Multiplication Grid

Switch on Diagonal. What do all the diagonal cells have in common? Why does this always create this kind of number?

Find 3 x 7 and 7 x 3. Then check 4 x 9 and 9 x 4. What stays the same? If the grid were folded along the diagonal, what would match up?

Show the 3 times table. Where do the highlighted row and column cross, and what product appears at that crossing?

Show the 4 and 6 times tables together. Which cells belong to both tables? What are those products, and why do they appear in both places?

Show the 9 times table. Look at 9, 18, 27, 36 and beyond. What happens when you add the digits of each product? Does the pattern keep working?

Show the 5 times table. What endings do its products have? Now add the 2 times table. Which products are shared, and what do those shared products end in?

Show the 11 times table. Describe the pattern from 11 to 99. What changes when you reach 11 x 10, 11 x 11 and 11 x 12?

Show the 1 times table. Why does this row and column simply copy 1, 2, 3, 4 and so on? What mathematical role does 1 have in multiplication?

Which times table shares the most products with the 6 times table? Try different tables alongside 6 and explain your answer using factors.

Show the even products. What fraction or proportion of the grid is highlighted? Now show the odd products instead. What proportion is odd, and why are the two amounts different?

Study where the odd products appear. Which rows contain them? Which columns contain them? What must be true about both factors for the product to be odd?

Before checking, predict how many products are odd on a 12 x 12 grid. Then test it. Can you create a rule for the number of odd products on an n x n grid?

Show the even products and then add the 3 times table. Are all multiples of 3 even? Are all even products multiples of 3? Use examples from the grid to justify your answer.

Show the square products. The diagonal should light up. Why does every diagonal cell give a square number?

Can you find square products that are not on the diagonal? For each one, explain why multiplying two different factors can still make a square number.

How many times does 36 appear on the grid? List the row and column pairs that make 36. Which of these are square facts, and what does this reveal about factor pairs?

Show both Square and Diagonal. Are there any diagonal cells that are not square products? Should that ever happen? Explain your reasoning.

Show the prime products. How many cells are highlighted, and where are they? Explain why a product in a multiplication grid can only be prime when it is in the 1 row or 1 column.

Could a prime product appear anywhere other than the first row or first column? Use the definition of a prime number to explain why or why not.

Show Prime and the 1 times table together. What do you notice? How are prime products connected to the 1 row and 1 column?

Use manual shading to mark every cell with product 12. How many times does 12 appear? Repeat for 24, then predict how many times 36 will appear before checking.

Which product appears most often on a 12 x 12 grid? Think about which number has the most factor pairs with both factors no greater than 12, then test your prediction by shading.

Which whole numbers from 1 to 144 do not appear anywhere on the 12 x 12 grid? What do those missing numbers tend to have in common?

How many different products appear on a 12 x 12 grid? Estimate first, then count systematically. Is the total nearer to 144 or to 72?

Change the grid to 5 x 5 and show the even products. Count them. Now try 6 x 6 and 7 x 7. What is happening to the proportion of even products as the grid grows?

On a 10 x 10 grid, show the 7 times table and count the highlighted cells. Now change to a 7 x 7 grid and do the same. What changes, and what mathematical idea does this connect to?

Try 1 x 1, then 2 x 2, then 3 x 3 and continue. Show Diagonal each time. How many diagonal cells are on an n x n grid, and why?

A number is abundant when the sum of its proper factors is greater than the number itself. Use manual shading to mark the abundant numbers that appear on the 12 x 12 grid. Where do they tend to appear?

Call two products neighbours if their cells touch horizontally or vertically. Find neighbouring cells whose products differ by 1. How many examples can you find, and where do they occur?

Create your own investigation question about the multiplication grid for another student. Test it yourself first, then swap with a partner.