Lesson 6.8Expanding and Factoring Expressions
Start hereWords, worked example, and sentence starters
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. | 3 + 7 = 7 + 3 = 10 both ways; 4 × 5 = 5 × 4 = 20 both ways — order does not matter |
| Associative PropertySpanish: Propiedad asociativa | You can change the grouping and get the same answer. | (2 + 3) + 4 = 5 + 4 = 9 and 2 + (3 + 4) = 2 + 7 = 9 — same answer, different grouping |
| Identity PropertySpanish: Propiedad de identidad | Adding 0 or multiplying by 1 keeps the same value. | 9 + 0 = 9 (zero is the identity for addition); 6 × 1 = 6 (one is the identity for multiplication) |
| 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 examplefactoring an expression with the gcf
These numbers are not on your problems. The steps are. Follow them with your own numbers.
Factor the expression 18x + 24 using the greatest common factor.
- Find what the terms share. The GCF of 18 and 24 is 6.
- Divide each term by the GCF. 18x ÷ 6 = 3x 24 ÷ 6 = 4
- Write the GCF outside parentheses. 6(3x + 4)
- Check by distributing. 6 × 3x = 18x and 6 × 4 = 24
Answer: The expression 18x + 24 factors as 6(3x + 4).
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
Expand the expression 7(n + 11) using the Distributive Property.
- Multiply the outside number by the first term inside. 7 × n =
- Multiply the outside number by the second term inside. 7 × 11 =
- Write the two products as a sum.
- Check: did every term inside get multiplied?
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- The greatest common factor of the terms is .
- Factored, the expression becomes .
- Expanded, the expression becomes .
- I checked my work by , and I got back .
Word bank Distributive Propertyexpandfactorgreatest common factorGCFtermproductparenthesesmultiplydividecheckequivalent
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)?
Lesson 6.8Expanding and Factoring Expressions
Apply Day · Version ASupported practice
Learning target I can use the commutative, associative, and identity properties to rewrite expressions.
-
1Complete each step, then write the final answer.
Use properties to find 6 + 8 + 4. Add the friendly pair first.
Show your workHint 6 and 4 make a ten.
Final answer: -
2Complete each step, then write the final answer.
Which property does 5 × 9 = 9 × 5 show? Type commutative or associative.
Show your workHint Did the order change, or only the parentheses?
Final answer: -
3Complete each step, then write the final answer.
Use properties to find 5 × 8 × 2.
Show your workHint 5 × 2 makes 10. Multiply that pair first.
Final answer:
Lesson 6.8Expanding and Factoring Expressions
Apply Day · Version BCore practice
Learning target I can use the commutative, associative, and identity properties to rewrite expressions.
-
1Circle the letter of the best answer. Show how you know.
Which equation shows the Identity Property?
- A9 × 1 = 9
- B9 × 0 = 0
- C9 + 1 = 1 + 9
- D(9 + 1) + 2 = 9 + (1 + 2)
Show your work -
2Complete each step, then write the final answer.
Use properties to find 2 × 17 × 50.
Show your workFinal answer: -
3Complete each step, then write the final answer.
Use properties to find 7 + 16 + 3. Add the friendly pair first.
Show your workFinal answer:
Lesson 6.8Expanding and Factoring Expressions
Apply Day · ChallengeExtension
Learning target I can use the commutative, associative, and identity properties to rewrite expressions.
-
1Circle the letter of the best answer. Show how you know.
Which property of operations is shown by the equation (7 + 4) + 6 = 7 + (4 + 6)?
- ACommutative Property of Addition
- BIdentity Property of Addition
- CDistributive Property
- DAssociative Property of Addition
Show your work -
2Complete each step, then write the final answer.
Eli says 12 ÷ 4 and 4 ÷ 12 have the same value, because the Commutative Property lets you switch the order. What is 12 ÷ 4?
Show your workFinal answer: -
3Complete each step, then write the final answer.
Use properties to find 25 × 11 × 4.
Show your workFinal answer: