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Year 4 Maths Whole Numbers & Place Value

Odd & even numbers

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Theory

Odd and even numbers are told apart by their ones (last) digit: even numbers end in 0, 2, 4, 6 or 8, and odd numbers end in 1, 3, 5, 7 or 9. This Year 4 maths page explains the last-digit rule, the pairing test, counting by twos, why every even number is 2 times a whole number, and how odd and even numbers behave when you add them. In the Australian Curriculum this is Year 4 content; in the Victorian Curriculum it is Level 3. Curriculum: VC2M3N01 (Level 3) / AC9M4N02.

An even number can be split into two equal groups, or into pairs, with none left over. An odd number always has one left over. For example, 12 counters make 6 pairs, or 2 equal rows of 6 (even), but 7 counters make 3 pairs with 1 left over (odd).

You do not have to count out the pairs every time. Just look at the ones (last) digit. Even numbers end in \(0, 2, 4, 6\) or \(8\). Odd numbers end in \(1, 3, 5, 7\) or \(9\). This works for any number, big or small.

Every even number is 2 times a whole number, such as \(14 = 2 \times 7\). As you count on, even and odd numbers take turns: \(0\) (even), \(1\) (odd), \(2\) (even), \(3\) (odd), and so on. Counting by twos from 2 gives the even numbers \(2, 4, 6, 8, 10, \dots\)

8 counters in pairs 8 counters make 4 pairs with none left over, so 8 is even. 8 = even
8 makes 4 pairs with none left over: even.
7 counters in pairs 7 counters make 3 pairs with one red counter left over, so 7 is odd. 7 = odd
7 makes 3 pairs and leaves 1 over: odd.
Odd and even numbers on a number line from 0 to 10 A number line from 0 to 10. The odd numbers 1, 3, 5, 7 and 9 have red dots marked O. The even numbers 0, 2, 4, 6, 8 and 10 have no dot. Even and odd take turns. 012345678910 O O O O O
Odd numbers (red, O) and even numbers take turns as you count.

Odd and even numbers do not need a formula, but these facts let you check any number quickly.

\[ \text{Even numbers end in } 0,\ 2,\ 4,\ 6 \text{ or } 8 \]
Even numbers end in 0246 or 8
\[ \text{Odd numbers end in } 1,\ 3,\ 5,\ 7 \text{ or } 9 \]
Odd numbers end in 1357 or 9
\[ \text{Every even number} = 2 \times \text{a whole number} \]
Every even number =2× a whole number
Adding odd and even. Even \(+\) even \(=\) even, odd \(+\) odd \(=\) even, and even \(+\) odd \(=\) odd. For example \(3 + 9 = 12\) (even) and \(6 + 9 = 15\) (odd).
Counting the evens. From 1 to 20 there are 10 even numbers: \(2, 4, 6, \dots, 20\). The 9th even number when you count by twos from 2 is \(2 \times 9 = 18\).

How to tell if a number is odd or even

  1. Find the ones digit. Look at the very last digit of the number and ignore all the others.
  2. Check that digit. If it is \(0, 2, 4, 6\) or \(8\) the number is even. If it is \(1, 3, 5, 7\) or \(9\) the number is odd.
  3. Or test by pairing. Split the amount into pairs, or into two equal rows. None left over means even; one left over means odd.

How to find the next even or odd number

  1. Count by twos. Even numbers go up in twos, and so do odd numbers. The next even number after an even number is 2 more.
  2. One step changes the type. The number just before or after an even number is odd.

How to make the largest even or odd number from digits

  1. Choose the last digit first. For an even number the ones digit must be even; for an odd number it must be odd.
  2. Put the other digits from largest to smallest in front of it.
Example 1 — Last-digit rule
Is \(3486\) odd or even?
Solution
\(\text{ones digit}\)\(=\)\(6\)
\(6\)\(\Rightarrow\)\(\text{even}\)
3486 even

So \(3486\) is even.

Example 2 — Next even number
What is the next even number after \(138\)?
Solution
\(\text{even numbers}\)\(:\)\(\text{go up in twos}\)
\(138 + 2\)\(=\)\(140\)
138+2=140

\(139\) ends in \(9\), so it is odd; the next even number is \(140\).

Example 3 — Equal rows
\(12\) counters are put into \(2\) equal rows. How many are in each row? Is \(12\) odd or even?
Solution
\(12\)\(=\)\(6 + 6\)
\(12\)\(=\)\(2 \times 6\)
12=6+6

There are \(6\) in each row with none left over, so \(12\) is even.

Example 4 — Adding odd and even
Ravi works out \(6 + 9\). Is the total odd or even?
Solution
\(6 + 9\)\(=\)\(15\)
\(15\)\(\Rightarrow\)\(\text{odd}\)
6+9=15

An even number plus an odd number is always odd.

Example 5 — Pairing up
\(7\) counters are put into pairs. How many are left over, and is \(7\) odd or even?
Seven counters put into pairs with one left over Seven counters make three pairs, with one red counter left over, so 7 is odd. 1 left over
Solution
\(3\text{ pairs}\)\(=\)\(6\)
\(7 - 6\)\(=\)\(1\text{ left over}\)
7-6=1

One is left over, so \(7\) is odd.

Example 6 — Counting by twos
(i) Count by twos: \(52,\ 54,\ 56,\ \ldots\) Write the next two numbers.
(ii) What is the 9th even number when you count by twos from 2?
Solution
\(56 + 2\)\(=\)\(58\)
\(58 + 2\)\(=\)\(60\)
\(2 \times 9\)\(=\)\(18\)
2×9=18

(i) \(58,\ 60\). They are all even.
(ii) \(18\): the evens are \(2, 4, 6, 8, 10, 12, 14, 16, 18\).

Example 7 — Largest even and odd from digits
Use the digits \(2,\ 5\) and \(9\), each once, to make three-digit numbers.
(i) What is the largest even number?
(ii) What is the largest odd number?
Solution
\(\text{even: ones digit}\)\(=\)\(2 \to 952\)
\(\text{odd: ones digit}\)\(=\)\(5 \text{ or } 9 \to 925\)
952 even

(i) \(952\): only 2 is even, so it must go last.
(ii) \(925\): keep 9 in front and end in the odd digit 5.

Common pitfalls

Use the last digit, not the first. Watch out for looking at the front of a number. \(72\,914\) is even because it ends in \(4\), even though it starts with \(7\).
Zero is even. A number ending in \(0\) is even, so \(40\), \(70\) and \(0\) itself are all even.
Odd and even take turns. Every second number is even. The number right after an even number is always odd, so you never get two evens in a row.
Odd plus odd is not odd. Two odd numbers add to an even number: \(3 + 9 = 12\). The two left-over ones make a pair.

Frequently asked questions

What is an even number?

An even number can be split into two equal groups, or into pairs, with none left over. It ends in \(0, 2, 4, 6\) or \(8\). Examples are \(8\), \(40\) and \(3486\).

What is an odd number?

An odd number always has one left over when you make pairs. It ends in \(1, 3, 5, 7\) or \(9\). Examples are \(7\), \(21\) and \(67\).

Is 0 odd or even?

Zero is even. It ends in \(0\), and zero counters make pairs with none left over.

How do you tell if a big number is odd or even?

Look only at the ones or last digit, not the first. If the last digit is \(0, 2, 4, 6\) or \(8\) the number is even. If it is \(1, 3, 5, 7\) or \(9\) it is odd. So \(72\,914\) is even because it ends in \(4\).

What do you get when you add two odd numbers?

You always get an even number. For example \(3 + 9 = 12\) and \(11 + 5 = 16\). The two odd numbers pair up with nothing left over.

Is 1 odd or even?

One is odd. It ends in \(1\), and a single counter cannot make a pair, so one is left over.

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