Mental Computation Examples: 2.75

Here are some examples to highlight how the number properties can be used to assist with computation.

Making addition easier by counting on from larger

Students who are still at the 'counting on' stage may attempt a problem like 3 + 37 by starting at 3 and trying to count on 37 more. After the counting on stage, comes the 'counting on from larger stage'.

Highlight that 3 + 37 can be calculated more easily if students first spin around to change it to 37 + 3, so they start at 37 and count on by 3. [Technically speaking, we are using the commutative property here.]

Try spin around on similar examples like:

4 + 12

7 + 15

2 + 29

6 + 33

5 + 64

Using known facts

As students begin to master their number facts, they can take advantage of knowing pairs of numbers that add to 10, which should have been explored using tens frames or similar. Knowing facts that add to 10 makes it much easier to work out a problem such as 3 + 27 without counting on. Instead, we see the 3 on its own and the 7 from the 27, add to 10. This 10 can be combined with the 20 part of 27 to give a total of 30.

A more complicated example is given by 26 + 54. We can add together the 6 and 4 to get 10, and then combine this with the 20 and the 50 to get 80. All the components to be added (the 20, the 6, the 50 and the 4) can be shuffled for our convenience. [Technically speaking both associative and commutative properties are involved here.]

Try similar examples like:

26 + 4

43 + 7

65 + 5

9 + 81

8 + 122

18 + 32

27 + 33

45 + 25

41 + 39

34 + 106

Of course, you should also include examples where tens facts are not needed (such as 43 + 32, where you just add the ones and add the tens and then combine the results), but the special examples given here can be useful for more complicated calculations as shown in the next section.

Adding strings of numbers

As students increase their familiarity with the number facts, and grow in confidence with examples like those in the previous section, introduce them to strategies for adding strings of numbers. For example, to add together mentally a set of numbers like 27 + 24 + 3, it can be a good idea to find numbers that combine to give 10. In this case, it makes sense to add the 27 and the 3 first, to get 30, and then add the 24 to get 54.

You can tell students that as long as it is only addition that is involved, then it doesn't matter in which order the numbers are added. They can be shuffled as we want. [Technically speaking both the associative and commutative properties get a good workout here.]

Try similar examples like:

24 + 5 + 35

43 + 22 + 7

15 + 22 + 25

13 + 12 + 27 + 8

28 + 13 + 42

Helping students learn their multiplication facts

As students start to learn their multiplication facts, highlight the fact that once they know one fact they also know the commuted fact as well. For example, if they know that 7 x 5 is 35, then they know the spin around of this, i.e. 5 x 7 is 35 as well. [Technically speaking, we are using the commutative property here.]

This means that if students have a hard multiplication, they can try the spin around to see if it is easier to work out. As an example, 6 x 9 might not be known, but 9 x 6 (the commuted fact) will give the same answer and may be worked out as one 6 less than 10 x 6, or 54. [Note that we actually used the distributive property here as well when we worked out 9 × 6 as 10 × 6 – 6.]

Mental computation and multiplication

As for addition, when it is only the multiplication operation that is involved, the order of the numbers does not matter. [In other words, multiplication is commutative and associative.] This means that we can shuffle computations to make them easier to carry out. For example, multiplying 9 x 10 x 6 is done more readily by working out 9 x 6 first to get 54, and then multiplying by 10 to get 540. Encourage students to look for easy-to-work-out combinations, and do the computation accordingly.

Try similar examples like:

10 x 3 x 2

2 x 10 x 4

4 x 10 x 3

3 x 2 x 5

5 x 4 x 2

15 x 7 x 2

20 x 6 x 5

4 x 8 x 5

4 x 9 x 25

50 x 14 x 2

Emphasise that shuffling is only appropriate if all the operations are addition, or all multiplication.

Doubling and halving

When working towards Level 4, students learn the ‘doubling and halving’ method of multiplication,

e.g. 24 × 16 = 48 × 8 = 96 × 4 = 192 × 2 = 384 × 1 = 384

This relies on the shuffling permitted by the associative and commutative properties,

e.g. 24 × 16 = 24 × (2 × 8) = 48 × 8 = 48 × (2 × 4) = (48 × 2) × 4 = 96 × 4 etc

Final Note

Whenever you do any sort of mental computation in class, it is a good idea to model and explain the strategies that you used in order to develop students’ repertoire of techniques.