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3 Acts - Broken Calculator

7/19/2012

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This is easily one of my favourite problems that I have come up with.  Mainly because of the back story.  I handed a student this calculator, and he told me that it didn't work.  The numbers weren't working.  I showed someone else and they decided to throw it out, but I couldn't help but think that something more than broken buttons was the problem.

Act 1 - The Brokenness

So ask the students: "What is wrong with this calculator?" or "What is this broken calculator going to give us for '433+233'?" 

Act 2 - Examples

Okay full disclosure here, this is not really what I want to give my students, but as a low tech version (and one that you can use as well), I have made these...
What I really want students to do is to explore their own numbers and find patterns on their own.  In order to do this, I want to program a base 5 calculator that kids can use on the school netbooks, BUT I don't know how to program.  If anyone has ideas about how I could put this calculator into my students hands without telling them that it is a different base please put them in the comments.

Update: The awesome Jed Butler programmed this amazing base five calculator on Geogebra. If you are not plugged into the MTBoS, you need to get on that!

The other option is I just put my calculator under the document camera and have students ask and record class wide.  That doesn't help you guys though, so this is what I have started with.  If you think I need some more/better examples please tell me in the comments and I will make them (groups of four look nice).

Update: I have also begun having students exploreJames Tanton's Exploding Dots. This is a great intro into different number bases, and really stretches students, but is not completely out of the range of grade seven students. So much love for this.

Act 3 - The Reveal

Sequels

This is a pretty pure mathematics WCYDWT so I can only think of standard sequels. (Please give me more ideas in the comments, these are pretty lame).
  • How does multiplication work in this number system?  Can you find some easy methods for solving basic multiplication statements?
  • Pick a random base (2,7,12,4.5(?), 16), and create some problems, and share them with a partner.  What is different and similar among different bases?
  • From @trianglemancsd How would you represent 1/2, 1/4, and 1/10 as a "decimal" number? What does 1.3, 1.021, and 0.033 become as a fraction? (All sorts of headaches happen here, clarify a fraction in base 10 or base 5; what does 1/10 mean?

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