Forgotten Number

I have forgotten the number of the combination of the lock on my briefcase. I did have a method for remembering it...

Man Food

Sam displays cans in 3 triangular stacks. With the same number he could make one large triangular stack or stack them all in a square based pyramid. How many cans are there how were they arranged?

Sam Again

Here is a collection of puzzles about Sam's shop sent in by club members. Perhaps you can make up more puzzles, find formulas or find general methods.

Triangle Numbers

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

This printable worksheet may be useful: Triangle Numbers .

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.