's also its own opposite. The distance from 0 to 0 is 0.

-7 x -2 = _____ Write the meaning of the multiplication problem. The Utah Middle School Math Project

This multiplication problem asks that you take the integer -7 and multiply it -2 times, so -7 x -2 can be thought of as the opposite of -7 x 2, which is -7 + -7. The opposite of -7 + -7 is 7 + 7

###Write an explanation to this problem: -7 x -2 = _____ Write the meaning of the multiplication problem. The Utah Middle School Math Project ###Explanation: This multiplication problem asks that you take the integer -7 and multiply it -2 times, so -7 x -2 can be thought of as the opposite of -7 x 2, which is -7 + -7. The opposite of -7 + -7 is 7 + 7

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.

Create a statistical question that could be answered by collecting numerical data. The Utah Middle School Math Project

A statistical question is one that can be answered by collecting a data set of varying answers. If the data is numerical, each piece of data is a number that represents a count or measure of something. One example of a statistical question that can be answered with numerical data: How many hours does each person in your class spend on the internet on a given day?

###Write an explanation to this problem: Create a statistical question that could be answered by collecting numerical data. The Utah Middle School Math Project ###Explanation: A statistical question is one that can be answered by collecting a data set of varying answers. If the data is numerical, each piece of data is a number that represents a count or measure of something. One example of a statistical question that can be answered with numerical data: How many hours does each person in your class spend on the internet on a given day?

The Utah Middle School Math Project

Choose four arbitrary values for x such as 0,1,2,3 and solve for y for each of these values, and put these four points in the chart. Then plot these points on the graph, and connect the points in a line.

###Write an explanation to this problem: The Utah Middle School Math Project ###Explanation: Choose four arbitrary values for x such as 0,1,2,3 and solve for y for each of these values, and put these four points in the chart. Then plot these points on the graph, and connect the points in a line.

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

The following situation represents a functional relationship between two quantities. Identify the independent variable and the dependent variable. The greater the distance a sprinter has to run the more time it takes to finish the race The Utah Middle School Math Project

Distance is independent variable, because the sprinter can go any reasonable distance within any time. The Time 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. The greater the distance a sprinter has to run the more time it takes to finish the race The Utah Middle School Math Project ###Explanation: Distance is independent variable, because the sprinter can go any reasonable distance within any time. The Time is the dependent variable.

The weather forecast says there is a 75% chance it will rain later today. Draw a spinner you could use to simulate this probability. Submit your drawing using the tools below.

There are different ways to represent the given information on a spinner. One possible answer is shown below. It is a circle spinner with 3/4 colored in blue and labeled rain. The other 1/4 is yellow and labeled no rain.

###Write an explanation to this problem: The weather forecast says there is a 75% chance it will rain later today. Draw a spinner you could use to simulate this probability. Submit your drawing using the tools below. ###Explanation: There are different ways to represent the given information on a spinner. One possible answer is shown below. It is a circle spinner with 3/4 colored in blue and labeled rain. The other 1/4 is yellow and labeled no rain.

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

Possible answers: -2, -10, -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, -10, -4/3, -5.75

Create a statistical question that could be answered by collecting categorical data. The Utah Middle School Math Project

A statistical question is one that can be answered by collecting a data set of varying answers. If the data is categorical, each piece of data is a word or category. One example of a statistical question that can be answered with categorical data: What color are the eyes of each person in your class?

###Write an explanation to this problem: Create a statistical question that could be answered by collecting categorical data. The Utah Middle School Math Project ###Explanation: A statistical question is one that can be answered by collecting a data set of varying answers. If the data is categorical, each piece of data is a word or category. One example of a statistical question that can be answered with categorical data: What color are the eyes of each person in your class?

Think about your work today, and write your best definition of area.

Area is the amount of space inside a shape or polygon. It is measured in square units.

###Write an explanation to this problem: Think about your work today, and write your best definition of area. ###Explanation: Area is the amount of space inside a shape or polygon. It is measured in square units.

Using circles and squares, create pictures that represent the ratio given. The ratio of circles to squares 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 circles to squares 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.

Write a story that matches the expressions and equations. World Languages Number of years: t of students in French: 160 - 9t of students in Spanish: 85 + 6t 160 - 9t = 85 + 6t The Utah Middle School Math Project

Initially at t = 0 years, there must be 160 students in French and 85 students in Spanish. Each year, the number of students in French will decrease by 9 and the number of students in Spanish will increase by 6. So 160 - 9t is the number of students in French after t years and 85 + 6t is the number of students in Spanish after t years. When 160 - 9t = 85 + 6t, there will be the same number f students in French class as there are in Spanish class. After how many years will the number of students in French class be the same as the number of students in Spanish class?

###Write an explanation to this problem: Write a story that matches the expressions and equations. World Languages Number of years: t of students in French: 160 - 9t of students in Spanish: 85 + 6t 160 - 9t = 85 + 6t The Utah Middle School Math Project ###Explanation: Initially at t = 0 years, there must be 160 students in French and 85 students in Spanish. Each year, the number of students in French will decrease by 9 and the number of students in Spanish will increase by 6. So 160 - 9t is the number of students in French after t years and 85 + 6t is the number of students in Spanish after t years. When 160 - 9t = 85 + 6t, there will be the same number f students in French class as there are in Spanish class. After how many years will the number of students in French class be the same as the number of students in Spanish class?

Decide which measure of center makes the most sense to use based on the distribution of your sample. Discuss your thinking with your partner. If you disagree, work to reach an agreement. copied for free from openupresources.org

I think the mean would be the best measure because there were no outliers in the data.

###Write an explanation to this problem: Decide which measure of center makes the most sense to use based on the distribution of your sample. Discuss your thinking with your partner. If you disagree, work to reach an agreement. copied for free from openupresources.org ###Explanation: I think the mean would be the best measure because there were no outliers in the data.

Invent a data set with five numbers that has a mean of 10 and a median of 12.

There are several data sets that could work for this scenario but here is an example of one that would work: 3, 11, 12, 12, 12 12 is the median because it is the middle number. 10 is the mean because the average add up all the values and divide by 5 you would get 10.

###Write an explanation to this problem: Invent a data set with five numbers that has a mean of 10 and a median of 12. ###Explanation: There are several data sets that could work for this scenario but here is an example of one that would work: 3, 11, 12, 12, 12 12 is the median because it is the middle number. 10 is the mean because the average add up all the values and divide by 5 you would get 10.

Ramon is collecting donations for a charity. He is hoping to raise 250. If he has already collected 187 in donations, how much more does he need to collect? Write an equation to solve the problem. The Utah Middle School Math Project

Ramon already has 187, and needs m more to get to 250. This means that 187+m=250

###Write an explanation to this problem: Ramon is collecting donations for a charity. He is hoping to raise 250. If he has already collected 187 in donations, how much more does he need to collect? Write an equation to solve the problem. The Utah Middle School Math Project ###Explanation: Ramon already has 187, and needs m more to get to 250. This means that 187+m=250

Create your own context or story that represents a relation that is 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 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.

A recipe for bread calls for 1 teaspoon of yeast for every 2 cups of flour. Name two quantities in this situation that are in a functional relationship. Copied for free from openupresources.org

The amount of yeast and the amount of flour are in a functional relationship. This means that if we know the amount of flour, then we can find the corresponding amount of yeast using a function, or the reverse.

###Write an explanation to this problem: A recipe for bread calls for 1 teaspoon of yeast for every 2 cups of flour. Name two quantities in this situation that are in a functional relationship. Copied for free from openupresources.org ###Explanation: The amount of yeast and the amount of flour are in a functional relationship. This means that if we know the amount of flour, then we can find the corresponding amount of yeast using a function, or the reverse.

The following situation represents a functional relationship between two quantities. Identify the independent variable and the dependent variable. Write your own relationship that contains an independent and dependent variable. The Utah Middle School Math Project

Possible Example, The size of a balloon depends on how long you are inflating it.

###Write an explanation to this problem: The following situation represents a functional relationship between two quantities. Identify the independent variable and the dependent variable. Write your own relationship that contains an independent and dependent variable. The Utah Middle School Math Project ###Explanation: Possible Example, The size of a balloon depends on how long you are inflating it.

You are starting a small business mowing lawns in your neighborhood for the summer. Create two statistical questions that will help make your lawn mowing business a success. The Utah Middle School Math Project

One example of an important question to ask would be: How many times in a summer does each house like to mow their lawn? How much is each person willing to pay someone to mow their lawn? You can collect data which will vary from house to house in order to answer these questions. They will help make the lawn mowing business a success by helping you figure out how many lawns you can mow, and if you know the amount of money you can make per lawn, you can estimate how much money you will make and make a budget from that.

###Write an explanation to this problem: You are starting a small business mowing lawns in your neighborhood for the summer. Create two statistical questions that will help make your lawn mowing business a success. The Utah Middle School Math Project ###Explanation: One example of an important question to ask would be: How many times in a summer does each house like to mow their lawn? How much is each person willing to pay someone to mow their lawn? You can collect data which will vary from house to house in order to answer these questions. They will help make the lawn mowing business a success by helping you figure out how many lawns you can mow, and if you know the amount of money you can make per lawn, you can estimate how much money you will make and make a budget from that.

The equation c=2.95g shows how much it costs to buy gas at a gas station on a certain day. In the equation, c represents the cost in dollars, and g represents how many gallons of gas were purchased. Write down at least four gallons of gas, cost pairs that fit this relationship.

1, 2.95 2,5.80 4, 10.60 0,0

###Write an explanation to this problem: The equation c=2.95g shows how much it costs to buy gas at a gas station on a certain day. In the equation, c represents the cost in dollars, and g represents how many gallons of gas were purchased. Write down at least four gallons of gas, cost pairs that fit this relationship. ###Explanation: 1, 2.95 2,5.80 4, 10.60 0,0

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

Example: A net is a two-dimensional figure that can be folded to create a three-dimensional solid.

###Write an explanation to this problem: What is a net? Describe it in your own words. ###Explanation: Example: A net is a two-dimensional figure that can be folded to create a three-dimensional solid.

Represent 56 / 8 = 7 using subtraction. Explain your reasoning.

56 - 7 - 7 - 7 - 7 - 7 - 7 - 7 - 7 = 0 because 56 - 7 = 49; 49 - 7 = 42; 42 - 7 = 35; 35 - 7 = 28; 28 - 7 = 21; 21 - 7 = 14; 14 - 7 = 7; 7 - 7 = 0 OR 56 - 8 - 8 - 8 - 8 - 8 - 8 - 8 = 0 because 56 - 8 = 48; 48 - 8 = 40; 40 - 8 = 32; 32 - 8 = 24; 24 - 8 = 16; 16 - 8 = 8; 8 - 8 = 0

###Write an explanation to this problem: Represent 56 / 8 = 7 using subtraction. Explain your reasoning. ###Explanation: 56 - 7 - 7 - 7 - 7 - 7 - 7 - 7 - 7 = 0 because 56 - 7 = 49; 49 - 7 = 42; 42 - 7 = 35; 35 - 7 = 28; 28 - 7 = 21; 21 - 7 = 14; 14 - 7 = 7; 7 - 7 = 0 OR 56 - 8 - 8 - 8 - 8 - 8 - 8 - 8 = 0 because 56 - 8 = 48; 48 - 8 = 40; 40 - 8 = 32; 32 - 8 = 24; 24 - 8 = 16; 16 - 8 = 8; 8 - 8 = 0

Write a situation that can be represented by the rational number. 3.25 The Utah Middle School Math Project

Any situation that could be represented by the rational number 3.25 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: 3.25 feet ahead

###Write an explanation to this problem: Write a situation that can be represented by the rational number. 3.25 The Utah Middle School Math Project ###Explanation: Any situation that could be represented by the rational number 3.25 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: 3.25 feet ahead

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

If they were not the same shape for example, if one was a triangle and one was a square, they could not be scaled copies. I could find an angle measure in one that was not an angle measure of the other. I could find that a different scale factor would have to be used on one part of the pair than on another.

###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: If they were not the same shape for example, if one was a triangle and one was a square, they could not be scaled copies. I could find an angle measure in one that was not an angle measure of the other. I could find that a different scale factor would have to be used on one part of the pair than on another.

In the problem below, a starting equation and transformed equation have been given. State the action that produced the equivalent equation. Original Equation: x = 6 Transformed Equation: x + 2 = 8 Action: ______ The Utah Middle School Math Project

On the left side of the equation, 2 has been added to x. 2 has also been added to 6 on the right side, to make 8. The action was: Add 2 to both sides of the equation.

###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 = 6 Transformed Equation: x + 2 = 8 Action: ______ The Utah Middle School Math Project ###Explanation: On the left side of the equation, 2 has been added to x. 2 has also been added to 6 on the right side, to make 8. The action was: Add 2 to both sides of the equation.

Choose a situation that could be described by the following ratio. Write a sentence to describe the ratio in the context of the situation you chose. 1 to 2

A sample could be for every one person there are 2 eyes.

###Write an explanation to this problem: Choose a situation that could be described by the following ratio. Write a sentence to describe the ratio in the context of the situation you chose. 1 to 2 ###Explanation: A sample could be for every one person there are 2 eyes.

Write a phrase for the following algebraic expression 15 - 2x The Utah Middle School Math Project

15 - 2x is 2 times a number x less than 15.

###Write an explanation to this problem: Write a phrase for the following algebraic expression 15 - 2x The Utah Middle School Math Project ###Explanation: 15 - 2x is 2 times a number x less than 15.

Sixth-grade students were asked, What grade are you in? Explain why this is not a statistical question.

This is NOT a statistical question because every answer would be 6th grade which will not produce a variety of answers. There would be no data to interpret with this question.

###Write an explanation to this problem: Sixth-grade students were asked, What grade are you in? Explain why this is not a statistical question. ###Explanation: This is NOT a statistical question because every answer would be 6th grade which will not produce a variety of answers. There would be no data to interpret with this question.

Simplify the following expressions using two different methods. Three multiplied by the difference of fifty and one First, show your work for each method. The Utah Middle School Math Project

The difference of fifty and one is 50 - 1, so three multiplied by the difference of fifty and one 350 - 1 One method is to first distribute the 3 to the terms within the parentheses: 350 - 1 = 350 - 31 = 150 - 3 = 147 Another method is to first combine the terms in the parentheses and then distribute the 3: 350 - 1 = 349 = 147

###Write an explanation to this problem: Simplify the following expressions using two different methods. Three multiplied by the difference of fifty and one First, show your work for each method. The Utah Middle School Math Project ###Explanation: The difference of fifty and one is 50 - 1, so three multiplied by the difference of fifty and one 350 - 1 One method is to first distribute the 3 to the terms within the parentheses: 350 - 1 = 350 - 31 = 150 - 3 = 147 Another method is to first combine the terms in the parentheses and then distribute the 3: 350 - 1 = 349 = 147

The height of the water in a tank decreases by 3.5 cm each day. When the tank is full, the water is 10 m deep. The water tank needs to be refilled when the water height drops below 4 m. Write a question that could be answered by solving the equation 10 - 0.035d = 4.

How many days will pass before the water tank needs to be refilled?

###Write an explanation to this problem: The height of the water in a tank decreases by 3.5 cm each day. When the tank is full, the water is 10 m deep. The water tank needs to be refilled when the water height drops below 4 m. Write a question that could be answered by solving the equation 10 - 0.035d = 4. ###Explanation: How many days will pass before the water tank needs to be refilled?

A carnival game has 160 rubber ducks floating in a pool. The person playing the game takes out one duck and looks at it. If there's a red mark on the bottom of the duck, the person wins a small prize. If there's a blue mark on the bottom of the duck, the person wins a large prize. Many ducks do not have a mark. After 50 people have played the game, only 3 of them have won a small prize, and none of them have won a large prize. Estimate the number of the 160 ducks that you think have red marks on the bottom. Then estimate the number of ducks you think have blue marks. Explain your reasoning.

If 3 out of 50 people won a small prize, that means that 3 out of 50 ducks had a red mark on them. We can scale this ratio of 3:50 by a factor of 3.2 to get the equivalent ratio of 9.6:160. This means that there are likely 9 to 10 ducks that have red marks out of 160 total ducks.

###Write an explanation to this problem: A carnival game has 160 rubber ducks floating in a pool. The person playing the game takes out one duck and looks at it. If there's a red mark on the bottom of the duck, the person wins a small prize. If there's a blue mark on the bottom of the duck, the person wins a large prize. Many ducks do not have a mark. After 50 people have played the game, only 3 of them have won a small prize, and none of them have won a large prize. Estimate the number of the 160 ducks that you think have red marks on the bottom. Then estimate the number of ducks you think have blue marks. Explain your reasoning. ###Explanation: If 3 out of 50 people won a small prize, that means that 3 out of 50 ducks had a red mark on them. We can scale this ratio of 3:50 by a factor of 3.2 to get the equivalent ratio of 9.6:160. This means that there are likely 9 to 10 ducks that have red marks out of 160 total ducks.

30 + -30 = 0. Enter another sum of two numbers that equals 0. copied for free from openupresources.org

Any two numbers that have the same absolute value but opposite signs will equal 0 when added together. Sample Answer: 45+-45 = 0

###Write an explanation to this problem: 30 + -30 = 0. Enter another sum of two numbers that equals 0. copied for free from openupresources.org ###Explanation: Any two numbers that have the same absolute value but opposite signs will equal 0 when added together. Sample Answer: 45+-45 = 0

Describe in your own words what it means for two figures to be similar. The Utah Middle School Math Project

Example; Two figures are said to be similar if there is a sequence of rigid motions and dilations that take one figure onto the other. Corresponding angles are congruent and corresponding sides are proportional the ratios of corresponding sides are all the same and are equal to the scale factor.

###Write an explanation to this problem: Describe in your own words what it means for two figures to be similar. The Utah Middle School Math Project ###Explanation: Example; Two figures are said to be similar if there is a sequence of rigid motions and dilations that take one figure onto the other. Corresponding angles are congruent and corresponding sides are proportional the ratios of corresponding sides are all the same and are equal to the scale factor.

Priya has completed 9 exam questions. This is 60% of the questions on the exam. Write an equation representing this situation. Explain the meaning of any variables you use.

0.6x=9 Change 60% to a decimal 0.6 of means to multiply

###Write an explanation to this problem: Priya has completed 9 exam questions. This is 60% of the questions on the exam. Write an equation representing this situation. Explain the meaning of any variables you use. ###Explanation: 0.6x=9 Change 60% to a decimal 0.6 of means to multiply

Create an algebraic expression that has four terms and meets the following requirements: the expression has no like terms, the expression contains a constant, one of the coefficients is 3. The Utah Middle School Math Project

Example: x+2y+3z+4

###Write an explanation to this problem: Create an algebraic expression that has four terms and meets the following requirements: the expression has no like terms, the expression contains a constant, one of the coefficients is 3. The Utah Middle School Math Project ###Explanation: Example: x+2y+3z+4

Explain why zero is considered its own opposite.

0 is a neutral number. The opposite of 0 is already 0.

###Write an explanation to this problem: Explain why zero is considered its own opposite. ###Explanation: 0 is a neutral number. The opposite of 0 is already 0.

In the problem below, a starting equation and transformed equation have been given. State the action that produced the equivalent equation. Original Equation: 5 = x Transformed Equation: 0 = x - 5 Action: ______ The Utah Middle School Math Project

On the right side of the equation, 5 has been subtracted from x. On the left side, 5 has been subtracted from 5, which results in 0. The action was: Subtract 5 from both sides of the equation.

###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: 5 = x Transformed Equation: 0 = x - 5 Action: ______ The Utah Middle School Math Project ###Explanation: On the right side of the equation, 5 has been subtracted from x. On the left side, 5 has been subtracted from 5, which results in 0. The action was: Subtract 5 from both sides of the equation.

Find 3 different ratios that are equivalent to 7 : 3

Sample response: Multiply both terms of the ratio, 7 and 3, by the same value, such as 14:6, 21:9, 28:12

###Write an explanation to this problem: Find 3 different ratios that are equivalent to 7 : 3 ###Explanation: Sample response: Multiply both terms of the ratio, 7 and 3, by the same value, such as 14:6, 21:9, 28:12

Sean was at batting practice. He swung 25 times but only hit the ball 15 times. Describe and write more than one ratio related to this situation.

Ratio of the number of hits to the total number of swings is 𝟏𝟓: 𝟐𝟓. Ratio of the number hits to the number of misses is 𝟏𝟓: 𝟏𝟎. Ratio of the number of misses to the number of hits is 𝟏𝟎: 𝟏𝟓. Ratio of the number of misses to the total number of swings is 𝟏𝟎: 𝟐5

###Write an explanation to this problem: Sean was at batting practice. He swung 25 times but only hit the ball 15 times. Describe and write more than one ratio related to this situation. ###Explanation: Ratio of the number of hits to the total number of swings is 𝟏𝟓: 𝟐𝟓. Ratio of the number hits to the number of misses is 𝟏𝟓: 𝟏𝟎. Ratio of the number of misses to the number of hits is 𝟏𝟎: 𝟏𝟓. Ratio of the number of misses to the total number of swings is 𝟏𝟎: 𝟐5

5. 60% of x is equal to 87. a. Write an equation that expresses the relationship between 60%, x, and 87. copied for free from openupresources.org

60% of x means to multiply 60% and x. In order to multiply by a percent, you will need to change the percent to a decimal first. 0.6x=87 would be the equation that expresses the relationship between 60%, x, and 87.

###Write an explanation to this problem: 5. 60% of x is equal to 87. a. Write an equation that expresses the relationship between 60%, x, and 87. copied for free from openupresources.org ###Explanation: 60% of x means to multiply 60% and x. In order to multiply by a percent, you will need to change the percent to a decimal first. 0.6x=87 would be the equation that expresses the relationship between 60%, x, and 87.

What causes a repeating decimal in the long division algorithm?

When following the long division algorithm, you basically divide, multiply, subtract and bring down. Many times, after you subtract, there is a remainder. If, after you subtract, you end up with the same remainder as the time before, and you are bringing down the same number as before, you will find that the algorithm will just keep producing the same digit in the quotient over and over again because you are dividing the same number over and over again. This causes a repeating decimal. As soon as you recognize this pattern, you can stop dividing because you will know that the digits in the quotient will continue to repeat on and on and on.

###Write an explanation to this problem: What causes a repeating decimal in the long division algorithm? ###Explanation: When following the long division algorithm, you basically divide, multiply, subtract and bring down. Many times, after you subtract, there is a remainder. If, after you subtract, you end up with the same remainder as the time before, and you are bringing down the same number as before, you will find that the algorithm will just keep producing the same digit in the quotient over and over again because you are dividing the same number over and over again. This causes a repeating decimal. As soon as you recognize this pattern, you can stop dividing because you will know that the digits in the quotient will continue to repeat on and on and on.

Determine if the relation or situation defines a function. Justify your answer. You know your cousin lives at the zip code 12345 so you type it in Google to find your cousin's full address. Is address a function of zip code? The Utah Middle School Math Project

No, the data does not describe a function More than one address relates to a zip code The zip code is the input and there is more than one address or output for each zip code

###Write an explanation to this problem: Determine if the relation or situation defines a function. Justify your answer. You know your cousin lives at the zip code 12345 so you type it in Google to find your cousin's full address. Is address a function of zip code? The Utah Middle School Math Project ###Explanation: No, the data does not describe a function More than one address relates to a zip code The zip code is the input and there is more than one address or output for each zip code

For the sample given, list two possible populations they could belong to. Sample: The prices for apples at two stores near your house.

There are many correct answers for this question. Remember that a population is the group of people/things that you are interested in asking questions about. The sample is a smaller part from that population that you think could represent that population. One example of a correct answer is: Population 1: Prices for apples at all stores in our state. Population 2: Prices for all fruit at these two stores.

###Write an explanation to this problem: For the sample given, list two possible populations they could belong to. Sample: The prices for apples at two stores near your house. ###Explanation: There are many correct answers for this question. Remember that a population is the group of people/things that you are interested in asking questions about. The sample is a smaller part from that population that you think could represent that population. One example of a correct answer is: Population 1: Prices for apples at all stores in our state. Population 2: Prices for all fruit at these two stores.

Write a context to match the numeric expression given. 10 - 30.75 - 20.50 The Utah Middle School Math Project

Tom has 10 dollars. He is at the store that sells books for 0.75 and cups for 0.50. He wants to buy one book for each of his 3 brothers, and one cup for his mom and one cup for his dad. How much money will he have left?

###Write an explanation to this problem: Write a context to match the numeric expression given. 10 - 30.75 - 20.50 The Utah Middle School Math Project ###Explanation: Tom has 10 dollars. He is at the store that sells books for 0.75 and cups for 0.50. He wants to buy one book for each of his 3 brothers, and one cup for his mom and one cup for his dad. How much money will he have left?

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

Since the number of tables is unknown, we can use a variable, t, to represent this value. There are 8 students at each table, and the total number of students is 120. This means that: 8 students * number of tables = 120 students total 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: Since the number of tables is unknown, we can use a variable, t, to represent this value. There are 8 students at each table, and the total number of students is 120. This means that: 8 students * number of tables = 120 students total 8t=120

Given information: x = Amount of money Jaime earned 1/5x = one-fifth of the money Jaime spent on clothes 2/5x = two-fifths of the money Jaime spent on shoes Equation: 1/5x + 2/5x = 165 Write a word problem to go with the information and the equation given above. The Utah Middle School Math Project

Jaime earned some money. She spent 1/5 of that money on clothes and 2/5 of it on shoes. If she spent a total of 165, how much money did she earn?

###Write an explanation to this problem: Given information: x = Amount of money Jaime earned 1/5x = one-fifth of the money Jaime spent on clothes 2/5x = two-fifths of the money Jaime spent on shoes Equation: 1/5x + 2/5x = 165 Write a word problem to go with the information and the equation given above. The Utah Middle School Math Project ###Explanation: Jaime earned some money. She spent 1/5 of that money on clothes and 2/5 of it on shoes. If she spent a total of 165, how much money did she earn?

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

x - 7 is seven less than a number x.

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

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 One: 1 cm * 1 cm * 50 cm Choice Two: 1 cm * 2 cm * 25 cm Choice Three: 1 cm * 5 cm * 10 cm Choice Four: 2 cm * 5 cm * 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 One: 1 cm * 1 cm * 50 cm Choice Two: 1 cm * 2 cm * 25 cm Choice Three: 1 cm * 5 cm * 10 cm Choice Four: 2 cm * 5 cm * 5 cm