Standard

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.

a. 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).

b. 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.)

c. 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?

Explanation

About the Math, Learning Targets, & Rigor

Students need many opportunities to work with problems in context to understand the connections between models and corresponding equations.

This standard builds on students’ work of adding fractions and extending that work into multiplication.(4.NF.4a)

This standard extends the idea of multiplication as repeated addition (4.NF.4b)

This standard calls for students to use visual fraction models (Area, Linear and Set Models) to solve word problems related to multiplying a whole number by a fraction. (4.NF.4c)

Resources

Videos

• Teaching Video (10:16-12:40)
• Khan Academy (scroll to find 4.NF.4 - click on a skill)
  • Videos (on left side in each skill)
• Learn Zillion
  • Videos

EngageNY Lessons

• 4.NF.4a
  • Module 5, Topic A, Lesson 3: Decompose non-unit fractions and represent them as a whole number times a unit fraction using tape diagrams.
• 4.NF.4b
  • Module 5, Topic G, Lesson 35: Represent the multiplication of n times a/b as (n × a)/b using the associative property and visual models.
  • Module 5, Topic G, Lesson 36: Represent the multiplication of n times a/b as (n × a)/b using the associative property and visual models.
  • Module 5, Topic G, Lesson 37: Find the product of a whole number and a mixed number using the distributive property.
  • Module 5, Topic G, Lesson 38: Find the product of a whole number and a mixed number using the distributive property.

Extra Practice

• North Carolina Tasks
• Illustrative Math Tasks
• Howard County Tasks
  • Click on 4th grade at bottom, then 4.NF.4
• Common Core Sheets
  • 4.NF.4a
  • 4.NF.4b
  • 4.NF.4c

Released Assessment Questions

PARCC

Common Core State Standards

4.NF.4a

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. a. 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 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4).

• Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

STUDENT SAMPLE RESPONSE

4.NF.4b-1

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. b. Understand a multiple of a/b as a multiple of 1/b. For example, use a visual fraction model to express 3 x (2/5) as 6 x (1/5).

• Tasks do not have a context.
• Prompts do not provide visual fraction models; students may at their discretion draw visual fraction models as a strategy.
• Results may equal fractions greater than 1 (including fractions equal to whole numbers limited to 0 through 5).
• Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

STUDENT SAMPLE RESPONSE

4.NF.4b-2

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. b. Use the understanding that a multiple of a/b is a multiple of 1/b to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x (2/5) as 6/5. (In general, n x (a/b) = (nxa)/b.)

• Tasks do not have a context.
• Prompts do not provide visual fraction models; students may at their discretion draw visual fraction models as a strategy.
• Tasks involve expressing a/b as a multiple of a/b as a fraction.
• Results may equal fractions greater than 1 (including fractions equal to whole numbers limited to 0 through 5).
• Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

STUDENT SAMPLE RESPONSE

4.NF.4c

Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. c. 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?

• Prompts do not provide visual fraction models; students may at their discretion draw visual fraction models as a strategy.
• Situations are limited to those in which the product is unknown (situations do not include unknown factors).
• Situations involve a whole number of fractional quantities—not a fraction of a whole-number quantity.
• Results may equal fractions greater than 1 (including fractions equal to whole numbers).
• Results may equal fractions greater than 1 (including fractions equal to whole numbers limited to 0 through 5). vi) Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

STUDENT SAMPLE RESPONSE

Performance Indicators: 4.NF.4a, 4.NF.4b-1, 4.NF.4b-2, 4.NF.4c

Level 5: Exceeds Expectations

• Describes a visual fraction model and solves mathematical and real-world problems by recognizing that fraction a/b is a multiple of 1/b and uses that construct to multiply a fraction by a whole number.

Level 4: Meets Expectations

• Using visual models and/or manipulatives, solves mathematical and real-world problems by recognizing that fraction a/b is a multiple of 1/b and uses that construct to multiply a fraction by a whole number.

Critiquing & Reasoning Standards

4.C.4-3

Base arithmetic explanations/reasoning on concrete referents such as diagrams (whether provided in the prompt or constructed by the student in her response), connecting the diagrams to a written (symbolic) method.

(Content Scope: Knowledge and skills articulated in 4.NF.4a only)

• Tasks have “thin context” or no context.
• Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• Tasks may include whole numbers. Whole numbers are limited to 0 through 5.

4.C.4-4

Base arithmetic explanations/reasoning on concrete referents such as diagrams (whether provided in the prompt or constructed by the student in her response), connecting the diagrams to a written (symbolic) method. (Content Scope: Knowledge and skills articulated in 4.NF.4b only)

• Tasks have “thin context” or no context.
• Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

4.C.5-4

Distinguish correct explanation/reasoning from that which is flawed, and – if there is a flaw in the argument – present corrected reasoning. (For example, some flawed ‘student’ reasoning is presented and the task is to correct and improve it.)

(Content Scope: Knowledge and skills articulated in 4.NF.B)

• Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• Results may equal fractions greater than 1 (including fractions equal to whole numbers limited to 0 through 5).

4.C.6-3

Present solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equals signs appropriately (for example, rubrics award less than full credit for the presence of nonsense statements such as 1 + 4 = 5 + 7 = 12, even if the final answer is correct), or identify or describe errors in solutions to multi-step problems and present corrected solutions.

(Content Scope: Knowledge and skills articulated in 4.NF.3d,4.NF.4c only)

• Denominators are limited to grade 3 possibilities (2, 3, 4, 6, 8) so as to keep computational difficulty lower.
• Multi-step problems must have at least 3 steps

4.C.7-4

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 4.NF.4a, 4.NF.4b only)

Performance Indicators: 4.C.4-3, 4.C.4-4, & 4.C.7-4

"Concrete diagrams"

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.

Performance Indicators: 4.C.5-4 & 4.C.6-3

"Distinguish correct explanation/reasoning from that which is flawed"

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 by:

• presenting and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately
• evaluating explanation/reasoning; if there is a flaw in the argument
• presenting and defending corrected reasoning

Response 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 by:

• presenting and defending solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equal signs appropriately
• distinguishing correct explanation/reasoning from that which is flawed
• identifying and describing the flaw in reasoning or describing errors in solutions to multi-step problems
• presenting corrected reasoning

Response 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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