Math 1 · Lesson 2
Commute, Associate, Distribute
The properties are named after ordinary words. If you know what the word means, you know what the numbers are allowed to do.
How to use this page
Try each problem yourself first. Use Hint for a nudge, Show next step to see one worked step at a time, and Reveal answer to check. Hide help / start over clears the card so you can try again.
Every property in this set is named after a word you already know. To commute is to travel back and forth, so the commutative property lets the numbers trade places. An associate is a partner, so the associative property lets you change which numbers are partnered up (the parentheses). To distribute is to hand one out to everybody, so the distributive property hands the multiplier to every term inside the parentheses. The number that moves is shown in this color.
Warm-up: three ordinary words
Before any math, write one sentence for each. What does it mean when a grown-up commutes to work? What is a business associate? What does a teacher do when she distributes the worksheets?
Check your sentences
Commute: go back and forth between two places. Associate: a partner, someone you pair up with. Distribute: hand one out to everybody. Keep those sentences. Every math rule today is named after one of them, and the name tells you exactly what the rule lets you do.
Words we use
| Property | The everyday word | What the numbers may do |
|---|---|---|
| Commutative (of addition, of multiplication) | To commute is to travel back and forth between two places. | Trade places. a + b = b + a and a x b = b x a |
| Associative (of addition, of multiplication) | An associate is a partner. | Change partners (move the parentheses). (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c) |
| Distributive (of multiplication over addition) | To distribute is to hand one out to everyone. | The multiplier goes to every term inside. a x (b + c) = a x b + a x c |
| Identity and Inverse (Lesson 1) | Identity = who you are. Inverse = turned around. | a + 0 = a, a x 1 = a; a + (-a) = 0, a x 1a = 1 |
The scoreboard: which operations pass?
Test each property with numbers. Trading places: 17 + 48 and 48 + 17 both make 65, but 10 - 4 = 6 while 4 - 10 = -6. Changing partners: (18 + 7) + 3 and 18 + (7 + 3) both make 28, but (24 / 6) / 2 = 2 while 24 / (6 / 2) = 8.
| Does it... | Add | Subtract | Multiply | Divide |
|---|---|---|---|---|
| commute? (trade places) | yes | no | yes | no |
| associate? (change partners) | yes | no | yes | no |
Multiplication also distributes over addition: 7 x (10 + 3) = 7 x 10 + 7 x 3 = 91. Watch the direction: 7 + (10 x 3) is not (7 + 10) x (7 + 3).
The big idea
Only addition and multiplication pass every test: they commute, they associate, and multiplication distributes. Subtraction and division do not. That is why algebra treats addition and multiplication as its two operations, and puts the other two in disguise: subtracting is adding the opposite, and dividing is multiplying by the reciprocal (Lesson 1). Once they are in disguise, every property works again.
Name that property
Eight quick ones. Type the property and whether it is about adding or multiplying, then check.
Practice: try it, then get help one step at a time
While you work, say the pair every time: the property name and the ordinary word. Which property? What does that word mean? So what are the numbers allowed to do?
Exit ticket
On paper or an index card. Use the math word and the everyday word.
- Why is it called the commutative property? What does "commute" mean?
- Show with your own numbers that subtraction is not commutative.
- Rewrite 15 - 6 so that it commutes, and show it works.
Check your exit ticket
- Because the numbers commute: they travel back and forth, trading places, and the answer does not change.
- Any pair works, for example 8 - 3 = 5 but 3 - 8 = -5.
- 15 + (-6) = 9 and (-6) + 15 = 9. Once subtraction is written as adding the opposite, it commutes.
Printables
The worksheet has Parts A to E with an answer key. The lesson plan has the 25-minute timeline, the vocabulary table, the scoreboard, and the misconceptions to listen for.