3 Acts - Broken Calculator - Embrace the Drawing Board

3 Acts - Broken Calculator

7/19/2012

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?

7 Comments

Fawn Nguyen link

8/5/2012 01:41:05 pm

Again, Timon, you spelled 'favorite' incorrectly. And you DO sound like you still play with Pokeman cards. (Score is still not even. You've sent me waaayyy too much iPad pics.)

It pains me to admit that I really like this 3Act too. But I don't see my kids recognizing it's base 5 or whatever base UNLESS we've played around with different bases prior. At this summer's Math Teachers' Circle, we got to investigate base 2/3 (or was it 3/2?) and that was a lot of fun. So, when would you do this lesson?

I'm confused. How did you get the calculators to do the "brokeness" then? Also, FYI: I'm this close to going with "Embrace the Nerd," but that would make you a #nguyener, and I'm still trying to even the score.

8/5/2012 01:43:09 pm

Damn. I meant too MANY, not too much. Great.

ihor Charischak link

4/12/2013 01:56:31 am

There is a classic lesson "What can you do with a broken calculator" developed by Judah Schwartz. It's page 8 and 9 of this newsletter
http://concord.org/sites/default/files/newsletters/2005/fall/at-concord-fall-2005.pdf
An applet also exists. See my lesson: http://ciese.org/math/db/broken2.html

James Taft

8/22/2013 04:01:16 am

Thank you so much! This rocked!
I used this as the first lesson today for my algebra 1 students, and I was amazed at how quickly and well the students dove into trying to figure out how to find the rule or pattern for the "brokenness."

I taught this to a group of students who had never been exposed to number systems of other bases (other than time) and I was amazed at how well they picked up on what was going on and what the "rule" was.

Daniel

7/28/2015 08:47:23 pm

Thanks for this. I was wondering what is the method for "breaking" your calculator? Can you only set Sharp brand calculators to different bases, or is that something that can be done with a TI? Thanks!

Andy link

8/6/2015 03:02:16 pm

I built a quick prototype of the broken calculator that you can try with your students: https://jsfiddle.net/mbnp27bm/1/embedded/result/. If you want to edit the original code, go to https://jsfiddle.net/mbnp27bm/1/. It isn't anything fancy, but you can go to the Javascript section and change the operation, number base, etc. Let me know if you have questions!

Timon Piccini

8/6/2015 04:22:15 pm

Interesting work Andy. My one thing is that it's not truly a base 5 calculator. It converts a decimal sum into a base 5 number, BIG difference. I don't know the first thing about programming that though so great start!