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'Triangle Numbers' printed from http://nrich.maths.org/
Why do 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.
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
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?
- 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
- What does moving one square down the multiplication grid do to
- What is the effect of moving across by one column in the
- Can you create a general algebraic expression for the $n$th
Providing grids for students to draw on might make the
Ask them to compare consecutive triangle numbers and explore