B. Number · Operations: Adding and Subtracting to 1000, Multiplying and Dividing

Multiplication Rules: Any Order, Ones, Zeros and Splitting

B2.1 B2.6 A1About 35 minutes

Learning goals

I can multiply in any order and get the same product.
I can multiply by 1 and by 0.
I can group three factors in a way that makes multiplying easier.
I can split a fact I don't know into two facts I do know.

Words to know

property
A math rule that is always true.

any-order rule
You can switch the factors and the product stays the same: 2 × 9 = 9 × 2. (Its math name is the commutative property.)

one rule
Any number times 1 is that same number: 1 × 8 = 8. (Its math name is the identity property.)

zero rule
Any number times 0 is 0: 0 × 6 = 0.

splitting
Breaking a factor into two parts to make two easier facts: 7 × 4 = 5 × 4 + 2 × 4. (Its math name is the distributive property.)

Remember from Grade 2

In Grade 2 you learned math rules for adding: you can add in any order, adding 0 keeps a number the same, and you can group numbers to make 10. Multiplication has rules too! They make hard facts easy.

Rule 1: Any order

Turn an array of 2 rows of 9 sideways, and you get 9 rows of 2. Same dots, same total.

2 × 9 = 9 × 2 = 18

So if 9 × 2 feels hard, think of it as 2 × 9 (double 9). This rule cuts the number of facts you need to learn almost in half!

2 rows of 9. Turned sideways, it is 9 rows of 2, still 18.

Rule 2: Times 1

1 × 8 means 1 group of 8. That is just 8. And 8 × 1 means 8 groups of 1. That is also 8.

Any number times 1 is the same number.

The same is true for dividing: 8 ÷ 1 = 8 (8 shared into 1 group). And 8 ÷ 8 = 1 (8 shared by 8 people: 1 each).

Rule 3: Times 0

Imagine 6 plates with 0 cookies on each plate. How many cookies? None!

6 × 0 = 0 and 0 × 6 = 0.

Any number times 0 is 0.

🍽️🍽️🍽️🍽️🍽️🍽️

6 empty plates. 6 groups of 0 cookies is 0 cookies. 6 × 0 = 0.

Don't mix up the rules

Adding 0 keeps a number the same: 5 + 0 = 5.

Multiplying by 0 makes 0: 5 × 0 = 0.

Multiplying by 1 keeps a number the same: 5 × 1 = 5.

Rule 4: Grouping three factors

When you multiply three numbers, you can multiply any two first. Pick the pair that is easiest!

2 × 3 × 5

Hard way: 2 × 3 = 6, then 6 × 5 = 30.
Easy way: 2 × 5 = 10, then 10 × 3 = 30.

The answer is the same, but making a 10 first is easier.

Worked example

Example 1: Group to make 10

Solve 5 × 7 × 2.

Look for a pair that makes 10: 5 × 2 = 10.
Then 10 × 7 = 70.

5 × 7 × 2 = 70

Rule 5: Splitting

Don't know 7 × 4? Split the 7 rows into 5 rows and 2 rows. Find each part, then add.

5 × 4 = 20
2 × 4 = 8
20 + 8 = 28

So 7 × 4 = 28.

An array of 7 rows of 4, split after 5 rows. The top part is 5 × 4 = 20. The bottom part is 2 × 4 = 8. Together: 28.

Worked example

Example 2: Split into facts you know

Solve 6 × 8.

Split 6 into 5 and 1.
5 × 8 = 40.
1 × 8 = 8.
Add: 40 + 8 = 48.

6 × 8 = 48

Worked example

Example 3: Use 10 and take one group away

Solve 9 × 5.

9 is 1 less than 10.
10 × 5 = 50.
9 groups is one group of 5 less than 10 groups: 50 − 5 = 45.

9 × 5 = 45

Worked example

Example 4: Use the rules to check

Kai says 4 × 0 × 9 = 36. Is he right?

Group first: 4 × 0 = 0.
Then 0 × 9 = 0.
Any time a 0 is one of the factors, the product is 0.

No. 4 × 0 × 9 = 0.

Rule	Example	Why it helps
Any order	2 × 9 = 9 × 2	Use the fact that's easier for you.
Times 1	1 × 8 = 8	Instant answer.
Times 0	6 × 0 = 0	Instant answer.
Grouping	2 × 3 × 5 = 10 × 3	Make 10 first.
Splitting	7 × 4 = 5 × 4 + 2 × 4	Turn a hard fact into easy ones.

Five multiplication rules.

Division rules are different

In division the order does matter: 12 ÷ 3 = 4, but 3 ÷ 12 is a very different question. Only multiplication (and addition) can be done in any order.

Just for fun

Zero is the hungriest number in math. Multiply anything by 0, even a million, and zero gobbles it all up. The answer is 0!

Hands-on activity

Split the array

You need: grid paper (or paper where you draw dots in rows), a pencil, and a piece of string or a ruler.

Draw an array of 8 rows of 3 (or build it with buttons or LEGO).
Lay the string across the array to split it into 5 rows and 3 rows.
Find each part: 5 × 3 = 15 and 3 × 3 = 9. Add them: 15 + 9 = 24. So 8 × 3 = 24.
Move the string to split it differently, like 4 rows and 4 rows: 4 × 3 = 12, 12 + 12 = 24. Same answer!
Try 7 × 6 and 9 × 4 the same way.

Math talk: "Where is the easiest place to split? Why?"

Your turn

9 × 2 = 2 × 9 = ?

1 × 47 = ?

0 × 9 = ?

5 × 2 × 4 = ? Make 10 first.

Split 6 × 5 into 5 × 5 and 1 × 5. What is the product?

You know 3 × 6 = 18. What is 6 × 3?

23 × 0 = ?

More practice: warm-up

4 × 1 = ?

0 × 3 = ?

Is 7 × 5 the same as 5 × 7?

10 × 1 = ?

8 ÷ 1 = ?

More practice: on your own

2 × 7 × 5 = ?

Use splitting to solve 8 × 4.

Solve 9 × 10 by starting with 10 × 10.

True or false: 6 × 0 = 6. Explain.

You know 3 × 7 = 21. Write two more facts you now know.

Challenge

Solve 4 × 9 using 2 × 9.

Solve 6 × 7 two different ways.

Find three different ways to multiply three numbers to make 60.

Check your understanding

Quiz

What is 1 × 9?

What is 0 × 8?

Which is equal to 4 × 7?

7 × 6 = 5 × 6 + ?

What is 5 × 3 × 2?

Which sentence is true?

For parents and teachers

Big idea: The Grade 3 curriculum asks students to use the properties of operations to solve problems and check calculations (B2.1). Children don't need the formal names (commutative, identity, zero, associative, distributive), but they should use the ideas: any order, times 1, times 0, grouping, and splitting a hard fact into easier ones.

Common difficulties:

Mixing up 0 and 1: 5 × 0 = 5 or 5 × 1 = 1. Use the plate story: 5 plates with nothing on them.
Mixing up adding and multiplying by 0: 5 + 0 = 5 but 5 × 0 = 0.
Splitting and forgetting to add the parts. Draw the array and point to both parts.
Thinking division can be done in any order. It can't.

How to help: When your child doesn't know a fact, ask "What fact do you know that's close?" For 6 × 8, "Do you know 5 × 8?" This builds flexible thinking rather than just memorizing.