Number Patterns and Sequences for PSLE Mathematics is the eduKateSingapore guide to recognising how a quantity changes from term to term, stage to stage or figure to figure. Pattern questions reward more than spotting a visual repetition: pupils must identify a rule, test it, extend it and sometimes connect the term number to the value directly.
Students searching for PSLE number patterns, sequences, figure patterns, growing patterns or nth-term Primary Mathematics often stop after finding “add 3 each time”. That works for simple arithmetic sequences but not for alternating, multiplicative, layered or shape-growth patterns. The stronger method is observe → record → compare → hypothesise → test → generalise.
This page is the canonical Number Patterns and Sequences owner under the PSLE Mathematics Heuristics hub.
Quick answer: the six pattern questions
- What changes?
- By how much?
- Is the change additive, multiplicative, alternating or layered?
- Does the same rule work for at least three transitions?
- Can I predict the next term?
- Can I connect term number directly to term value?
1. Constant-difference sequences
Sequence: 7, 12, 17, 22, …
Difference is +5 each time.
Term n = 7 + (n − 1) × 5.
At Primary level, pupils may use repeated addition or a table before formalising the direct rule.
2. Constant-ratio sequences
Sequence: 3, 6, 12, 24, …
Each term is ×2.
The important distinction is that the difference is not constant. Always test multiplicative change when additive differences do not stabilise.
3. Alternating patterns
Sequence: 5, 8, 6, 9, 7, 10, …
There are two interleaved sequences: 5,6,7,… and 8,9,10,…
Another way to describe it is +3, −2, +3, −2. Both descriptions are useful; choose the one that makes the next step clearest.
4. Growing-difference patterns
Sequence: 2, 5, 9, 14, 20, …
Differences are +3, +4, +5, +6. The first differences themselves follow a pattern.
When first differences are not constant, inspect the differences as a new sequence.
5. Square and rectangular growth
Sequence: 1, 4, 9, 16, 25, …
These are square numbers: 1², 2², 3², 4², 5².
A figure pattern may show growing squares or arrangements where each stage adds an odd number of objects.
6. Triangular growth
Sequence: 1, 3, 6, 10, 15, …
Differences are +2, +3, +4, +5. Each new stage adds one more object than the previous stage.
Visual arrangements often make this structure easier to see than the number list alone.
7. Repeating cycles
Pattern: red, blue, green, red, blue, green, …
Cycle length = 3. To find the colour at position 50, divide 50 by 3. Remainder 2 means the second item in the cycle: blue.
Cycle problems are really remainder problems.
8. Figure patterns
For growing shapes, create a table with:
- stage number
- total objects
- objects added from previous stage
- fixed part
- repeating group
- direct rule if visible
The table separates what grows from what stays fixed.
Worked figure example
Pattern: Stage 1 has 5 tiles. Each new stage adds 3 tiles.
Stage 1 = 5.
Stage 2 = 8.
Stage 3 = 11.
Stage 4 = 14.
Direct rule: 5 + 3(n − 1), or equivalently 3n + 2.
The +2 is the fixed offset after accounting for 3 tiles per stage.
From recursive rule to direct rule
A recursive rule tells how to get the next term: “add 4”. A direct rule tells the value of any term without generating all previous terms.
For 6,10,14,18,… the recursive rule is +4. The direct rule is 6 + 4(n − 1) = 4n + 2.
Direct rules become especially useful when the question asks for the 50th or 100th stage.
Pattern tables prevent visual guessing
Visual patterns can trick the eye. Record numbers. If the figure seems to “grow by two sides”, translate that into actual counts. A pattern claim is reliable only if it survives the table.
Common pattern errors
- finds one difference and assumes it stays constant
- does not test the rule across enough terms
- mixes term number with term value
- counts a shared edge or shared object twice in a figure
- uses repeated addition when a direct rule is required
- ignores an alternating pattern
- forgets remainder 0 means the last item in a repeating cycle
The term-number check
Always write two columns: term number and term value. Many pupils accidentally substitute the value into the position or vice versa. Keeping the columns explicit prevents that confusion.
A mixed practice set
Practice 1
Pattern: 4, 9, 14, 19, …
Rule: Add 5. Direct rule 5n − 1.
Practice 2
Pattern: 2, 6, 18, 54, …
Rule: Multiply by 3.
Practice 3
Pattern: 1, 4, 2, 5, 3, 6, …
Rule: Two interleaved sequences or +3,−2 cycle.
Practice 4
Pattern: 3, 7, 12, 18, 25, …
Rule: Differences +4,+5,+6,+7.
Practice 5
Pattern: A,B,C,D repeating; position 37
Rule: 37 ÷ 4 leaves remainder 1 → A.
How pattern work prepares for algebra
Number patterns teach pupils to move from examples to general relationships. The symbol n is simply a placeholder for the term number. This is an important bridge into Secondary algebra, where relationships are expressed directly rather than generated one step at a time.
Frequently asked questions
What if the differences are not constant?
Inspect the differences themselves, test multiplication, or look for alternating/interleaved rules.
How do I solve large term numbers?
Find a direct rule instead of generating every term.
How do repeating cycles work?
Use division and remainders to locate the position within the cycle.
Should I trust the visual pattern?
Translate it into a number table and verify the rule.
Where does this sit in Atlas?
This is the canonical Number Patterns and Sequences owner under PSLE Mathematics Heuristics.
The final Patterns rule
Do not guess the next term from appearance alone. Record the values, compare changes, test the rule across several stages, then generalise only after the pattern survives the evidence.