Practice: Rule-Finder

Look at the geometric patterns and find the algebraic rule that describes them.

Pattern A: A Growing Row of Squares

Stage 1: □
Stage 2: □□
Stage 3: □□□

1. Complete the table:

Stage (n)	1	2	3	4	5
Squares	1	2	3	?	?

2. What is the rule? Rule: _________

Pattern B: L-Shapes

Stage 1: □

Stage 2: □□
    □

Stage 3: □□□
        □
        □

1. Complete the table:

Stage (n)	1	2	3	4	5
Squares	1	3	5	?	?

2. What is the rule? Rule: _________

🔑 Kaiako answer key

Pattern A — a growing row of squares

Table: Stage 4 → 4 squares; Stage 5 → 5 squares.
Rule: n — the number of squares is the same as the stage number.

Pattern B — L-Shapes

Table: Stage 4 → 7 squares; Stage 5 → 9 squares.
Rule: 2n − 1. Check it: 2(1) − 1 = 1, 2(2) − 1 = 3, 2(3) − 1 = 5 — all three match the drawings.

Watch for: Pattern A is deliberately the easy one, and some ākonga will distrust an answer as plain as “n”. Accept “it goes up by one” as the words and then push for the symbol. For Pattern B the common wrong answer is 2n, which gives 2, 4, 6 — have them test it against Stage 1 and see it fail.

📋 Teacher Planning Snapshot

Ngā Whāinga Ako — Learning Intentions

Practice spanning Lessons 1 and 3: recovering the rule from a table of values.

Ngā Paearu Angitū — Success Criteria

✅ I can complete a table of values from a drawn pattern.
✅ I can write the rule as an expression in n.
✅ I can test my rule against Stage 1 before I commit to it.

Differentiation & Inclusion

Ākonga who cannot see the rule should build the next stage physically with counters before writing anything. Pattern A is deliberately easy and can be used as the worked example for the whole class. For extension, ask for the 100th stage rather than the 5th.