Put them in pairs
To tell whether a number is odd or even, take that many things and put them into pairs, groups of 2. Six counters make 3 pairs, and no counter is left over. A number that makes pairs with none left over is even, so 6 is even.
6 counters make 3 pairs, with none left over: 6 is even.
One left over
Now try 7 counters. They make 3 pairs, and 1 counter is left over with no partner. A number that leaves one over when you make pairs is odd, so 7 is odd.
An odd number always leaves exactly 1 over, never 2 or more. Two counters left over would simply make another pair.
7 counters make 3 pairs, and 1 counter has no partner: 7 is odd.
Odd and even take turns
Add one more counter to 7, and it becomes the partner of the one that was left over. Now there are 4 pairs and none left over, so 8 is even. Add one more again, and 9 has a counter left over, so 9 is odd.
So as you count, the numbers take turns: 1 is odd, 2 is even, 3 is odd, 4 is even, and on forever. One more than an odd number is even, and one more than an even number is odd.
7 = 3 pairs + 1: one has no partner, so 7 is odd
Add one at a time until every one has a partner
The counters go into bags of 2. With 7 counters the last bag holds only 1. Add one more and the last bag fills: 8 is even.
The last digit decides
The even numbers are the numbers you land on when you count in twos from 0: 2, 4, 6, 8, 10, 12, and on. They end in 0, 2, 4, 6 or 8. The odd numbers are the ones in between, and they end in 1, 3, 5, 7 or 9. 0 is even, because nothing is left over.
For a bigger number, look only at its last digit, the one on the right. Every ten pairs up exactly into 5 pairs, so the tens never leave one over; only the last digit can. 34 ends in 4, so it is even. 27 ends in 7, so it is odd, even though its first digit, 2, is even.
The even numbers are ringed. Every one of them ends in 0, 2, 4, 6 or 8.
Worked example: Children Walking in Pairs
Question 17 children line up to walk to the hall in pairs. (a) How many pairs can they make? (b) How many children have no partner? Is 17 odd or even?
1.Draw one dot for each child. There are 17 dots.
One dot is one child. There are 17 dots. 2.Ring the dots in twos. Count in twos as you ring: 2, 4, 6, 8, 10, 12, 14, 16.
Ring the dots in twos and count in twos: 2, 4, 6, 8, 10, 12, 14, 16. 3.(a) There are 8 rings, so the children make 8 pairs. The pairs use 16 children.
(a) There are 8 rings, so there are 8 pairs. The pairs use 16 children. 4.(b) 17 − 16 = 1, so 1 child has no partner. A number with one left over after pairing is odd, so 17 is odd.
(b) 17 − 16 = 1 child has no partner. One is left over, so 17 is odd.
Answer: (a) 8 pairs; (b) 1 child, so 17 is odd
Common mistakes
- Looking at the tens digit to decide. The ones digit decides whether a number is odd or even. The ones digit of 17 is 7, and 7 is odd, so 17 is odd.
- Saying there are 9 pairs. The ninth ring would have only 1 child in it, and a pair needs 2 children.