Wednesday, November 14, 2012

Chutes and Ladders (or Tangents and Cotangents)

Chutes and Ladders - a game for ages 4 and up
Tangents and Cotangents - a game for ages 16 and up
Last week I was playing Chutes and Ladders with my daughter (well, I was playing and she was arguing with me about why I kept moving my piece three spaces forward and didn't just put it on a space that had a three in the number, like 3, 13, 23, etc like she must have thought were the rules) and I made the amazingly smart observation that there were six spaces in the spinner, and there were six basic trig functions. Before long I had gotten board and wanted to find ways to add strategy to the game, and found that at times it was more beneficial to go backwards several spaces instead of forward, if by doing so I could avoid a chute or land on a ladder.

It wasn't long before I had decided upon a few rules, and come up with a game we could play in precalculus, which I decided to call "Tangents and Cotangents".  I thought the name was a nerdy way of describing Chutes and ladders, because the graph of cotangents is always decreasing like chutes, and the graphs of tangents are always increasing like ladders.

I brought the game to school, and described the following rules to the students:

  • Students (in teams) would answer estimation questions like sin(25) or tan(258).  (NO CALCULATORS ALLOWED!) Each trig function was worth a different amount of points. If the students estimated within 10% of the correct value, they moved forward that many spaces. If not, the other teams would be able to steal the points if they had estimated correctly -- so everyone was interested in every question. The points were defined as follows:
    • sine = 1pt
    • cosine = 2pt
    • tangent = 3pt
    • secant = 4pt
    • cotangent = 5pt
    • cosecant = 6pt
  • If the team guessed the value within 10%, but had the opposite sign, they went backwards that many spaces. While this might seem like an annoying penalty, eventually one of the teams caught on that it could be used their advantage to land on the coveted 28 square, and it wasn't long before teams were "purposely" getting the sign wrong to their advantage. I didn't mind, because it made them think about and practice the signs of the values too -- part of my objective for the day.
  • Once per game, I allowed the teams to choose what point value they wanted to play for -- typically they were chosen at random like normally in the game. This added an element of strategy for the students who would wait to the opportune time to try to hit a big ladder. I think I would change this rule next time to reward REALLY close guesses (say, within 1%) with an additional free choice.  
  • Occasionally we would play a question that was available for all four teams. Any team that guessed within range was awarded the points (forward or backwards as necessary)
In order to help facilitate the random choosing of the game, I created an excel file that gave me a random trig estimation question every time I pressed F9, and then displayed the answer and the acceptable range of values when I pressed F9 again. When students chose a particular point value, I had to press F9 repeatedly until the appropriate value came up -- for instance, when one group wanted a 4 point question in order to land on space 80 for the win, I had to press F9 over and over again until a secant question came up. You may download this excel file here, and you can read about the underlying workings of it at my post here.  For more on estimating trig functions, I suggest you read on the unit circle definitions of trig functions.

1 comment:

  1. This is so cool!! You are such a COOL teacher!! this is awesome! im so proud of my Nerdy brother!


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