Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.AT.7

Lesson 6.8Generate Equivalent Expressions

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can use the commutative, associative, and identity properties to rewrite expressions.

Start hereWords for this lesson

Read each word before you begin. Say it out loud.

Word	What it means	Example
Properties of OperationsSpanish: Propiedades de las operaciones	Rules that let you reorder or regroup numbers without changing an expression's value.	3 + 5 = 5 + 3 (commutative property).
Commutative PropertySpanish: Propiedad conmutativa	You can change the order and get the same answer.	a + b = b + a
Associative PropertySpanish: Propiedad asociativa	You can change the grouping and get the same answer.	(a + b) + c = a + (b + c)
Identity PropertySpanish: Propiedad de identidad	Adding 0 or multiplying by 1 keeps the same value.	a + 0 = a · a × 1 = a
PropertySpanish: Propiedad	A rule that is always true in math.	Commutative works for + and ×, but NOT for − or ÷ (since 5 − 3 = 2 but 3 − 5 = −2)
Distributive PropertySpanish: Propiedad distributiva	Multiplying a number by everything inside the parentheses: a(b + c) = ab + ac.	a(b + c) = ab + ac

How it worksWorked examplerewriting an expression with properties

These numbers are not on your problems. The steps are. Follow them with your own numbers.

Write an expression equivalent to 6(y + 3) + 9y in simplest form.

Distribute the 6 to each term inside. 6(y + 3) = 6y + 18
Rewrite the whole expression. 6y + 18 + 9y
Reorder so like terms sit together. 6y + 9y + 18
Combine like terms. 15y + 18

Answer: 6(y + 3) + 9y is equivalent to 15y + 18.

Now you trySame steps, your turn

The steps are the same as the worked example. The numbers are yours. Do the work.

Write an expression equivalent to 4(m + 7) + 3m in simplest form.

Distribute the 4 to each term inside. 4(m + 7) = +
Rewrite the whole expression.
Reorder so like terms sit together.
Combine like terms.

Answer:equivalent expression in simplest form

Say it and write itSentence starters

Finish each sentence out loud with a partner. Then use them in your writing.

I used the Property so that I could .
After distributing, the expression became .
The like terms and combine to make .
My rewritten expression is equivalent because .

Word bank CommutativeAssociativeIdentityDistributivepropertyequivalentreorderregroupdistributecombinelike termsvalue

Watch outA common mistake

A common mistake in Properties of Operations is applying the commutative property where it does not belong, or confusing it with the associative property. Students often assume 15 − 8 = 8 − 15 (it does not: 7 ≠ −7 — commutative only works for addition and multiplication), or they see parentheses move and call it "associative" even when the numbers inside just swapped places, like (2 + 3) + 4 = (3 + 2) + 4 (that is commutative — the order inside the parentheses changed, not the grouping). Before you label a property, ask: did only the order change (commutative), did only the grouping change (associative), or was a 0 or 1 involved (identity)?

Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.AT.7

Lesson 6.8Generate Equivalent Expressions

Version ASupported practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice

Name Date Period

Learning target I can use the commutative, associative, and identity properties to rewrite expressions.

Part 1Understand the ideamodels and meaning

1Write the name of the correct group on each line.
Sort each equation by the property it shows.

Groups: Commutative Associative Identity
- 9 + 4 = 4 + 9
- 7 × 1 = 7
- (5 + 1) + 3 = 5 + (1 + 3)
Hint For each equation ask: did the order swap, did the parentheses move, or did a 0 or 1 leave the number unchanged?

Part 2Apply itreal-world problems

2Circle the letter of the best answer. Show how you know.
Which property is shown? 5 + 13 = 13 + 5
1. ACommutative Property
2. BAssociative Property
3. CIdentity Property
4. DDistributive Property
Hint Compare the two sides: same numbers, same operation. What changed between left and right?
Show your work
3Circle the letter of the best answer. Show how you know.
Which property is shown? (6 × 3) × 2 = 6 × (3 × 2)
1. ACommutative Property
2. BIdentity Property
3. CAssociative Property
4. DDistributive Property
Hint Read both sides carefully: 6, 3, 2 stay in the same order. Only the parentheses moved.
Show your work
4Circle the letter of the best answer. Show how you know.
Which property is shown? 47 + 0 = 47
1. ACommutative Property
2. BAssociative Property
3. CDistributive Property
4. DIdentity Property
Hint Notice what was added: 0. Did adding it change 47 at all?
Show your work
5Circle the letter of the best answer. Show how you know.
Which property lets you rearrange 8 × 5 to 5 × 8?
1. AAssociative Property
2. BIdentity Property
3. CDistributive Property
4. DCommutative Property
Hint The factors 8 and 5 swapped positions. Nothing else changed — no parentheses, no new numbers.
Show your work

Part 3Explain and correctreasoning

6Find the mistake. Explain it, then show the correct work.
Spot the Common Mistake
1. 1Statement4 + 0 = 4
2. 2Student's labelThis shows the Commutative Property.
3. 3ReasoningThe numbers 4 and 0 switched places, so it must be commutative.
Which step has the mistake? Explain what went wrong, then show the correct work.

Hint What does adding 0 to a number do to its value?
Correct work
Correct answer:

Explain your thinkingHow can you tell the difference between the commutative and associative properties? What clue helps you decide?

Sentence starter The ___ property changes the ___, while the ___ property changes the ___. I can tell by looking at whether the ___ moved or the ___ changed.

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help

Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.AT.7

Lesson 6.8Generate Equivalent Expressions

Version BCore practice

Mastery check☐ Exceeds☐ Meets target☐ Needs practice

Name Date Period

Learning target I can use the commutative, associative, and identity properties to rewrite expressions.

Part 1Understand the ideamodels and meaning

1Write the name of the correct group on each line.
Match each equation with the property it demonstrates.

Groups: Commutative Associative Identity
- a + b = b + a
- x × 1 = x
- (m + n) + p = m + (n + p)
- c × d = d × c
- y + 0 = y
- (a × b) × c = a × (b × c)
2Write the letter of the matching item on each line.
Match each equation to its property.
1. 7 + 9 = 9 + 7
2. (2 × 5) × 3 = 2 × (5 × 3)
3. 15 × 1 = 15
4. 0 + 23 = 23
5. 6 × 4 = 4 × 6
6. (8 + 3) + 7 = 8 + (3 + 7)
- ACommutative (Add)
- BAssociative (Mult)
- CIdentity (Mult)
- DIdentity (Add)
- ECommutative (Mult)
- FAssociative (Add)
3Complete the table. Show how you found each value.

Use properties to rewrite each expression in a way that makes mental math easier. Then calculate.

Original Rearranged (using properties) Value
25 × 7 × 4 25 × 4 × 7 = 100 × 7
19 + 36 + 1 19 + 1 + 36 = 20 + 36
50 × 9 × 2 50 × 2 × 9 = 100 × 9

Show your work
4Decide whether each pair balances. Check Yes or No, then explain.

Associative Property of Addition: same value (13), different grouping

Left side Right side Balanced?

Explain your choice

Original	Rearranged (using properties)	Value
25 × 7 × 4	25 × 4 × 7 = 100 × 7
19 + 36 + 1	19 + 1 + 36 = 20 + 36
50 × 9 × 2	50 × 2 × 9 = 100 × 9

Part 2Explain and correctreasoning

5Find the mistake. Explain it, then show the correct work.
Spot the Common Mistake
1. 1Statement(2 + 3) + 4 = (3 + 2) + 4
2. 2Student's labelThis shows the Associative Property.
3. 3ReasoningThere are parentheses, so the grouping must have changed.
Which step has the mistake? Explain what went wrong, then show the correct work.

Correct work
Correct answer:

Explain your thinkingHow can you tell the difference between the commutative and associative properties? What clue helps you decide?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help

Grade 6 Mathematics · Unit 6: Numerical and Algebraic ExpressionsStandard 6.AT.7

Lesson 6.8Generate Equivalent Expressions

ChallengeExtension

Mastery check☐ Exceeds☐ Meets target☐ Needs practice

Name Date Period

Learning target I can use the commutative, associative, and identity properties to rewrite expressions.

Part 1Understand the ideamodels and meaning

1Complete the table. Show how you found each value.

For each expression, use the commutative and/or associative properties to rearrange and simplify the mental math. Show which property you used.

Expression Rearranged Property Used Value
2 × 13 × 50
45 + 27 + 55

Show your work
2Complete the table. Show how you found each value.

6.AT.7 asks you to use properties to GENERATE an equivalent expression, not just to name a property — try it with variables instead of just numbers. For each expression, apply the property named to write an equivalent expression.

Expression Property to apply Equivalent expression
y + y + y Repeated addition → multiplication
(x + 4) + 6 Associative Property
5 + n Commutative Property

Show your work

Expression	Property to apply	Equivalent expression
y + y + y	Repeated addition → multiplication
(x + 4) + 6	Associative Property
5 + n	Commutative Property

Part 2Explain and correctreasoning

3Find the mistake. Explain it, then show the correct work.
Find the Property Error
1. 1Statement15 − 8 = 8 − 15
2. 2Student's claimThis shows the Commutative Property of Subtraction.
3. 3ReasoningI switched the order, so it must be commutative.
Which step has the mistake? Explain what went wrong, then show the correct work.

Correct work
Correct answer:
4Answer in complete sentences.

Explain why the commutative property works for addition and multiplication but NOT for subtraction and division. Give a specific number example for each operation to prove your point.
Write your answer

Explain your thinkingHow can you tell the difference between the commutative and associative properties? What clue helps you decide?

How did it go? ☐ 4 · I can teach it☐ 3 · I've got it☐ 2 · I need a hint☐ 1 · I need help