Extending Patterns and Relations

1. What it means to extend a pattern

A pattern is a set of shapes or numbers that follows a rule. Each member is called a term. To extend a pattern means to continue it — to work out the terms that come next — by using the same rule that made the first few terms.

The best first question to ask about any pattern is: how does each term differ from the one before it? Subtract each term from the next one. If the answer is the same every time, you have found the pattern's difference, and extending it is easy: just keep adding (or subtracting) that difference.

Patterns can also go down. A family water tank holds 200 litres and 15 litres are used each day: 200, 185, 170, 155, … Here each term is 15 less than the one before, so the difference is −15 and the next term is 155 − 15 = 140.

2. Growing patterns made with matchsticks

Patterns made with real objects such as matchsticks, bottle tops or seeds are called symbolic or concrete patterns. Look at the row of squares below. Pattern 1 is one square made from 4 sticks. To make Pattern 2 we do not build a brand-new square; we share one side with the first square, so we only add 3 more sticks.

▶ Watch how each new pattern adds one more square using only 3 new sticks, because one side is shared. Notice the counts: 4, 7, 10, 13, 16.

We record the counts in a table of values. The top row (pattern number) is the domain and the bottom row (number of sticks) is the co-domain.

To draw the fifth pattern, copy the fourth pattern (4 squares in a row) and add one more square on the right using 3 sticks: one on top, one at the bottom and one closing the side. Count to check: 13 + 3 = 16 sticks.

3. Extending relations in a table

A relation is often given as an incomplete table. Your job is to find the change from one co-domain value to the next, fill the gaps, and keep going.

1. 1Find the change between known neighbours: 7 − 4 = 3 and 10 − 7 = 3. So each co-domain value is 3 more than the one before.
2. 2Fill the gaps in order: 10 + 3 = 13, then 16 is given, 16 + 3 = 19, 19 + 3 = 22.
3. 3Check against a given value: 13 + 3 = 16 ✓ — the rule agrees with the table.
4. 4Keep extending to answer questions further along: domain 8 → 25, domain 9 → 28.

4. Patterns that are not just 'add the same number'

Most patterns in this sub-strand change by the same amount each time. But always check — some patterns grow differently. For example 2, 4, 8, 16, … doubles each time (×2), so the next term is 32, not 18. And 1, 4, 9, 16, … are square numbers (1 × 1, 2 × 2, 3 × 3, 4 × 4), so the next term is 5 × 5 = 25.

• Same difference (add or subtract the same number): 3, 8, 13, 18 → +5 each time.
• Same multiplier (multiply by the same number): 3, 6, 12, 24 → ×2 each time.
• Growing difference: 1, 4, 9, 16 → +3, +5, +7, so the next difference is +9.

Once you know how each term differs from the one before, you can extend the pattern as far as you like, and you can explain your answer to anyone — which is exactly what a BECE examiner wants to see.

Worked examples

Try each one yourself first, then open the solution and compare every step.