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'Triangle Numbers' printed from http://nrich.maths.org/

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Why do this problem?

This problem allows students to investigate the relationship between triangle numbers expressed as a sum and triangle numbers expressed visually. Students may focus on either representation to arrive at a conclusion. The problem can be concrete with students focusing on small numbers or can be extended to give a general result.

Possible approach

You could discuss the structure of triangle numbers as a sum (i.e. add $2$, then $3$, then $4$, then $5$, etc.) and then ask students to describe the visual layout of the triangle numbers coloured in the grid. Students can then experiment on paper to try to spot patterns from which a general conjecture might be made.

It can be helpful to consider this problem as a process by which we move from one triangle number to the next. What changes with each step? Is this different for odd and even numbers?

Key questions

  • What can you see in the grid?
  • Where would you expect the next pair of triangle numbers to go? Were you correct?
  • Can you describe how to move from one triangle number to the next?
  • What does moving one square down the multiplication grid do to the number?
  • What is the effect of moving across by one column in the grid?

Possible extension

  • Can you create a general algebraic expression for the $n$th triangle number?

Possible support

Providing grids for students to draw on might make the connections clearer.
Ask them to compare consecutive triangle numbers and explore the differences.