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.

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

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.

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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So this one hasn't done absolutely amazing on 101qs.com, but I think that the number of questions that line up together mean that I think we have a worthy entrance into the 3 acts database.

The most interesting aspect of this 101qs is not so much the commercial itself, but the outcome of the commercial.  The student that actually earned the number of Pepsi points to get the harrier and what happened to that.  So let's check it out.

So when I was a kid and first saw this, I instantly wondered, how much Pepsi would it take to get that Harrier? Also I always wondered, "How much more will it cost to get the Harrier via Pepsi rather than just purchasing it?" Youmight notice that the new version of this video (based on some comments) that I blocked out the points so that students can come up with the number themselves.  How could you offer this prize and still make a profit?

A couple of ways I would do this is have students look up some of these items to get a general sense of price to point conversion.  Then we can compare that to point to price in Pepsi.  This task is more about making reasonable decisions rather than finding the exact correct answer.

I am a bit worried about this reveal, because being right is always so much more fun, but seeing this we can talk about how completely unreasonable this deal is.  There is no way Pepsi would give away this kind of prize for so little. Enter the...

The whole reason I choose this task is that a 21 year old student actually came up with the points necessary for the Harrier.  These are some of the questions I would ask.

• You are a lawyer for Pepsi, trying to show that this commercial is clearly a joke, and not a real offer.  Prepare a statement for the judge.
• What is the least amount of money that the 21 year old could pay for the Harrier, assuming he purchased bottles and cans to obtain the points?
• Pepsi offered a deal, that you could buy a point for 10 cents.  How much could you get the Harrier for? (Yes this is an easy sequel for math teachers, but I know a few students that would have to think about it).