Triangular numbers with dots
Stack dots in rows of 1, 2, 3, 4 and more to build triangles. Count the dots in each triangle, find the rule n(n + 1) ÷ 2, and see how two neighbouring triangles fit together into a square.
Core concept
The nth triangular number is 1 + 2 + … + n = n(n + 1) ÷ 2, and a triangle with the one before it makes a square: Tₙ + Tₙ₋₁ = n²
Topic: Maths in everyday life
Put 1 dot in the top row, 2 in the next, 3 in the next, and so on. The dots make a triangle, and the numbers of dots, 1, 3, 6, 10, 15, …, are the triangular numbers. Bowling pins, a stack of oranges and the balls in a carrom-like rack all make triangles. Adding 1 + 2 + … + n gets slow, but there is a shortcut: put two equal triangles together to make an n by (n + 1) rectangle, so one triangle has n(n + 1) ÷ 2 dots. And a triangle placed next to the one before it makes a perfect square of n × n dots.
What you will learn
- Build triangular numbers from rows of dots.
- Use the rule Tₙ = n(n + 1) ÷ 2.
- Show that two neighbouring triangular numbers make a square number.
Before you start
- Know square numbers.
Materials
- Bottle caps, coins or dots on squared paper
Equipment
- None
Procedure
- Record the triangle of 4 rows.
- Add a row to make 5 rows, and record.
- Build a triangle of 10 rows, and record.