G6 U5 Constructed Response Rubric
Grade 6 Unit 5 Constructed Response Inequalities Scoring Rubric Task
1. Writing and Solving Expressions and Equations; Independent and Dependent Variables
2. Writing and Solving Inequalities
Common Core State Standard for Mathematical Content (MC) 6.EE.5: Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 6.EE.6: Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. 6.EE.7: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. 6.EE.8: Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. 6.EE.3: Apply the properties of operations to generate equivalent expressions. 6.EE.8: Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 1
Standards for Mathematical Practice (MP)
MP.1, MP.2, MP.4, MP.6, MP.7, MP.8
MP.1, MP.2, MP.3, MP.4, MP.6, MP.7
Grade 6 Unit 5 Constructed Response Inequalities
3. Graphing Inequalities on a Number Line
6.EE.8: Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
MP.1, MP.2, MP.3, MP.4, MP.6, MP.7
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.
Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 2
Grade 6 Unit 5 Constructed Response Inequalities Scoring Rubric Question 1. Student gives correct representations for the expression: (Answers may vary, check student’s work for accuracy) Sample Answers: a. Model: 1
1
1
1
1
s
s
1
1
1
1
1
s
s
b. Phrase: Fifteen more than four times a number c. Word Problem: Christie collected 15 red leaves on Tuesday, and Jillian collected 4 times as many yellow leaves on Thursday. How many leaves did they collect altogether? 2. a. Student gives correct answers: (Variables may vary) Ex: $15ℎ + 10 = $250; h = number of hours Marshall worked during the week. b. Student gives an accurate explanation. Wording may vary. Sample explanation: In order to write the equation I had to use several pieces of important information given to me in the problem. The first piece of information I am given is Marshall works at a rate of $15 per hour. This means he gets $15 for every hour he works, so if I want to find how much he makes for a certain amount of time I would multiply the rate by the number of hours he works. I know based on this information the first part of the equation would be $15 times h, or $15h, where h equals the number of hours he worked. The next piece of information I was given was Marshall also charges an additional $10 fee for supplies. So on top of his hourly rate he adds a supply fee. For the equation I know I would have to add this fee to the hourly rate, so the next part of the equation is + $10. The first portion of the equation became $15h + $10 . The last piece of information I am given is at the end of the week he makes a total of $250. So the final equation I need to write to find the total hours he worked is $15h + $10 = $250 c. Student gives correct answer and shows work: (Check student’s work for accuracy); 16 hours
Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 3
Points 1
1 1
1
2
1
Total
Grade 6 Unit 5 Constructed Response Inequalities
3. a. Student gives correct answer: Independent Variable: number of packages of cookies sold; Dependent Variable: Profit made; Equation: $8𝑐 = 𝑝 b. Student creates an accurate table of values to find the profit from given values: Packages sold ( c ) Profit (p) 4 32 8 64 12 96 16 128 20 160 24 192 28 224
1
1
1
c. Student creates an accurate graph: (x and y axis should be labeled, straight line connecting points) 250
Profit (p)
200 150 100
50 0 0
4
8 12 16 20 24 Packages of Cookies Sold (c)
28
32
1 d. Student gives correct answer and shows work: 70 packages of cookies 4. a. Student gives correct answer: 2𝑙 + 14𝑓𝑡 ≤ 38𝑓𝑡 b. Student gives an accurate explanation. Wording may vary. Sample explanation: In order to write the inequality I had to use several pieces of Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 4
1 2
Grade 6 Unit 5 Constructed Response Inequalities important information given to me in the problem and the formula for the perimeter of a rectangle. The first piece of information I am given is the perimeter of the rectangle can be no more than 38 ft, and from the diagram I know the width of the rectangle is 7ft, and the missing length is L. I know opposite sides of a rectangle are equal in length and that the formula for the perimeter of a rectangle is equal to the sum of the lengths of all the sides. So with this I was able to write the first portion of the inequality, 𝑙 + 𝑙 + 7𝑓𝑡 + 7𝑓𝑡. Since the question is asking me to find the value of L if the perimeter is no more than 38ft, I know my inequality needs to have a less than or equal to sign. With this information I can then complete my inequality which is, 2𝐿 + 14𝑓𝑡 ≤ 38𝑓𝑡. c. Student gives correct answer and shows work: 𝐿 ≤ 12𝑓𝑡 ; The maximum length of the rectangle is 12ft. 5. a. Student gives correct answer: $9.50ℎ ≥ $437 Student gives an accurate explanation. Wording may vary. Sample explanation: b. Student gives correct answer and shows work: ℎ ≥ 46 ℎ𝑜𝑢𝑟𝑠; Marilyn needs to work a minimum of 46 hours
6. a. Student gives correct answer: 26 𝑚𝑖𝑙𝑒𝑠 + 𝑚 > 31 𝑚𝑖𝑙𝑒𝑠 b. Student gives correct answer and shows work: 𝑚 > 5 𝑚𝑖𝑙𝑒𝑠; Stephanie needs to run more than 5 miles on Saturday to exceed her weekly average. c. Student gives an accurate explanation and shows process. Wording may vary. Sample explanation: : Student should insert any value greater than 5 into the inequality. Sample Explanation: In order to prove my answer is correct I could substitute values greater than 5 into the inequality I wrote. If the inequality is still true after substituting these Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 5
1 1 1
1 1
1
Grade 6 Unit 5 Constructed Response Inequalities values then I know my answer is correct.
7. a. Student gives correct answer and an accurate explanation. Wording may vary. Sample explanation: After analyzing the inequality and the number line Cora created, I was able to determine she was incorrect. One of the errors she made was she represented the statement with the wrong inequality sign. The statement says the number of swimmers in class must be at least 25 students, so because it says “at least” the inequality should have been 𝑠 ≥ 25. She made the same error in her number line. Her number line was correct in having the shaded region move to the left of 25 to represent values larger, but she made her error in placing an open circle on the number 25. Since the inequality involves a greater than or equal to sign it should have been a solid circle on the number 25 . b. Student gives correct inequality and an accurate number line:
2
1
Inequality: s ≥ 25; 23 24 25
26 27 28 29
8. a. Student gives correct answers: 𝑔 < $300 ; 𝑠 ≥ $275 b. Student creates accurate number lines:
0
50
270
275
100
280
150
285
200
250
300
350
1 2
300
400
9. a. Student gives correct answer: 𝑑 ≤ 45 b. Student creates an accurate number line:
0
5
10
15 20 25 30 35 40 45
1 1
50
Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 6
Grade 6 Unit 5 Constructed Response Inequalities c. Student gives an accurate explanation. Wording may vary. Sample explanation: My number line represents the scenario and the inequality because I show the number of days they can take to build the boat and still be under the 45 days. The number line represents the inequality by having a closed circle over the number 45 and a shaded line going all the way up to zero where it stops. The line stops here and does not continue into the negative numbers because there can’t be a negative number of days. At the zero I have an open circle because the job needs to take longer than zero days. So the region that is shaded on my number line represents all the days from zero to 45 they can take to build the boat.
1
d. Student gives correct answers and an accurate explanation. Answers may vary; Student should give any three number of days that are greater than 0 and less than or equal to 45 Total
1
Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 7
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1. Writing and Solving Expressions and Equations; Independent and Dependent Variables
2. Writing and Solving Inequalities
Common Core State Standard for Mathematical Content (MC) 6.EE.5: Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 6.EE.6: Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. 6.EE.7: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. 6.EE.8: Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. 6.EE.3: Apply the properties of operations to generate equivalent expressions. 6.EE.8: Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 1
Standards for Mathematical Practice (MP)
MP.1, MP.2, MP.4, MP.6, MP.7, MP.8
MP.1, MP.2, MP.3, MP.4, MP.6, MP.7
Grade 6 Unit 5 Constructed Response Inequalities
3. Graphing Inequalities on a Number Line
6.EE.8: Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
MP.1, MP.2, MP.3, MP.4, MP.6, MP.7
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.
Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 2
Grade 6 Unit 5 Constructed Response Inequalities Scoring Rubric Question 1. Student gives correct representations for the expression: (Answers may vary, check student’s work for accuracy) Sample Answers: a. Model: 1
1
1
1
1
s
s
1
1
1
1
1
s
s
b. Phrase: Fifteen more than four times a number c. Word Problem: Christie collected 15 red leaves on Tuesday, and Jillian collected 4 times as many yellow leaves on Thursday. How many leaves did they collect altogether? 2. a. Student gives correct answers: (Variables may vary) Ex: $15ℎ + 10 = $250; h = number of hours Marshall worked during the week. b. Student gives an accurate explanation. Wording may vary. Sample explanation: In order to write the equation I had to use several pieces of important information given to me in the problem. The first piece of information I am given is Marshall works at a rate of $15 per hour. This means he gets $15 for every hour he works, so if I want to find how much he makes for a certain amount of time I would multiply the rate by the number of hours he works. I know based on this information the first part of the equation would be $15 times h, or $15h, where h equals the number of hours he worked. The next piece of information I was given was Marshall also charges an additional $10 fee for supplies. So on top of his hourly rate he adds a supply fee. For the equation I know I would have to add this fee to the hourly rate, so the next part of the equation is + $10. The first portion of the equation became $15h + $10 . The last piece of information I am given is at the end of the week he makes a total of $250. So the final equation I need to write to find the total hours he worked is $15h + $10 = $250 c. Student gives correct answer and shows work: (Check student’s work for accuracy); 16 hours
Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 3
Points 1
1 1
1
2
1
Total
Grade 6 Unit 5 Constructed Response Inequalities
3. a. Student gives correct answer: Independent Variable: number of packages of cookies sold; Dependent Variable: Profit made; Equation: $8𝑐 = 𝑝 b. Student creates an accurate table of values to find the profit from given values: Packages sold ( c ) Profit (p) 4 32 8 64 12 96 16 128 20 160 24 192 28 224
1
1
1
c. Student creates an accurate graph: (x and y axis should be labeled, straight line connecting points) 250
Profit (p)
200 150 100
50 0 0
4
8 12 16 20 24 Packages of Cookies Sold (c)
28
32
1 d. Student gives correct answer and shows work: 70 packages of cookies 4. a. Student gives correct answer: 2𝑙 + 14𝑓𝑡 ≤ 38𝑓𝑡 b. Student gives an accurate explanation. Wording may vary. Sample explanation: In order to write the inequality I had to use several pieces of Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 4
1 2
Grade 6 Unit 5 Constructed Response Inequalities important information given to me in the problem and the formula for the perimeter of a rectangle. The first piece of information I am given is the perimeter of the rectangle can be no more than 38 ft, and from the diagram I know the width of the rectangle is 7ft, and the missing length is L. I know opposite sides of a rectangle are equal in length and that the formula for the perimeter of a rectangle is equal to the sum of the lengths of all the sides. So with this I was able to write the first portion of the inequality, 𝑙 + 𝑙 + 7𝑓𝑡 + 7𝑓𝑡. Since the question is asking me to find the value of L if the perimeter is no more than 38ft, I know my inequality needs to have a less than or equal to sign. With this information I can then complete my inequality which is, 2𝐿 + 14𝑓𝑡 ≤ 38𝑓𝑡. c. Student gives correct answer and shows work: 𝐿 ≤ 12𝑓𝑡 ; The maximum length of the rectangle is 12ft. 5. a. Student gives correct answer: $9.50ℎ ≥ $437 Student gives an accurate explanation. Wording may vary. Sample explanation: b. Student gives correct answer and shows work: ℎ ≥ 46 ℎ𝑜𝑢𝑟𝑠; Marilyn needs to work a minimum of 46 hours
6. a. Student gives correct answer: 26 𝑚𝑖𝑙𝑒𝑠 + 𝑚 > 31 𝑚𝑖𝑙𝑒𝑠 b. Student gives correct answer and shows work: 𝑚 > 5 𝑚𝑖𝑙𝑒𝑠; Stephanie needs to run more than 5 miles on Saturday to exceed her weekly average. c. Student gives an accurate explanation and shows process. Wording may vary. Sample explanation: : Student should insert any value greater than 5 into the inequality. Sample Explanation: In order to prove my answer is correct I could substitute values greater than 5 into the inequality I wrote. If the inequality is still true after substituting these Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 5
1 1 1
1 1
1
Grade 6 Unit 5 Constructed Response Inequalities values then I know my answer is correct.
7. a. Student gives correct answer and an accurate explanation. Wording may vary. Sample explanation: After analyzing the inequality and the number line Cora created, I was able to determine she was incorrect. One of the errors she made was she represented the statement with the wrong inequality sign. The statement says the number of swimmers in class must be at least 25 students, so because it says “at least” the inequality should have been 𝑠 ≥ 25. She made the same error in her number line. Her number line was correct in having the shaded region move to the left of 25 to represent values larger, but she made her error in placing an open circle on the number 25. Since the inequality involves a greater than or equal to sign it should have been a solid circle on the number 25 . b. Student gives correct inequality and an accurate number line:
2
1
Inequality: s ≥ 25; 23 24 25
26 27 28 29
8. a. Student gives correct answers: 𝑔 < $300 ; 𝑠 ≥ $275 b. Student creates accurate number lines:
0
50
270
275
100
280
150
285
200
250
300
350
1 2
300
400
9. a. Student gives correct answer: 𝑑 ≤ 45 b. Student creates an accurate number line:
0
5
10
15 20 25 30 35 40 45
1 1
50
Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 6
Grade 6 Unit 5 Constructed Response Inequalities c. Student gives an accurate explanation. Wording may vary. Sample explanation: My number line represents the scenario and the inequality because I show the number of days they can take to build the boat and still be under the 45 days. The number line represents the inequality by having a closed circle over the number 45 and a shaded line going all the way up to zero where it stops. The line stops here and does not continue into the negative numbers because there can’t be a negative number of days. At the zero I have an open circle because the job needs to take longer than zero days. So the region that is shaded on my number line represents all the days from zero to 45 they can take to build the boat.
1
d. Student gives correct answers and an accurate explanation. Answers may vary; Student should give any three number of days that are greater than 0 and less than or equal to 45 Total
1
Copyright © Swun Math Grade 6 Unit 5 Constructed Response Rubric, Page 7
30
