"The Grasshopper and the Ant" teach Exponents

What do you notice, what do you wonder when you see this image?

It looks cold, how do you know which one is the ant? Which one is the grasshopper? Which one looks warmer? How do you think the discussion is going?

Next have the students read the fable that is associated with it:

A Grasshopper gay Sang the summer away,
And found herself poor By the winter's first roar.
Of meat or of bread, Not a morsel she had!
So a begging she went, To her neighbour the ant,
For the loan of some wheat, Which would serve her to eat,
Till the season came round. "I will pay you," she saith,
"On an animal's faith, Double weight in the pound
Ere the harvest be bound." The ant is a friend
(And here she might mend) Little given to lend.
"How spent you the summer?" Quoth she, looking shame
At the borrowing dame. "Night and day to each comer
I sang, if you please." "You sang! I'm at ease;
For 'tis plain at a glance, Now, ma'am, you must dance."

What do you notice? What do you wonder now?

Where math comes in to play is the idea of where the grasshopper says, "I will pay you, she saith,

On an animal's faith, Double weight in the ." When you can't afford something say, the full price of a car, how do you afford it?  Most students will talk about saving money or a loan. If the grasshopper wants to survive, she wants a cut of the ants food savings and next season the grasshopper will give double back. Ask your students is this fair? Is this how a bank works?

There is a way to calculate it mathematically, but right now I want students getting use to the idea of exponents. The equation I would have them use is the compound interest equation.

A=P(1+r/n)^(nt)

Where A is the amount
P is the principle
r is the interest rate
n is the number of times it is compounded per year
t is the time in years.

I would have students think about what each of them means and how the rate effects how much the grasshopper would way in the long run.

Exponential Growth through TAG

After a great day of teaching exponential growth and decay, I felt like my students really knew the topic forwards and backwards. We did this Desmos activity as some students were finish up their MAP test.

It was a great Desmos activity, almost all the students wanted 100$ at the beginning and very insightful finishing questions at the end and I was pleased overall. Later that day I began to wonder if students would notice if something was exponential or not exponential if I gave it to them. I thought to myself how could I find a question that I could do that would model exponential growth or decay.

Then I found this game: 

I wanted an activity that got students out of the classroom. Since I have two periods at the beginning of the day we did inside in the gym, but the last time we went outside and played it on the Football field.

If you didn't watch the video, it is a simple game where one person is a shark, they yell "Minnows come out to play." the minnows job is to make it to the other side without getting tagged. The sharks job is to tag people, once a minnow is tagged they become a shark.

Here comes the math:

I had them start out with one shark, I made all the others line up and asked them how easy it was going to be this time down. All of them were confident that they could make it down without any real sweat, then I asked them what about the 5th time down? I was surprised how most of them thought it was still going to be easy, thinking of it linearly instead of exponential. We played it through the first time here are the pictures and the charts we did at the end.

Here is one of the charts that I made after each run down and back.

After all the students were tagged on the 4th down and back with ease. I asked them to estimate how many would get tagged on the fourth time back. Then I asked about the whole school, how many down and backs would there be playing with 576 students?

We came back after playing 2-3 more times. Then asked them how you could write an equation to model the graph. We did a short mini-lesson on finding equations of a exponential graph.