Neft Teacher · Learn It

Unit 6 · Standard 6.AT.7

Generate Equivalent Expressions

Lesson 6-8 · Learn It — how the math works

Why we're learning this

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!

✱ Math Workbench

Learn It

What are the properties of operations?

Properties are math rules that are always true. The Commutative Property lets you change the order. The Associative Property lets you change the grouping (the parentheses).

Picture itSee what the math looks like before you read about it

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).
Where you see this in real lifeA 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).

Watch me solve itI do — read one step at a time

The problem looks like this
Tape diagram: 3 guitarists, 5 drummers and 2 keyboard players make 10 in either order, 3 + 5 + 2 or 5 + 3 + 2.103 + 5 + 23guitar5drums2keys105 + 3 + 25drums3guitar2keys
  1. I have 3 guitarists, 5 drummers, and 2 keyboard players.
  2. I can add them as 3 + 5 + 2, which gives 10.3 + 5 + 2
  3. The says order does not matter: 5 + 3 + 2 also gives 10.5 + 3 + 2
  4. The says grouping does not matter: (3 + 5) + 2 = 3 + (5 + 2), both equal 10.(3 + 5) + 2 = 3 + (5 + 2)

Watch out — mistakes students make here

  • Commutative changes the ORDER (8 + 5 vs. 5 + 8). Here 8, 5, 2 stay in order — only the parentheses moved.
  • Identity adds 0 without changing the value (8 + 0 = 8). Here the grouping changed — no zero involved.
  • Distributive multiplies across parentheses: a(b + c) = ab + ac. There's no multiplication here — just regrouped addition.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Look at 6 + 9 and 9 + 6. Do they give the same total?6 + 9
  2. 6 + 9 = 15 and 9 + 6 = 15. Yes, the same.6 + 9 = 15
  3. Only the order changed, so this shows the .

On your ownYou do — solve this one by yourself

Before you start, remember:

  • Sort each example into the it shows: commutative or associative.
  • Ask: did the order change, or did the grouping change?

Solve: Which is shown? 5 + 13 = 13 + 5

My answer

Stuck? Open one hint at a time.

Hint 1

Compare the two sides: same numbers, same operation. What changed between left and right?

Hint 2

Match the change to the property: order swapping, regrouping with parentheses, or adding 0 / multiplying by 1.

Hint 3

Left is 5 + 13; right is 13 + 5. The addends traded places — find the property about changing order.

Check my answer
✅ Answer: Commutative Property
The order of the addends changed, so this is the Commutative Property of Addition.

Turn & TalkSay it out loud with a partner

The band has 3 guitarists, 5 drummers, and 2 keyboard players. Whether you seat guitarists or drummers first, you still have 10 musicians. Which explains this, and why does it work?

Try starting with:

  • Changing the order still gives 10 because of the ___ property.Cambiar el orden todavía da 10 por la propiedad ___.
  • It works because order does not change ___.Funciona porque el orden no cambia ___.
Use these wordsCommutative PropertyAssociative PropertyIdentity PropertyProperty