Patterns — Grade 3 Mathematics
In Grade 3, patterns move from simple skip-counting to two-step function machines and more complex sequences. We also explore square number and triangular number patterns.
Function Machines
A two-step machine applies two operations in sequence.
Two-step machine: ×3 then +4
5
19
5 × 3 = 15; 15 + 4 = 19
| Input | × 3 | + 4 (Output) |
|---|---|---|
| 2 | 6 | 10 |
| 5 | 15 | 19 |
| 7 | 21 | 25 |
| 10 | 30 | ? |
10 × 3 + 4 = 34
Special Number Patterns
Triangular numbers: 1, 3, 6, 10, 15, 21 … (each adds one more than the last: +1, +2, +3, +4 …)
Worked Examples
A machine doubles the input then subtracts 3. Complete: input=6, input=9, input=12.
- 6 × 2 − 3 = 12 − 3 = 9
- 9 × 2 − 3 = 18 − 3 = 15
- 12 × 2 − 3 = 24 − 3 = 21
Find the pattern and fill in the missing terms: 4, 9, 16, 25, ___, 49, ___
- 4 = 2², 9 = 3², 16 = 4², 25 = 5² → pattern: square numbers
- Next: 6² = 36 and 8² = 64
A cinema has rows. Row 1: 3 seats, Row 2: 5 seats, Row 3: 7 seats, Row 4: 9 seats. How many seats in Row 8?
- Pattern: 3, 5, 7, 9 … rule: +2 each row
- Row 8: 3 + 7 × 2 = 3 + 14 = 17 seats
- Or: nth row = 2n + 1 → Row 8 = 2(8)+1 = 17
Practice Questions
1. A machine multiplies by 4 then adds 2. What is the output for input = 7?
2. What are the next two square numbers after 25?
3. Find the missing number: 1, 3, 6, 10, ___, 21
4. A machine halves then adds 10. Output = 17. What was the input?
5. The pattern of tiles: Row 1=5, Row 2=8, Row 3=11. How many in Row 10?
Key Points to Remember
- A two-step function machine applies operations in order, left to right.
- To reverse a function machine, use inverse operations in reverse order.
- Square numbers: 1, 4, 9, 16 … gaps increase by odd numbers: +3, +5, +7 …
- Triangular numbers: 1, 3, 6, 10 … gaps increase by consecutive integers.
- Always verify the rule works for every term in the sequence.