A square number makes a square of dots: 5 rows of 5 is 5 × 5 = 25. A triangular number makes a triangle: rows of 1, 2, 3, 4 add to 10. Square numbers go 1, 4, 9, 16, 25. Triangular numbers go 1, 3, 6, 10, 15. To get the next triangular number, add a new bottom row one longer than before. To get the next square, add a new row and a new column.
Explore this idea with maths tools
Maths you can move
Build 8 + 5
Tap two gold counters to fill the ten-frame.
8 in the frame + 5 waiting. The total stays the same.
Words to know
- square number
- a number times itself, like 5 × 5 = 25
- triangular number
- the total of 1 + 2 + 3 + … up to a number
Worked examples
What is the 5th square number?
- 1.Draw 5 rows of 5 dots.
- 2.5 × 5 = 25.
= 25
What is the 4th triangular number?
- 1.Rows of 1, 2, 3 and 4 dots.
- 2.1 + 2 + 3 + 4 = 10.
= 10
The 9th triangular number is 45. What is the 10th triangular number?
- 1.The 10th triangle adds a new row of 10 dots.
- 2.45 + 10 = 55.
= 55
For parents
Method: Dot patterns
What to say: What shape do the dots make? What gets added to make the next one?
Common mistakes:
- Doubles the number instead of multiplying it by itself (the 5th square number as 10).
- Counts only the dots around the edge of the pattern.
- Mixes up square and triangular numbers or their growth rules.
- Uses k × (k + 1) for a triangular number and forgets to halve it.
- Counts positions from zero, so gives the number one place too early.
- To find the next number, adds the current position instead of the growth of the next pattern.
- Assumes each number in the pattern is double the one before.