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Picturing Triangular Numbers

Age 11 to 14 Challenge Level:

Why do this problem?

This problem offers students an opportunity to relate numerical ideas to spatial representation, and vice versa.

The interactivity allows students to explore large triangle numbers. Visualising the combination of two identical triangle numbers can lead to an understanding of the general formula for triangle numbers.

Possible approach

This printable worksheet may be useful: Picturing Triangle Numbers.

This problem works very well in conjunction with Mystic Rose and Handshakes. The whole class could work on all three problems together, or small groups could be allocated one of the three problems to work on, and then report back to the rest of the class.

Write the sequence $1, 3, 6, 10, 15 ... $ on the board and ask the students to work out what's going on. Can they continue the sequence?
"Can you work out the tenth number in this sequence? What about the 20th?"

Suggest to students that it would be very helpful to have a quicker method for working out numbers at any point in this sequence.

Introduce the terminology "triangle numbers" if it has not been met before, and show the pictorial representation of the fifth triangle number. Ask students to visualise how they could fit together two copies of the same triangle number to make a rectangle.

Give students time to work together in pairs to explain what happens when they combine other pairs of identical triangle numbers, keeping a record of what they do. Then bring the class together and write on the board the dimensions of the rectangles they found for different triangle numbers, using the interactivity to check.

"What is special about the dimensions of the rectangles? Why?"

"Can you write down the dimensions of the rectangle made from two copies of the 250th triangle number?"
"Can you use this to work out the 250th triangle number?"

Ask students to explain a method for finding any triangle number. This may be in words, using diagrams, or algebraically.

"Can we use our insights to help us to determine whether a number is a triangle number?"
Give the class a selection of numbers such as the ones suggested in the problem, and allow some time for them to work in pairs to determine which are triangle numbers.

Key questions

What is special about the dimensions of the rectangles made from two identical triangle numbers?
Can you devise a method for working out any triangle number?

Possible extension

Students could write their method for calculating triangle numbers using algebra. This formula could then be used to gain insight into facts about triangle numbers such as:
$T_{n+1} - T_{n} = n+1$
$T_{n} + T_{n-1} = n^2$

Why can triangle numbers end with the digit $8$ but never with the digit $9$?
What other digits can never appear in the units column of a triangle number?

Will there ever be two consecutive triangle numbers ending with the digits 000? What about 0000?

Possible support

Make multilink cubes available and encourage use of diagrams to aid with visualisation of the triangle numbers.

For another problem that uses a similar idea go to Picturing Square Numbers