1. Describe what changes
Imagine a row of connected squares made from matchsticks. One square uses 4 sticks. Each additional square shares one side with the previous square, so it needs only 3 new sticks.
| Number of squares, n | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Matchsticks, M | 4 | 7 | 10 | 13 |
A useful description is: “The number of matchsticks forms a linear sequence with a first term of 4 sticks and a common difference of 3 sticks per additional square.” This states the type, starting value, change and units.
2. Explain the general rule
There are three sticks for each square, plus one extra stick at the end of the row. Therefore:
Another way to see it is to start with 4 sticks for the first square and add 3 for each of the remaining n − 1 squares:
M = 4 + 3(n − 1) = 3n + 1.
This explanation connects the rule to the structure of the diagram. Merely spotting a difference of 3 suggests the rule; explaining the shared sides justifies it for this construction.
3. Test the rule against given terms
Check at least two values already present in the table:
n = 2: M = 3(2) + 1 = 7
“I tested the rule using n = 1 and n = 2. Both results match the numbers of matchsticks in the table.”
Write the substitution and the conclusion. The reader should be able to see both the calculation and what you learned from it.
4. Verify independently predicted terms
First predict the next terms using the construction: add 3 sticks to 13 to get 16 for 5 squares, then add 3 again to get 19 for 6 squares. Now check these predictions using the formula.
n = 6: M = 3(6) + 1 = 19
“I verified the rule using n = 5 and n = 6. Both results match my independently predicted values of 16 and 19 matchsticks.”
Your turn: investigate a claim
A student says: “Ten connected squares need 40 matchsticks because each square has four sides.” Explain the mistake and find the correct number.
Reveal the explanation
The student counts every square separately and counts shared sides twice. There are 9 shared sides, so 40 − 9 = 31 matchsticks. The rule confirms this: M = 3(10) + 1 = 31.
Extension: if a row uses 76 matchsticks, solve 3n + 1 = 76. This gives n = 25 squares. Check the answer in the original rule.
These are original teaching examples. Your school’s syllabus and task-specific instructions take priority. Ask about personalised support.