We used a problem from the Calculus Nspired activity Infestation to Extermination recently during our unit on differential equations:

The rate of increase of bugs is proportional to the number of bugs in a certain area. When t=0, there are 2 bugs and they are increasing at a rate of 3 bugs/day.

What does this mean?

I set the mode to individual and watched as students worked.

Many recognized that the rate of change changes.

Several used the initial condition to write a statement about the rate of change.

Eventually, we went back to the given information to decipher what it was saying.

And then we (anti)derived the model for exponential growth, which of course students recognized using in a previous math course.

So what are the constants for this particular model?

I sent a poll to collect their model honestly having no idea that the bell was going to ring in less than two minutes. A few students correctly answered before the end of class.

Productive struggle isn’t fast.

I should have paid better attention to the time … I really had no idea it had taken us as long as it did. But students were engaged in “grappling with mathematical ideas and relationships” the entire time. That’s got to be better for their learning than them watching me tell them how to work the problem.

What opportunities are you giving your students to struggle productively? Even if you don’t “cover” as much as you think you should?

And so the #AskDontTell journey continues …

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howardat58

March 9, 2015 at 6:42 am

There’s much more going on here !

Questions that need answers are to do with the real world.

1. How do they know that the initial rate is 3 per day?

2. how in practice could this fact have been established?

3. Initial population 2 / So 1 male and one female ! or two hermaphrodites !!

As differential equations always arise from real world situations there is much scope for real inquiry into “modeling with math” in this example.

ps This is NOT a criticism, but just suggestions. Reality must get a look in sometimes !