Tables and Division - A Reasoning Approach using Key Facts

Cat Eadle (Endorsed by Steve Chinn)

Pre- Skills

It is important that a learner starting to use this method for 3, 4, 6, 7, 8 11- and 12-times tables (and related division facts) has the required pre-skills. If we don’t wait until these are secure and move too quickly to trying to learn times tables, no matter how well-meaning we are, it will not be beneficial to the learner and the method will not be successful in the longer term.

The key times tables facts a learner needs to know to use this method are 2x 5x and 10x. These should be taught using a multi-sensory reasoning-based approach. These are generally the facts an older learner will tell you they know if asked so this is for many a confidence building starting point.

The rest of the times tables facts are derived from these key facts -2x 5x and 10x. We are using the principle that multiplication is linked to addition as ‘groups of’ so for example 6 x 3 is 3+3+3+3+3 (5 x 3) plus another 3 (1x3). This shows the learner how multiplication is linked to addition and sets a great foundation understanding for future work – it is developmental for further maths topics, including algebra. Bonus learning!

By using multi-sensory resources and reasoning/thinking about an answer, rather than just trying to learn the facts by verbal association, the learner will have a much better chance of internalising the times tables, and in the longer term will be able to improve their knowledge and speed. It does, however, take repeated practice so don’t rush and play plenty of games to make it fun!

This method takes away a huge amount of anxiety about having to ‘remember or know’ times tables facts as the learner has a method to fall back on for each question- this is hugely reassuring for the learner and greatly reduces anxiety.

This is a written version of the verbal reasoning we often begin with. It should be accompanied by ‘groups of 3, 4, 6, 7, 8, 9, 11, 12’ using the learners preferred manipulative (e.g., Cuisenaire rods, counters, toys). We do not always practice the times tables in order -we give random questions covering all the required facts: this prevents the leaner from using the fact before to help them and encourages them to think through each question using the relevant ‘key fact’.

The teacher must talk through their reasoning strategy as they work with the learner to model thinking and encourage the learner to do the same – it is surprising how quickly we hear back ‘ummm.. 9 x 4 well 10 x 4 is 40 so it must be 4 less than that… so 40 take away 4 is … 36!

Example:

3 Times Tables and Related Division Facts

Remind the learner by building the groups of three that we are thinking in threes 😊 We like to use some ‘key fact’ cards to put on the key fact groups of 3 – highlighted here -

0 x 3

If there are zero groups of three - I have none so the answer is 0

1 x 3

One group of 3 is 3

2 x 3

A key Fact –

Double three is 6- pre skill so should be known before starting this method

3 x 3

I know that 2 x 3 (double three) is 6 so I need one more group of 3, so 3 x 3 is 6+3 which is 9.

4 x 3

I know that 5 x 3 is 15, so 4 x 3 is one group of 3 less than this….15-3 is 12. So 4 x 3 is 12.

5 x 3

A key fact – Based on pre-skill of knowing that 5 x 3 is half of 10 x 3

6 x 3

I know that 5 x 3 is 15. I need one more group of 3. So 6 x 3 must be 15+3 (5 x 3 + 1 x 3) which is 18.

7 x 3

I know that 5 x 3 is 15, so 6 x 3 is 18. So 7 x 3 is one group of 3 more than this so 7 x 3 is 18+3 which is 21 or 5 x 3=15 + 2 x 3=6 so 15+6=21

8 x 3

There are two ways we can get to 8 x 3 -

Method 1- I know that 5 x 3 is 15, 6 x 3 is 18, 7 x 3 is 21 so 8 x 3 is one group of 3 more so 21+3 is 24 (5 x 3 + 3 x 3)

OR

Method 2- I know that 10 x 3 is 30, so 9 x 3 is 27, so 8 x 3 is one group of 3 less than 9 x 3. 8 x 3 is 27-3 which is 24 (10 x 3 – 2 x 3)

9 x 3

I know that 10 x 3 is 30. So, 9 x 3 is one group of three less than that. 30-3 (10 x 3 – 1 x 3) is 27 so 9 x 3 is 27.

10 x 3

A key fact – a pre- skill of place value understanding of ‘groups of 10’

Once the learner has used this method for one times table they can start to see how they can relate this to the next which inspires confidence e.g. I know 10 x 3 = 30 and 9 x 3 is 27 as it is one group of 3 less…. so 10 x 4 is 40 and to get to 9 x 4 that would be 1 group of 4 less so 40-4= 36.

As they progress through looking at each set of facts, the learner often starts to use commutativity (for example that 4 x 6 is the same as 6 x 4) and develop further reasoning strategies and links e.g., noticing that 3 x 4 is related to 6 x 4 as there are twice as many groups of 4. The multi-sensory ‘groups’ really help the learner ‘see’ these patterns.

When the learner reasons times tables facts in this way the more they practice ‘good thinking’ the better they get at it - repeated practice using games and activities but verbalizing thinking really makes a big difference.