Triangular Numbers

Problem

Triangular numbers are made by forming triangular patterns with counters. Riwa has made the first four triangular numbers with blue counters.

Riwa didn’t think that the first triangular number really looked like a triangle but it seemed a good place for the pattern to start. The first triangular number is made with just one counter and so is one. The second triangular number is 3. The 3rd triangular number is 6 and the 4th triangular number is 10.

What is the 10th triangular number?

What is the 20th triangular number?

Teaching sequence

1. Introduce the problem to the class. Brainstorm ideas for approaching the problem and keeping track of what has been done.
2. As the students work on the problem in pairs you might ask the following questions to extend their thinking:
   What strategies might help you to find the answer?
   How can you use your knowledge about numbers here?
   Can you see any patterns that might help?
3. Share the students’ answers. Ask them to explain their reasoning.
4. Ask students to write up their method of solution.
5. Use the extension problem for the faster students or as a problem on another day.

Extension to the problem

Which triangular number equal 120?

 Solution

Most students will add new rows of counters, and make the 6th, 7th, 8th, 9th and 10th triangular numbers by construction. But get the students who do this to stop and think about what is going on. It is not difficult to see that each new row requires one more counter than the previous one. This leads to the idea that the 10th triangular number is the 4th triangular number plus 5 + 6 + 7 + 8 + 9 + 10. That is, 10 + 5 + 6 + 7 + 8 + 9 + 10. These can be added in order to give the 10th triangular number as 55.

But they can do better than this. Note that the 1st triangular number has one on the bottom, the 2nd two on the bottom, the 3rd three and so on. So clearly the 4th triangular number is made up of 1 + 2 + 3 + 4 counters. So the result for the 10th triangular number can be written as 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.

Now there is a quick way to add consecutive numbers like this. (See Algebra Information.) When a string of numbers like this are added it is useful to ask yourself whether adding them in a different order makes the task more interesting. In this case, since the students will be familiar with ‘making ten’, it is natural for them to suggest adding (1+ 9) + (2 + 8) + (3 + 7) + (4 + 6) leaving only 5 and 10 to be added later. So, the 10th triangular number is 10 + 10 + 10 + 10 + 5 + 10.

Another interesting way of adding the numbers is to add the first and the last, then the second and the second to last, and so on. This leads to (1+ 10) + (2 + 9) + (3+ 8) + (4 + 7) + (5 + 6). This simplifies to 11 + 11 + 11 + 11 + 11 = 55.

The students should be encouraged to think like this when they work out the 20th triangular number. So they have to add 1 + 2 + 3 + … + 20 = (1 + 20) + (2 + 19) + … + (10 + 11) = 10 x 21 = 210.

Solution to the Extension:

Now we have 120 counters and we want to see which triangular number they’ll make. Of course, one way to do this is to take 120 counters and make up a triangle. The number of the counters in the bottom row is the number of the triangular number.

A slightly better way is to guess that it is the 12th triangular number. (By what we have already done, it must be between the 10th and the 20th triangular number.) Checking using the method above shows that the 12th triangular number is 78. This is still not enough so try again. Eventually the students will hit on the right answer.

But again, they could draw up a table. They know that they have to add the number of the triangular number so it should be easy to finish the table off.

The answer is clearly the 15th triangular number.

There are other ways though …

Let’s draw the triangular numbers slightly differently – in the shape of a staircase.