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Multiplying as equal groups

Numbers · level 2, lesson 4 of 7

A question to start

A muffin tray has 4 rows, with 6 muffins in each row.

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Lily counts them one by one: "one, two, three, ..., twenty-four". Her brother just looks at the rows and says "6, 12, 18, 24". He is much faster.

Then they turn the tray around, so it has 6 rows of 4. "Now there are more rows," says Lily, "so there must be more muffins." Is she right?

In this lesson you will learn a new way to count things that come in equal groups. It is called multiplication, and it answers both of these puzzles.

The idea

Equal groups

Here are 4 bags, with 3 apples in each bag.

3 6 9 12

How many apples? You could count all 12 one by one. But each bag has the same number, so you can count by threes: "3, 6, 9, 12". There are 12 apples.

Groups that all have the same number in them are called equal groups. Counting equal groups is so useful that it has its own sign and its own name. We write

4 × 3 = 12

and we say "four times three is twelve", or "four groups of three is twelve". The sign × is called the times sign, and this kind of sum is called multiplying. The answer, 12, is called the product.

In this lesson, 4 × 3 means "4 groups of 3": the first number says how many groups, and the second says how many in each group. (Some books use the other order. It does not change the answer, as you will soon see.)

Three ways to find a product

1. Repeated adding. 4 groups of 3 is 3 + 3 + 3 + 3 = 12. Multiplying is a short way of writing an adding sum where every number is the same. 4 × 3 is "add 3, four times".

2. Skip counting. Count by the size of the group, once for each group: "3, 6, 9, 12". You said four numbers, one for each bag. On a number line, it is 4 equal jumps of 3, starting at 0.

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3. An array. Put the things in neat rows and columns. A row goes across, like people in a line. A column goes up and down, like the floors of a building. A picture like this is called an array. The muffin tray is an array with 4 rows and 6 columns: 4 × 6 = 24 muffins.

Groups must be equal

Multiplying only works when the groups are equal. If one bag has 3 apples, another has 2 and another has 5, you cannot say "3 bags of 3" or "3 bags of 5". There is no single size for the groups. You have to add: 3 + 2 + 5 = 10. Before you multiply, always check: are the groups the same size?

Turn the array: 3 × 4 = 4 × 3

Here is an array with 3 rows of 4. Now turn it a quarter of the way around, like turning a page on its side.

3 rows of 4: 3 × 4 = 12turn it4 rows of 3: 4 × 3 = 12

Now it has 4 rows of 3. But it is the same dots! Nothing was added or taken away. So

3 × 4 = 4 × 3 = 12.

This works for any two numbers, because every array can be turned. So 6 × 4 = 4 × 6, and 2 × 9 = 9 × 2. You can multiply two numbers in either order and get the same product.

That answers Lily: 4 rows of 6 and 6 rows of 4 are the same tray turned around, so both have 24 muffins. More rows, but fewer muffins in each row.

This is very handy. 2 × 9 means "2 groups of 9", which is easy: 9 + 9 = 18. But 9 × 2 means "9 groups of 2", which takes longer to add: 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2. Since they are equal, you can always pick the easier one.

Special numbers: 1, 0, 2, 5 and 10

  • Times 1. 7 × 1 is 7 groups of 1, which is 7. And 1 × 7 is one group of 7, which is also 7. Multiplying by 1 changes nothing.
  • Times 0. 5 × 0 is 5 empty bags: 0 apples. And 0 × 5 is no bags at all: 0 apples. Multiplying by 0 always gives 0.
  • Times 2. 2 groups of something is a double. 2 × 8 = 8 + 8 = 16.
  • Times 5. Count by fives: 5, 10, 15, 20, 25, ... The answers always end in 5 or 0. A clock face helps: the minute hand counts by fives as it goes from number to number.
  • Times 10. Count by tens: 10, 20, 30, ... 7 × 10 is 7 tens, which is 70. The 7 just moves into the tens place.

Building a new fact from one you know

Suppose you know 5 × 4 = 20, but not 6 × 4. Six groups of 4 is five groups of 4 and one more group of 4. So 6 × 4 = 20 + 4 = 24.

Or you know 2 × 7 = 14. Then 4 × 7 is double that, because 4 groups is two lots of 2 groups: 14 + 14 = 28.

In level 3 you will use this idea to build all the times tables from just a few easy ones: Times tables with understanding.

Worked examples

1. Equal groups. There are 5 spiders. Each spider has 8 legs. How many legs?

5 groups of 8: 5 × 8. Skip count by 8: 8, 16, 24, 32, 40. So there are 40 legs. (Or turn it around: 8 × 5 is counting by fives eight times: 5, 10, 15, 20, 25, 30, 35, 40. Same answer.)

2. Reading an array. A window has 3 rows of 5 panes. How many panes?

3 × 5. Count by fives, three times: 5, 10, 15. 15 panes.

3. Write it as multiplying. Write 6 + 6 + 6 + 6 + 6 as a multiplication.

There are five 6s, so it is 5 groups of 6: 5 × 6 = 30.

4. Choosing the easy order. Find 9 × 2.

Turn it around: 2 × 9 = 9 + 9 = 18.

5. Building a fact. Find 6 × 5 if you know 5 × 5 = 25.

6 groups of 5 is 5 groups of 5 and one more 5: 25 + 5 = 30.

6. Mars wheels. A Mars rover has 6 wheels. How many wheels do 3 rovers have?

3 groups of 6: 6, 12, 18. 18 wheels.

Try it

  1. Write as a multiplication, then find the answer: 4 + 4 + 4.
  2. There are 6 cars with 4 wheels each. How many wheels?
  3. Draw an array with 2 rows of 7. How many dots? Then turn it: how many rows, and how many in each row?
  4. Find 7 × 10 and 10 × 7.
  5. Find 0 × 9 and 9 × 1.
  6. A box holds 3 bags of 5 marbles and 1 bag of 4 marbles. Can you find the total as one multiplication? Why or why not? Find the total.
  7. You know 5 × 7 = 35. Use it to find 6 × 7.
  8. Which is bigger: 4 × 8 or 8 × 4? Explain without working out either one.
Answers
  1. Three 4s: 3 × 4 = 12.
  2. 6 × 4: skip count by 4 six times, 4, 8, 12, 16, 20, 24. 24 wheels.
  3. 2 × 7 = 7 + 7 = 14 dots. Turned, it has 7 rows of 2: 7 × 2 = 14, the same dots.
  4. Both are 70. 7 groups of 10 is 7 tens; 10 groups of 7 is the same array turned.
  5. 0 × 9 = 0 (no groups, so nothing). 9 × 1 = 9 (nine groups of one).
  6. Not as one multiplication, because the groups are not equal (three 5s and a 4). Do it in two parts: 3 × 5 = 15, then 15 + 4 = 19 marbles.
  7. 6 groups of 7 is 5 groups of 7 and one more 7: 35 + 7 = 42.
  8. Neither; they are equal. An array of 4 rows of 8 turned on its side is 8 rows of 4, with the same dots. (Both are 32.)

Common mistake

Adding instead of multiplying. "3 bags of 4" is not 3 + 4 = 7. It is 4 + 4 + 4 = 12. Ask yourself: "Do I have 3 things and 4 things, or 3 groups of 4 things?" Drawing the groups for a moment usually makes it clear.

Multiplying unequal groups. If the groups are different sizes, multiplying gives the wrong answer. Add them instead.

Where it leads