Deriving multiplication and division facts (PDF, 33 KB)
*Multiplicative Strategies progression, 3rd step; *

*Number Facts progression, 3rd step *

## The purpose of the activity

In this activity, the learners extend their repertoire of multiplication and division facts and gain an understanding of the multiplication and division facts they need to practise for quick recall.

Traditionally, multiplication tables have been taught by rote, without an emphasis on learner understanding. This activity, however, follows international best practice and promotes an understanding of multiplication and division facts by using already-known facts to derive unknown facts. This activity builds on the ‘Understanding multiplication’ activity.

## The teaching points

- The inverse of multiplication is division.
- The multiplication ‘basic’ facts are the facts made from the digits 0 to 9.
- Any number multiplied by 0 has an answer of 0.
- Any number multiplied by one has the original number as the answer.
- Multiplication and division facts can be directly connected. For example, 6 x 7 = 42, 7 x 6 = 42, 42 ÷ 7 = 6 and 42 ÷ 6 = 7 are connected facts.
- Unknown multiplication and division facts can be derived from known facts. For example if 6 x 7 = 42 then 7 x 7 = 42 + 7 = 49.
- Discuss with the learners the importance of being able to recall the basic multiplication and division facts. These facts are the ‘building blocks’ for estimation and more complex calculations.

## Resources

- A set of multiplication and division fact cards, preferably one set for each learner.

## The guided teaching and learning sequence

1. Write 8 x 9 = on the board.

2. Ask for a volunteer to give you the answer and record 72 on the board.

3. Ask the learners to think about how they might work out 8 x 9 if they had forgotten (or didn’t know) it automatically.

4. Let volunteers share their strategies for working out 8 x 9.

“I know that 8 x 10 is 80 and 8 less is 72.”

“I know 4 x 9 is 36, and I double it to get 72.”

“I know 8 x 8 is 64 and 8 more is 72.”

“I know the nines pattern.” (18, 27, 36 …)

5. Encourage the learners to notice that you can learn unknown facts by deriving (or figuring) them from already known facts.

6. Write 72 ÷ 8 = on the board and ask the learners if they know the answer. Encourage them to notice that this problem is directly connected to 8 x 9 = 72.

7. Ask the learners to state the other connected facts (72 ÷ 9 = 8 and 9 x 8 = 72.)

8. Ask the learners to think about the multiplication facts they have the most trouble with recalling. List these on the board.

9. Take one of the listed facts and ask the learners to think of all the ways they could work out that fact from other known multiplication or division facts. For example if 7 x 7 was recorded.

“I know that 6 x 7 is 42, and 7 more is 49.”

“I know 7 x 8 is 56, and 7 less is 49.”

“I know 5 x 7 is 35, and 14 more is 49.”

10. Ask:

“If you know your 2 times facts, what other facts can you easily figure out?”

Ensure all the learners understand the connections between the twos, fours and eights by looking at examples such as:

3 x 2 = 6 so 3 x 4 = 12 and 3 x 8 = 24 5 x 2 = 10 so 5 x 4 = 20 and 5 x 8 = 40

Repeat this, using other linked facts. For example, derive the 6 facts from the 3 facts by doubling (for example, 3 x 5 = 15 so 6 x 5 = 30) and derive the 9 facts by subtracting from the 10 facts (for example, 10 x 3 = 30 so 9 x 3 = 30 – 3 = 27).

## Follow-up activity

1. Ask the learners to spread out their known facts (from the “Understanding multiplication” activity above).

2. Next, ask them to take a fact from their unknown fact pile and see if it is closely connected to a known fact.

3. Ask the learners to explain how the unknown fact can be derived from the known fact. If the fact is not closely connected, ask the learner to take another fact from the unknown pile and repeat until they find a fact they can closely connect.

4. Ask the learners whether they think this fact now belongs in their known fact pile. If not, suggest they need to focus on practising/ learning it.