Multiplying by 0 and 1

Step 1: Let's Learn

Read it, or press Listen and follow the words.

Watch the idea first — 54 seconds. Then read on, and try it yourself in the next step.

Multiplying by 1

7 × 1 means one group of 7. So the answer is 7. Nothing changed.

Multiplying by 0

7 × 0 means seven groups with nothing in them. Seven lots of nothing is still nothing.

And the other way round

0 × 7 means no groups at all. Also nothing.

Dividing by 1

7 ÷ 1 means sharing 7 between one person. They get all 7.

Step 2: Try It Yourself

Tap and try it out.

Set the columns to 1, then to 0, and watch the array.

1 + 1 + 1 + 1 + 1 + 1 + 1 = 7

7 × 1 = 7

7 rows of 1 makes 7.

Rows

Columns

The same total, laid out every way it goes. What will not fit sits outside.

7 × 1 = 7. 7 and 1 are a factor pair of 7.

Pairs you have found

1 × 7

That is all of them: 7 has 1 factor pair, so it is a prime number.

How many squares

How many rows

Step 3: In Real Life

An empty muffin tray

A tray with 6 spaces and 1 muffin in each holds 6. Six spaces and 0 muffins holds 0. Any number of empty spaces is still nothing.

Step 4: Watch an Example

One step at a time.

Watch Ana Explain Zero

Ana is asked why 6 × 0 is 0.

Step 1
She reads it as six groups of nothing.

Step 5: Your Turn

Practice makes it stick.

Empty Plates

Problem 1 of 2

5 plates have no biscuits on them. How many biscuits are there?

biscuits

One Bag

Problem 2 of 2

One bag holds 9 marbles. How many marbles in all?

marbles

Zero and One

1 of 4

8 × 1 = ?

2 of 4

8 × 0 = ?

3 of 4

0 × 6 = ?

4 of 4

9 ÷ 1 = ?

Build the Array

Build it, then check.

Build 7 rows of 1.

Set the rows and the columns.

1 = 1

1 row of 1 makes 1.

Rows

In each row

Step 6: Quick Check

Show what you know.

Question 1 of 2

7 × 0 = ?

Question 2 of 2

Why does 6 × 1 = 6?

What You Learned

Multiplying by 1 leaves a number alone.
Multiplying by 0 always gives 0.
Reading it as "groups of" explains both.

For the grown-up

This lesson is where the child multiplies and divides by 0 and 1 and explains why the answers are what they are. Multiplication is equal groups before it is a fact to recall. Three fours is three groups of four, and division is the same picture asked from the other end.

Watch for multiplication and division being learned as separate subjects. They are one relationship, and a child who sees that halves what there is to remember.

When they are stuck on a division, ask the multiplication instead: "what times 4 gives 24?" Nearly every child can answer that when they cannot answer 24 divided by 4.

Why zero and one behave oddly — 7 × 0 means seven groups with nothing in each, so the total is nothing: 0. And 0 × 7 means no groups of seven at all, which is also 0. Meanwhile 7 × 1 means seven groups of one, which is 7, and 1 × 7 means one group of seven, also 7.

What these two are called — One is the identity for multiplication — multiplying by it changes nothing. Zero is the annihilator: multiplying by it destroys everything and leaves zero. You do not need those words yet, but the behaviors are worth recognizing as rules rather than facts to memorize one at a time.

The confusion with addition — Adding zero changes nothing, but multiplying by zero wipes everything out. Many mistakes come from carrying the addition habit into multiplication and writing 7 × 0 = 7. Say the meaning out loud — "seven groups of nothing" — and the answer is obvious.

Groups of nothing, and one group — 5 × 0 means five groups with nothing in each. Nothing at all. 0 0 × 5 means no groups. Also nothing. 0 5 × 1 means five groups of one. 5 1 × 5 means one group of five. 5 Both make sense the moment you say what the groups are.