Standard

Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Student language:

• "I can find the answer to a division problem by thinking of the missing factor in a multiplication problem. (I can figure out 32/8 because I know that 8 x 4 = 32.)"

Explanation

About the Math, Learning Targets, and Rigor

This standard refers to Table 2. Since multiplication and division are inverse operations, students are expected to solve problems and explain their processes of solving division problems that can also be represented as unknown factor in multiplication problems.

Students use their understanding of the meaning of the equal sign as “the same value as” to interpret an equation with an unknown. When given 32 ÷ ∎ = 4, students may think:

• 4 groups of some number is the same as 32
• 4 times some number is the same as 32
• I know that 4 groups of 8 is 32 so the unknown number is 8
• The missing factor is 8 because 4 times 8 is 32.

Equations in the form of a ÷ b = c and c = a ÷ b need to be used interchangeably, with the unknown in different positions.

Resources

Videos

• Instructional Video (1:05 - 4:44)
• Khan Academy
  • scroll to 3.OA.6, click on a skill, watch videos on left side in each skill
• Learn Zillion
  • Videos
• Virtual Nerd

EngageNY Lessons

Extra Practice

• North Carolina Tasks
• Illustrative Math Tasks
• Howard County Tasks
  • Click on 3rd grade at bottom, then 3.OA.6
• Common Core Sheets

Released Assessment Questions

PARCC

3.OA.6

Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

• All products and related quotients are from the harder three quadrants of the times table (a × b where a > 5 and/or b > 5).

Performance Indicators: 3.OA.6

Level 5: Exceeds Expectations

• Understands and interprets products and quotients of whole numbers.
• Determines the unknown whole number in a multiplication or division problem by relating multiplication and division.
• Both factors are greater than 5 and less than or equal 10.
• Represents a multiplication or division situation as an equation.
• Accurately multiplies and divides within 100, using strategies relating multiplication and division or properties of operations

Level 4: Meets Expectations

• Interprets products and quotients of whole numbers.
• Determines the unknown whole number in a multiplication or division problem by relating multiplication and division. One factor is greater than or equal to 5.
• Accurately multiplies and divides within 100, using strategies relating multiplication and division or properties of operations.

Critiquing Standards

3.C.2

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

(Content Scope: Knowledge and skills articulated in 3.OA.6)

• Products and related quotients are limited to the 10 x 10 multiplication table.

3.C.4-2

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 3.OA.6)

• Products and related quotients are limited to the 10x10 multiplication table.

Performance Indicators: 3.C.2

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 complete written response based on explanations/reasoning using the:

• properties of operations
• relationship between addition and subtraction
• relationship between multiplication and division
• identification of arithmetic patterns

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
• determination 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 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 complete written response based on explanations/reasoning using the:

• properties of operations
• relationship between addition and subtraction
• relationship between multiplication and division
• identification of arithmetic patterns

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
• evaluating, interpreting and critiquing the validity of other’s responses, reasonings, and approaches, utilizing mathematical connections (when appropriate).

Performance Indicators: 3.C.4-2

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 counter-example 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
• evaluating, interpreting and critiquing the validity of other’s responses, approaches and reasoning.