Standard

Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.) Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Student language:

• "I can use the Commutative property of multiplication. (I know that if 6 x 4 = 24, then 4 x 6 = 24.)"
• "I can use the Associative property of multiplication. (To figure out 3 x 5 x 2 I can multiply 3 x 5 = 15, then 15 x 2 = 30 OR multiply 5 x 2 = 10, then 3 x 10 = 30.)"
• "I can use the Distributive property of multiplication. (To figure out 8 x 7, I can think of 8 x (5 + 2) which means (8 x 5) + (8 x 2) = 40 + 16 = 56.)"

Explanation

About the Math, Learning Targets, and Rigor

This standard references properties of multiplication. While students DO NOT need to use the formal terms of these properties, student should understand that properties are rules about how numbers work.

Commutative Property of Multiplication

• The order of numbers does not matter when adding or multiplying numbers.
• For example, if a student knows that 5 x 4 = 20, then they also know that 4 x 5 = 20.
• Students change the order of numbers to determine that the order of numbers does not make a difference in multiplication (but does make a difference in division).

The array below could be described as a 5 x 4 array for 5 columns and 4 rows, or a 4 x 5 array for 4 rows and 5 columns. (There is no “fixed” way to write the dimensions of an array as rows x columns or columns x rows.) Students should have flexibility in being able to describe both dimensions of an array.

Associative Property of Multiplication

• The product stays the same when the grouping of addends or factors is changed.
• For example, when a student multiplies 7 x 5 x 2, a student could rearrange the numbers to first multiply 5 x 2 = 10 and then multiply 10 x 7 = 70.

Distributive Property of Multiplication

• Decomposing numbers to find products
• For example, to solve 6 x 12, a student could decompose 12 into 10 and 2. Then multiply 6 by each 10 and 2 (6 x 12 = 6 x 10 + 6 x 2 = 60 + 12 = 72).
• Multiple ways exist to decompose numbers using the distributive property to solve equations.

Resources

Videos

• Instructional Video (4:45 - 13:50)
• Khan Academy
  • scroll to 3.OA.5, 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.5
• Common Core Sheets

Released Assessment Questions

PARCC

Common Core State Standards

3.OA.5

Apply properties of operations as strategies to multiply and divide.²

Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Critiquing Standards

3.C.1-1

Base explanations/reasoning on the properties of operations.

(Content Scope: Knowledge and skills articulated in 3.OA.5)

• Students need not use technical terms such as commutative, associative, distributive, or property.
• Products and related quotients are limited to the 10x10 multiplication table
• These tasks may not exceed the content limits of grade 3. For example, 2 x 4 x 5, would be acceptable as students can use the associative property to rewrite the expression as 8 x 5 which falls within the content limits of grade 3. The problem 7 x 4 x 5 would exceed the content limits of grade 3 because any use of the associative property would result in a 2-digit multiplier.

3.C.4-1

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.5)

• Students need not use technical terms such as commutative, associative, distributive, or property. i
• Products and related quotients are limited to the 10x10 multiplication table.

Performance Indicators: 3.C.1-1

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.4-1

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.