### 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.