G5 U6 Constructed Response Rubric
Grade 5 Unit 6 Constructed Response Multiplying Fractions Scoring Rubric Task
1. Fractions as Division; Multiplying Fractions by One
2. Reasoning About Multiplying Fractions by Fractions, and Multiplying Fractions Using an Area Model
Common Core State Standard for Mathematical Content (MC) 5.NF.3: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, or decimal fractions, e.g., by using visual fraction models or equations to represent the problem. 5.NF.5a: Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. 5.NF.5b: Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a) / (n×b) to the effect of multiplying a/b by 1. 5.NF.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 5.NF.4b: Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 5.NF.5b: Explaining why multiplying a given Copyright © Swun Math Grade 5 Unit 6 Constructed Response Rubric, Page 1
Standards for Mathematical Practice (MP)
MP.1, MP.2, MP.4, MP.7, MP.8
MP.1, MP.3, MP.4, MP.6, MP.7, MP.8
Grade 5 Unit 6 Constructed Response Multiplying Fractions
3. Word Problems with Fractions
number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a) / (n×b) to the effect of multiplying a/b by 1. 5.NF.4a: Interpret the product (a/b) × q as a part of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) 5.NF.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
MP.1, MP.2, MP.6, MP.7, MP.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.
Copyright © Swun Math Grade 5 Unit 6 Constructed Response Rubric, Page 2
Grade 5 Unit 6 Constructed Response Multiplying Fractions Scoring Rubric Question 12
1. a. Student gives correct answers: (Variables may vary) 12 ÷ 8 = 𝑥; 8 = 𝑥 b. Student gives correct answer and shows work by creating an accurate 12 model: (Check student’s area model for accuracy in using the strategy); 8 c. Student gives an accurate explanation. Wording may vary. Sample explanation: The relationship between division and fractions is the fraction bar in a fraction represents division. So in the problem when I 12 12 wrote the equation 8 = 𝑥, the fraction 8 could also be read as twelve divided by eight. Reading the fraction as twelve divided by eight represented the word problem, because Ricky had to divide 12 pounds of apples equally among his eight friends. So as can be seen through the problem and my work fractions are a way of representing division. 2. a. Student creates correct answer and shows work: Billy’s room has an area of 416 ft2 which is half the size of his parents’ room with an area of 832 ft2 b. Student creates an accurate visual model: Sample answer:
Points 1 1 2
1
2
26 feet 16 feet
16 feet
Billy’s Room
Parent’s
Room
52 feet Student gives an accurate explanation. Wording may vary. Sample explanation: 3. a. Student gives correct answer and shows work using an accurate area model: (Check student’s area model for accuracy in using the strategy); The total amount of yards will be greater than 8 yards. b. Student gives an accurate explanation. Wording may vary. Sample explanation: When a whole number is being multiplied by a fraction greater than 1, the answer will be greater than the whole number. By using Copyright © Swun Math Grade 5 Unit 6 Constructed Response Rubric, Page 3
1
2
Total
Grade 5 Unit 6 Constructed Response Multiplying Fractions 3
the Identity property you can see 8 x 1 is equal to 8. Since 1 4 yards is 3
greater than 1 yard, then 8 × 1 4 will be greater than 8 yards. 4. a. Student gives correct answer and shows work using an accurate diagram: (Check student’s diagram for accuracy); The remaining amount 1 will be less than 3 of the gallon. b. Student gives correct answer and shows work using an accurate diagram: (Check student’s diagram for accuracy); The remaining amount 5 will be less than 8 of the gallon. c. Student gives an accurate explanation. Wording may vary. Sample explanation: I know my answers from question 4a and 4b are reasonable because when you multiply a fraction by a fraction, the product will be less 5 5 1 than either fraction. Since 8 is less than 1, it makes sense 8 × 3 is also less
1
1
1
1
than 3. 5. a. Student gives correct answer and shows work using an accurate area 9 model: (Check student’s model for accuracy in using the strategy); 7 b. Student gives an accurate explanation. Wording may vary. Sample explanation: An area model helped me make sense of the problem because I could visualize what was happening to the fraction as I multiplied 3 3 it by the whole. Since 3 × 7 is equal to 3 groups of 7, I illustrated this in my area model by drawing three rectangles partitioned into seven equal parts. In each rectangle I shaded 3 out of the seven parts. Looking at my 3 3 3 3 area model I saw 3 × 7 could also be represented as 7 + 7 + 7 where the numerator of each fraction represents the number of shaded pieces. By 9 3 adding all the fractions together I got an answer of 7. So 3 × 7 is equal to 9
1
3
.
7
6. a. Student gives correct answer and an accurate explanation. Wording may vary. Sample explanation: After analyzing Gianna’s answer, I was able to determine it was incorrect. I know she is incorrect because based on her answer she multiplied the whole numbers first and then multiplied the fractions separately. This is not how you multiply mixed numbers. What she should have done first was convert the mixed numbers into improper fractions and then multiply the fractions separately. The new 20 31 expression to multiply would be 7 × 9 . b. Student gives correct answer and shows work using an algorithm: Copyright © Swun Math Grade 5 Unit 6 Constructed Response Rubric, Page 4
620 63
2
1
Grade 5 Unit 6 Constructed Response Multiplying Fractions 5
7. a. Student gives correct answer: 3 × 6 b. Student gives correct answer and shows work using an algorithm and an accurate area model: (Check student’s work for accuracy in using both
1 1
1
strategies); 2 pounds of soil 2
c. Student gives correct answer and shows work:
𝟏 𝟐
1
pound of soil 5
8. a. Student gives correct answers and shows work: Carly= 30 6 minutes; 1
11
Julian= 22 5 minutes; Robby= 17 32 minutes b. Student gives correct answer: The order in which Paula and her friends 11 finished the race is: 1st place Robby with 17 32 minutes, 2nd place Paula 1
2
1
1
with 18 2 minutes, 3rd place Julian with 22 5 minutes, and 4th place Carly 5
with 30 6 minutes. The winner of the race was Robby who came in with the fastest time. 9. a. Student gives an accurate explanation. Wording may vary. Sample explanation: The problem is asking me to find the total number of hours Pete and Molly work during the month based on the fraction of the day they each work. To find these total hours I first need to find how many hours they work a day based on the fraction of hours in a day. To find the number of hours they work in a day I would have to multiply the fraction I am given for each person by 24. I multiply by 24 because this is the number of hours in a day. My next step, once I have the number of hours, is to multiply each number by two because they each work 2 times a week. Once I have the number of hours per week for Pete and Molly I need to multiply both numbers by 4 because the problem is asking me to find the number of hours they each work in 4 weeks. b. Student gives correct answers and shows work: Pete works 48 hours and Molly works 64 hours during the four weeks. Based on the numbers Molly works the most hours during the fours weeks. Total
Copyright © Swun Math Grade 5 Unit 6 Constructed Response Rubric, Page 5
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1
30
1. Fractions as Division; Multiplying Fractions by One
2. Reasoning About Multiplying Fractions by Fractions, and Multiplying Fractions Using an Area Model
Common Core State Standard for Mathematical Content (MC) 5.NF.3: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, or decimal fractions, e.g., by using visual fraction models or equations to represent the problem. 5.NF.5a: Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. 5.NF.5b: Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a) / (n×b) to the effect of multiplying a/b by 1. 5.NF.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 5.NF.4b: Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 5.NF.5b: Explaining why multiplying a given Copyright © Swun Math Grade 5 Unit 6 Constructed Response Rubric, Page 1
Standards for Mathematical Practice (MP)
MP.1, MP.2, MP.4, MP.7, MP.8
MP.1, MP.3, MP.4, MP.6, MP.7, MP.8
Grade 5 Unit 6 Constructed Response Multiplying Fractions
3. Word Problems with Fractions
number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a) / (n×b) to the effect of multiplying a/b by 1. 5.NF.4a: Interpret the product (a/b) × q as a part of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) 5.NF.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
MP.1, MP.2, MP.6, MP.7, MP.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.
Copyright © Swun Math Grade 5 Unit 6 Constructed Response Rubric, Page 2
Grade 5 Unit 6 Constructed Response Multiplying Fractions Scoring Rubric Question 12
1. a. Student gives correct answers: (Variables may vary) 12 ÷ 8 = 𝑥; 8 = 𝑥 b. Student gives correct answer and shows work by creating an accurate 12 model: (Check student’s area model for accuracy in using the strategy); 8 c. Student gives an accurate explanation. Wording may vary. Sample explanation: The relationship between division and fractions is the fraction bar in a fraction represents division. So in the problem when I 12 12 wrote the equation 8 = 𝑥, the fraction 8 could also be read as twelve divided by eight. Reading the fraction as twelve divided by eight represented the word problem, because Ricky had to divide 12 pounds of apples equally among his eight friends. So as can be seen through the problem and my work fractions are a way of representing division. 2. a. Student creates correct answer and shows work: Billy’s room has an area of 416 ft2 which is half the size of his parents’ room with an area of 832 ft2 b. Student creates an accurate visual model: Sample answer:
Points 1 1 2
1
2
26 feet 16 feet
16 feet
Billy’s Room
Parent’s
Room
52 feet Student gives an accurate explanation. Wording may vary. Sample explanation: 3. a. Student gives correct answer and shows work using an accurate area model: (Check student’s area model for accuracy in using the strategy); The total amount of yards will be greater than 8 yards. b. Student gives an accurate explanation. Wording may vary. Sample explanation: When a whole number is being multiplied by a fraction greater than 1, the answer will be greater than the whole number. By using Copyright © Swun Math Grade 5 Unit 6 Constructed Response Rubric, Page 3
1
2
Total
Grade 5 Unit 6 Constructed Response Multiplying Fractions 3
the Identity property you can see 8 x 1 is equal to 8. Since 1 4 yards is 3
greater than 1 yard, then 8 × 1 4 will be greater than 8 yards. 4. a. Student gives correct answer and shows work using an accurate diagram: (Check student’s diagram for accuracy); The remaining amount 1 will be less than 3 of the gallon. b. Student gives correct answer and shows work using an accurate diagram: (Check student’s diagram for accuracy); The remaining amount 5 will be less than 8 of the gallon. c. Student gives an accurate explanation. Wording may vary. Sample explanation: I know my answers from question 4a and 4b are reasonable because when you multiply a fraction by a fraction, the product will be less 5 5 1 than either fraction. Since 8 is less than 1, it makes sense 8 × 3 is also less
1
1
1
1
than 3. 5. a. Student gives correct answer and shows work using an accurate area 9 model: (Check student’s model for accuracy in using the strategy); 7 b. Student gives an accurate explanation. Wording may vary. Sample explanation: An area model helped me make sense of the problem because I could visualize what was happening to the fraction as I multiplied 3 3 it by the whole. Since 3 × 7 is equal to 3 groups of 7, I illustrated this in my area model by drawing three rectangles partitioned into seven equal parts. In each rectangle I shaded 3 out of the seven parts. Looking at my 3 3 3 3 area model I saw 3 × 7 could also be represented as 7 + 7 + 7 where the numerator of each fraction represents the number of shaded pieces. By 9 3 adding all the fractions together I got an answer of 7. So 3 × 7 is equal to 9
1
3
.
7
6. a. Student gives correct answer and an accurate explanation. Wording may vary. Sample explanation: After analyzing Gianna’s answer, I was able to determine it was incorrect. I know she is incorrect because based on her answer she multiplied the whole numbers first and then multiplied the fractions separately. This is not how you multiply mixed numbers. What she should have done first was convert the mixed numbers into improper fractions and then multiply the fractions separately. The new 20 31 expression to multiply would be 7 × 9 . b. Student gives correct answer and shows work using an algorithm: Copyright © Swun Math Grade 5 Unit 6 Constructed Response Rubric, Page 4
620 63
2
1
Grade 5 Unit 6 Constructed Response Multiplying Fractions 5
7. a. Student gives correct answer: 3 × 6 b. Student gives correct answer and shows work using an algorithm and an accurate area model: (Check student’s work for accuracy in using both
1 1
1
strategies); 2 pounds of soil 2
c. Student gives correct answer and shows work:
𝟏 𝟐
1
pound of soil 5
8. a. Student gives correct answers and shows work: Carly= 30 6 minutes; 1
11
Julian= 22 5 minutes; Robby= 17 32 minutes b. Student gives correct answer: The order in which Paula and her friends 11 finished the race is: 1st place Robby with 17 32 minutes, 2nd place Paula 1
2
1
1
with 18 2 minutes, 3rd place Julian with 22 5 minutes, and 4th place Carly 5
with 30 6 minutes. The winner of the race was Robby who came in with the fastest time. 9. a. Student gives an accurate explanation. Wording may vary. Sample explanation: The problem is asking me to find the total number of hours Pete and Molly work during the month based on the fraction of the day they each work. To find these total hours I first need to find how many hours they work a day based on the fraction of hours in a day. To find the number of hours they work in a day I would have to multiply the fraction I am given for each person by 24. I multiply by 24 because this is the number of hours in a day. My next step, once I have the number of hours, is to multiply each number by two because they each work 2 times a week. Once I have the number of hours per week for Pete and Molly I need to multiply both numbers by 4 because the problem is asking me to find the number of hours they each work in 4 weeks. b. Student gives correct answers and shows work: Pete works 48 hours and Molly works 64 hours during the four weeks. Based on the numbers Molly works the most hours during the fours weeks. Total
Copyright © Swun Math Grade 5 Unit 6 Constructed Response Rubric, Page 5
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1
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