Linear relationships:
how to find the rule
A pattern tells you what changes. A rule lets you find any value without writing out every step. Learn how to move from a matchstick pattern to a table, an equation and a graph.
Worked examples adapted from BEAM’s Year 8 Linear Relationships lesson. Published 24 October 2026.
1. Start with what changes
Imagine a row of joined squares made from matches. One square uses four matches. Each new square shares one side with the previous square, so it adds only three matches.
Let n be the number of squares and m the number of matches. Put the two quantities in a table:
| Squares, n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Matches, m | 4 | 7 | 10 | 13 | 16 |
Every increase of 1 in n gives an increase of 3 in m. This constant rate of change is the clue that the relationship is linear. Always compare equal changes in the input before deciding that the output has a constant step.
2. Turn the pattern into a rule
The step is 3, so begin with m = 3n. But at n = 1, that gives 3 matches. The table says 4. We need one extra match:
m = 3n + 1
There is a physical reason for that extra 1: count the leftmost vertical match once, then add three matches for every square. The rule describes how the whole row is built.
Check more than the first row. At n = 2, the rule gives 3 × 2 + 1 = 7. At n = 5, it gives 3 × 5 + 1 = 16. Both agree with the table.
How many matches for 25 squares?
This worked example comes from our lesson. Substitute 25 for n:
m = 3 × 25 + 1 = 76
So 25 joined squares need 76 matches. You do not need to draw all 25 squares or extend the table one row at a time.
A method you can reuse
- Identify the input and output, including their units.
- Find the change in output for each increase of 1 in the input.
- Multiply the input by that rate.
- Use a known pair of values to find what must be added or subtracted.
- Check the rule against the other values.
In general, a linear rule has the form y = ax + b. Here a is the rate of change and b is the value predicted when x = 0. The first value listed in a table is only b if its input is 0.
3. Check your understanding
Our lesson also uses a row of triangles: one triangle uses 3 matches, and each new triangle adds 2 more. Before opening the explanation, think about why the rule needs more than just “multiply by 2”.
Show the triangle explanation
The first five match counts are 3, 5, 7, 9 and 11. The rate is 2 matches per triangle. At n = 1, multiplying by 2 gives 2, so add 1. The rule is m = 2n + 1. Checking n = 5 gives 11 matches.
Two mistakes to watch for
“It adds 3, so the rule is m = n + 3.” In the square pattern, adding 3 describes the change from one output to the next. The input must be multiplied by 3. The incorrect rule gives 5 matches for two squares, but the table gives 7.
“The first value is 4, so add 4.” The square table starts at n = 1, not 0. The rule 3n + 4 gives 7 for the first square. Substitute a known input to check the constant term.
What if the inputs skip numbers?
Our lesson includes the pairs (2, 11), (5, 17) and (9, 25). The outputs rise by 6 and then 8, but the inputs also have different gaps. Divide the output change by the input change:
(17 − 11) ÷ (5 − 2) = 2
(25 − 17) ÷ (9 − 5) = 2
The rate is 2 in both cases. Use (2, 11): 11 = 2 × 2 + b, so b = 7. The rule is v = 2n + 7. At position 14, the value is 35. Unequal gaps in the output do not automatically mean a relationship is non-linear.
4. Connect the rule to a graph
Put the number of squares on the horizontal axis and the number of matches on the vertical axis. Plot (1, 4), (2, 7), (3, 10), (4, 13) and (5, 16). The points line up because each step of 1 to the right corresponds to 3 upwards.
The equation m = 3n + 1 describes a straight line. For actual rows of squares, only positive whole-number inputs make sense. There is no row of 2.5 complete squares in this model. Keep the plotted points discrete when representing the physical pattern.
The extended line crosses the vertical axis at 1. That is the equation’s intercept, not a claim that an empty row of squares contains one match. A mathematical rule and the situation it models need to be interpreted together.
Where to go next
If finding the rate is straightforward but writing the rule is difficult, practise using one known pair to find the constant term. If the rule is correct but the answer is wrong, write the substitution step before calculating.
At BEAM, this topic sits within our Year 8 work on tables, rules and graphs. Explore Year 8 Maths lessons