1. Understand it

Times Tables · Lesson 1

Multiplication is compressed counting

Before trying to remember multiplication facts, make sure the answers are not just arbitrary numbers attached to questions. Multiplication is a shortcut for counting equal groups.

Try it interactively

Choose the number of groups and the size of each group. Then notice that the same total can be described by a picture, repeated addition, skip-counting in either direction, and a multiplication sentence.

Open the multiplication interactive →

See the counting happen

In this example, we have 4 groups of 6. Watch one row of 6 being highlighted at a time. This helps show why we can count in 6s: 6, 12, 18, 24.

Animation showing a 4 by 6 dot array with one row highlighted at a time for counting in 6s

At the same time, we have 6 groups of 4. Watch one column of 4 being highlighted at a time. This helps show why we can count in 4s: 4, 8, 12, 16, 20, 24.

Animation showing a 4 by 6 dot array with one column highlighted at a time for counting in 4s
4 rows of 6
6 + 6 + 6 + 6 = 24
Count in 6s: 6, 12, 18, 24
4 × 6 = 24
6 rows of 4
4 + 4 + 4 + 4 + 4 + 4 = 24
Count in 4s: 4, 8, 12, 16, 20, 24
6 × 4 = 24

What changed — and what stayed the same?

The dots have been rearranged, but none have been added or removed. So the total stays the same: 24.

You can see the same array in two ways:

  • 4 rows of 6, so you can count in 6s: 6, 12, 18, 24
  • 6 rows of 4, so you can count in 4s: 4, 8, 12, 16, 20, 24

That is why 4 × 6 = 6 × 4.

A useful mindset: if one route feels awkward, try the other. Some learners find counting in 6s easier here. Others find counting in 4s easier once they see the same dots arranged differently.
Try it before moving on

Draw five groups of seven dots.

First count in 7s: 7, 14, 21, 28, 35, and write 5 × 7 = 35.

Then rearrange the same dots as seven groups of five and count in 5s: 5, 10, 15, 20, 25, 30, 35.

You have just seen why 5 × 7 = 7 × 5.