ACARA v9 CONTENT DESCRIPTION “recognise, continue and create patterns formed from numbers, and describe the rule that generates a pattern, including patterns that increase or decrease by a constant amount”
A number pattern is not a random list; it is built by a rule applied over and over. The pattern 3, 5, 7, 9 grows by adding two each time, and that constant step is its rule. Year 4 work is to see the rule behind a pattern, because once the rule is known the pattern holds no surprises: it can be continued forever, and any term can be worked out. Patterns can grow by a constant amount or shrink by one, and they can repeat a core over and over. In every case the goal is the same — find the rule, and the whole pattern is yours.
A growing pattern
A growing pattern adds the same amount each step. That step is its rule.
Start at 3 and add 2 each step. Reveal the next term.
Growing by a constant step
The most common pattern grows by adding the same amount each step. Starting at three and adding two gives 3, 5, 7, 9, and the rule is simply add two. Seeing the pattern build, one constant step at a time, makes the rule obvious: it is the gap between any term and the next. This is the skip-counting of earlier years recognised as a rule, and it is the foundation for everything else in the unit. A pattern that grows by a constant amount is the simplest kind, and the constant step is the whole of its rule.
Find the rule
The rule of a growing pattern is the constant gap between terms.
What is added between each pair of terms? That constant gap is the rule.
Reading the rule from the gaps
To find a pattern's rule, look at the gap between consecutive terms. In 4, 7, 10, 13 the gap from one term to the next is always three, so the rule is add three. The test is to subtract each term from the one after it: if the gap is the same all along, that gap is the rule. This works for shrinking patterns too — 20, 17, 14, 11 has a constant gap of three going down, so its rule is subtract three. Finding the rule is the key skill, because it turns a list of numbers into something a child can extend and predict.
The next term
Find the rule, then apply it once more to get the next term.
Work out the rule, then use it to find the next term in the pattern.
Continuing a pattern
Once the rule is found, continuing the pattern is easy: apply the rule one more time to the last term. If 3, 6, 9, 12 grows by three, the next term is 12 plus 3, which is 15. Continuing a pattern is the most natural test of whether a child has really found the rule, because a wrong rule gives a wrong next term. This is also where patterns become useful — predicting what comes next, whether it is the next number, the next tile, or the next step in a sequence, all rest on knowing and applying the rule.
The repeating core
A repeating pattern is built from a core unit that repeats.
A repeating pattern is made of a core that repeats. Find the part that repeats.
Patterns that repeat
Not all patterns grow; some repeat. A repeating pattern is built from a core unit that repeats over and over, like red-blue-red-blue, whose core is red-blue, two long. Finding the core is the rule for a repeating pattern: once you know the part that repeats and how long it is, you can say what colour or shape sits at any position. Repeating patterns appear in borders, music, days of the week and tiling, and the skill is the same as for growing patterns — find the rule, here the repeating core, and the whole pattern is determined.
Rule to a term
A position rule finds any term directly, without listing them all.
Use the rule to find the term at position 5.
A rule that jumps to any term
The most powerful kind of rule links a term to its position directly, so you can jump straight to the hundredth term without listing the first ninety-nine. A rule like multiply the position by two gives the term at position five as ten, at once. Some rules combine steps, like times two then add one. A position rule is stronger than a step rule because it does not need the previous term — just the position. With patterns continued by their step, repeating cores found, and position rules used to jump to any term, a child can describe, continue and create number patterns, the heart of algebraic thinking, and the groundwork for the arrays and multiplicative patterns of the next Algebra unit.
Position and term
A table pairs each position with its term, exposing the rule between them.
A table lines up each position with its term. Reveal each term and watch the rule appear.
Quick self-check
1. In the pattern 4, 7, 10, 13, ... the rule is...
2. What is the next term after 7, 11, 15, 19?
3. A repeating pattern red, blue, red, blue, ... has a core that is...
4. A pattern follows the rule "multiply the position by 2". The term at position 5 is...
5. In a table, position 1 gives 4, position 2 gives 7, position 3 gives 10. The rule is...