Generate Equivalent Expressions

Pick up where we left off

Name: Date: Group:

Where we left off

Our goal: 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.

The big idea: Order and grouping can change for addition and multiplication without changing the value, but not for subtraction or division.

Model to copy — Example 1 · Make friendly numbers

Find 25 × 9 × 4 without a calculator. Use properties to make the work easier.
Switch the order of 9 and 4: 25 × 9 × 4 = 25 × 4 × 9
Group 25 and 4 together: (25 × 4) × 9
Multiply inside the parentheses. Then multiply by 9: 100 × 9 = 900
25 × 9 × 4 = 900.
Does the Associative Property work for subtraction? Compare (12 − 6) − 2 and 12 − (6 − 2).
Subtract inside the parentheses first: (12 − 6) − 2 = 6 − 2 = 4
Now group the last two numbers: 12 − (6 − 2) = 12 − 4 = 8
Compare the two answers: 4 ≠ 8
No. (12 − 6) − 2 = 4, but 12 − (6 − 2) = 8.

Start here

You did these with your group. Now do them on your own — the model above is yours to copy.

1
Same stepsFinish what the group started. Use the model above, step for step.
Try it together — then prove it: Look at 6 + 9 and 9 + 6. Do they give the same total? Explain why. Find both sums. (6 + 9 = 15 and 9 + 6 = 15) Which property does this show? (The Commutative Property. Only the order changed.) Why can you add 6 and 9 in either order? (Adding puts two amounts together. 6 and then 9, or 9 and then 6, makes the same total.)
Show your work
2
Warm restartWhat is the GCF of 18 and 24?
1. A2
2. B3
3. C6
4. D72
How do you know?
3
Warm restartWhat is the LCM of 4 and 6?
1. A2
2. B10
3. C12
4. D24
How do you know?
4
Warm restartOne bell rings every 8 minutes and another every 12 minutes. They just rang together. After how many minutes will they ring together again?
1. A4 minutes
2. B20 minutes
3. C24 minutes
4. D96 minutes
How do you know?

6.AT.7 Group 2 · Challenge

Practice Set · Part 2

Generate Equivalent Expressions

Keep going

You answered these out loud. Now write them.

Answer, then say how you know — the reason is the part that counts.

5
Think it throughWhich property lets you change the ORDER of the factors?
1. ACommutative
2. BAssociative
3. CDistributive
4. DIdentity
Why is that the answer?
6
Think it throughWhich property lets you change the GROUPING of the factors?
1. ACommutative
2. BDistributive
3. CIdentity
4. DAssociative
How do you know? Give a second reason as well.
7
Think it throughWhat is 4 × 17 × 25?
1. A170
2. B425
3. C1,000
4. D1,700
Explain your thinking. What would have to change for a different choice to be right?
8
Back to the modelHow can you tell the difference between the commutative and associative properties? What clue helps you decide?

6.AT.7 Group 2 · Challenge

Practice Set · Part 3

Generate Equivalent Expressions

Words and reasoning

Word bank · Banco de palabras

Properties of Operations (Propiedades de las operaciones)Commutative Property (Propiedad conmutativa)Associative Property (Propiedad asociativa)Identity Property (Propiedad de identidad)Property (Propiedad)Distributive Property (Propiedad distributiva)Equivalent expressions (Expresiones equivalentes)Expand (Desarrollar)

Use the words

Fill each blank with a word from the bank.

Rules that let you reorder or regroup without changing the ___.
The rule that order does not change a sum or product, like a + b = b + a, is the ___.
The rule that grouping does not change a sum or product, like (a + b) + c = a + (b + c), is the ___.
The rule that adding 0 or multiplying by 1 keeps a number the same is the ___.

Explain the thinking

9
Find the mistakeAnother student made this exact mistake in this lesson. Explain why it is wrong, then write what they should have done.
The 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.
10
Say moreHow can you tell the commutative property from the associative property? Use 6 + 9 = 9 + 6 and (3 + 5) + 2 = 3 + (5 + 2) to explain.
11
Write your ownWrite your own problem about this situation, then solve it and show the answer. The situation: A sound engineer adds up the volume levels of three mixing-board channels: guitar at 15, bass at 22, and drums at 8. She writes the sum four ways: 15 + 22 + 8, 8 + 15 + 22, (15 + 22) + 8, and 15 + (22 + 8).
Show your work

6.AT.7 Group 2 · Challenge

Practice Set · Part 4

Generate Equivalent Expressions

Show what you know

Last check

These two come from Lesson 6.7. If they are shaky, that is the lesson to revisit — not this one.

12
Show what you knowExplain your thinking — Which property is shown? (8 + 5) + 2 = 8 + (5 + 2)
1. AAssociative Property
2. BCommutative Property
3. CIdentity Property
4. DDistributive Property
Explain your choice.
13
From Lesson 6.7Explain your thinking — Two supply runs repeat every 6 weeks and every 8 weeks, and both happen this week. When will they next happen in the same week?
1. AWeek 2, the greatest common factor of 6 and 8
2. BWeek 14, because 6 + 8 = 14
3. CWeek 24, the least common multiple of 6 and 8
4. DWeek 48, because 6 × 8 = 48
How do you know?
14
From Lesson 6.7A problem asks when two repeating schedules meet again. Which do you use?
1. AThe least common multiple
2. BThe greatest common factor
3. CEither one works
4. DNeither — you list factors
How do you know?

Be honest — this tells your teacher what to do next

I can…	Not yet	Getting there	Got it
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.
I can explain why it works: Order and grouping can change for addition and multiplication without changing the value…
I can justify my answer to a skeptic and connect it to a second strategy or representation.

One question I want to ask my group next time