Patterns are at the heart of mathematics. Recognising regularities in numbers and shapes, and expressing them as rules, allows us to predict, extend, and solve problems efficiently. This area of Foundation Mathematics connects number skills with algebraic thinking.
KEY TAKEAWAY: A pattern is a rule that repeats. Once you find the rule, you can find any term in the sequence — without listing every value in between.
A pattern is a sequence or arrangement where something repeats in a predictable way.
Types of patterns:
- Repeating patterns: The same unit repeats — ABABAB…
- Growing patterns: Each step increases by a consistent amount or factor
- Number patterns: A sequence of numbers following a rule
Each term is obtained by adding (or subtracting) the same value — the common difference $d$.
$\$3, 7, 11, 15, 19, \ldots \quad d = +4$$
General term: $t_n = t_1 + (n-1) \times d$
Worked Example:
Find the 10th term of \$5, 8, 11, 14, \ldots$
$$d = 3, \quad t_1 = 5$$
$$t_{10} = 5 + (10-1) \times 3 = 5 + 27 = 32$$
Each term is obtained by multiplying by the same value — the common ratio $r$.
$\$2, 6, 18, 54, \ldots \quad r = \times 3$$
| Pattern | Example | Rule |
|---|---|---|
| Square numbers | \$1, 4, 9, 16, 25, \ldots$ | $n^2$ |
| Triangular numbers | \$1, 3, 6, 10, 15, \ldots$ | $\frac{n(n+1)}{2}$ |
| Fibonacci-type | \$1, 1, 2, 3, 5, 8, \ldots$ | Each term = sum of two before |
EXAM TIP: To find the rule for a number sequence: (1) check if the differences are constant (arithmetic); (2) check if the ratios are constant (geometric); (3) look for other patterns in the differences.
A relationship shows how two quantities depend on each other.
Direct proportion: As one quantity increases, the other increases at the same rate.
$$y = kx \quad \text{(} k \text{ is the constant of proportionality)}$$
Example: Cost of apples at $\$3.50$ per kg:
$$\text{Cost} = 3.50 \times \text{weight}$$
| Weight (kg) | Cost (\$) |
|---|---|
| 1 | 3.50 |
| 2 | 7.00 |
| 3 | 10.50 |
| 4 | 14.00 |
| Representation | Example |
|---|---|
| Table of values | As above |
| Rule (formula) | $\text{Cost} = 3.50 \times w$ |
| Graph | Straight line through origin |
| Words | “Cost increases by $\$3.50$ for each extra kilogram” |
Worked Example:
A gardener plants 4 seedlings in week 1. Each week she plants 6 more than the previous week. How many seedlings does she plant in week 8?
$$t_1 = 4, \quad d = 6$$
$$t_8 = 4 + (8-1) \times 6 = 4 + 42 = 46 \text{ seedlings}$$
Total planted over 8 weeks:
$$S = \frac{n}{2}(t_1 + t_n) = \frac{8}{2}(4 + 46) = 4 \times 50 = 200 \text{ seedlings}$$
COMMON MISTAKE: Confusing the value of a term with the position (term number). In $t_n = 4 + (n-1) \times 6$, the variable $n$ is the position, not the value.
REMEMBER: Patterns and relationships connect number, algebra and graphs. A table of values, a formula, and a straight-line graph can all describe the same relationship — being able to move between representations is a key Foundation Mathematics skill.
