Numbers & the number system
Counting and sequences
A sequence is a list of numbers that follow a pattern. When you know the pattern, you can count on or back to find the next number every time.
Counting on and back in steps
You can count on or back in steps of tens, hundreds and thousands, starting from any number. Only the digits that change need to change. Count on 100 from 946 and you get 1046. Count back 1000 from 9999 and you get 8999.
The term-to-term rule
Each number in a sequence is called a term. The term-to-term rule tells you how to get from one term to the next, such as 'add 3' or 'subtract 5'. For 185, 180, 175, ... the rule is 'subtract 5'. Once you know the rule you can extend the sequence.
Linear and non-linear sequences
In a linear sequence you add or subtract the SAME amount each time, so the gaps are equal — like 4, 9, 14, 19 (add 5). In a non-linear sequence the gaps change. Multiply by three (2, 6, 18, 54) and 'add one more each time' (2, 3, 5, 8) are both non-linear.
Counting back through zero
When you count back in ones past zero you reach negative numbers. Tara counts 3, 2, 1, 0, and the next number is −1. Numbers can carry on below zero: −1, −2, −3, and so on.
Square numbers
Some spatial patterns make square numbers. A 1 by 1 square has 1 dot, a 2 by 2 square has 4 dots, a 3 by 3 square has 9 dots. The square numbers are 1, 4, 9, 16, 25, ... — you get them by multiplying a number by itself.
Worked examples
Counting on in hundreds
- Mia starts at 946 and counts on in steps of 100.
- Counting on 100 changes the hundreds: 946 + 100.
- This crosses into the thousands.
Answer: The next number is 1046.
Finding the fifth term
- Carlos starts at 8 with the rule 'subtract 2'.
- Subtract 2 each time until you have five terms.
- 8, 6, 4, 2, 0.
Answer: The fifth term is 0.
Linear or non-linear?
- Look at 2, 6, 18 with the rule 'multiply by three'.
- The gaps are 4 then 12 — they are not equal.
- Unequal gaps mean the sequence is non-linear.
Answer: The sequence 2, 6, 18, 54 is non-linear.
Counting back past zero
- Tara counts back in ones: 3, 2, 1, 0.
- Count back one more from 0.
- You go below zero to a negative number.
Answer: The next number is −1.
The next square number
- The square numbers are 1, 4, 9, 16, ...
- The next one comes from 5 × 5.
- 5 × 5 = 25.
Answer: The next square number is 25.
Where you see this in real life
- Saving the same amount of money each week makes a linear sequence.
- Counting temperatures back through zero on a very cold day.
- Seeing square patterns in floor tiles laid out in a square.
Key words
- Sequence
- A list of numbers that follow a pattern.
- Term
- One of the numbers in a sequence.
- Term-to-term rule
- How to get from one term to the next, such as 'add 3'.
- Linear sequence
- A sequence where you add or subtract the same amount each time.
- Non-linear sequence
- A sequence where the gaps between terms change.
- Square number
- A number you get by multiplying a whole number by itself, like 9 = 3 × 3.