In the orientation phase of the inquiry, the teacher directs pupils to use positive integers only. After pupils have discussed the meaning of the prompt, they might start by testing the contention with their own examples.

Example 1          3 + 4 + 5 = 12          3 x 4 x 5 = 60

Example 2          5 + 1 + 6 = 12          5 x 1 x 6 = 30

Example 3          6 + 4 + 2 = 12          6 x 4 x 2 = 48

Example 4          2 + 3 + 7 = 12          2 x 3 x 7 = 42

Reasoning and proof

The prompt is always true. A proof is accessible to older pupils in primary school:

• If you choose an odd integer first (examples 1 and 2), there is an odd number left. You can split an odd number into an even and an odd (example 1) or an odd and an even (example 2). So there is an odd (O1), even (E) and odd (O2) or an odd (O1), odd (O2) and even (E). For the first way, O1 x E x O2 = O because O1 x E = E and then E x O2 = E. For the second, O1 x O2 = O and then O x E = E.

• If you choose an even integer first, there is an even number left. You can split an even number into an even and an even (example 3) or an odd and an odd (example 4), which gives: E1 x E2 x E3 or E x O1 x O2. For the first way, E1 x E2 = E and E x E3 = E. For the second, E x O1 = E and E x O2 = E.

In all four cases (OEO, OOE, EEE and EOO), the product is even. 

When pupils go on to inquire into other even numbers, they find that the products are also even when the number is partitioned into three.

This discovery can give rise to the conjecture that even numbers give an even product and odd numbers give an odd product. However, inquiring into eleven, for example, shows that the product could be odd or even: 4 + 4 + 5 = 13 and 4 x 4 x 5 = 80, while 3 + 3 + 7 = 13 and 3 x 3 x 7 = 63.

Guided inquiry

The slides contain lines of inquiry that develop from the four regulatory cards shown below.