G4 U6 Constructed Response Rubric
Grade 4 Unit 6 Constructed Response Multiplying Fractions Scoring Rubric Task
Common Core State Standard for Mathematical Content (MC)
Standards for Mathematical Practice (MP)
1. Decompose Fractions
4.NF.4a: Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
MP.1, MP.2, MP.4, MP.6, MP.7, MP.8
2. Multiplying Fractions
4.NF.4b: Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
MP.1, MP.2, MP.4, MP.5, MP.7
3. Word Problems with Fractions
4.NF.4b: Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) 4.NF.4c: Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
MP.1, MP.2, MP.3, MP.5, 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 4 Unit 6 Constructed Response Rubric, Page 1
Grade 4 Unit 6 Constructed Response Multiplying Fractions
Copyright © Swun Math Grade 4 Unit 6 Constructed Response Rubric, Page 2
Grade 4 Unit 6 Constructed Response Multiplying Fractions Scoring Rubric Question 1. a. Student gives correct answer and shows work by an accurate creating a number line: (Check Student’s number line for accuracy in using the 1 strategy); 5 × 8 b. Student gives an accurate explanation. Wording may vary. Sample explanation: Using a number line helped me to solve the problem because 1 5 I could visualize how many sets of 8 are in the fraction 8. To make my number line, first I had to determine how many equal pieces to divide the number line. To find this I looked at the denominator in the fraction, and it was eight. So I proceeded to break up my number line into eight equal parts and labeled each part with the fraction it represented. My next step was to look at the numerator in the fraction, because that number represents how many of the equally divided pieces are represented by the fraction. In this fraction the numerator was five, so starting from zero on the number line I used arrrows to show 5 equal jumps. My number line 5 1 1 1 1 1 showed that 8 = 8 + 8 + 8 + 8 + 8. From this equation and my number line 1
5
Points 1 2
1
I could see there were five sets of 8 , so I could conclude that 8 = 5 × 8. 2. a. Student gives correct answer and shows work using an accurate area model: (Check Student’s area model for accuracy in using the strategy); 7 × 1
2
10
b. Student gives an accurate explanation. Wording may vary. Sample explanation: Using an area model helped me solve the problem because I 1 7 could visualize how many sets of 10 are in the fraction 10. To make my area model, first I had to determine how many equal pieces I had to divide my model. To find this I looked at the denominator in the fraction, and it was ten. So I proceeded to break up my area model into ten equal parts and 1 labeled each part with the fraction 10. My next step was to look at the numerator in the fraction, so I shaded seven of the ten parts in my model. 7 1 The total shaded region of my model could be represented by 10 = 10 + 1
1
1
1
1
1
1
+ 10 + 10 + 10 + 10 + 10. From this equation and my area model I could 10 1
7
1
see there were seven sets of 10 , so I could conclude that 10 = 7 × 10.
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Total
Grade 4 Unit 6 Constructed Response Multiplying Fractions 3. a. Student creates an accurate area model: Sample answer
2
b. Student gives an accurate explanation. Wording may vary. Sample 1 1 explanation: My visual model shows 3 5 can also be written as 16 × 5.
2
1
When I decomposed 3 5 in my model I had to draw four wholes broken up into five equal parts. Of those four wholes, three were shaded completely to represent the whole number three in the mixed number and the fourth 1 whole only had one part shaded to represent the fraction 5. If you count the number of shaded parts there are a total of 16, so represented as a 1 1 1 1 1 1 1 1 1 1 1 fraction sentence this would be 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 1
1
1
1
1
+ 5 + 5 + 5 + 5. If I decompose this to a whole number multiplied by a 5 1
1
fraction I would get 16 × 5. So as can be seen in my model 3 5 can also be 1
written as 16 × 5. Student gives an accurate explanation. Wording may vary. Sample explanation: 4. a. Student gives correct answer and an accurate explanation. Wording may vary. Sample explanation: After analyzing Lewis’ work, I was able to determine that his work and his answer are both incorrect. I know he is incorrect because he did not move along the number line like he should 5 have according to the problem. From the point 8 he moved forward a distance of one whole three times. So what he represented on his line was 5 5 adding three wholes to 8. What he should have done from the point 8 was move forward two times and each move forward should have been a 5 distance of 8. This would have accurately represented the product of 3 ×
2
5
.
8
1 Copyright © Swun Math Grade 4 Unit 6 Constructed Response Rubric, Page 4
Grade 4 Unit 6 Constructed Response Multiplying Fractions b. Student gives correct answer and accurately shows work using a number line: (Check Student’s number line for accuracy in using the strategy);
15 8
5. a. Student gives correct answer and accurately shows work using area 20 model: (Check students area model for accuracy in using the strategy); 9 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 4 4 it by the whole. Since 5 × 9 is equal to 5 groups of 9, I illustrated this in
1 2
4
my area model by adding one whole at a time with 9 of each whole shaded. When you are multiplying a fraction by a whole number you are basically adding the fraction by itself the number of times stated in the whole number. So when I made it to 5 wholes, I counted up all the shaded portions and got a total of 20. The 20 was the numerator of the fraction. 20 So my final fraction as represented by the model was 9 . 6. a. Student gives correct answer and accurately shows work using a 21 number line or area model: (Check students work for accuracy); 10 b. Student gives an accurate explanation: When you multiply fractions by a whole number, the whole number in the expression is telling you how many times the fraction is being added to itself. So when you actually use repeated addition, you are adding the fraction by itself as many times as stated in the whole number. The fraction you get from using repeated addition represents the product of the original whole number and the fraction. 1 3 7. a. Student gives correct answers: 84 × 3 ; 84 × 7 b. Student gives correct answers: 28 campers in archery class, 36 campers in rowing class
1
8. a. Student gives an accurate explanation. Wording may vary. Sample explanation: In order to find the solution I need to first analyze the information given to me. In the problem I am told 7 kids including Samantha’s little sister will be at the party. Samantha bought a bag of 1 candy that will be shared equally so each child’s goodie bag will receive 7 of the total candy. Looking at the denominator of the fraction, I know there are seven bags to be filled, because Samantha is splitting the candy into sevenths and each child will receive one of these portions. She also buys 21 toys to place in the bags equally. Since I am trying to find how
2
Copyright © Swun Math Grade 4 Unit 6 Constructed Response Rubric, Page 5
1
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Grade 4 Unit 6 Constructed Response Multiplying Fractions 1
many toys each child got in their bag, I would multiply 21 by 7 because that is the portion each child will get based on how many there will be at the party. To find how many toys Samantha’s little sister will get I would take the number of toys she got in her own bag and multiply it by two because she received two bags. 1 b. Student gives correct answer: 21 × 7 c. Student gives correct answers and shows work by accurately using either a number line, area model, or properties of operations: 3 toys per child, 6 toys for Samantha’s little sister 9. Student gives correct answers: a. False b. True c. False d. True e. Student creates an accurate model to prove answers are correct. Models may vary. Sample models: White Chocolate Macadamia
1 2
2
2
Peanut Butter Cookies
Total
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Common Core State Standard for Mathematical Content (MC)
Standards for Mathematical Practice (MP)
1. Decompose Fractions
4.NF.4a: Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
MP.1, MP.2, MP.4, MP.6, MP.7, MP.8
2. Multiplying Fractions
4.NF.4b: Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
MP.1, MP.2, MP.4, MP.5, MP.7
3. Word Problems with Fractions
4.NF.4b: Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) 4.NF.4c: Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
MP.1, MP.2, MP.3, MP.5, 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 4 Unit 6 Constructed Response Rubric, Page 1
Grade 4 Unit 6 Constructed Response Multiplying Fractions
Copyright © Swun Math Grade 4 Unit 6 Constructed Response Rubric, Page 2
Grade 4 Unit 6 Constructed Response Multiplying Fractions Scoring Rubric Question 1. a. Student gives correct answer and shows work by an accurate creating a number line: (Check Student’s number line for accuracy in using the 1 strategy); 5 × 8 b. Student gives an accurate explanation. Wording may vary. Sample explanation: Using a number line helped me to solve the problem because 1 5 I could visualize how many sets of 8 are in the fraction 8. To make my number line, first I had to determine how many equal pieces to divide the number line. To find this I looked at the denominator in the fraction, and it was eight. So I proceeded to break up my number line into eight equal parts and labeled each part with the fraction it represented. My next step was to look at the numerator in the fraction, because that number represents how many of the equally divided pieces are represented by the fraction. In this fraction the numerator was five, so starting from zero on the number line I used arrrows to show 5 equal jumps. My number line 5 1 1 1 1 1 showed that 8 = 8 + 8 + 8 + 8 + 8. From this equation and my number line 1
5
Points 1 2
1
I could see there were five sets of 8 , so I could conclude that 8 = 5 × 8. 2. a. Student gives correct answer and shows work using an accurate area model: (Check Student’s area model for accuracy in using the strategy); 7 × 1
2
10
b. Student gives an accurate explanation. Wording may vary. Sample explanation: Using an area model helped me solve the problem because I 1 7 could visualize how many sets of 10 are in the fraction 10. To make my area model, first I had to determine how many equal pieces I had to divide my model. To find this I looked at the denominator in the fraction, and it was ten. So I proceeded to break up my area model into ten equal parts and 1 labeled each part with the fraction 10. My next step was to look at the numerator in the fraction, so I shaded seven of the ten parts in my model. 7 1 The total shaded region of my model could be represented by 10 = 10 + 1
1
1
1
1
1
1
+ 10 + 10 + 10 + 10 + 10. From this equation and my area model I could 10 1
7
1
see there were seven sets of 10 , so I could conclude that 10 = 7 × 10.
Copyright © Swun Math Grade 4 Unit 6 Constructed Response Rubric, Page 3
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Grade 4 Unit 6 Constructed Response Multiplying Fractions 3. a. Student creates an accurate area model: Sample answer
2
b. Student gives an accurate explanation. Wording may vary. Sample 1 1 explanation: My visual model shows 3 5 can also be written as 16 × 5.
2
1
When I decomposed 3 5 in my model I had to draw four wholes broken up into five equal parts. Of those four wholes, three were shaded completely to represent the whole number three in the mixed number and the fourth 1 whole only had one part shaded to represent the fraction 5. If you count the number of shaded parts there are a total of 16, so represented as a 1 1 1 1 1 1 1 1 1 1 1 fraction sentence this would be 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 1
1
1
1
1
+ 5 + 5 + 5 + 5. If I decompose this to a whole number multiplied by a 5 1
1
fraction I would get 16 × 5. So as can be seen in my model 3 5 can also be 1
written as 16 × 5. Student gives an accurate explanation. Wording may vary. Sample explanation: 4. a. Student gives correct answer and an accurate explanation. Wording may vary. Sample explanation: After analyzing Lewis’ work, I was able to determine that his work and his answer are both incorrect. I know he is incorrect because he did not move along the number line like he should 5 have according to the problem. From the point 8 he moved forward a distance of one whole three times. So what he represented on his line was 5 5 adding three wholes to 8. What he should have done from the point 8 was move forward two times and each move forward should have been a 5 distance of 8. This would have accurately represented the product of 3 ×
2
5
.
8
1 Copyright © Swun Math Grade 4 Unit 6 Constructed Response Rubric, Page 4
Grade 4 Unit 6 Constructed Response Multiplying Fractions b. Student gives correct answer and accurately shows work using a number line: (Check Student’s number line for accuracy in using the strategy);
15 8
5. a. Student gives correct answer and accurately shows work using area 20 model: (Check students area model for accuracy in using the strategy); 9 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 4 4 it by the whole. Since 5 × 9 is equal to 5 groups of 9, I illustrated this in
1 2
4
my area model by adding one whole at a time with 9 of each whole shaded. When you are multiplying a fraction by a whole number you are basically adding the fraction by itself the number of times stated in the whole number. So when I made it to 5 wholes, I counted up all the shaded portions and got a total of 20. The 20 was the numerator of the fraction. 20 So my final fraction as represented by the model was 9 . 6. a. Student gives correct answer and accurately shows work using a 21 number line or area model: (Check students work for accuracy); 10 b. Student gives an accurate explanation: When you multiply fractions by a whole number, the whole number in the expression is telling you how many times the fraction is being added to itself. So when you actually use repeated addition, you are adding the fraction by itself as many times as stated in the whole number. The fraction you get from using repeated addition represents the product of the original whole number and the fraction. 1 3 7. a. Student gives correct answers: 84 × 3 ; 84 × 7 b. Student gives correct answers: 28 campers in archery class, 36 campers in rowing class
1
8. a. Student gives an accurate explanation. Wording may vary. Sample explanation: In order to find the solution I need to first analyze the information given to me. In the problem I am told 7 kids including Samantha’s little sister will be at the party. Samantha bought a bag of 1 candy that will be shared equally so each child’s goodie bag will receive 7 of the total candy. Looking at the denominator of the fraction, I know there are seven bags to be filled, because Samantha is splitting the candy into sevenths and each child will receive one of these portions. She also buys 21 toys to place in the bags equally. Since I am trying to find how
2
Copyright © Swun Math Grade 4 Unit 6 Constructed Response Rubric, Page 5
1
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Grade 4 Unit 6 Constructed Response Multiplying Fractions 1
many toys each child got in their bag, I would multiply 21 by 7 because that is the portion each child will get based on how many there will be at the party. To find how many toys Samantha’s little sister will get I would take the number of toys she got in her own bag and multiply it by two because she received two bags. 1 b. Student gives correct answer: 21 × 7 c. Student gives correct answers and shows work by accurately using either a number line, area model, or properties of operations: 3 toys per child, 6 toys for Samantha’s little sister 9. Student gives correct answers: a. False b. True c. False d. True e. Student creates an accurate model to prove answers are correct. Models may vary. Sample models: White Chocolate Macadamia
1 2
2
2
Peanut Butter Cookies
Total
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