Exponential Potential

It really struck me listening to Shawn Cornally in this week's #globalmath session on SBG (click on the Recording tab) how his perspective as a physics teacher leads him to approach his math lessons as experiments. Starting with an experience that makes us want to model or makes modeling useful is definitely the start of some of my favorite math lessons. (While I'm writing this he tweets: "Leaf-Blower soccer went over *really* well today in physics. (vectors, f=ma)")

Starting exponential functions with my preservice teachers, I love to use this lesson adapted from a 5th grade Math in Art lesson. (From my pre-blog webpage.) The idea is the multiplicative patterns present in a Sierpinski Carpet.


(Also in Word format if you want to edit.)

One of the interesting discussions in the initial exploration is the 9 or 17 issue. 9 squares if we count the number of squares as distinct shapes, 17 if we unitize to the smallest level square. For algebra students there's some good opportunities for equivalent expressions, regression and even deduction of function rules. This is a good opportunity for sharing how recording how you're getting your answer can be more powerful than recording answers. The 17, for example, is 1·9+8, then the next level is 9·(17)+8·8. But later, most write it as \( 9^n–8^n \) - which can lead to a pretty neat binomial expansion. Maybe even more interesting and accessible is the 9=1+8, so the next step is 73=1+8+8·8, and gives them a way to generalize this pattern besides recursion.

Once we get to the design your own carpet, there are so many new patterns to find. Here are some samples from this week:

Note that these last two aren't really Sierpinski patterns - but they still raise interesting patterning questions that are extensions of what we already noticed.

Probably easy to see why this is one of my favorite lessons. I also have seen the power of adding in places where students who have not traditionally been strong in math class can do amazing work.

The next lesson to follow up this one has some other opportunities to gather/generate multiplicative data.



It always strikes me how even college math majors find things to be surprised about in this data. Especially the penny balancing one. This class made some neat displays of their data - but I haven't taken the pictures yet. (Didn't know I would be blogging this, as I thought I already had! Maybe I was thinking of the quadratic simulations?) I'll add them at first opportunity.

First Opportunity:

Next we'll look at modeling this data symbolically using technology, and asking questions that raise the need for logarithms. Since, of course, every exponential data set is logarithmic when seen through the looking glass.

Angle Acquisition

Quick game idea. I've had a few bustling around, and I've got to get started writing them down.

Observing student teachers is a great job. I get to see and have in depth teaching discussions with lots of hard-working, talented teachers.  And see a broad range of content. We use a coaching model, which helps these be positive exchanges and ratchet up the interest level of the dialogue.

I was observing Terry Austen (the class he writes about here) and realized that once students knew the terminology well for parallel lines they were correctly identifying and applying the relevant properties.

That gave me the idea for a game. I thought it would be neat if students used the terminology to capture points. My first idea was to have a parallel line puzzle - I like those as practice, too - and have the players build it and then take the pieces. That's a lot of set up, though, so I decided on cards to make for less preparation. There's name cards to cut out, still.



Let me know what you think. I'll try it out with the PSTs this semester and post an update.

3rd Degree

I Love Charts, constant source of fun graphics, had a fun temperature comparison chart. temperature graph, complete with fun. I did share my graphs finally. I used GeoGebra to make them, of course, because that's the best accurate mathematical image maker. Easy enough to make all three variations. Which, of course, lead to wondering what features would you put in a dynamic sketch for it? How could you make the sketch switch between the three ranges?
But it wasn't to scale, so I started thinking about what that would look like. Just lengths? What's the norm? Then Shawn Cornally shared his oneupped

So I had to make it then. Obviously. It's on GeoGebraTube: for download or use as an applet.

Instead of checkboxes, I knew I wanted a slider to switch amongst Kelvin, Fahrenheit and Celsius as the input. I thought about adding an input box for conversion, but those still don't work on the iPad, and I'm trying to think about that more.

The idea came to make the segments for the 0 to 100 degree temperature range where the endpoints were a function of the slider. Then I just needed a particular value for three different inputs, so it would only have to be a quadratic. I started to go to Wolfram|Alpha to do the regression, when I realized that, of course, GeoGebra would do it more easily. So with the slider at values 0, 1 and 2, the range for Fahrenheit, for example, is just determined by

F0(x)=FitPoly[{(0, -459.67), (1, 0), (2, 32)}, 2]
F100(x)=FitPoly[{(0, -279.67), (1, 100), (2, 212)}, 2]
a=Segment[(3, F0(n)), (3, F100(n))]

I was struck again by how nice GeoGebra is as a source of activities for students, but also how rich the task of making things in GeoGebra is, and wonder: how do I encourage more sketch creation from students?

Having sunk the time into making the sketch, I did think about what activity could go with it, and whipped this up:


I'd be interested in feedback on the sketch. Does it have the features you'd want for your students to use? Should it have more cute doodads? What do you think about the design of such a sketch as a task? How far should the students be past the material before trying to design the sketch?

Why Questioning?

Found at From-Student-to-Teacher Tumblr
From Joy of Literacy

Last week became questioning week with student teachers. It came up in two action plans and we were able to have some really interesting discussions about it.

A lot of what I'm writing about here is from work with David Coffey, inspired by Kathy Coffey, and processed from Mosaic of Thought (link includes Chap.1 as a sample) among other books. Neither Dave nor I can remember the actual origin... which is sometimes symptomatic of having done it yourself.

In my own growth as a teacher questioning is definitely one of the places where effort and reflection have helped me improve. The first level was just asking better math problems. More open-ended, that required more problem solving. When the problems are better, there's more interesting things to ask the students about later. The next was to ask more appropriate questions. Some of those excellent problems I gave to students were too much for students. This is a zone of proximal development idea. When it was too much, I needed to scaffold. Later I learned about rephrasing the question instead of narrowing it. Later still I learned about demonstrations of how I think about a problem, sometimes more appropriate than guiding students through it. Then you ask 'what did you notice?', which is - of course - one of the all time great math questions.

One of the best things I ever learned was to stop being the authority. I don't say what is right or wrong. I ask the students if they 'agree or disagree?' That might start an actual conversation. 

One of the first things I really noticed about teaching was that students would tell me that they couldn't do it (whatever it was at the time) but when I asked them questions they could get from beginning to end with no difficulty. So, obviously, I needed to teach them to ask those questions of themselves.   It was difficult. Really difficult. Shifting from asking them 'how to do (next step)?' to 'now what?' and 'how do you know?'

Literacy learning experts are good about this idea of questioning as a process, with the idea that questions are how we move ourselves forward. I like this framework to help me think about the kinds of questions that I'm asking.


Obviously, we have had a tendency to ask too many literal and application questions in math class.  I think about inference questions being predictions, reading between the lines, hypothetical questions and the like. Analysis questions are reflections, synthesis, connections, recommendations and so on.

What I shared that seemed to tie it together for the student teachers is a simple idea: ask to find out what I want to know about. I don't need to ask for answers - I know those. I don't need to ask right or wrong. I need to ask about what they are thinking. Students know when a question is genuine, and this simple idea has improved my assessment more than anything else.  I'm more persistent in getting answers when I really want to know them, also.

From A Softer World

Some good student teacher writing on questioning:

• Caitlin - Open Ended Questions
• Stephanie - Questioning Techniques