5.1 Number & Shape Patterns

Patterns are the very first algebra skill ICAS tests, and they appear right from the Introductory paper (Year 2) through Paper A (Year 3) and beyond. The official skills list for these papers includes skip counting, place value, and simple linear number and shape patterns — and questions about them reward careful looking, not fast guessing. Because ICAS questions get harder as you move through the paper, a pattern question near the start might just ask for the next number, while one near the end might ask you to describe the rule in words or find a term much further along.

What Is a Pattern?

A pattern is a list of numbers, shapes or pictures that follows a rule. The rule tells you how to get from each term to the next one. Your job in an ICAS question is usually one of three things:

• Find the next term — work out what comes immediately after the last one shown
• Find a later term — work out a term several steps ahead
• Describe the rule — say in words what the pattern is doing

The golden rule: a pattern rule must work for every step you can see, not just the first one. If 3, 6, 9 … looks like 'add 3', check: 3 + 3 = 6 ✓ and 6 + 3 = 9 ✓. Only then trust it.

Skip-Counting Patterns

Skip counting means counting forwards (or backwards) in equal jumps. It is the most common number pattern in the early papers.

Notice the last pattern: not every pattern adds the same amount. Always check the gap between two different pairs of terms. If the gaps are the same (like +5, +5, +5), it is a simple adding pattern. If the gaps grow (like +1, +2, +4), the rule is different — doubling, or adding one more each time.

Worked example: Find the missing number: 27, 24, 21, __, 15.

1. Find the gap: 27 − 24 = 3, and 24 − 21 = 3. The rule is 'subtract 3'.
2. Apply it: 21 − 3 = 18.
3. Check forwards: 18 − 3 = 15 ✓ — it matches the term after the gap, so 18 is correct.

That final check is a powerful habit. ICAS distractors are often the answer you get if you add instead of subtract (24) or subtract the wrong amount (20).

Growing Shape Patterns

Shape patterns use matchsticks, tiles, dots or blocks. Figure 1 might be a triangle made of 3 matchsticks, Figure 2 two triangles made of 5 matchsticks, Figure 3 three triangles made of 7 matchsticks.

The smart move is to turn the picture into numbers. Count the pieces in each figure and write the number pattern underneath:

Now it is just a number pattern: 3, 5, 7, … — add 2 each time, so Figure 4 needs 9 matchsticks.

Finding a later term

Suppose the question asks how many matchsticks Figure 10 needs. You could keep adding 2 nine times, but there is a faster way. Each new figure adds 2 matchsticks. From Figure 1 to Figure 10 there are 9 steps, so: 3 + (9 × 2) = 3 + 18 = 21 matchsticks. Counting the steps carefully matters — from Figure 1 to Figure 10 is 9 steps, not 10. A very common ICAS trap answer is 3 + (10 × 2) = 23.

Describing the rule in words

Paper A sometimes asks 'Which sentence describes this pattern?' Write your own description first, before reading the options: 'Start at 3 and add 2 each time.' Then match it to the closest option. Good descriptions always include where to start and what to do each step.

Patterns in Tables

Sometimes the pattern is given in a table instead of a list:

Read the table horizontally: 4, 8, 12, 16 … is counting in 4s, so term 5 is 20. Tables like this also appear with tally marks in the early papers — count the tallies first (remember a gate of five, 卌, counts as 5), then look for the pattern in the totals.

'Which Shape Comes Next?' Reasoning

For picture sequences, ask yourself three questions:

1. What changes? The number of dots, the shading, the direction an arrow points, the size.
2. What stays the same? Often the outside shape or the starting position never changes.
3. Does the change repeat or grow? A pattern might rotate a shape 90° each step (repeating every four terms), or add one more dot each step (growing).

Worked example: A flag points up, right, down, left, up, right, … Which way does it point in position 9? The cycle is 4 long. Position 9 is two full cycles (8) plus 1 more, so it matches position 1: up. Dividing by the cycle length and looking at the remainder is the reliable method — much safer than drawing all nine flags.

Common Traps

• Only checking the first gap. 2, 4, 8, … is not 'add 2' — check the second gap too.
• Ignoring direction. A pattern can decrease (subtract) or alternate (add, subtract, add, subtract).
• Miscounting figures. In matchstick patterns, shared sides mean you cannot just multiply one figure's count by the number of figures.
• Guessing from the picture. Always convert shapes to numbers; the numbers never lie about the rule.

Since personal calculators are not allowed and the on-screen calculator only appears when needed, these questions expect mental skip counting — practise counting in 2s, 3s, 4s, 5s and 10s forwards and backwards until it is automatic.