Multiplication Grid Explorer — Interactive Tool

1

Toggle the Diagonal button to highlight where row equals column. What do you notice about all the numbers on the diagonal? Why are they all this type of number?

2

Find 3 × 7 and 7 × 3 on the grid. What do you notice? Now check 4 × 9 and 9 × 4. If you folded the grid along the diagonal, what would happen? What property of multiplication does this show?

3

Toggle the 3 times table. You should see a cross shape (a highlighted row and column). Where do the row and the column overlap? What product appears there?

4

Toggle the 4 times table and the 6 times table at the same time. Find all the cells that are highlighted by both. What are these products, and why?

5

Toggle the 9 times table. Look at the products in the 9 row: 9, 18, 27, 36… What happens when you add the digits of each product? Does this pattern continue? Can you explain why?

6

Toggle the 5 times table. What do all the products end in? Now toggle the 2 times table as well. Which products appear in both? What do all these shared products end in?

7

Toggle the 11 times table. Look at the products 11, 22, 33… up to 99. What pattern do you see? What happens after 99 (you may need to look at 11 × 10, 11 × 11, 11 × 12)?

8

Toggle the 1 times table. Why does this row/column just repeat 1, 2, 3, 4…? What is the mathematical name for the role that 1 plays in multiplication?

9

Which times table has the most products that are also in the 6 times table? Try toggling different tables alongside 6 to investigate. Can you explain your answer using factors?

10

Toggle Even products. What proportion of the grid is shaded? Now toggle Odd products instead. What proportion is odd? Why aren’t they equal?

11

Look carefully at where the odd products appear. Which rows contain odd products? Which columns? What must be true about BOTH the row number and the column number for the product to be odd?

12

Without using the grid, predict: on a 12 × 12 grid, how many products are odd? Now check your answer. Can you write a formula for the number of odd products on an n × n grid?

13

Toggle Even products and then toggle the 3 times table. Are all multiples of 3 even? Are all even numbers multiples of 3? Use the grid to find examples that show the difference.

14

Toggle Square products. All the diagonal cells should be highlighted. Why does the diagonal always contain square numbers?

15

Are there any square products that are NOT on the diagonal? Find them. For each one, can you explain why the product of those two numbers is a perfect square?

16

How many times does the number 36 appear on the grid? List all the pairs of row and column numbers that give 36. For how many of these pairs is 36 a square product — and what does this tell you about factor pairs?

17

Toggle both Square and Diagonal. Are there any diagonal cells that are NOT highlighted as squares? Should there be? Explain.

18

Toggle Prime products. How many cells are highlighted? Where do they all appear? Can you explain why a product in a multiplication grid can only be prime if it’s in the 1 row or the 1 column?

19

Is it possible for a product to be prime and appear somewhere other than the first row or first column? Explain why or why not using the definition of a prime number.

20

Toggle both Prime and the 1 times table. What do you notice? What is the connection between these two categories on a multiplication grid?

21

Using manual shading, find every cell that contains the number 12. How many times does 12 appear? Now do the same for 24. Can you predict how many times 36 appears without looking?

22

What is the most common product on a 12 × 12 grid? (Hint: think about which number has the most factor pairs where both factors are ≤ 12.) Use manual shading to test your prediction.

23

Which products between 1 and 144 do NOT appear anywhere on the 12 × 12 grid? What do these missing numbers have in common?

24

How many distinct (different) products appear on a 12 × 12 grid? First estimate, then use manual shading to count systematically. Is the answer closer to 144 or closer to 72?

25

Change the grid to 5 × 5. Toggle Even and count the even products. Now try 6 × 6, then 7 × 7. What is happening to the proportion of even products as the grid grows?

26

On a 10 × 10 grid, toggle the 7 times table. How many cells are highlighted? Now try the same on a 7 × 7 grid. What changes, and what mathematical idea does this connect to?

27

Change the grid to 1 × 1, then 2 × 2, then 3 × 3 and so on. Toggle Diagonal each time. How many diagonal cells are there on an n × n grid? Can you explain why?

28

A product is called ‘abundant’ if the sum of its proper factors (factors less than itself) is greater than the number. Using manual shading, find all the abundant numbers that appear on the 12 × 12 grid. What do you notice about where they tend to appear?

29

Two products are ‘neighbours’ if they are next to each other (horizontally or vertically). Find a pair of neighbouring cells whose products differ by 1. How many such pairs can you find? Is there a pattern to where they occur?

30

Design a question of your own about the multiplication grid that you think would be interesting for another student to investigate. Test it yourself first, then exchange with a partner.