Here is a surprise that saves a lot of work. An array of 3 rows of 5 has 15 dots. Turn it on its side, and you get 5 rows of 3. How many dots now?
Why it works
Turning the array does not change the dots, only how we read them. Read across, it is 3 rows of 5: $3 \times 5$. Read down, it is 5 columns of 3: $5 \times 3$. They count the same dots, so $3 \times 5 = 5 \times 3$. That is also why $6 \times 7 = 7 \times 6$: turn a 6-by-7 array and it becomes 7-by-6.
Key idea
You can multiply two numbers in either order: $a \times b = b \times a$. These are turn-around facts.
Worked example: Use a fact you know
You know $6 \times 7 = 42$. What is $7 \times 6$?
- $7 \times 6$ uses the same two numbers in the other order.
- Turning the array does not change the total.
Answer: $7 \times 6 = 42$
Worked example: Fill in the box
$4 \times 9 = 9 \times \square$
- Both sides must use the same two factors.
- So the box is 4. Check: both sides are 36.
Answer: $4$
Common mistake
Learning $3 \times 5$ and $5 \times 3$ as two separate facts. They are the same fact, turned. Once you know one, you know the other, so there are about half as many facts to learn.
Turning works for adding and multiplying, but not for subtracting or dividing: $8 - 2$ is 6, but $2 - 8$ is not.