Lesson 6.8Generate Equivalent Expressions
Start hereWords, worked example, and sentence starters
Learning target I can use the commutative, associative, and identity properties to rewrite expressions.
Why This Matters You're a band director arranging musicians on stage. You have 3 guitarists, 5 drummers, and 2 keyboard players. Whether you seat the guitarists first or the drummers first, you still have 3 + 5 + 2 = 10 musicians. The order you arrange them doesn't change the total!
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 example
These numbers are not on your problems. The steps are. Follow them with your own numbers.
Simplify 4(m + 2) + 3m.
- Distribute the 4: 4m + 8 + 3m.
- Move like terms together: 4m + 3m + 8.
- Combine: 7m + 8. Test m = 1: 4(3) + 3 = 15 and 7 + 8 = 15.
Answer: 7m + 8
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
Look at 6 + 9 and 9 + 6. Do they give the same total?
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 out!A 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
Lesson 6.8Generate Equivalent Expressions
Level 0Most support · every hint printed
Learning target I can use the commutative, associative, and identity properties to rewrite expressions.
Why This Matters You're a band director arranging musicians on stage. You have 3 guitarists, 5 drummers, and 2 keyboard players. Whether you seat the guitarists first or the drummers first, you still have 3 + 5 + 2 = 10 musicians. The order you arrange them doesn't change the total!
How it worksWorked example
These numbers are not on your problems. The steps are. Follow them with your own numbers.
Simplify 4(m + 2) + 3m.
- Distribute the 4: 4m + 8 + 3m.
- Move like terms together: 4m + 3m + 8.
- Combine: 7m + 8. Test m = 1: 4(3) + 3 = 15 and 7 + 8 = 15.
Answer: 7m + 8
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1Circle the letter of the best answer. Show how you know.
Which property is shown? 5 + 13 = 13 + 5
- ACommutative Property
- BAssociative Property
- CIdentity Property
- DDistributive Property
Steps to try- Compare the two sides: same numbers, same operation. What changed between left and right?
- Match the change to the property: order swapping, regrouping with parentheses, or adding 0 / multiplying by 1.
- Left is 5 + 13; right is 13 + 5. The addends traded places — find the property about changing order.
Show your work -
2Circle the letter of the best answer. Show how you know.
Which property is shown? (6 × 3) × 2 = 6 × (3 × 2)
- ACommutative Property
- BIdentity Property
- CAssociative Property
- DDistributive Property
Steps to try- Read both sides carefully: 6, 3, 2 stay in the same order. Only the parentheses moved.
- Ask: did the ORDER change, or did the GROUPING change? Each property handles one of those.
- Left groups (6 × 3) first; right groups (3 × 2) first. Pick the property that's about regrouping.
Show your work -
3Circle the letter of the best answer. Show how you know.
Which property is shown? 47 + 0 = 47
- ACommutative Property
- BAssociative Property
- CDistributive Property
- DIdentity Property
Steps to try- Notice what was added: 0. Did adding it change 47 at all?
- Think about what each property does. Which one is about a number staying itself?
- 47 + 0 gives back 47 unchanged. Find the property named for keeping a number's identity.
Show your work -
4Circle the letter of the best answer. Show how you know.
Which property lets you rearrange 8 × 5 to 5 × 8?
- AAssociative Property
- BIdentity Property
- CDistributive Property
- DCommutative Property
Steps to try- The factors 8 and 5 swapped positions. Nothing else changed — no parentheses, no new numbers.
- Swapping order is one property; moving parentheses is another. Which happened here?
- Compare: 8 × 5 became 5 × 8, like trading seats. Choose the property that lets factors change order.
Show your work
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.