Lesson Plan: Triangular Numbers

When it comes to math, many characteristics are applied to numbers. One such characteristic that a teacher may want students to understand is if a number is a triangular number.

Identifying Triangular Numbers

At first, this may seem odd. One is a triangular number, but is has nothing to do with the shape of the number. Instead, it has to do with making the shape of a triangle.

It is true that one dot will not make a triangle. However, the idea becomes clearer if an array of bowling pins is imagined. Bowling pins are arranged in a triangle. The first line has one pin. The second line has two pins. The third line has three pins. The fourth line has four pins. This equals ten pins.

If the last row is taken away, there are only six pins. However, there is still a triangle. If a fifth row is added, there are 15 pins. There is still a triangle. This is how triangular numbers are identified. The triangle always starts with one and each row grows by one number. Then the number of items in the entire array is added.

This makes the first ten triangular numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, and 55. Go ahead and get some poker chips and try making triangles with any amount of these poker chips.

Determining if Two Triangular Numbers Are Consecutive

When the row of "1, 3, 6, 10, 15, 21, 28, 36, 45, 55" is seen, it is obvious that 3 and 6 are consecutive triangular numbers. It is also obvious that 28 and 36 are consecutive triangular numbers.

The long way of working out each triangle can be used to try to determine consecutive triangular numbers. However, a test can be done that will help to know if two numbers are consecutive triangular numbers.

Consecutive triangular numbers will always add up to a square number. If two numbers do not add up to a square number, they are not consecutive triangular numbers. As you can see, 1 and 3 add up to 4. Four is the square of 2. Forty-five and 55 add up to 100. One hundred is the square of ten.

If you watch the pattern of squares when the first ten triangular numbers is added, it is 4, 9, 16, 25, 36, 49, 64, 81, and 100. These are the squares of 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

Because the numbers are in order, with none missing, this means that it is easy to know when a set of two numbers are not consecutive triangular numbers. Nothing else will add up to a square number except a consecutive triangular number.

Finding a Missing Triangular Number

Now that the properties of triangular numbers are known, you can easily find missing triangular numbers in a sequence.

If you are given 1, _, _, 10, 15, 21, there is an easy way to figure out the solution. Ten plus 15 is 25. Twenty-five is the square of five. This means that 10 plus the previous number has to equal the square of four. The square of four is 16. Sixteen minus ten is 6. This makes the number before ten 6. Knowing this, it means that six plus the second number is the square of three or that one plus the second number is the square of two. Either way works.

You can figure out that the square of three is nine, so the number has to be six minus nine which is three.

You can also figure out that the square of two is four, so four minus one equals three. Either way, the solution to the last missing number is three.