A pattern is not just a string of numbers to memorise; it is a rule for making them. Tell the rule and you can build the pattern as far as you like, and read backwards to check where it began. Patterns grow in two ways that turn out to be the same idea: a shape can grow visually, gaining a fixed group of squares each stage, and a sequence can grow numerically, gaining a fixed amount each term. The squares you can see and the numbers you count rise together, step for step, which is why a growing picture and a growing list are two views of one rule.
A pattern is a rule, not a list
Add the same group each stage and a shape grows by a steady step.
Each new row adds 3 squares, so the counts climb 3, 6, 9, 12 — a constant step of 3 is the rule.
The step between the terms
The simplest rule says what to add to get from one term to the next. When every jump along the line is the same size, the pattern has a constant step, and reading that single jump is enough to continue forever. The sequence one, three, five, seven jumps by two each time, so the rule is add two and the next term must be nine. This is a recursive rule: it tells you how to take one more step from where you are. Spotting the constant step is the first move in working with any growing pattern.
The step between the terms
Equal jumps mean a constant rule. Read the size of one jump.
The numbers 1, 3, 5, 7, 9 climb by the same jump each time — the rule is add 2.
Patterns can step by fractions
A rule does not have to step by whole numbers. A pattern can climb by a half, by a quarter or by a tenth, and the rational numbers involved sit on the very same line as the whole ones, just at finer marks. The pattern a half, one, three halves, two steps by a half each time, and because a half is also 0.5, those marks can wear two coats — fraction or decimal — without moving. Reading rational steps this way means a pattern of fractions is no harder than a pattern of whole numbers; only the size of the step has changed.
Patterns can step by fractions
A pattern need not jump by whole numbers. Step along in halves.
0.0 — each step adds a half. The same marks read as decimals; a half and 0.5 are one point in two coats.
Position can tell you the term
A recursive rule is patient: to reach the twentieth term it walks every step in between. A position rule is faster, because it ties each term directly to its position. In the pattern four, eight, twelve, the term is always four times its position, so the fifth term is four times five, twenty, found without listing the ones before it. This leap — from a rule about the step to a rule about the position — is the heart of the unit. It lets you predict a far-off term in a single calculation, and it is the first shape of an algebraic formula.
Position can tell you the term
A position rule jumps straight to any term without listing the ones before it.
Position 1 gives term 4, because the rule is 4 times the position — no need to list every step to reach it.
The rule sets the speed of growth
Two patterns can start at the same number and end up wildly apart, because the rule decides the speed. A rule that adds three each stage climbs steadily, in even steps; a rule that doubles each stage starts level, then races away as each term feeds the next. Seeing the two side by side makes the difference between adding and multiplying concrete: addition grows by a fixed amount, multiplication grows by a fixed factor, and the second soon overtakes the first. The rule, not the starting number, governs the shape of the growth.
The rule sets the speed of growth
Two patterns start together. The rule decides how steeply each one climbs.
By stage 1: adding 3 reaches 3, doubling reaches 3. Same start, but the rule sets how fast the pattern climbs.
Use the rule to look ahead
The point of finding a rule is to use it. Once you know how a pattern grows, you can predict the next term before it appears and then check the prediction against the rule. Patterns three, six, twelve might tempt you toward eighteen by adding six, but testing the rule shows the step is a doubling, so the next term is twenty-four. Predicting and checking is how mathematicians treat any sequence: propose the rule, extend it, and confirm it holds. A rule that survives the check is one you can trust for terms you have not yet drawn.
Use the rule to look ahead
Spot the rule, then predict the next term before you reveal it.
The pattern 2, 5, 8 grows by a rule. Which value comes next?
From patterns to algebra
Growing patterns are early algebra in disguise. A visual pattern, its number sequence, the constant step and the position rule are all describing one relationship between position and term — the same relationship a formula captures with a letter. When the position rule term equals four times position becomes term equals four times n, the pattern has turned into algebra. From here the unknown position becomes a variable, and the rules that grow these patterns become the equations of the years ahead.
Quick self-check
1. A tile pattern grows 3, 6, 9, 12, ... Which rule generates the next term?
2. A pattern starts at 2 and follows the rule add 0.5. What is the third term?
3. In the pattern 1/2, 1, 3/2, 2, ... what is added at each step?
4. A pattern begins at 4 and adds 4 each step: 4, 8, 12, ... What is the 5th term?
5. Which rule generates the pattern 10, 8, 6, 4, ...?