BTT Mathematics / Primary Mathematics Learning Hub / Visual Sequences

Primary Mathematics: Matchstick Patterns, Figurate Numbers and Visual Sequences | Worked Learning Guide

A visual pattern becomes mathematics when the learner can say what changes, what stays fixed, and how the next figure can be predicted without redrawing everything.

This Primary Mathematics enrichment guide develops growing patterns, matchstick structures, first differences, recursive rules, direct rules, triangular numbers, square numbers, rectangular arrays, staircase patterns, reverse problems and proof-by-structure. It extends the existing Patterns, Early Algebra and Equations guide by focusing on visual growth and figurate structure rather than general equations.

Wait, What? The Picture Is Not the Rule

A learner may correctly draw Figure 4 after seeing Figures 1, 2 and 3 without knowing the mathematical rule. Recognition is useful, but transfer requires more: how many new objects are added each stage, where they are added, whether the growth is constant, and whether a direct expression can predict Figure 50 without drawing forty-nine earlier figures.

1. Describe the Growth Before Counting

Ask three questions:

1. What is present in every figure?
2. What new part is added from one figure to the next?
3. Does the amount added stay constant or itself change?

Worked Example 1: Constant Growth

A sequence contains 4, 7, 10, 13 dots.

Each new figure adds 3 dots. Recursive rule: add 3.

Direct rule for Figure n: 3n + 1.

Worked Example 2: Check the Direct Rule

For n=1, 3(1)+1=4. For n=4, 3(4)+1=13. Testing known cases supports the rule.

2. Matchstick Squares in a Row

One isolated square needs 4 matchsticks. Two joined squares do not need 8 because the shared side is counted once.

Worked Example 3: One, Two and Three Joined Squares

Figure 1: 4 sticks.

Figure 2: add 3 new sticks → 7.

Figure 3: add 3 new sticks → 10.

Rule: Figure n uses 3n+1 sticks.

Why 3n+1?

Start with one vertical side, then each square contributes three new exposed edges: top, bottom and new outer vertical side. The formula records construction, not just pattern matching.

3. Reverse Matchstick Problems

Worked Example 4: Which Figure Uses 31 Sticks?

3n+1=31.

3n=30, so n=10.

Check: ten joined squares need 31 sticks.

Worked Example 5: Is 29 Possible?

3n+1=29 gives 3n=28, not a whole number. Therefore no figure in this pattern uses exactly 29 sticks.

4. Triangular Numbers

Arrange counters in rows of 1, then 2, then 3, and so on. Totals are 1,3,6,10,15,… These are triangular numbers.

Worked Example 6: Fifth Triangular Number

1+2+3+4+5=15.

Worked Example 7: Sixth Triangular Number From the Fifth

Add the new sixth row: 15+6=21.

5. Why Triangular Numbers Grow Faster

The difference sequence is 2,3,4,5,6,… The amount added is not constant; it increases by one each time. This distinguishes triangular growth from a simple arithmetic sequence.

Worked Example 8: Direct Rule by Pairing

Take two copies of a triangular arrangement with n rows. Fit them together to form an n by (n+1) rectangle. The rectangle contains n(n+1) counters, so one triangle contains half:

Tₙ = n(n+1)/2.

For Primary enrichment, the visual pairing matters more than memorising the formula.

6. Square Numbers

Square numbers count equal rows and columns: 1,4,9,16,25,…

Worked Example 9: Build the Next Square

A 4×4 square has 16 counters. To make a 5×5 square, add an L-shaped border of 9 counters: 5 along one side and 4 along the other without double-counting the corner.

16+9=25.

Odd Differences

Square numbers differ by consecutive odd numbers: +3,+5,+7,+9,…

This gives the identity: n² − (n−1)² = 2n−1.

7. Rectangular Numbers

Not every figurate sequence is square or triangular. A rectangle with n rows and n+1 columns contains n(n+1) objects.

Worked Example 10

Figure n forms n rows of n+1 dots. Figure 4 has 4×5=20 dots.

8. Staircase Patterns

A staircase with rows 1,2,3,…,n is another triangular-number representation.

Worked Example 11: Staircase of 8 Steps

1+2+…+8 = 8×9/2 = 36.

9. Border Growth

Some visual patterns grow by adding a border around the previous figure.

Worked Example 12: Growing Square Frame

A filled 3×3 square has 9 tiles. A 4×4 square has 16. The new border contributes 7 tiles, not 8, because one corner would otherwise be counted twice when adding a new row and column.

10. Constant First Difference

If a number sequence has a constant first difference, a linear rule is likely.

Worked Example 13

Sequence: 5,9,13,17.

Difference = +4.

Rule: 4n+1.

11. Changing First Difference

Sequence: 2,5,10,17,26.

First differences: 3,5,7,9.

The differences themselves increase by 2. This suggests square-like structure.

Worked Example 14

Observe n²+1: 1²+1=2, 2²+1=5, 3²+1=10, 4²+1=17.

So Figure n can be described by n²+1.

12. Multiple Rules Can Fit a Few Terms

From only 2,4,6, a learner may guess “add 2”, but many more complicated rules can also produce the first three terms. Pattern rules should be justified by the visual construction or stated generating process, not only by a short numerical list.

Worked Example 15: Picture Beats Guessing

If each figure is built by adding one row of two tiles, then “add 2” is structurally justified. If the figure changes by a different geometric rule after Figure 3, the numerical guess may fail.

13. Recursive Rule Versus Direct Rule

A recursive rule tells how to get the next term. A direct rule tells how to get Figure n immediately.

Worked Example 16

Sequence 7,12,17,22,…

Recursive: add 5.

Direct: 5n+2.

To find Figure 100, the direct rule is much more efficient.

14. Visual Decomposition

A complex figure can often be split into familiar pieces.

Worked Example 17: Cross Pattern

Suppose Figure n contains one centre square plus four arms each of length n. Total squares = 1+4n.

Figure 6 = 1+24=25.

Worked Example 18: Two Rectangles Minus Overlap

A plus-shaped tile pattern can be viewed as a horizontal rectangle plus vertical rectangle minus the central overlap counted twice. This is the same double-count correction used in Venn diagrams.

15. Perimeter Patterns

The number of tiles and the perimeter may follow different sequences.

Worked Example 19: Row of Joined Unit Squares

n joined squares in a row have area n square units but perimeter 2n+2 units.

For n=5, perimeter = 12.

Worked Example 20: Why Perimeter Adds 2

Each new square shares one side with the previous shape. It contributes four sides but hides two boundary copies along the shared edge, so net perimeter increase is 2.

16. Reverse Perimeter Pattern

Worked Example 21

A row of joined unit squares has perimeter 26. Solve 2n+2=26, giving n=12.

17. Visual Pattern and Ratio

If one quantity grows linearly while another grows quadratically, their ratio changes.

Worked Example 22

A square n×n has area n² and perimeter 4n. For n=2, area/perimeter=4/8=1/2. For n=10, 100/40=2.5. Scaling a shape changes area faster than perimeter.

18. Common Errors

• Drawing the next picture correctly but not stating the rule.
• Counting shared matchsticks twice.
• Assuming every pattern has constant growth.
• Using a direct rule without checking Figure 1 and another known figure.
• Confusing term number with term value.
• Assuming the first few numerical terms uniquely determine the intended visual rule.
• Counting a new row and column without correcting the shared corner.
• Using area growth rules to predict perimeter.

19. Error Repair

Error A: Two Joined Squares Need 8 Sticks

The shared side belongs to both squares but exists physically once. Correct count = 4+3=7.

Error B: Figure 5 of 4,7,10,… Is 16

The pattern adds 3. Figure 4=13, Figure 5=16. If the learner wrote 15, check whether the start value or step count was confused.

Error C: T₆ = 15+5

The sixth triangular figure adds a row of 6, not 5. Correct T₆=21.

20. Practice: 24 Original Questions

Questions 1–8: Linear and Matchstick Growth

1. Find the next two terms: 4,7,10,13,…
2. Give a direct rule for Question 1.
3. How many sticks are needed for 8 joined squares in a row?
4. Which joined-square figure uses 31 sticks?
5. Can a joined-square figure use exactly 29 sticks? Explain.
6. Sequence 7,12,17,22,… Find Figure 20.
7. A cross pattern has 1+4n tiles. Find Figure 9.
8. A row of n unit squares has perimeter 2n+2. Find perimeter for n=14.

Questions 9–16: Figurate Numbers

9. Find the fifth triangular number.
10. Find the eighth triangular number.
11. Find the sixth square number.
12. What odd number is added to 25 to get the next square number?
13. Find Figure 7 of n(n+1) rectangular numbers.
14. Sequence 2,5,10,17,26 follows n²+1. Find Figure 12.
15. A staircase has 15 rows. How many tiles does it contain?
16. Explain why two triangular arrangements can form an n by n+1 rectangle.

Questions 17–24: Reverse and Transfer

17. A row of joined unit squares has perimeter 26. How many squares?
18. A triangular number is 36. Which figure number is it?
19. A square number is 81. Which square figure?
20. Figure n contains n²+1 dots and has 50 dots. Is there a whole-number n?
21. A pattern grows 6,10,14,18,… Give recursive and direct rules.
22. A 12×12 square grows to 13×13. How many new unit squares are added?
23. Compare area and perimeter of a 6×6 square.
24. Explain why a visual construction is stronger evidence for a pattern rule than merely matching three early terms.

21. Worked Answers

1. 16,19. 2. 3n+1. 3. 25 sticks. 4. n=10. 5. No; 3n+1=29 gives non-whole n. 6. 5(20)+2=102. 7. 37. 8. 30.

9. 15. 10. 36. 11. 36. 12. 11, because 25+11=36. 13. 56. 14. 145. 15. 120. 16. Two matching staircases fill every cell of the rectangle, so one staircase has half n(n+1).

17. 12. 18. Figure 8, because T8=36. 19. Figure 9. 20. No; n²=49 gives n=7, so n²+1 would be 50 actually yes—n=7. 21. Recursive +4; direct 4n+2. 22. 25 new squares, since 13²−12²=169−144. 23. Area 36 square units; perimeter 24 units. 24. A construction explains why the relationship continues, not merely that a few terms happen to fit.

22. Checking Ladder

• What changes from one figure to the next?
• Is the amount added constant?
• Are shared edges or corners double-counted?
• Does the direct rule reproduce Figure 1?
• Does it reproduce another known figure?
• Am I finding term number or term value?
• Does the visual construction justify continuation?

23. Transfer Test

Build a pattern from joined equilateral triangles where each new triangle shares one complete side with the previous figure. Count edges for Figures 1–5, find the first difference, propose a direct rule, then explain structurally why it works. Next create a different visual pattern with the same first five numerical totals and explain why identical early totals do not mean identical constructions.

24. Delayed Return

Three days later, solve one matchstick rule, one triangular-number question, one square-border question and one reverse pattern problem without redrawing every earlier figure.

25. Parent and Tutor Guide

Ask “What did the new figure add?” before asking “What is the next number?” If the child guesses a formula, request a physical explanation of every term in it. For example, in 3n+1 for joined squares, what does 3n represent and what does the extra 1 represent?

26. Mastery Receipt

• I distinguish a picture from its growth rule.
• I count shared matchsticks only once.
• I recognise constant and changing differences.
• I connect triangular and square numbers to visual constructions.
• I distinguish recursive and direct rules.
• I solve reverse figure-number problems.
• I compare area and perimeter growth separately.
• I justify a pattern from structure rather than only from early numerical terms.

Official Reference Route

Singapore Ministry of Education — Primary Mathematics Syllabus P1–P6, updated October 2025

This guide is enrichment and extends pattern reasoning beyond required school scope where appropriate.

Continue the Next Enrichment Batch

• BTT Primary Mathematics Learning Hub
• Sets, Venn Diagrams and Inclusion–Exclusion
• Route Networks, Euler Trails and Traversal Puzzles
• Number Bases, Place-Value Codes and Numeral Systems

The Quiet Return

A strong pattern rule does more than predict the next picture. It explains why the same structure must continue.