Exploring nines

Write the numeral 9 high on the board, and ask the students to call out the next number, counting up by 9. Write the numbers (18, 27 etc.) underneath, keeping the units and tens columns straight. Stop at 54.

Ask the students to describe any patterns they see, and for each pattern ask why that would happen. Prompt with questions, if necessary. Students should recognise the following patterns.

• Units go down by 1, and tens go up by 1.
  Ask the reason. What number is needed before the 9? Ask a student to continue the pattern.

• The unit and ten digits in each number add up to 9.
  What happens at 99? Use the facts that 9 + 9 = 18 and 1 + 8 = 9.

• The numeral in the tens place of the answer is one less than the number being multiplied by 9.
  Why? How far would that pattern continue, and why?

Do the students know how to make the nine times table on their fingers? Watch this video. Why is this possible with the nine times table?

Given one number fact (e.g. 3 \(\times\) 9 = 27), how many number facts could we write using addition, multiplication, division and subtraction? For example, students could write:

• 27 ÷ 3 = 9
• 9 + 9 + 9 = 27
• 27 – 2 \(\times\) 9 = 9

Discuss the number facts in terms of associative, distributive and commutative properties — not using those terms, but expecting students to use meaningful language such as 'swap places', 'adding these two first' and 'working backwards'.