Sum and product of integers inquiry

The prompt

Mathematical inquiry processes: Explore; generate other examples; conjecture, generalise, reason and prove. Conceptual field of inquiry: Sum and product of integers.

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

The prompt is always true. A proof is accessible to older primary students:

  • 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. This can give rise to the conjecture that even numbers give an even product and odd numbers give an odd product.

Lines of inquiry

(1) Split another even number into three integers

Pupils create some examples and then explain why the product is always even for any even number.

(2) Split an odd number into three integers

Pupils create their own examples. Is the product odd or even? Prove your conjecture is true by reasoning with odd and even numbers.

(3) Split 12 into four integers and more

Is the product odd or even or could it be both? Can you spot a pattern in your results (see table below)? Why are the results different depending on whether you split the number into an odd or even number of integers?

Now split an odd number into four and more integers. Record your results and explain your findings.