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Dartboard and Equations

While reading Dan Pearcy’s blog entry about Rich Questions I could not agree more about his statement

I love it when people share rich questions. Generally because I find that I can’t keep track of all the resources which come my way but I do seem to remember great questions.

For me, piling up resources is so easy to do, while actually implementing them is the difficult part. So, I am trying a new strategy next semester. Instead of having the “oh this would be a cool problem and oh this would be a cool problem too, oh-oh-M-G and I have to do this”, which ends up in only doing a few problems, I will focus on one problem and see how much I can get out it. Can a whole chapter be focused around one problem? I don’t know but if I find one, it will be placed in the “Rich Question” Category.

After seeing Dan’s post about linear equations and shapes being shaded, I wondered if the two could be combined. Here is my idea. First, I would have students look at this graph made in Desmos and ask the basic question, “What part is shaded in each square?”

(Click on the above picture to see graph at desmos.com.) After we agree that 50% is shaded in each region, I would ask the main question of the project.

If each square is an individual dartboard, then which square is the easiest to hit a shaded region? Which square is the hardest to hit a shaded region?

Can you prove your answer mathematically?

Hopefully, this would hook their attention to the project. Next, each student would create four different dartboards. One dartboard they think is too easy, one dartboard that is too hard, and one dartboard that is just right (fair). The fourth would be one that is a cheat – that is it looks like 50% shaded but actually it is less. Once each student completes their four graphs they can go around assessing each other graphs with HARD, EASY, FAIR, and CHEAT. Side note: The student that gets aways the least amount of shaded region should earn extra points. This would give more of an incentive to create unique ways of shading 50% of a square. Finally, the general case be examined. How should one shade 50% of a square so that it is the easiest/hardest dartboard?

Where are the linear equations? The students would create the dartboards in Desmos by inputing linear equations. For example y={-9 \leq x \leq -1;x+9} would graph the linear equation x+9 with a domain restriction -9 to -1. I think this could provide some good opportunity to solve equations, create equations, and practice domain restrictions. Of course the shading part, which uses a dampening function, would have to be explained. But once that is shown, that would create a bridge of transformations, a concept that seems only start with quadratics and not linear. Also, students might find a reason to know how to find the distance between two points.

Overall, this is still really fresh in my head. There seems to be lots potential I have heard that less is more but with so much great stuff on the internet, focusing on less is tough. But, I think finding a Rich Question can help me do more with less. If you land a bullseye with a Rich Question then let me know. Thanks!

Categories: Rich Question
  1. November 20, 2012 at 9:05 pm

    Hi Luke,

    I’m glad that my post inspired a new project idea for you. I’ve never thought about the level of difficulty in hitting regions of dartboards before- a unique and interesting perspective.

    I’d be interested to see some student work on this when the project is completed and to hear your reflections on how it goes. Good luck!

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