Addition & subtraction
Odd & Even Numbers
Odd and even numbers follow neat patterns when you add or subtract them. A rule that works for every example is called a generalisation. Let's find some and test them.
Odd and even dots
An even number of dots can be split into two equal rows to make a rectangle. An odd number always has 'a bit left over' when you divide it by 2. Put two matching odd L-shapes together and they make an even rectangle — that shows odd + odd = even.
Adding odd and even numbers
There are three patterns to remember: odd + odd = even (3 + 3 = 6), even + even = even (4 + 6 = 10), and even + odd = odd (4 + 3 = 7). Add three odd numbers and you get odd, because odd + odd = even, then even + odd = odd.
Subtracting odd and even numbers
The difference between two even numbers is even (8 - 4 = 4). The difference between two odd numbers is also even (9 - 5 = 4). So if someone says 'the difference between two odd numbers is odd', you can prove them wrong.
Generalisations and counter-examples
A generalisation is a statement that is true for all examples, like 'odd + odd = even'. To show a statement is wrong, you only need one counter-example. For 'two odd numbers add to an odd number', the example 3 + 5 = 8 breaks the rule, because 8 is even.
Worked examples
Testing odd + odd
- Try 3 + 3 = 6, 5 + 5 = 10 and 7 + 7 = 14.
- Each answer is even.
- So odd + odd = even.
Answer: The sum of two odd numbers is always even.
Adding three odd numbers
- Try 1 + 3 + 5 = 9 and 11 + 23 + 35 = 69.
- odd + odd = even, then even + odd = odd.
- Both answers are odd.
Answer: The sum of three odd numbers is always odd.
Difference of two even numbers
- Try 8 - 4 = 4 and 10 - 6 = 4.
- Each answer is even.
- So even - even = even.
Answer: The difference between two even numbers is always even.
Using a counter-example
- Hassan says two odd numbers add to an odd number.
- Try 3 + 5 = 8.
- 8 is even, which breaks his rule.
Answer: 3 + 5 = 8 is a counter-example, so Hassan is wrong.
Where you see this in real life
- Pairing up socks: an even number pairs perfectly, an odd number leaves one over.
- Sharing sweets equally between two friends.
- Splitting a team into two equal halves.
- Checking if a total could be right using an odd-or-even rule.
Key words
- Even number
- A whole number that splits into two equal parts, like 2, 4 or 6.
- Odd number
- A whole number with one left over when split in two, like 1, 3 or 5.
- Generalisation
- A statement that is true for all examples, such as 'odd + odd = even'.
- Counter-example
- One example that shows a statement is not true.
- Difference
- The result of subtracting one number from another.