Turn it a quarter turn

Before you read anything

Lay your twenty-four stones out as four straight rows of six. No heaps.

Now walk round the table and look at the same stones from the side. Say what you see from there.

The same stones, from the side

Four rows of six turns a quarter turn and becomes six rows of four. The twenty-four at the top never changes.

Nobody added a stone and nobody took one away. So $4 \times 6$ and $6 \times 4$ are not two facts. They are one fact and two places to stand.

Why straight rows

:x nutshell-array

Things in straight rows, all the same length. A heap has to be counted. An array can be read.

Four heaps of six and six heaps of four cannot be told apart by looking. Made into an :array, they are one rectangle seen twice, and nobody has to count anything to see it.

The half that disappears

Every fact in the times table has a twin on the other side of the diagonal. $7 \times 8$ and $8 \times 7$ are the same tray. Learn one and you have both, which takes the work you are facing and halves it before you start.

Halved, and still a lot

Seventy-two is still seventy-two. And you cannot stop in the middle of a longer question to work out $7 \times 8$ without losing the question.

Where this does not work

Otim

tried the same turn on $9 - 4$

Turning a tray is a fact about rectangles, not a rule about order. There is no rectangle behind $9 - 4$, so nothing turns and $4 - 9$ is a different question.

Check yourself

Six to try

Draw the rectangle if you cannot see it. Most of these are asking which two are the same tray.

1

Which of these has the same answer as $4 \times 5$?

2

Which of these has the same answer as $9 \times 4$?

3

Which of these has the same answer as $9 \times 6$?

4

Which of these has the same answer as $6 \times 5$?

5

Which of these has the same answer as $8 \times 7$?

6

$4 \times 6$

Rows either way · page 1 of 2 Print it

Rows either way · page 2 of 2 Print it

Show that you have it

One fact, asked from the other side

These two are the same tray. Work out the first one. Then read the second one off it without counting anything.

1

$5$ groups of $7$. How many altogether?

2

$7$ groups of $5$. How many altogether?

Next: How few facts there are The whole thing on one page