MISSION · Number sense
Patterns and Rules
Spotting what stays the same when everything is changing
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
A pattern is a rule applied over and over, and finding the rule matters far more than guessing the next number.
Skip counting is the first pattern
Counting by 2s, 5s, and 10s is a rule in disguise: add the same amount each time.
\[5,\; 10,\; 15,\; 20,\; 25, \ldots\]Every step adds 5. Once you see that, you can continue forever, jump ahead, and — this is the part that pays off later — recognise the same structure when it turns up as the five times table, as counting coins, or as the minutes on a clock face.
Patterns that add a constant are the beginning of a long road. In secondary school the same idea is called an arithmetic sequence, and later still it becomes the straight-line graph. It’s the same pattern the whole way; only the name changes.
Two kinds of rule
Adding rules grow steadily: \(3, 7, 11, 15\) adds 4 each time.
Multiplying rules grow explosively: \(3, 6, 12, 24\) doubles each time.
To tell them apart, look at the gaps. If the gaps are equal, it’s an adding rule. If the gaps themselves keep growing, try dividing consecutive terms and see whether that is constant.
Not every pattern is one of these two. \(1, 4, 9, 16\) is the square numbers, and \(1, 1, 2, 3, 5, 8\) adds the two previous terms. Part of the skill is checking rather than assuming.
Function machines
A rule can also work on inputs that aren’t in order. Picture a machine: a number goes in, the rule is applied, a number comes out.
| In | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Out | 5 | 8 | 11 | 14 |
The outputs go up by 3 each time, so the rule multiplies by 3 — and \(3 \times 1 = 3\), not 5, so something is added too. The rule is times 3, add 2, and you can check it against every column before trusting it.
That checking step is the whole discipline. A rule that works for the first pair and fails on the third isn’t the rule.
Describing rather than continuing
“What comes next?” is the easy question. “What would the 100th term be?” is the useful one, and it’s answerable only if you’ve written the rule down properly.
For the table above, the 100th output is \(3 \times 100 + 2 = 302\), found without listing anything. Turning a pattern into a rule is what turns counting into mathematics.
Common mix-ups
- Guessing from two terms. Two terms fit endless rules; check with a third and fourth.
- Only describing the step. “Add 3” continues the pattern but won’t reach the 100th term — you need the rule in terms of the position.
- Assuming every pattern adds. Check for multiplying and for stranger rules too.
- Ignoring where the counting starts. A rule beginning at term 1 differs from one beginning at term 0.
Check yourself
What is the rule for 4, 9, 14, 19? What would the 10th term be?
Each term adds 5, starting at 4, so the rule is \(5n - 1\). The 10th term is \(5 \times 10 - 1 = 49\) — and checking, the 1st is \(5 - 1 = 4\). Correct.
LEVEL 3 · REFLECT
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