I began this year by having my students tackle one of Dan Meyer‘s 3-act math tasks. In fact, all of the teachers in my department did. We wanted to pick a problem that wasn’t too difficult mathematically so we could focus on the problem solving process. We picked “The Incredible Shrinking Dollar” and we had all of our students solving the problem on day 1. I teach one 10th grade integrated math class, two 11th grade integrated math classes, and one 12th grade pre-calculus class. Here’s what happened:
- Watched the video a couple of times.
- Asked the question, “How big is the dollar after 2 (or 3) shrinks?” (The class determined how many times Dan shrank the dollar in the video.) Groups worked and presented their process on white boards.
- Some groups interpreted “big” as referring to side dimensions, while other groups determined that “big” meant area. Several groups correctly calculated the new dimensions (using four different strategies). One group determined that losing 25% two times meant that the dollar would now be 50% of it’s original size. What a fabulous opportunity to address this common misconception! Groups interpreting “big” as area were split depending on how they chose to calculate the reduction. Some shrunk the area (typically incorrectly), while others shrunk the dimensions and then calculated the area. The differences in these results made for some very rich discussion.
- Asked the questions, “How big is the dollar after Dan shrinks it 9 times? Draw your guess. Will you still be able to see it?” This time all groups made correct calculations. Some of the “area” groups multiplied the original area by 0.75^18, while others used (0.75^2)^9. Cool, huh? But, I wasn’t convinced that they saw that these dimensions weren’t even close to the estimated pictures they had drawn. In fact, several drawings suggested that the groups thought that the dollar would be 1/9th the size in area. So I gave them rulers and made them draw their results. This gave us another great opportunity to address misconceptions: units were in millimeters (“which side is millimeters”), is 1/9th of the dollar (in area) the same as shrinking it 9 times, and so on.
- Asked the question, “How many times does Dan have to copy the dollar for it to become invisible?” A couple of groups thought this was a trick question: “The dollar will never be invisible – it will always exist.” Other’s asked, “What do you mean by invisible?” I turned the question back around to them so they had to define the term and then answer the question accordingly. One group looked up “too small to see with the naked eye” while others made more arbitrary decisions. We had a great discussion about how different assumptions can lead to different, valid results.
- The pre-calculus class had to use a different problem because one student had already solved the dollar problem. I chose a different problem on the fly. I looked down Dan’s list of problems and Coke vs Sprite caught my eye. This was true problem solving for this class. The students looked to me for help, but I hadn’t done the problem, either. All I could do was point them to the problem solving guidance that our department created. The problem poses a very simple question that my students spent 75 minutes trying to answer. The best part of the whole process was that they didn’t believe their results – because the results didn’t agree with their initial prediction. How awesome is that?
I’m really happy that we decided to start the year this way. It really set the tone:
- Different approaches can lead to equivalent results
- Different assumptions can lead to equally valid results
- Equally valid doesn’t necessarily mean the same
- We can learn from everyone and from everyone’s mistakes
- Mistakes are opportunities for discussion and clarification
- Initial thinking can be wrong; you have to do the math to find out
What a terrific way to start the year!