In this topic you will…

• Build the multiplication tables for 3, 6, and 9.
• Connect the multiplication tables for 3, 6, and 9 to see how they relate.
• Count in threes, sixes, and nines starting from any number.

🧠 Key Words

• counting stick

💡 Quick Math Tip

Link your times tables: If you know one multiplication table, you can find others by doubling, adding, or subtracting — and use number patterns to count forwards or backwards from any starting number.

❓ EXERCISES

1. Write the multiples of $3$ and $6$.

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$3$ times table (from $1$ to $10$): $3, 6, 9, 12, 15, 18, 21, 24, 27, 30$.

$6$ times table (from $1$ to $10$): $6, 12, 18, 24, 30, 36, 42, 48, 54, 60$.

2. Which multiplication facts from the multiplication tables for $3$ and $6$ have the same value? The first one is done for you.
$3 \times 2 = 6 \times 1$

👀 Show answer

$3 \times 2 = 6 \times 1$

$3 \times 4 = 6 \times 2$

$3 \times 6 = 6 \times 3$

$3 \times 8 = 6 \times 4$

$3 \times 10 = 6 \times 5$

3. Write the missing multiplication facts.

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$3 \times 5 = 15$

$6 \times 7 = 42$

$3 \times 8 = 24$

$6 \times 9 = 54$

4. Colour the multiples of $9$ on this $100$ square. Describe the pattern you make.

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Multiples of $9$ up to $100$: $9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99$.

Pattern: each time you add $9$, you move one row down and one square to the left, making a diagonal line across the $100$ square.

5. Doubling the multiplication table for $6$ does not give the multiplication table for $9$. What could you do instead? Give an example.

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Use that $9 = 6 + 3$, so you can add the $6$ times table and the $3$ times table.

Example: $9 \times 7 = 6 \times 7 + 3 \times 7 = 42 + 21 = 63$.

6. Multiply the numbers in the bricks next to each other to find the number for the brick above.
Where have you seen walls like this before? What were they used for?

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Left wall: middle bricks are $3 \times 3 = 9$ and $3 \times 3 = 9$, top brick is $9 \times 9 = 81$.

Right wall: middle bricks are $1 \times 9 = 9$ and $9 \times 1 = 9$, top brick is $9 \times 9 = 81$.

Examples of where you see walls like this: brick walls in buildings, garden walls, and old fortifications; they were used to build structures, make boundaries, and provide protection.

7. The term-to-term rule is add $9$. Choose a start number and write the first five numbers in your sequence.

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One example (start at $1$): $1, 10, 19, 28, 37$.

8. $9 \times 3 = 5 \times 3 + 4 \times 3 = 15 + 12 = 27$
Draw a picture to show how you can add the multiplication tables for $5$ and $3$ together to find $8 \times 4$.

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Split $8$ into $5 + 3$ and use an array (or dots) with $4$ rows (or columns): one part is $5 \times 4$ and the other is $3 \times 4$.

$8 \times 4 = 5 \times 4 + 3 \times 4 = 20 + 12 = 32$.

9. If you know that $6 \times 9 = 54$, what other connected facts do you know?

👀 Show answer

$9 \times 6 = 54$

$54 \div 6 = 9$

$54 \div 9 = 6$

$6 \times 8 = 48$ so $48 + 6 = 54$

$9 \times 5 = 45$ so $45 + 9 = 54$

$6 \times 10 = 60$ so $60 - 6 = 54$

🧠 Think like a Mathematician

The digit sum of a number is found by adding the digits together; for example, the digit sum of 24 is 2 + 4 = 6.

If the sum is more than 9, repeat until you have a single digit; for example, the digit sum of 48 is 4 + 8 = 12, then 1 + 2 = 3.

Challenge: Is there a pattern in the digit sums of the multiples of 3, 6 and 9?

Method:

1. Write down the first 10–15 multiples of 3, then do the same for 6 and 9.
2. For each number, add its digits. If the result is bigger than 9, add the digits again until you get one digit.
3. Compare the final one-digit answers and look for repeating cycles.

Quick examples (to get you started):

• Multiples of 3: 3→3, 6→6, 9→9, 12→1+2=3, 15→1+5=6, 18→1+8=9, ...
• Multiples of 6: 6→6, 12→3, 18→9, 24→6, 30→3, 36→9, ...
• Multiples of 9: 9→9, 18→9, 27→9, 36→9, 45→9, ...