Standard

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1

a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Student Language:

"I can divide fractions by whole numbers and whole numbers by fractions."

Explanation

About the Math, Learning Targets, and Rigor

This is the first time that students are dividing with fractions. In fourth grade students divided whole numbers, and multiplied a whole number by a fraction. In fifth grade, students experience division problems with whole number divisors and unit fraction dividends (fractions with a numerator of 1) or with unit fraction divisors and whole number dividends.

5.NF.7a

This standard asks students to work with story contexts where a unit fraction is divided by a non-zero whole number. Students should use various fraction models and reasoning about fractions.

Example: You have 1/8 of a bag of pens and you need to share them among 3 people. How much of the bag does each person get?

5.NF.7b

This standard calls for students to create story contexts and visual fraction models for division situations where a whole number is being divided by a unit fraction.

Example: Create a story context for 5 ÷ 1/6 . Find your answer and then draw a picture to prove your answer and use multiplication to reason about whether your answer makes sense. How many 1/6 are there in 5?

5.NF.7c

Extends students’ work from other standards in 5.NF.7. Students should continue to use visual fraction models and reasoning to solve these real-world problems.

Example: Angelo has 4 lbs of peanuts. He wants to give each of his friends 1/5 𝑙b. How many friends can receive 1/5 𝑙b of peanuts?

Resources

Videos

• Instructional Video (11:30 - 16:17)
• Khan Academy
  • Scroll to 5.NF.7, click on skill, find videos on left side of each skill
• Learn Zillion
  • 5.NF.7a
  • 5.NF.7b
  • 5.NF.7c

EngageNY Lessons

Extra Practice

• North Carolina Tasks
• Illustrative Math Tasks
• Howard County Tasks
  • Click on 5th grade at bottom, then 5.NF.7
• Common Core Sheets

Released Assessment Questions

PARCC

Common Core State Standards

5.NF.7a

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.

STUDENT SAMPLE RESPONSE

5.NF.7b

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.

STUDENT SAMPLE RESPONSE

5.NF.7c

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

• Tasks involve equal group (partition) situations with part size unknown and number of parts unknown. (See Table 2, Common multiplication and division situations, CCSS p 89)
• Prompts do not provide visual fraction models; students may at their discretion draw visual fraction models as a strategy.

STUDENT SAMPLE RESPONSE

Performance Indicators: 5.NF.7a, 5.NF.7b, 5.NF.7c

Level 5: Exceeds Expectations

Describes a model to represent and/or solve real-world problems, by multiplying a mixed number by a fraction, a fraction by a fraction and a whole number by a fraction; dividing a fraction by a whole number and a whole number by a fraction using visual fraction models and creating context for the mathematics and equations, including rectangular areas; and interpreting the product and/or quotient.

Level 4: Meets Expectations

Multiplies a fraction or a whole number by a fraction and divides a fraction by a whole number – or whole number by a fraction – using visual fraction models and creating context for the mathematics, including rectangular areas.

Critiquing/Reasoning Standards

5.C.2-4

Base explanations/reasoning on the relationship between multiplication and division.

(Content Scope: Knowledge and skills articulated in 5.NF.7)

5.C.5-3

Base explanations/reasoning on a number line diagram (whether provided in the prompt or constructed by the student in her response).

(Content Scope: Knowledge and skills articulated in 5.NF.7a, 5.NF.7b)

Performance Indicators: 5.C.2-4

Level 5: Exceeds Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student constructs and communicates a well-organized and complete written response based on explanations/reasoning using the:

• properties of operations
• relationship between addition and subtraction
• relationship between multiplication and division

Response may include:

• a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
• an efficient and logical progression of steps with appropriate justification
• precision of calculation
• correct use of grade-level vocabulary, symbols and labels
• justification of a conclusion
• evaluation of whether an argument or conclusion is generalizable
• evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate). Provides a counter-example where applicable.

Level 4: Meets Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student constructs and communicates a well-organized and complete written response based on explanations/reasoning using the:

• properties of operations
• relationship between addition and subtraction
• relationship between multiplication and division

Response may include:

• a logical/defensible approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
• a logical progression of steps
• precision of calculation
• correct use of grade-level vocabulary, symbols and labels
• justification of a conclusion
• evaluation of whether an argument or conclusion is generalizable
• evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate).

Performance Indicators: 5.C.5-3

Level 5: Exceeds Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response based on operations using concrete referents such as diagrams--including number lines (whether provided in the prompt or constructed by the student) and connecting the diagrams to a written (symbolic) method, which may include:

• a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
• an efficient and logical progression of steps with appropriate justification
• precision of calculation
• correct use of grade-level vocabulary, symbols and labels
• justification of a conclusion
• evaluation of whether an argument or conclusion is generalizable
• evaluating, interpreting, and critiquing the validity of other’s responses, approaches, and reasoning, and providing a counterexample where applicable

Level 4: Meets Expectations

In connection with the content knowledge, skills, and abilities described in Sub-claim A, the student clearly constructs and communicates a well-organized and complete response based on operations using concrete referents such as diagrams--including number lines (whether provided in the prompt or constructed by the student) and connecting the diagrams to a written (symbolic) method, which may include:

• a logical approach based on a conjecture and/or stated assumptions, utilizing mathematical connections (when appropriate)
• a logical progression of steps
• precision of calculation
• correct use of grade-level vocabulary, symbols and labels
• justification of a conclusion
• evaluation of whether an argument or conclusion is generalizable
• evaluating, interpreting, and critiquing the validity of other’s responses, approaches, and reasoning.

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