G8 U3 Constructed Response Rubric
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations Scoring Rubric Task 1. Graphing and Identifying Proportional Relationships 2. Slope, Unit Rate, and Comparing Liner Forms
3. Similar Triangles with Slope and Deriving Slope-Intercept Form
Common Core State Standard 8.EE.5[m]: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. 8.EE.5 [m]: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. 8.EE.5 [m]: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. 8.EE.6 [m]: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx+b, for a line intercepting the vertical axis at b.
Standards for Mathematical Practice SMP.1, SMP. 2, SMP.3, SMP.4, SMP.5
SMP.1, SMP.3, SMP.4, SMP.6, SMP.8 SMP.1, SMP.2, SMP.3, SMP.4, SMP.5, SMP.7, SMP.8
Note to Teacher: The following scoring rubric should be used as a guide to determine points given to students for each question answered. Students are required to show the process through which they arrived at their answers for every question involving problem solving. For questions involving a written answer, full points should be given to answers that are written in complete sentences, which address each component of the questions being asked. [m]: major work for grade level [a/s]: additional/supporting standard
Copyright © Swun Math Grade 8 Unit 3 Constructed Response Rubric, Page 1
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations Scoring Rubric Question 1. a. Student gives correct answer and shows work: Proportional b. Student gives correct answer and shows work: 5 meters per minute c. Student gives explanation: The unit rate I found in question 2b tells us the distance over a certain amount of time, or their average. In this case we are using meters per minute to describe the distance covered and the time they are covering that distance in. So for their hike they are traveling at a constant unit rate of five meters per minute, based on the amount of ground they have covered at certain points on their hike. d. Student creates graph:
Points 0.25 0.25 0.5
1 450 400
Distance (meters)
350 300 250 200 150 100 50 0 -10
10
30
50
70
90
Time (minutes)
2. a. Student gives correct answers and shows work (0.5 pt for each correct answer): Quality Foods= $3.95/lb; Trader’s Market= $4.30/lb b. Student gives explanation: The unit rate I found for the trail mix at the two stores tells me the price that is paid in dollars for every pound of trail mix. In comparing the two unit rates, I can conclude that Quality Foods offers a better price per pound, or unit rate, for the trail mix. When I calculate the unit price for the trail mix at the two stores, I find that Quality Foods sells their trail mix at $3.95 per pound and Trader’s Market sells their trail mix for $4.30 per pound. By finding the difference, I know that Quality Foods sells their trail mix $0.35 cheaper and, therefore offers the better deal. c. Student gives correct answer and shows work: $35.55 Copyright © Swun Math Grade 8 Unit 3 Constructed Response Rubric, Page 2
1 0.75
0.5
Total
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations 3. a. Student gives correct answer, shows work, and explanation: Of the two children’s boats, Karen has the fastest boat, which travels at a unit rate of 29 feet per minute. Billy’s boat travels at 25 feet per minute. Karen’s boat travels approximately 5 feet per minute. b. Student gives explanation: Answers may vary
0.25 0.25 1
4. a. Student gives correct answer and shows work: 2/3 b. Student creates table with correct answer: x -3 -2 -1 0 1 y
- 5/3
-1
- 1/3
1/3
1
0.5
2 5/3
5. a. Student gives explanation: After analyzing the problem, I know that my first
0.75
step is to determine if there is a proportional relationship between the quantities being described. In this case we are looking at the gallons of water pumped per minute. So the best place to start is to create a proportion comparing gallons of water pumped to minutes using the data I am. If I determine that there is a proportional relationship, then I can proceed using the unit rate to multiply or divide to find my answer, or if there is no proportional relationship, I need to solve the problem by setting up a proportion using slope to determine how much to add to the endpoints.
b. Student gives correct answer and shows work: 156 gallons c. Student gives explanation: To find my solution I had to check if the quantities I was given were proportional. In doing this I would be able to decide if I had to multiply or divide with the unit rate or set up a proportion with the slope. I decided that I needed to set up a proportion with the slope to find my answer because when I checked the two quantities to see if they were proportional, I found that there was no proportional relationship between the gallons pumped per minute. For example, 36 gallons in five minutes had a unit rate of 7.2 gallons per minute, whereas 96 gallons per 15 minutes had a unit rate of 6.4 gallons. These two rates are not equal and not proportional. Knowing this I used the equation to find slope to set up a proportion of 6 gallons per minute. This value was then added to my last time endpoint of 96 gallon in 15 minutes. However, since the new time I was looking for was gallons pumped in 25 minutes, which is ten more than the last time she pumped water, I had to multiply the rate of 6 gallons per minute by ten and then add it to the 96 gallons she pumped the second time. After 25 minutes she would pump 156 gallons of water.
6. a. Student gives correct answer and shows work: y2 has the larger unit rate, unit rate = 3.3 b. Student gives explanation: A mathematical tool I could use to prove that my answer is a coordinate grid. I could graph the lines using the slopes and then compare the actual lines to verify that my answer is correct. Copyright © Swun Math Grade 8 Unit 3 Constructed Response Rubric, Page 3
0.75 1
1 0.25
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations 0.5 0.25
1
d
35 ft
7. a. Student gives correct answers and shows work: s= 8 in; t= 9 in b. Student gives explanation: In order to verify that my answers are correct I could set up the sides of the triangles into proportions and simplify those fractions. If the simplest form of all three proportions is equal, then I know that the values I found for the missing sides of the two larger triangles are correct. 8. a. Student creates drawing: Drawings may vary; Look for an understanding of the information given. Student should have the height of the larger tree and its shadow, plus the shadow of the smaller tree labeled correctly. See example:
14 ft 8ft b. Student gives explanation: In my drawing you will find that I represented the trees and their shadows. The larger tree is labeled by its height of 35 ft. and its shadow is labelled by 14 ft. For the smaller tree I was only given the length of the shadow, which is labeled as 8 ft. in my drawing, and the unknown height that the question is asking me to find is labeled alongside the tree with the variable d. This drawing allows me to visualize the information I have and the information I need to find. c. Student gives correct answer and shows work: 20 feet d. Student completes table: Student should show and explain the process of checking his/her answer by comparing writing the height of each tree and its shadow into a proportion. Simplification of the fraction should yield equivalent fractions. 9. a. Student gives correct answer and shows work: linear b. Student gives explanation: I was able to determine that the relation was not proportional and was linear because when I found the slope of the line created by the two points I found that the line did not cross the origin at (0, 0). I came to this conclusion because I used the slope I found to move towards the x-axis from the point (2, 5), and I stopped at the new point (-2, 0).
0.5
0.5 0.25 0.5 0.5
1 Copyright © Swun Math Grade 8 Unit 3 Constructed Response Rubric, Page 4
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations 5
c. Student shows work and give correct answer: 𝑦 = 4 𝑥 + 5 Test Total
Copyright © Swun Math Grade 8 Unit 3 Constructed Response Rubric, Page 5
15
3. Similar Triangles with Slope and Deriving Slope-Intercept Form
Common Core State Standard 8.EE.5[m]: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. 8.EE.5 [m]: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. 8.EE.5 [m]: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. 8.EE.6 [m]: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx+b, for a line intercepting the vertical axis at b.
Standards for Mathematical Practice SMP.1, SMP. 2, SMP.3, SMP.4, SMP.5
SMP.1, SMP.3, SMP.4, SMP.6, SMP.8 SMP.1, SMP.2, SMP.3, SMP.4, SMP.5, SMP.7, SMP.8
Note to Teacher: The following scoring rubric should be used as a guide to determine points given to students for each question answered. Students are required to show the process through which they arrived at their answers for every question involving problem solving. For questions involving a written answer, full points should be given to answers that are written in complete sentences, which address each component of the questions being asked. [m]: major work for grade level [a/s]: additional/supporting standard
Copyright © Swun Math Grade 8 Unit 3 Constructed Response Rubric, Page 1
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations Scoring Rubric Question 1. a. Student gives correct answer and shows work: Proportional b. Student gives correct answer and shows work: 5 meters per minute c. Student gives explanation: The unit rate I found in question 2b tells us the distance over a certain amount of time, or their average. In this case we are using meters per minute to describe the distance covered and the time they are covering that distance in. So for their hike they are traveling at a constant unit rate of five meters per minute, based on the amount of ground they have covered at certain points on their hike. d. Student creates graph:
Points 0.25 0.25 0.5
1 450 400
Distance (meters)
350 300 250 200 150 100 50 0 -10
10
30
50
70
90
Time (minutes)
2. a. Student gives correct answers and shows work (0.5 pt for each correct answer): Quality Foods= $3.95/lb; Trader’s Market= $4.30/lb b. Student gives explanation: The unit rate I found for the trail mix at the two stores tells me the price that is paid in dollars for every pound of trail mix. In comparing the two unit rates, I can conclude that Quality Foods offers a better price per pound, or unit rate, for the trail mix. When I calculate the unit price for the trail mix at the two stores, I find that Quality Foods sells their trail mix at $3.95 per pound and Trader’s Market sells their trail mix for $4.30 per pound. By finding the difference, I know that Quality Foods sells their trail mix $0.35 cheaper and, therefore offers the better deal. c. Student gives correct answer and shows work: $35.55 Copyright © Swun Math Grade 8 Unit 3 Constructed Response Rubric, Page 2
1 0.75
0.5
Total
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations 3. a. Student gives correct answer, shows work, and explanation: Of the two children’s boats, Karen has the fastest boat, which travels at a unit rate of 29 feet per minute. Billy’s boat travels at 25 feet per minute. Karen’s boat travels approximately 5 feet per minute. b. Student gives explanation: Answers may vary
0.25 0.25 1
4. a. Student gives correct answer and shows work: 2/3 b. Student creates table with correct answer: x -3 -2 -1 0 1 y
- 5/3
-1
- 1/3
1/3
1
0.5
2 5/3
5. a. Student gives explanation: After analyzing the problem, I know that my first
0.75
step is to determine if there is a proportional relationship between the quantities being described. In this case we are looking at the gallons of water pumped per minute. So the best place to start is to create a proportion comparing gallons of water pumped to minutes using the data I am. If I determine that there is a proportional relationship, then I can proceed using the unit rate to multiply or divide to find my answer, or if there is no proportional relationship, I need to solve the problem by setting up a proportion using slope to determine how much to add to the endpoints.
b. Student gives correct answer and shows work: 156 gallons c. Student gives explanation: To find my solution I had to check if the quantities I was given were proportional. In doing this I would be able to decide if I had to multiply or divide with the unit rate or set up a proportion with the slope. I decided that I needed to set up a proportion with the slope to find my answer because when I checked the two quantities to see if they were proportional, I found that there was no proportional relationship between the gallons pumped per minute. For example, 36 gallons in five minutes had a unit rate of 7.2 gallons per minute, whereas 96 gallons per 15 minutes had a unit rate of 6.4 gallons. These two rates are not equal and not proportional. Knowing this I used the equation to find slope to set up a proportion of 6 gallons per minute. This value was then added to my last time endpoint of 96 gallon in 15 minutes. However, since the new time I was looking for was gallons pumped in 25 minutes, which is ten more than the last time she pumped water, I had to multiply the rate of 6 gallons per minute by ten and then add it to the 96 gallons she pumped the second time. After 25 minutes she would pump 156 gallons of water.
6. a. Student gives correct answer and shows work: y2 has the larger unit rate, unit rate = 3.3 b. Student gives explanation: A mathematical tool I could use to prove that my answer is a coordinate grid. I could graph the lines using the slopes and then compare the actual lines to verify that my answer is correct. Copyright © Swun Math Grade 8 Unit 3 Constructed Response Rubric, Page 3
0.75 1
1 0.25
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations 0.5 0.25
1
d
35 ft
7. a. Student gives correct answers and shows work: s= 8 in; t= 9 in b. Student gives explanation: In order to verify that my answers are correct I could set up the sides of the triangles into proportions and simplify those fractions. If the simplest form of all three proportions is equal, then I know that the values I found for the missing sides of the two larger triangles are correct. 8. a. Student creates drawing: Drawings may vary; Look for an understanding of the information given. Student should have the height of the larger tree and its shadow, plus the shadow of the smaller tree labeled correctly. See example:
14 ft 8ft b. Student gives explanation: In my drawing you will find that I represented the trees and their shadows. The larger tree is labeled by its height of 35 ft. and its shadow is labelled by 14 ft. For the smaller tree I was only given the length of the shadow, which is labeled as 8 ft. in my drawing, and the unknown height that the question is asking me to find is labeled alongside the tree with the variable d. This drawing allows me to visualize the information I have and the information I need to find. c. Student gives correct answer and shows work: 20 feet d. Student completes table: Student should show and explain the process of checking his/her answer by comparing writing the height of each tree and its shadow into a proportion. Simplification of the fraction should yield equivalent fractions. 9. a. Student gives correct answer and shows work: linear b. Student gives explanation: I was able to determine that the relation was not proportional and was linear because when I found the slope of the line created by the two points I found that the line did not cross the origin at (0, 0). I came to this conclusion because I used the slope I found to move towards the x-axis from the point (2, 5), and I stopped at the new point (-2, 0).
0.5
0.5 0.25 0.5 0.5
1 Copyright © Swun Math Grade 8 Unit 3 Constructed Response Rubric, Page 4
Grade 8 Unit 3 Constructed Response Proportional Relationships, Lines, and Linear Equations 5
c. Student shows work and give correct answer: 𝑦 = 4 𝑥 + 5 Test Total
Copyright © Swun Math Grade 8 Unit 3 Constructed Response Rubric, Page 5
15
