Penny Pyramid

After reading about Dan Meyer's three act Penny Pyramid, I was keen and ready to give it a shot.  I had read through the acts and watched the videos.  I had watched one of the workshop presentations he had given on this topic.  I had also read Fawn Nguyen's posting on her site.  In addition, I thought my students were ready for the estimation part because we had begun some work with the Estimation 180 site, and my students were familiar with making reasonable estimates and making reasonable high and low estimates.

And it was fantastic.

I began by showing the video immediately following the bell.  The video is pretty impressive, and so it got their attention right away, and I immediately heard lots of comments, including "cool" and "why would someone do that".  I showed it a second time, and asked them to write down some of their questions.  Then I collected them on the board.

Unlike Dan, who said he wanted the first question, I wanted all of them, and I spent some time wringing them out.  We then went through the list to determine which were mathematically related, and checked them off.  I told them that we would get to some of the "who" and "why" questions later (and I did), and that I'd like them to focus on how many pennies there were in the pyramid.

I then put the photo of the pyramid back up and asked the students for their estimates, including their reasonable estimates for the high and low bounds, and I got this:

Next was to ask them what else they needed from me to be able to figure out how many pennies there were in the pyramid.  They wanted the height of the stack, and the dimensions of the bottom layer.

I split them into random groups of three (thanks Fawn!), and they were off.

Many of the groups went straight into number crunching with the calculator, and when I visited their groups, complained that it was taking a long time.  Some tried to find some sort of pattern to make things easier:

Some groups calculated what they were calling the area of each triangle, and when I asked them if there were pennies in the middle of the pyramid, there was recognition that this needed to be calculation of the inside of the thing, not the outside.

We wrapped up and I told them that next time we'd be going to the computer lab to have a look at what could have made this calculation much more efficient and faster.

Extra Time

One of the yearly struggles I have is with students who are done early.  Whether they are working individually and finished the practice, or whether it is a group of students who are finished after working on a group problem, I have never come up with a constructive way for them to spend their time when they are done early.  I have experimented with different options over the years, but I have never been satisfied with any I have tried.

One of the options I have tried is to let the students who were finished early get up and move around to "help others".  This, of course, turned into giving others the answers, showing them how to do it, or fooling around, none of which is satisfactory to me.

I recently read that in some countries, the classroom culture is that of having students help others when they are done.  It is expected, and part of what goes on in a classroom.  This makes sense to me.

Then I thought about how I help students.  I ask questions.

With these ideas in mind, I thought that if I came up with a list of questions that I ask, and posted these, then the student helpers could ask these questions while they are helping others.

1. What is the math concept/topic for this problem?
2. What information is given?
3. What is the meaning of the word ___________?
4. What do you think you need to do?
5. Have we already done a problem like this one?
6. What have we learned that may be helpful here?
7. What drawing/diagram could you make to show this problem?
8. What other strategy (table, equation, etc.) could you use here?
9. How would you describe the problem in your own words?
10. What would be your best guess/estimate for the answer?

I will model these questions.  I will post them.  When I explain how I will expect students to help others when they are done, I will refer to the posted list of questions and encourage them to use only these. 

Maybe this will work.