G5 U1 Constructed Response Rubric
Grade 5 Unit 1 Constructed Response Place Value System Scoring Rubric Task
1. Place Value
2. Powers of 10 With Whole Numbers and Decimals
3. Comparing and Rounding Decimals
Common Core State Standard 5.NBT.1: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 5.NBT.2: Explain patterns in the number of zeroes of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. 2.NBT.3a: Count within 1000; skip-count by 2s, 5s, 10s, and 100s. 5.NBT.1: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 5.NBT.3a: Read and write decimals to thousandths using base-ten numerals, number names, and expanded form. 5.NBT.3b: Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. 5.NBT.4: Use place value understanding to round decimals to any place.
Standards for Mathematical Practice SMP.1, SMP.4, SMP.5, SMP.7, SMP.8
SMP.2, SMP.6, SMP.7, SMP.8
SMP.1, SMP.2, SMP.4, SMP.5, SMP.6, 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.
Copyright © Swun Math Grade 5 Unit 1 Constructed Response Rubric, Page 1
Grade 5 Unit 1 Constructed Response Place Value System Scoring Rubric Question 1. a. Student completes table:
Points 1
Thousandths
6 0 0 0 0
Hundredths
One
3 6 0 0 0
Tenths
Tens
3 6 0
3 6 0 0
.
Hundreds
3 6
THOUSANDTHS
One-thousands
Start 36 × 10 360 × 10 3,600 × 10 36,000 × 10 3
ONES
Ten-thousands
Hundred-thousand
THOUSANDS
b. Student gives explanation: A place value chart. In a place value chart each move to the left on the table results in the value of the number getting ten times bigger, and with each move to the right the value gets smaller by ten times. Since this was a problem involving multiplication, the table showed the value of 36 getting ten times bigger until it reached a value in the hundred thousandths. This tool is useful in this situation because you can see how the number increases as you multiply it by 10. So in this case 36 is ten times bigger than 36, and 3600 is 10 times bigger than 360. 2. a. Student completes table:
0 3 3
b. Student gives correct answer: 33 Copyright © Swun Math Grade 5 Unit 1 Constructed Response Rubric, Page 2
Thousandths
Hundredths
0 0 0 3
Tenths
0 0 3 3
.
One
3 3
1
THOUSANDTHS
Tens
3
Hundreds
One-thousands
Start 33,000 ÷ 10 3,300 ÷ 10 330 ÷ 10
ONES
Ten-thousands
Hundred-thousand
THOUSANDS
1
0.5
Total
Grade 5 Unit 1 Constructed Response Place Value System c. Student gives explanation: After completing the table, I observed that the quotient in each line had one zero less than the dividend. This pattern was important because each zero taken from the quotient meant that it was 10 times less than the dividend. 3. a. Student gives correct explanation: The six in the thousands place is different because it is 100 times larger than the six in the tens place. The reason for this is because each place value after the first six is ten times less than the place value next to it on the right. The tens place where the second six is located, is 2 spaces over so 10 × 10 is equal to 100. 4. a. Student gives explanation: After looking at the problem I know that the expression is asking me to multiply the number 224 by ten with an exponent of five. The exponent tells me how many times to multiply the base number, 10, by itself. So in this case it is 10 × 10 × 10 × 10 × 10. Finally, I will multiply the product by 224. b. Student gives explanation: By looking at the expression I know that I am dealing with powers of ten, so I can use the powers of ten facts as a shortcut to help me solve this problem. The powers of ten facts will help me because these facts indicate the number of spaces the digits need to be shifted to the left. In this case since I am working with 10 with an exponent of 5 I know that this is the same as 10 ×10 ×10 ×10 ×10 which is equal to 100,000. This means the digits in the number 224 will shift five spaces to the left. This makes my process easier because I will not have to work out the problem 224 × 100,000. c. Student gives correct answer and shows work: 22,400,000 5. a. Student completes equation: 70 × 4 × 106 = 28,000,000 b. Student gives explanation: I know that my number selections made the equation true because I used my knowledge of exponents and the powers of ten facts to solve it. When I looked at the answer of the equation, I knew that best route to find the missing numbers was to break apart the number itself. Two things that I noticed first were that the number 28 was in the beginning of the whole number and that there was six zeroes after this number. I worked on finding the multiples of 28 from the number bank first, and I concluded that they were 7 and 4. From here I moved on to work with the six zeroes in the number. Using knowledge of the power of ten facts I knew that the digits were moved six paces to left when multiplied by 1,000,000. This number is the same as saying 10 × 10 × 10 × 10 × 10 × 10 or 106. With these pieces of information I was able to correctly put together the equation as 7 × 4 × 106 = 28,000,000. 6. a. Student provides explanation: After analyzing the two expressions I Copyright © Swun Math Grade 5 Unit 1 Constructed Response Rubric, Page 3
0.5
1
0.5
1
0.5 0.5 1.5
1.5
Grade 5 Unit 1 Constructed Response Place Value System
Hundreds
Tens
Ones
.
Tenths
Hundredths
Thousandths
was able to find some similarities and differences in the process necessary to solve each expression. I know that for both I will be able to use my knowledge of power of base ten facts as a shortcut to find the solutions. The quantities in each expression are the same however there are slight differences in how those quantities will be worked with because of the use of division in one expression and multiplication in the other. By looking at the base ten with an exponent of 4 present in both expressions I know that the decimal will be moved four spaces in both. However, because the first expression involves division, the decimal will be moved four spaces to the left. On the other hand, in the second expression, which involves multiplication the decimal will be moved four spaces to the right. This difference in the direction that the decimal will be moved will also make very different answers despite both problems containing the same numbers. b. Student gives correct answers: Expression #1: 0.056782, Expression #2: 5678200.0 7. a. Student gives explanation: In order to write a number in expanded form and word form I could use a place value chart. This math tool would be useful to write a number in expanded form because it shows the period, and place value of each digit in the standard form of the number. Knowing this information would make it easy to know how much each digit is worth which I necessary when writing out a number in expanded form and word form. b. Student shows work on place value chart and gives correct answers: ONES Thousandths
3
3
5
.
6
9
8
Expanded form: (3 × 100) + (3 × 10) + (5 × 1) + [6 × (1/10)] + [9 × (1/100)] + [8 × (1/1000)] Word form: three hundred thirty-five and six hundred ninety-eight thousandths
Copyright © Swun Math Grade 5 Unit 1 Constructed Response Rubric, Page 4
0.5
0.5
0.5
Grade 5 Unit 1 Constructed Response Place Value System 8. a. Student gives correct answer: > b. Student gives correct answer: In order to compare the two numbers I first wrote out both numbers in standard form to make them easier to compare. So the new numbers to work with were 79.32 and 70.8. Writing them in standard form made it easier to place them into a place value chart. I lined up the decimals in both numbers to the decimal in the place value chart. Reading from left to write I compared the digits in each place value. By doing this I was able to conclude that 79.32 was larger than 70.8 because of the 9 in the tens place. c. Student creates number line: 70.8 70
0.5 1
0.5
79.32 80
9. a. Student completes table with missing values: Possible answers: 6.866, 6.871, 6.869, 6.873 b. Student gives explanation: In order to find the four numbers that round to 6.87, I had to first identify the place value of the number that would have to be rounded. In this case I determined the place value to be the number in the hundredths place. This number would either be rounded up or stay the same depending on the number I chose for the thousandths place. For example, in the number 6.866, since there is a six in the thousandths place I knew I could round up the six in the hundredths place to seven. This gave me an answer 6.87. Test Total
Copyright © Swun Math Grade 5 Unit 1 Constructed Response Rubric, Page 5
1 1
15
1. Place Value
2. Powers of 10 With Whole Numbers and Decimals
3. Comparing and Rounding Decimals
Common Core State Standard 5.NBT.1: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 5.NBT.2: Explain patterns in the number of zeroes of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. 2.NBT.3a: Count within 1000; skip-count by 2s, 5s, 10s, and 100s. 5.NBT.1: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 5.NBT.3a: Read and write decimals to thousandths using base-ten numerals, number names, and expanded form. 5.NBT.3b: Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. 5.NBT.4: Use place value understanding to round decimals to any place.
Standards for Mathematical Practice SMP.1, SMP.4, SMP.5, SMP.7, SMP.8
SMP.2, SMP.6, SMP.7, SMP.8
SMP.1, SMP.2, SMP.4, SMP.5, SMP.6, 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.
Copyright © Swun Math Grade 5 Unit 1 Constructed Response Rubric, Page 1
Grade 5 Unit 1 Constructed Response Place Value System Scoring Rubric Question 1. a. Student completes table:
Points 1
Thousandths
6 0 0 0 0
Hundredths
One
3 6 0 0 0
Tenths
Tens
3 6 0
3 6 0 0
.
Hundreds
3 6
THOUSANDTHS
One-thousands
Start 36 × 10 360 × 10 3,600 × 10 36,000 × 10 3
ONES
Ten-thousands
Hundred-thousand
THOUSANDS
b. Student gives explanation: A place value chart. In a place value chart each move to the left on the table results in the value of the number getting ten times bigger, and with each move to the right the value gets smaller by ten times. Since this was a problem involving multiplication, the table showed the value of 36 getting ten times bigger until it reached a value in the hundred thousandths. This tool is useful in this situation because you can see how the number increases as you multiply it by 10. So in this case 36 is ten times bigger than 36, and 3600 is 10 times bigger than 360. 2. a. Student completes table:
0 3 3
b. Student gives correct answer: 33 Copyright © Swun Math Grade 5 Unit 1 Constructed Response Rubric, Page 2
Thousandths
Hundredths
0 0 0 3
Tenths
0 0 3 3
.
One
3 3
1
THOUSANDTHS
Tens
3
Hundreds
One-thousands
Start 33,000 ÷ 10 3,300 ÷ 10 330 ÷ 10
ONES
Ten-thousands
Hundred-thousand
THOUSANDS
1
0.5
Total
Grade 5 Unit 1 Constructed Response Place Value System c. Student gives explanation: After completing the table, I observed that the quotient in each line had one zero less than the dividend. This pattern was important because each zero taken from the quotient meant that it was 10 times less than the dividend. 3. a. Student gives correct explanation: The six in the thousands place is different because it is 100 times larger than the six in the tens place. The reason for this is because each place value after the first six is ten times less than the place value next to it on the right. The tens place where the second six is located, is 2 spaces over so 10 × 10 is equal to 100. 4. a. Student gives explanation: After looking at the problem I know that the expression is asking me to multiply the number 224 by ten with an exponent of five. The exponent tells me how many times to multiply the base number, 10, by itself. So in this case it is 10 × 10 × 10 × 10 × 10. Finally, I will multiply the product by 224. b. Student gives explanation: By looking at the expression I know that I am dealing with powers of ten, so I can use the powers of ten facts as a shortcut to help me solve this problem. The powers of ten facts will help me because these facts indicate the number of spaces the digits need to be shifted to the left. In this case since I am working with 10 with an exponent of 5 I know that this is the same as 10 ×10 ×10 ×10 ×10 which is equal to 100,000. This means the digits in the number 224 will shift five spaces to the left. This makes my process easier because I will not have to work out the problem 224 × 100,000. c. Student gives correct answer and shows work: 22,400,000 5. a. Student completes equation: 70 × 4 × 106 = 28,000,000 b. Student gives explanation: I know that my number selections made the equation true because I used my knowledge of exponents and the powers of ten facts to solve it. When I looked at the answer of the equation, I knew that best route to find the missing numbers was to break apart the number itself. Two things that I noticed first were that the number 28 was in the beginning of the whole number and that there was six zeroes after this number. I worked on finding the multiples of 28 from the number bank first, and I concluded that they were 7 and 4. From here I moved on to work with the six zeroes in the number. Using knowledge of the power of ten facts I knew that the digits were moved six paces to left when multiplied by 1,000,000. This number is the same as saying 10 × 10 × 10 × 10 × 10 × 10 or 106. With these pieces of information I was able to correctly put together the equation as 7 × 4 × 106 = 28,000,000. 6. a. Student provides explanation: After analyzing the two expressions I Copyright © Swun Math Grade 5 Unit 1 Constructed Response Rubric, Page 3
0.5
1
0.5
1
0.5 0.5 1.5
1.5
Grade 5 Unit 1 Constructed Response Place Value System
Hundreds
Tens
Ones
.
Tenths
Hundredths
Thousandths
was able to find some similarities and differences in the process necessary to solve each expression. I know that for both I will be able to use my knowledge of power of base ten facts as a shortcut to find the solutions. The quantities in each expression are the same however there are slight differences in how those quantities will be worked with because of the use of division in one expression and multiplication in the other. By looking at the base ten with an exponent of 4 present in both expressions I know that the decimal will be moved four spaces in both. However, because the first expression involves division, the decimal will be moved four spaces to the left. On the other hand, in the second expression, which involves multiplication the decimal will be moved four spaces to the right. This difference in the direction that the decimal will be moved will also make very different answers despite both problems containing the same numbers. b. Student gives correct answers: Expression #1: 0.056782, Expression #2: 5678200.0 7. a. Student gives explanation: In order to write a number in expanded form and word form I could use a place value chart. This math tool would be useful to write a number in expanded form because it shows the period, and place value of each digit in the standard form of the number. Knowing this information would make it easy to know how much each digit is worth which I necessary when writing out a number in expanded form and word form. b. Student shows work on place value chart and gives correct answers: ONES Thousandths
3
3
5
.
6
9
8
Expanded form: (3 × 100) + (3 × 10) + (5 × 1) + [6 × (1/10)] + [9 × (1/100)] + [8 × (1/1000)] Word form: three hundred thirty-five and six hundred ninety-eight thousandths
Copyright © Swun Math Grade 5 Unit 1 Constructed Response Rubric, Page 4
0.5
0.5
0.5
Grade 5 Unit 1 Constructed Response Place Value System 8. a. Student gives correct answer: > b. Student gives correct answer: In order to compare the two numbers I first wrote out both numbers in standard form to make them easier to compare. So the new numbers to work with were 79.32 and 70.8. Writing them in standard form made it easier to place them into a place value chart. I lined up the decimals in both numbers to the decimal in the place value chart. Reading from left to write I compared the digits in each place value. By doing this I was able to conclude that 79.32 was larger than 70.8 because of the 9 in the tens place. c. Student creates number line: 70.8 70
0.5 1
0.5
79.32 80
9. a. Student completes table with missing values: Possible answers: 6.866, 6.871, 6.869, 6.873 b. Student gives explanation: In order to find the four numbers that round to 6.87, I had to first identify the place value of the number that would have to be rounded. In this case I determined the place value to be the number in the hundredths place. This number would either be rounded up or stay the same depending on the number I chose for the thousandths place. For example, in the number 6.866, since there is a six in the thousandths place I knew I could round up the six in the hundredths place to seven. This gave me an answer 6.87. Test Total
Copyright © Swun Math Grade 5 Unit 1 Constructed Response Rubric, Page 5
1 1
15
