Standard

Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Student language:

• "I can find patterns in addition and multiplication tables and explain them using what I know about how numbers work."

Explanation

About the Math, Learning Targets, and Rigor

This standard calls for students to examine arithmetic patterns involving both addition and multiplication.

Arithmetic patterns are patterns that change by the same rate, such as adding the same number. For example, the series 2, 4, 6, 8, 10 is an arithmetic pattern that increases by 2 between each term.

Examples

• Even numbers are always divisible by 2. Even numbers can always be decomposed into 2 equal addends (14 = 7 + 7).
• Multiples of even numbers (2, 4, 6, and 8) are always even numbers.
• On a multiplication chart, the products in each row and column increase by the same amount (skip counting).
• On an addition chart, the sums in each row and column increase by the same amount.
• Using a multiplication table, highlight a row of numbers and ask students what they notice about the highlighted numbers.
• Any sum of two even numbers is even.
• Any sum of two odd numbers is even.
• Any sum of an even number and an odd number is odd.
• The multiples of 4, 6, 8, and 10 are all even because they can all be decomposed into two equal groups.
• The doubles (2 addends the same) in an addition table fall on a diagonal while the doubles (multiples of 2) in a multiplication table fall on horizontal and vertical lines.
• The multiples of any number fall on a horizontal and a vertical line due to the commutative property.
• All the multiples of 5 end in a 0 or 5 while all the multiples of 10 end with 0. Every other multiple of 5 is a multiple of 10.

Students should investigate addition and multiplication tables in search of patterns and explain why these patterns make sense mathematically.

Resources

Videos

• Khan Academy
  • scroll to 3.OA.9, click on a skill, watch videos on left side in each skill
• Learn Zillion
  • Videos

EngageNY Lessons

Extra Practice

• North Carolina Tasks
• Illustrative Math Tasks
• Howard County Tasks
  • Click on 3rd grade at bottom, then 3.OA.9
• Common Core Sheets

Released Assessment Questions

PARCC

Common Core State Standards

3.OA.9

Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

Critiquing Standards

3.C.1-2

Base explanations/reasoning on the properties of operations.

(Content Scope: Knowledge and skills articulated in 3.OA.9)

• Students need not use technical terms such as commutative, associative, distributive, or property

3.C.4-6

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.9)

Performance Indicators: 3.C.1-2

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.4-6

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.