G7 U3 Constructed Response Rubric
Grade 7 Unit 3 Constructed Response Multiplying and Dividing Rational Numbers Scoring Rubric Task
1. Multiplication of Signed Rational Numbers
2. Division of Signed Rational Numbers
Common Core State Standard 7.NS.2a[m]: Understand multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1)=1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. 7.NS.2c [m]: Apply properties of operations as strategies to multiply and divide rational numbers. 7.NS.2b [m]: Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. Interpret quotients of rational numbers by describing real world contexts. 7.NS.2c [m]: Apply properties of operations as strategies to multiply and divide rational numbers. 7.NS.2d [m]: Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
Standards for Mathematical Practice SMP.1, SMP. 2, SMP.4, SMP.7
SMP.1, SMP.2, SMP.3, SMP.4, SMP.6, SMP.7, SMP.8
SMP.1, SMP.2, 3. Converting Rational SMP.4, SMP.6, Numbers to Decimals SMP.7 and Decimal Forms of Rational Numbers 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 7 Unit 3 Constructed Response Rubric, Page 1
Grade 7 Unit 3 Constructed Response Multiplying and Dividing Rational Numbers Scoring Rubric Question 1. a. Student gives explanation: After analyzing the two expressions I was able to make some observations about the similarities and differences in the possible answers of the two expressions. At first glance I noticed that the two numbers that are being multiplied are the same for both Expression A and B, so the product of the two expressions will be the same. However, because the symbols in front of each number in the expressions are different, their products will have a different sign. For example, in Expression A, a negative number is multiplied by a positive number, so the product is negative. In Expression B, a negative number is multiplied by a negative number, so the product is positive. b. Student gives correct answers and shows work: Student should display work on a number line as shown in lessons; Expression A= -36, Expression B= 36. 2. a. Student gives explanation: After reading and analyzing the information given, I was able to conclude that the problem is asking me to find how much money Cassidy will save up in 8 weeks, after her weekly earnings and expenses. In order to find this I first need to find out how much money she is saving a week. To do this I need to find how much money she is making a week and subtract from this number how much money she is spending in that same week. Since the problem does not state that the money she earns and spends weekly changes, I know that I can multiply her weekly earnings by 8 weeks to get her total savings. b. Student gives correct answer: 8 ∙ (30 + 15 + 20 − 10) c. Student gives correct answer and shows work: $440.00 d. Student gives explanation: In order to find the solution to this problem, I had to use the Distributive Property. I used this property in order to solve the expression I wrote in question 2b, 8 ($30 + $15 + $20 - $10). To solve this expression using the Distributive Property I first had to find Cassidy’s weekly savings in the parentheses, which was equal to $55. I then multiplied this number by 8 on the outside of the parentheses, which gave me $440. 3. a. Student gives explanation: After analyzing the information given in the problem I was able to identify pieces of information that are important to finding the solution as well as extra information not necessary. Important pieces of information that I found in the scenario were: 1) the number of Copyright © Swun Math Grade 7 Unit 3 Constructed Response Rubric, Page 2
Points 0.5
0.5
0.75
0.25 0.5 0.5
1
Total
Grade 7 Unit 3 Constructed Response Multiplying and Dividing Rational Numbers trees that Walter and his brother planted per day, 2) the amount their grandfather is going to pay them per tree planted, and 3) the number of trees planted is the same each day. This information is important because the question is asking me to find how much money the boys will earn if they plant 12 trees a day for four days, at the rate that their grandfather will be paying them, $7 per tree. Their daily ten minute break and their hours worked are two unnecessary pieces of information in the problem. Since their grandfather is paying them per tree and not per hour, the time worked does not really matter. b. Student gives circles correct answer: (12 × 4 × 7) c. Student gives correct answer and shows work: $336 d. Student gives explanation: The mathematical operation used to find the solution of this problem was multiplication. I knew that this was the necessary operation because I had to combine groups of quantities or, in this case, total money earned per day over four days. 4. a. Student gives explanation: My first step in solving this problem is to change the expression by flipping the second fraction. By changing the second fraction to its reciprocal, I changed the problem from division to multiplication. My next step is to cross cancel any common factors to simplify the problem. Finally, I multiply across to get my answer. However, since there is a positive multiplied by a negative, I need to make my answer a negative number. 14 b. Student gives correct answer and shows work: − 15 c. Student gives explanation: In order to prove that my answer is correct I could use multiplication to check that my answer is correct since it is the inverse operation of division. To do this I would multiply the second fraction in the expression by my answer from above. If I get a product of – 3/5 then I know that my answer is correct. d. Student shows work for checking answer: Should follow the explanation given in question 4c. 5. a. Student gives explanation: In order to find how many miles she will be running in fifty minutes she needs to use Expression 1. I know that this is 1 the correct answer because what she is trying to find is how many 8 2 minute miles are in 50 minutes. What she is doing is breaking down the 50 1 minutes into groups of 8 2 minutes and to do this she needs to divide 50 by 1
1
8 2, so the correct expression is 50 ÷ 8 2. Expression 2 is incorrect because 1
what is happening in this situation is that the 8 2 minutes are being divided Copyright © Swun Math Grade 7 Unit 3 Constructed Response Rubric, Page 3
0.25 0.5 0.25
0.5
0.5 0.25
0.25 1
Grade 7 Unit 3 Constructed Response Multiplying and Dividing Rational Numbers by 50 minutes, which does not make sense in this situation. 100 b. Student gives correct answer and shows work: 17 155
6. a. Student gives correct answers and shows work: 32 b. Student creates number line: This process to solve this expression flipping
0.5 1 1
the two last fractions to their reciprocals and reducing the fractions to the simplest form to find the solution. Despite this being a three fraction expression the same rules of dividing fractions applied as they did to the expression in question 4, because the first fraction remained unchanged while the second was turned into its reciprocal. One more difference between the expressions was the second expression had a mixed number that I needed to convert to an improper fraction before I could solve.
7. Student gives correct answers, shows work, and gives explanation: a. 0.51470; irrational; The decimal in this problem is irrational because the decimal continues without stopping and has no pattern. b. 0.33333…..; repeating; The decimal in this problem is repeating because the decimal numbers follow a sequence that does not end. c. 2.75; terminating; The decimal in this problem is terminating because the decimal ends at 5 and does not continue. 8. a. Student gives correct answer: 2.25 + 2.125 ÷ (0.6 ∙ 0.6666) b. Student gives correct answer and shows work: 7.56 c. Student gives explanation: While solving this problem, a rule that was important for finding the correct solution was the Order of Operations rule. This was important because according to this rule you need to solve for things in the parentheses first and then work from the left depending on the operation signs. If these rules are not followed chances are you will get the wrong answer, and since this expression involved a few different signs it was important. d. Student gives explanation: Answers may vary. 7 9. a. Student gives correct answer and shows work: 10; 0.7 b. Student gives explanation: In order to find the solution, I had to add the two fractions that represented the amount of cake that was eaten. To do this I needed to find a common denominator so I could add the fractions. After finding the common denominator I was able to add the fractions, which gave me the total fraction of the cake eaten. To convert this fraction to a decimal I had to divide the numerator by the denominator. So since my 7 fraction of cake eaten was 10, I had to solve the expression 7 ÷ 10, and this gave me 0.7. My strategy was effective because if I had converted the fractions to decimals first, I may have had to work with very large decimals. Copyright © Swun Math Grade 7 Unit 3 Constructed Response Rubric, Page 4
0.25 0.25 0.25 0.25 0.5 0.5
0.25 1 1.25
Grade 7 Unit 3 Constructed Response Multiplying and Dividing Rational Numbers This way I worked with the original fractions and then converted my final answer to a decimal. Test Total
Copyright © Swun Math Grade 7 Unit 3 Constructed Response Rubric, Page 5
15
1. Multiplication of Signed Rational Numbers
2. Division of Signed Rational Numbers
Common Core State Standard 7.NS.2a[m]: Understand multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1)=1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. 7.NS.2c [m]: Apply properties of operations as strategies to multiply and divide rational numbers. 7.NS.2b [m]: Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. Interpret quotients of rational numbers by describing real world contexts. 7.NS.2c [m]: Apply properties of operations as strategies to multiply and divide rational numbers. 7.NS.2d [m]: Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
Standards for Mathematical Practice SMP.1, SMP. 2, SMP.4, SMP.7
SMP.1, SMP.2, SMP.3, SMP.4, SMP.6, SMP.7, SMP.8
SMP.1, SMP.2, 3. Converting Rational SMP.4, SMP.6, Numbers to Decimals SMP.7 and Decimal Forms of Rational Numbers 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 7 Unit 3 Constructed Response Rubric, Page 1
Grade 7 Unit 3 Constructed Response Multiplying and Dividing Rational Numbers Scoring Rubric Question 1. a. Student gives explanation: After analyzing the two expressions I was able to make some observations about the similarities and differences in the possible answers of the two expressions. At first glance I noticed that the two numbers that are being multiplied are the same for both Expression A and B, so the product of the two expressions will be the same. However, because the symbols in front of each number in the expressions are different, their products will have a different sign. For example, in Expression A, a negative number is multiplied by a positive number, so the product is negative. In Expression B, a negative number is multiplied by a negative number, so the product is positive. b. Student gives correct answers and shows work: Student should display work on a number line as shown in lessons; Expression A= -36, Expression B= 36. 2. a. Student gives explanation: After reading and analyzing the information given, I was able to conclude that the problem is asking me to find how much money Cassidy will save up in 8 weeks, after her weekly earnings and expenses. In order to find this I first need to find out how much money she is saving a week. To do this I need to find how much money she is making a week and subtract from this number how much money she is spending in that same week. Since the problem does not state that the money she earns and spends weekly changes, I know that I can multiply her weekly earnings by 8 weeks to get her total savings. b. Student gives correct answer: 8 ∙ (30 + 15 + 20 − 10) c. Student gives correct answer and shows work: $440.00 d. Student gives explanation: In order to find the solution to this problem, I had to use the Distributive Property. I used this property in order to solve the expression I wrote in question 2b, 8 ($30 + $15 + $20 - $10). To solve this expression using the Distributive Property I first had to find Cassidy’s weekly savings in the parentheses, which was equal to $55. I then multiplied this number by 8 on the outside of the parentheses, which gave me $440. 3. a. Student gives explanation: After analyzing the information given in the problem I was able to identify pieces of information that are important to finding the solution as well as extra information not necessary. Important pieces of information that I found in the scenario were: 1) the number of Copyright © Swun Math Grade 7 Unit 3 Constructed Response Rubric, Page 2
Points 0.5
0.5
0.75
0.25 0.5 0.5
1
Total
Grade 7 Unit 3 Constructed Response Multiplying and Dividing Rational Numbers trees that Walter and his brother planted per day, 2) the amount their grandfather is going to pay them per tree planted, and 3) the number of trees planted is the same each day. This information is important because the question is asking me to find how much money the boys will earn if they plant 12 trees a day for four days, at the rate that their grandfather will be paying them, $7 per tree. Their daily ten minute break and their hours worked are two unnecessary pieces of information in the problem. Since their grandfather is paying them per tree and not per hour, the time worked does not really matter. b. Student gives circles correct answer: (12 × 4 × 7) c. Student gives correct answer and shows work: $336 d. Student gives explanation: The mathematical operation used to find the solution of this problem was multiplication. I knew that this was the necessary operation because I had to combine groups of quantities or, in this case, total money earned per day over four days. 4. a. Student gives explanation: My first step in solving this problem is to change the expression by flipping the second fraction. By changing the second fraction to its reciprocal, I changed the problem from division to multiplication. My next step is to cross cancel any common factors to simplify the problem. Finally, I multiply across to get my answer. However, since there is a positive multiplied by a negative, I need to make my answer a negative number. 14 b. Student gives correct answer and shows work: − 15 c. Student gives explanation: In order to prove that my answer is correct I could use multiplication to check that my answer is correct since it is the inverse operation of division. To do this I would multiply the second fraction in the expression by my answer from above. If I get a product of – 3/5 then I know that my answer is correct. d. Student shows work for checking answer: Should follow the explanation given in question 4c. 5. a. Student gives explanation: In order to find how many miles she will be running in fifty minutes she needs to use Expression 1. I know that this is 1 the correct answer because what she is trying to find is how many 8 2 minute miles are in 50 minutes. What she is doing is breaking down the 50 1 minutes into groups of 8 2 minutes and to do this she needs to divide 50 by 1
1
8 2, so the correct expression is 50 ÷ 8 2. Expression 2 is incorrect because 1
what is happening in this situation is that the 8 2 minutes are being divided Copyright © Swun Math Grade 7 Unit 3 Constructed Response Rubric, Page 3
0.25 0.5 0.25
0.5
0.5 0.25
0.25 1
Grade 7 Unit 3 Constructed Response Multiplying and Dividing Rational Numbers by 50 minutes, which does not make sense in this situation. 100 b. Student gives correct answer and shows work: 17 155
6. a. Student gives correct answers and shows work: 32 b. Student creates number line: This process to solve this expression flipping
0.5 1 1
the two last fractions to their reciprocals and reducing the fractions to the simplest form to find the solution. Despite this being a three fraction expression the same rules of dividing fractions applied as they did to the expression in question 4, because the first fraction remained unchanged while the second was turned into its reciprocal. One more difference between the expressions was the second expression had a mixed number that I needed to convert to an improper fraction before I could solve.
7. Student gives correct answers, shows work, and gives explanation: a. 0.51470; irrational; The decimal in this problem is irrational because the decimal continues without stopping and has no pattern. b. 0.33333…..; repeating; The decimal in this problem is repeating because the decimal numbers follow a sequence that does not end. c. 2.75; terminating; The decimal in this problem is terminating because the decimal ends at 5 and does not continue. 8. a. Student gives correct answer: 2.25 + 2.125 ÷ (0.6 ∙ 0.6666) b. Student gives correct answer and shows work: 7.56 c. Student gives explanation: While solving this problem, a rule that was important for finding the correct solution was the Order of Operations rule. This was important because according to this rule you need to solve for things in the parentheses first and then work from the left depending on the operation signs. If these rules are not followed chances are you will get the wrong answer, and since this expression involved a few different signs it was important. d. Student gives explanation: Answers may vary. 7 9. a. Student gives correct answer and shows work: 10; 0.7 b. Student gives explanation: In order to find the solution, I had to add the two fractions that represented the amount of cake that was eaten. To do this I needed to find a common denominator so I could add the fractions. After finding the common denominator I was able to add the fractions, which gave me the total fraction of the cake eaten. To convert this fraction to a decimal I had to divide the numerator by the denominator. So since my 7 fraction of cake eaten was 10, I had to solve the expression 7 ÷ 10, and this gave me 0.7. My strategy was effective because if I had converted the fractions to decimals first, I may have had to work with very large decimals. Copyright © Swun Math Grade 7 Unit 3 Constructed Response Rubric, Page 4
0.25 0.25 0.25 0.25 0.5 0.5
0.25 1 1.25
Grade 7 Unit 3 Constructed Response Multiplying and Dividing Rational Numbers This way I worked with the original fractions and then converted my final answer to a decimal. Test Total
Copyright © Swun Math Grade 7 Unit 3 Constructed Response Rubric, Page 5
15
