G8 U3 Constructed Response
Name: ________________________________________ Date:_______________
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations TASK 1: Graphing and Identifying Proportional Relationships (CCSS: 8.EE.5 [m]; SMP: 1, 2, 3, 4, 5)
1. Bobby and his dad are avid hikers. On a hike up Mt. Mitchell (NC), they go 150 meters in 30 minutes and then travel another 450 meters in 90 minutes. a. Determine if the relationship between meters and minutes is proportional or linear. Show your work in detailed and organized steps.
Answer: ___________________ b. Find the unit rate Bobby and his dad are traveling up Mt. Mitchell. Show your work in detailed and organized steps.
Answer: ___________________ c. Write a brief explanation that communicates the meaning of the unit rate you found in question 1b. What does this unit rate tell us about their hike?
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Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations d. On the coordinate grid below, graph the relationship between meters and minutes. Clearly label your axes with an appropriate scale and the dependent and independent variables.
2. Steven goes to two different local markets to purchase trail mix. Use the information in the table below to answer the following questions. Store Quality Foods Trader’s Market
Pounds Purchased 5 4
Amount Paid $19.75 $17.20
a. Find the unit rate of the trail mix at both stores in dollars per pound. Show your work in detailed and organized steps.
Answer: Quality Foods: ___________________ Trader’s Market: ___________________
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Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations b. Interpret the unit rates you found for the trail mix at the two stores. Which of the two stores offers a better price on the trail mix? Justify your answer by referencing your work and answers from above.
c. Steven wants to buy another 9 pounds of trail mix to distribute among his family and friends. He decides to purchase the trail mix from the store offering him the better price. Using your answers from above find the total price he will pay for the 9 pounds of trail mix he wants to buy. Show your work in detailed and organized steps.
Answer: ________________________
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Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations 3. Karen and Billy are sailing their toy boats at a nearby park in preparation for an upcoming race. Karen’s times and distances are recorded in the table below, and Billy’s numbers are displayed on the graph below. Use the data from the graph and table to answer the following questions. 250 225
DISTANCE (FEET)
200 175 150 125 100 75 50 25 0 0
1
2
3
4
5
6
7
8
9
10
Time (minutes) 0 1 2 3 4 5 6 7 8 9 10
Distance (feet) 0 29 58 87 116 145 174 203 232 261 290
TIME (MINUTES)
a. Determine which of the two children has the faster boat and by how much. Show your work in detailed and organized steps. Give and explain your answer in complete sentences.
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Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations b. Use your knowledge of unit rate to explain how this concept is important in real world applications. Why is it beneficial to know how to find the unit rate?
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Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations TASK 2: Slope, Unit Rate, and Comparing Linear Forms (CCSS: 8.EE.5 [m]; SMP: 1, 2, 3, 4, 5)
4. a. Find the slope of the line that passes through (−2, −1) and (1, 1). Show your work in detailed and organized steps.
Answer: __________________ b. Find the following points listed below and create a chart to show your answers. Show your work in detailed and complete steps. Points in between the given points One point to the left of (−2, −1). One point to the right of (1, 1).
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Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations 5. Tracy is using a water pump to remove water from her small pool. On her first use, the water pump extracted 36 gallons of water in 5 minutes. The second time she used the pump it extracted 96 gallons in 15 minutes. How many gallons of water will it extract if she runs it for 25 minutes? a. After analyzing the problem, describe the starting point for solving the problem. How will this starting point help you determine the best strategy to find the solution? Justify your reasoning.
b. Find the solution to the problem. Show your work in detailed and organized steps.
Answer: ________________
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Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations c. Explain the strategy you used to find your solution. How did you decide that it was the correct strategy to use in this situation? Justify your answer by referencing your work and answers from above.
6. Analyze the two relations below: 𝑦1 7.5 = 𝑥 3 X 𝑦2
0 0
6 19.8
8 26.4
a. Determine which relation, 𝑦1 or 𝑦2 , has a larger unit rate, or if they are equal. Give the value of the unit rate. Show your work in detailed and organized steps.
Answer: ___________________ Copyright © Swun Math Grade 8 Unit 3 Constructed Response, Page 8
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations b. Identify and describe a mathematical tool you could use to prove that your answer from above is correct. How would this tool prove that your answer is correct?
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Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations TASK 3: Similar Triangles with Slope and Deriving Slope-Intercept Form (CCSS: 8.EE.5, 8.EE.6 [m]; SMP: 1, 2, 3, 4, 5, 7, 8)
7. Christina has four triangular slides that are proportional. She knows the measurements of the smallest but only one measurement of each of the larger triangles. The unknown sides are labeled s and t. Triangle A
Triangle B
Triangle C
12 in 4in
s 3 in
6 in
t
a. Find the length of each missing side of the triangles. Show your work in detailed and organized steps.
Answer: s = ______________ t = ______________ b. Using your knowledge of proportional relationships, how would you be able to verify that your answers are correct? Explain your reasoning.
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Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations 8. George has a 35 foot tree near his house, which casts a shadow that is 14 feet from the base of the tree. He also has a smaller tree near his house, which casts a shadow that is 8 feet from the base of the tree. What is the height of the smaller tree? a. Before solving the problem, create a drawing that best represents the situation described in the problem. Label your drawing with the correct measurements and units, and label the unknown side with the letter d.
b. Explain the meaning of your drawing in terms of what the question is asking you find as your solution.
c. Using your drawing and the information given, find the height of the smaller tree. Show your work in detailed and organized steps.
Answer: _______________
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Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations d. Show and explain how you can prove that your answer is correct in the table below. Steps
Explanation of Steps
9. A line goes through the points (2, 5) and (7, 10). a. Determine if the relationship is proportional or linear. Show your work in detailed and organized steps.
Answer: ____________________ b. Explain how you determined whether the relationship was proportional or linear. Justify your reasoning using your work and answers from above.
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Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations c. Using your answers from above find the equation of the line that goes through the points (2, 5) and (7, 10). Show your work in detailed and organized steps.
Answer: __________________
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Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations TASK 1: Graphing and Identifying Proportional Relationships (CCSS: 8.EE.5 [m]; SMP: 1, 2, 3, 4, 5)
1. Bobby and his dad are avid hikers. On a hike up Mt. Mitchell (NC), they go 150 meters in 30 minutes and then travel another 450 meters in 90 minutes. a. Determine if the relationship between meters and minutes is proportional or linear. Show your work in detailed and organized steps.
Answer: ___________________ b. Find the unit rate Bobby and his dad are traveling up Mt. Mitchell. Show your work in detailed and organized steps.
Answer: ___________________ c. Write a brief explanation that communicates the meaning of the unit rate you found in question 1b. What does this unit rate tell us about their hike?
Copyright © Swun Math Grade 8 Unit 3 Constructed Response, Page 1
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations d. On the coordinate grid below, graph the relationship between meters and minutes. Clearly label your axes with an appropriate scale and the dependent and independent variables.
2. Steven goes to two different local markets to purchase trail mix. Use the information in the table below to answer the following questions. Store Quality Foods Trader’s Market
Pounds Purchased 5 4
Amount Paid $19.75 $17.20
a. Find the unit rate of the trail mix at both stores in dollars per pound. Show your work in detailed and organized steps.
Answer: Quality Foods: ___________________ Trader’s Market: ___________________
Copyright © Swun Math Grade 8 Unit 3 Constructed Response, Page 2
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations b. Interpret the unit rates you found for the trail mix at the two stores. Which of the two stores offers a better price on the trail mix? Justify your answer by referencing your work and answers from above.
c. Steven wants to buy another 9 pounds of trail mix to distribute among his family and friends. He decides to purchase the trail mix from the store offering him the better price. Using your answers from above find the total price he will pay for the 9 pounds of trail mix he wants to buy. Show your work in detailed and organized steps.
Answer: ________________________
Copyright © Swun Math Grade 8 Unit 3 Constructed Response, Page 3
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations 3. Karen and Billy are sailing their toy boats at a nearby park in preparation for an upcoming race. Karen’s times and distances are recorded in the table below, and Billy’s numbers are displayed on the graph below. Use the data from the graph and table to answer the following questions. 250 225
DISTANCE (FEET)
200 175 150 125 100 75 50 25 0 0
1
2
3
4
5
6
7
8
9
10
Time (minutes) 0 1 2 3 4 5 6 7 8 9 10
Distance (feet) 0 29 58 87 116 145 174 203 232 261 290
TIME (MINUTES)
a. Determine which of the two children has the faster boat and by how much. Show your work in detailed and organized steps. Give and explain your answer in complete sentences.
Copyright © Swun Math Grade 8 Unit 3 Constructed Response, Page 4
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations b. Use your knowledge of unit rate to explain how this concept is important in real world applications. Why is it beneficial to know how to find the unit rate?
Copyright © Swun Math Grade 8 Unit 3 Constructed Response, Page 5
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations TASK 2: Slope, Unit Rate, and Comparing Linear Forms (CCSS: 8.EE.5 [m]; SMP: 1, 2, 3, 4, 5)
4. a. Find the slope of the line that passes through (−2, −1) and (1, 1). Show your work in detailed and organized steps.
Answer: __________________ b. Find the following points listed below and create a chart to show your answers. Show your work in detailed and complete steps. Points in between the given points One point to the left of (−2, −1). One point to the right of (1, 1).
Copyright © Swun Math Grade 8 Unit 3 Constructed Response, Page 6
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations 5. Tracy is using a water pump to remove water from her small pool. On her first use, the water pump extracted 36 gallons of water in 5 minutes. The second time she used the pump it extracted 96 gallons in 15 minutes. How many gallons of water will it extract if she runs it for 25 minutes? a. After analyzing the problem, describe the starting point for solving the problem. How will this starting point help you determine the best strategy to find the solution? Justify your reasoning.
b. Find the solution to the problem. Show your work in detailed and organized steps.
Answer: ________________
Copyright © Swun Math Grade 8 Unit 3 Constructed Response, Page 7
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations c. Explain the strategy you used to find your solution. How did you decide that it was the correct strategy to use in this situation? Justify your answer by referencing your work and answers from above.
6. Analyze the two relations below: 𝑦1 7.5 = 𝑥 3 X 𝑦2
0 0
6 19.8
8 26.4
a. Determine which relation, 𝑦1 or 𝑦2 , has a larger unit rate, or if they are equal. Give the value of the unit rate. Show your work in detailed and organized steps.
Answer: ___________________ Copyright © Swun Math Grade 8 Unit 3 Constructed Response, Page 8
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations b. Identify and describe a mathematical tool you could use to prove that your answer from above is correct. How would this tool prove that your answer is correct?
Copyright © Swun Math Grade 8 Unit 3 Constructed Response, Page 9
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations TASK 3: Similar Triangles with Slope and Deriving Slope-Intercept Form (CCSS: 8.EE.5, 8.EE.6 [m]; SMP: 1, 2, 3, 4, 5, 7, 8)
7. Christina has four triangular slides that are proportional. She knows the measurements of the smallest but only one measurement of each of the larger triangles. The unknown sides are labeled s and t. Triangle A
Triangle B
Triangle C
12 in 4in
s 3 in
6 in
t
a. Find the length of each missing side of the triangles. Show your work in detailed and organized steps.
Answer: s = ______________ t = ______________ b. Using your knowledge of proportional relationships, how would you be able to verify that your answers are correct? Explain your reasoning.
Copyright © Swun Math Grade 8 Unit 3 Constructed Response, Page 10
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations 8. George has a 35 foot tree near his house, which casts a shadow that is 14 feet from the base of the tree. He also has a smaller tree near his house, which casts a shadow that is 8 feet from the base of the tree. What is the height of the smaller tree? a. Before solving the problem, create a drawing that best represents the situation described in the problem. Label your drawing with the correct measurements and units, and label the unknown side with the letter d.
b. Explain the meaning of your drawing in terms of what the question is asking you find as your solution.
c. Using your drawing and the information given, find the height of the smaller tree. Show your work in detailed and organized steps.
Answer: _______________
Copyright © Swun Math Grade 8 Unit 3 Constructed Response, Page 11
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations d. Show and explain how you can prove that your answer is correct in the table below. Steps
Explanation of Steps
9. A line goes through the points (2, 5) and (7, 10). a. Determine if the relationship is proportional or linear. Show your work in detailed and organized steps.
Answer: ____________________ b. Explain how you determined whether the relationship was proportional or linear. Justify your reasoning using your work and answers from above.
Copyright © Swun Math Grade 8 Unit 3 Constructed Response, Page 12
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations c. Using your answers from above find the equation of the line that goes through the points (2, 5) and (7, 10). Show your work in detailed and organized steps.
Answer: __________________
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