Starting the Math Club
Mr. Power was a Math teacher who liked the students to be engaged. He talked to a few students and inquired if they would be interested in forming an extracurricular Math club. Students would make monthly presentations in the club. They may present anything as long as it was related to Math in some way or the other, and later on the club could add other activities. He suggested that any students from the school could participate. He asked if there was any one who wanted to start the club. Two students named Kirsten and Jonathan agreed to take the initiative. Kirsten and Jonathan had average grades but they were extremely social. They liked talking to everyone in the school. They met with several students and talked about Mr. Power’s idea. They also got signatures of the students who wanted to join. Some of the students signed because they were really interested. There were also others who signed just because they had nothing to lose. There were no fees and no hard and fast commitments.
After talking to their schoolmates and getting these signatures, Kirsten and Jonathan met with Mr. Power and told him that about 50 students wanted to join – mostly from grades 11 and 12, and a few from grade 10. Mr. Power was unhappy that they were no grade 9 students but he let that go thinking that they might be asked later by an announcement by their Math teacher.
Math club T-shirts
Kirsten and Jonathan also told him that the students suggested to get T-shirts made for those joining the club. In fact they gave the impression that the T-shirts might increase the interest of the students. They also wanted to know if and how much the school could contribute to these shirts. Mr. Power thought about it, talked to the Principal, and then he said that the school would contribute 60% of the cost of the shirts and that the students would have to share the remainder 40%. This was conditional upon approval of the Math based logo to be printed on the shirts. Someone from the group would have to show the design and make a presentation to him and to the Principal for this fund to be approved. He also cautioned them that they would have to work on a good Math concept and not simply something like two plus three equals five.
Kirsten and Jonathan called a meeting of the Math club to discuss any suggestions for the logo design. At first, there was a complete silence but then the first suggestion came. It was to draw a circle with the writing πr2 in the middle. Another was to draw shapes with different number of sides. These ideas may not have been very interesting or highly appealing but they started the conversation.
Logo design and Pythagoras theorem
One student then suggested a right angle triangle for the proof of Pythagoras theorem. It would use four different colours – one for the right angle triangle, and a different colour for each of the squares that could be drawn from the three sides of the triangle. Kirsten and Jonathan picked up on it and said that they were thinking of this idea but in a more advanced way. They called it a Pythagorean Triple. A Pythagorean Triple is a right angle triangle with all its sides being whole numbers. The first triple has the sides 3, 4, 5 because 32 + 42 = 52. The next triple is 5, 12, 13 but it would give a triangle of a weird shape. Besides, the school address was 345 Triangle Park. They canvassed for this idea (Fig. 4.1).
Connecting trigonometry and geometry
Every one liked this idea but one student raised a hand and said, “This may pass but we should improve on it to make sure that we get the money. In our algebra class, our Math teacher challenged us to connect arithmetic, geometry and algebra. Sara gave the solution for that. Sara, can you come up with something that connects this geometrical figure with another Math subject?”
Sara: Johnny and I were talking about Pythagoras theorem the other day while studying Trig. Those of you in the senior years must have studied Trig identities and you would know that some of them are derived from sin2 x + cos2 x = 1.
One of the students said: yes I remember two identities based on sin2 x + cos2 x = 1. I remember sec2 x = 1 + tan2 x and cosec2 x = 1 + cot2x.
Sara: Sin2 x + Cos2 x =1 is a Trigonometric restatement of the Pythagoras theorem.
Many students said how they had used Pythagoras theorem to prove this identity.
Kirsten and Jonathan agreed that this was a good idea to connect Geometry and Trig but they still wanted to use the idea of the Pythagorean Triple.
Sara raised her hand and said: I just checked on my calculator that sin 37° = 0.6018 which is approximately 3/5 and cos 37° = 0.7986 which is approximately 4/5. That means one angle of the first Pythagorean Triple with the sides 3 and 4 will be 37°. The hypotenuse will be 5. So, how about we keep the same design but with some writing on it. She went to the board and wrote some things, and the final picture that looked like this (Fig. 4.2).
Kirsten and Jonathan asked for a vote on this design. It passed rapidly and unanimously.
Kirsten: That’s great that we have a design but I have two more issues.
Jonathan: What are they?
Kirsten: The first concern is that an image with the triangle of sides 3, 4 and 5 centimeters will be too small and will not look good on a shirt.
One student shouted that the size could be 3, 4 and 5 inches.
Kirsten smiled and said: Thank you. That will solve this problem.
Jonathan: Kirsten you said that you had two concerns.
Who will make presentation to the principal?
Kirsten: Yes, who will make the presentation to Mr. Power and to the Principal?
Many students pointed towards Sara.
Sara: Kirsten, you and Jonathan did all the work. You should make the presentation. If you want me to tag along, I will be glad to do that.
Kirsten: What does everyone think of this idea?
They loved it.
Kirsten and Jonathan went to a local T-shirt shop to ask for an estimate for 50 shirts. They said that the total cost for 50 T-shirts was $600 plus taxes but as a discount, the store could pay the taxes.
Kirsten, Johnathan and Sara impressed the teachers with their presentation of the design and the Math concepts behind it. The school agreed to pay 60% of the cost of the shirts. That meant that out of the $600, the school would pay $360 and the students would be the remaining $240. Then the shirts would be only $240/50 or $4.80 per student. So the Math Club was started.
Challenge: Sara wanted to know how many presents she would get on her birthday. Her mom gave this straight forward answer: 4 sin2 x + 2 – 2 tan2 x + 4 cos2x + 2 sec2 x. Sara said, “thanks mom.”
I guess Sara figured it out. Can you?
Solution: : Objective: Simplify 4 sin2 x + 2 – 2 tan2 x + 4 cos2x + 2 sec2x
From the Pythagoras theorem sin2 x + cos2x = 1
Dividing both sides by cos2x, one gets tan2 x + 1 = sec2x or sec2x = 1+ tan2 x.
Now, 4 sin2 x + 2 – 2 tan2 x + 4 cos2x + 2 sec2x = 4(sin2 x + cos2x) +2 – tan2 x +2(1+ tan2 x) = 4 because the remaining terms cancel out.
So should get 4 presents on her birthday. Her mom just likes to make things complicated.