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

Lesson 6.86.8 Small Group · Group 2

Start hereWords, worked example, and sentence starters

Name Date Period

Learning target I can use the commutative, associative, and identity properties to rewrite expressions, explain why the method works, and use it on a problem I have not seen before.

Start hereWords for this lesson

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

WordWhat it meansExample
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.

  1. Distribute the 6 to each term inside. 6(y + 3) = 6y + 18
  2. Rewrite the whole expression. 6y + 18 + 9y
  3. Reorder so like terms sit together. 6y + 9y + 18
  4. 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.

  1. Distribute the 4 to each term inside. 4(m + 7) = +
  2. Rewrite the whole expression.
  3. Reorder so like terms sit together.
  4. 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 6.86.8 Small Group · Group 2

Group 2Challenge 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, explain why the method works, and use it on a problem I have not seen before.

  1. 1Complete each step, then write the final answer.
    20 × 7 × 5(20 × 5) × 7=
    Both sides have the same value.

    Find 20 × 7 × 5 without a calculator. Use properties to make the work easier.

    1. 1Switch the order of 7 and 5. Which property is this?
    2. 2Group the friendly pair. What is 20 × 5?
    3. 3Multiply by 7. What is the product?
    Final answer:
  2. 2Complete each step, then write the final answer.
    ExpressionValue
    (24 ÷ 4) ÷ 2?
    24 ÷ (4 ÷ 2)?
    Find each value. Compare them.

    Does the Associative Property work for division? Compare (24 ÷ 4) ÷ 2 and 24 ÷ (4 ÷ 2).

    1. 1Find (24 ÷ 4) ÷ 2.
    2. 2Find 24 ÷ (4 ÷ 2).
    3. 3Why can you not move the parentheses in division?
    Final answer:
  3. 3Complete each step, then write the final answer.

    Use properties to find 2 × 17 × 50.

    Show your work
    Final answer:
  4. 4Complete each step, then write the final answer.

    Rewrite 7 + x + 13 as x + ☐. What number goes in the box?

    Show your work
    Final answer:
  5. 5Complete each step, then write the final answer.

    A store sells 46 shirts, 38 hats and 54 jackets. Use properties to find the total number of items.

    Show your work
    Final answer:
  6. 6Complete 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 work
    Final answer:

Explain your thinkingWhen can you change the order or the grouping of numbers? When can you not? Give one example of each.

ChallengeWrite your own problem

Write a new word problem that uses this lesson's idea. Then solve it and show every step.

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