The modeling method I’ve been using to teach physics has really made me think about way I teach Algebra. Here’s a good example:

**The topic:** Exponential Functions

**The way I taught it last year:**

* Step 1:* Introduce the topic with some notes and some example problems. I used a PPT presentation to show the students the equation y = a*b^x and explain what all the parts mean. Then we looked at some example problems.

* Step 2:* Explore the topic with an activity. We did the classic M&M’s activity to explore a “real world” example of exponential growth.

* Step 3:* Practice. We used problems from the text, most of which did not have a context.

* Step 4:* Assess. We had a test after about a week or so. Done.

**The way I taught it this year (modeling):**

* Step 1:* Explore the topic starting with this clip from the movie Contagion. I got the idea from Leslie Macfarlane on 101 Questions. I used this handout with it. We started by discussing what factors would effect the spread of a virus. We came up with a bunch, but most importantly: the type of virus (R-0 in the video clip) and the number of people who currently carry it. Here’s what their handouts looked like filled in:

* Step 2:* The kids figured out

*on their own*(in groups) that the flu virus doubles each cycle and the smallpox virus quadruples each cycle. They also were able (with a little help) to figure out that the cycle # (x) needed to be the exponent in the function. Then, it wasn’t hard to get them to write the function like this:

**(total # of people infected) = (starting number of people) * (mutiplier)^x**

* Step 3:* Generalize it! We then discussed a general form for the equation. They needed to make a connection between the R-0 number from the video and the multiplier they used to generate their charts. As soon as I brought up the question, they noticed that it was just (1 + R). At this point, my head almost exploded from awesomeness. Now, they understood that the “multiplier” they used was given by one plus the growth rate. How did that happen?! I didn’t “teach” it to them!! So now they’ve got y = a * b^x, they know what the “b” means, and since we graphed each initial scenario, it was simple for them to see that “a” is the y-intercept.

* Step 4:* Expand the generalization to new scenarios (exponential decay). We used this handout to figure out that if you are losing a certain percentage each cycle, then it’s just (1 – R) instead.

* Step 5:* Practice. I gave them 3 or 4 scenarios of each type to practice. For each scenario, they made a chart and a graph (using Desmos or by hand). The answer to any questions asked was secondary to showing me that they could analyze the situation using the exponential model.

* Step 6:* Assess. Two short multiple choice quizzes, some questions with contexts, some without. Then an open ended task in which they had to explain to me the differences between two functions. Retakes are available on all three assessments as needed. They also take the assessments when they are ready.

**Why is this new way better?**

I have students now who are willing to try figuring things out on their own. They see that it can be done. Especially in Step 4, I saw a bunch of students working to find a function that defined an exponential decay scenario. They didn’t say “Mr. Owen, what’s the formula?” or “I don’t know how to do this because you didn’t teach us this yet.” Check out the example below. This girl decided that (hmw * .10) = y and that the next value in the table was given by (original amount – y). I just had to suggest using substitution to combine those two expressions. From there, it wasn’t too hard for her to see that to do that 10 times, you just raise the expression to the tenth power.

Here’s an example of a practice quiz done by a student (this one has no context):

How did they know to write terms in exponential form if they had never been shown that form? It’s just a convention. This is not a trivial question for me – is each child supposed to invent its own math conventions, surely somewhere along the way we say, “This is how the rest of the world writes this.” It’s where I have philosophical difficulties with discovery learning – I hear it is being applied to linear algebra (really? students are going to come up with matrices, determinants and all the conventions on their own? Everyone has to be a genius?) Seems kind of unfair. I am all about standing on the shoulders of giants, frankly.

I totally stole your worksheets. sorry.

Totally flattered that you would steal my worksheets! I must be getting better at it.

Getting them to write the function using an exponent was the hardest part. But that’s why we started with the example of something doubling each cycle, and starting at one. They were struggling (sweet!) to find a function that defined the data, so I gave them a hint. I wrote next to the chart 2, then 2*2, then 2*2*2 and so on. They got the idea that I was leading them towards exponents and then it was only a matter of time before they noticed that the “x” value was the exponent.

I also teach them physics using modeling, so they’re used to the method, that could be another factor that helps them to pick things up.

I’d love a copy of the handout. It is no longer available through your link.

Sorry about that! I moved all my stuff from Dropbox to GDrive and forgot about changing some links here. I fixed it though, it should work fine now. Thanks for the heads up!