Multiplying by 1
7 × 1 means one group of 7. So the answer is 7. Nothing changed.
Step 1: Picture It
Start with something you already know.
A copy machine
- A copy machine makes as many copies as you ask for.
- Ask for 1 copy of your drawing: you get it once.
- Ask for 0 copies: nothing comes out at all.
- Ask for 3 copies of a blank page: still no drawing.
Times 1 is one copy: you get back just what you started with.
Times 0 is no copies, or copies of nothing: there is nothing to count.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Priya has 4 boxes. Each box holds 1 crayon. How many crayons in all?
crayons
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
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 3: Try It Yourself
Tap and try it out.
1 + 1 + 1 + 1 + 1 + 1 + 1 = 7
7 × 1 = 7
7 rows of 1 makes 7.
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.
Step 4: 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 5: Watch an Example
One step at a time.
Watch Ana Explain Zero
Ana is asked for 8 × 0.
- Step 1
Ana reads it as groups of. Eight groups, with 0 in each.
Watch Leo Multiply by One
Leo is asked for 9 × 1. Leo starts, and you finish.
- Step 1
Leo reads it as groups of. Nine groups, with 1 in each.
Step 6: Your Turn
Practice makes it stick.
The Pencil Pots
Problem 1 of 12
A desk has 5 pencil pots. Every pot is empty. How many pencils are in the pots?
pencils
The Bookmark Stall
Problem 2 of 12
Bookmarks cost $1 each. Priya buys 10 bookmarks. How much does she pay?
dollars
Zero and One
Say it as groups of. Then answer.
Problem 3 of 12
0 × 3 = ?
Problem 4 of 12
6 ÷ 1 = ?
Problem 5 of 12
Which multiplication matches this array?
Find the Mistake
One line on each page is a slip. Find it, then fix it.
Problem 6 of 12
Dev works out 4 × 0. One line is a slip. Which?
Dev's answer for 4 × 0
- 14 × 0
- 2= 4 + 0
- Answer= 4
Problem 7 of 12
What should 4 × 0 be?
Problem 8 of 12
Maya works out 5 × 1. One line is a slip. Which?
Maya's answer for 5 × 1
- 15 × 1
- 2= 5 + 1
- Answer= 6
Problem 9 of 12
What should 5 × 1 be?
Same Answer?
Work out the first one. Then find the one that matches.
Problem 10 of 12
Which has the same answer as 9 × 0?
Problem 11 of 12
Which has the same answer as 8 × 1?
Build the Groups
Build it, then check.
Problem 12 of 12
Build 6 rows of 0. Each row is a group with nothing in it.
1 = 1
1 × 1 = 1
1 row of 1 makes 1.
Step 7: Quick Check
Show what you know.
Question 1 of 2
7 × 0 = ?
Question 2 of 2
Why does 6 × 1 = 6?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
Noor multiplies a number by 1 and gets 12. Then she multiplies the same number by 0. What does she get?
Stretch 2 of 2
Which is bigger: 10 × 0 or 1 × 1?
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.
The slip this lesson targets is carrying the adding habit into multiplying: 4 × 0 read as 4 + 0 = 4, and 5 × 1 read as 5 + 1 = 6. Dev's and Maya's pages are both of that kind. Ask "four groups of what?" and have them say the long way out loud: four zeros, five ones. The other trap is thinking the 0 is what disappears rather than the groups being empty. The build with six rows of nothing is there so they can watch the array vanish and still read 6 × 0 = 0 underneath it.
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.