Mathematics is all about the beauty of patterns and their reasonable connections. How about connecting patterns from seemingly different domains of the subject? This is a note borne out of a Geometry workshop at Cheenta where we tried exactly that. The audience comprised of 9 to 11 years old students.
The purpose of this note is to share some teaching methods in mathematics. A detailed discussion on this method is available in another note.
We begin the discussion with triangular numbers.
One dot gives the number 1. We may have think of this as the 1-dotted triangle. The shape of this triangle is still not very triangular.
Next we add a 2-dotted row to make a 3-dotted triangle. Thus the second triangle has 3 dots. Now it looks like a triangle!
Next step makes the evolution of the triangular shape apparent. We add 3-dot row beneath in the 3-dotted triangle to get the 6-dotted triangle. Thus the third triangle has 6 dots.
Can you guess how to create the next triangle?
Obviously we add a 4-dot row beneath the 6-dotted triangle. This gives us the fourth triangle in the sequence which is a 10-dotted triangle.
Students quickly catch on and they create fifth triangle which is 15-dotted, sixth triangle which is 21-dotted and seventh triangle which is 28-dotted. By this time, the process of designing the next triangle is understood by most students. It takes only one more indulgence to expose the series form of the number of dots.
- 1st triangle has 1 dot
- 2nd triangle has 1+2 dots
- 3rd triangle has 1+2+3 dots
- 4th triangle has 1+2+3+4 dots
Can you find the number of dots in the 20th triangle? Well it must be 1+2+3+…+20 dots. How do we sum these numbers quickly and efficiently?
At this juncture we let the cat out of the bag and introduce the students to the genius of Carl Freidrich Gauss. Write the sum backward beneath the original sum. Each column adds up to 21. There are 20 columns. Hence the sum of twenty 21’s is 420. But we added each number twice hence the sum we are looking for is 210!
As the kids get marvelled by this little trick, we quickly switch gears and look at a more geometric problem.
If you put 1 line in the plane how many regions do we have? Clearly two.
Next put another line in the plane. This second line must cut through the first line. How many regions do we have now? Four.
Let us put another line in the plane. This line must cut the other two lines and must not pass through the previous intersection point. How many regions do we have now? Students take a little time to label the regions and come up with the right answer: seven.
We continue the process of drawing by adding the fourth, fifth and the sixth line. Each time we ensure that the new line cuts all previous line. Moreover the new line must not pass through any of the old intersection points. How many regions are produced in each step?
- 1st line produces 2 regions
- 2nd line produces 2 more regions. In total we have 4 regions now.
- 3rd line produces 3 more regions. In total we have 7 regions now.
- 4th line produces 4 more regions. In total we have 11 regions now.
Students quickly notice that 5th line produces 5 new regions, 6th line produces 6 new regions and so on.
The punch line is this: total number of regions produced by n lines is exactly 1 more than the nth triangular number.
The splitting of the plane by lines (which is of more universal appeal) has this striking connection with a sequence of integers related to dotted triangles.
The spirit of the discussion should be experimental in nature. We constantly ask the students questions like:
- Can you draw the next dotted triangle?
- How many dots are there in the 10th dotted triangle?
- Can you observe a pattern?
- Can you find the number of regions created by four lines?
Draw draw draw… observe observe observe… analyze and conclude.
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