Month: October 2013

Do They Get It? The Instantaneous Rate of Change Exactly

Today in calculus I wanted to check if students really understood what they were doing when they were finding the instantaneous rate of change. (We haven’t learned the word derivative yet, but this is the formal definition of the derivative.)

So I handed out this worked out problem.

And I had them next to each of the letters write a note answering the following individually (not as a group):

A: write what the expression represents graphically and conceptually

B: write what the notation \lim_{h\rightarrow0} actually means. Why does it need to be there to calculate the instantaneous rate of change. (Be sure to address with h means.)

C: write what mathematical simplification is happening, and why were are allowed to do that

D: write what the reasoning is behind why were are allowed to make this mathematical move

E: explain what this number (-1) means, both conceptually and graphically

It was a great activity. I had them do it individually, but I should have had students (after completing it) discuss in groups before we went to the whole group context. Next time…

Anyway, the answers I was looking for (written more drawn out):

A: the expressions represents the average rate of change between two points, one fixed, and the other one defined in relation to that first point. The average rate of change is the constant rate the function would have to go at to start at one point and end up at the second. Graphically, it is the slope of the secant line going through those two points.

B: the \lim_{h\rightarrow0} is simply a fancy way to say we want to bring h closer and closer and closer to zero (infinitely close) but not equal zero. That’s all. The expression that comes after it is the average rate of change between two points. As h gets closer and closer to 0, the two points get closer and closer to each other. We learned that if we take the average rate of change of two points super close to each other, that will be a good approximation for the instantaneous rate of change. If the two points are infinitely close to each other, then we are going to get an exact instantaneous rate of change!

C: we see that \frac{h}{h} is actually 1. We normally would not be allowed to say that, because there is the possibility that h is 0, and then the expression wouldn’t simplify to 1. However we know from the limit that h is really close to 0, but not equal to 0. Thus we can say with mathematical certainty that \frac{h}{h}=1

D: as we bring h closer and closer to 0, we see that h-1 gets closer and closer to -1. Thus if we bring h infinitely close to 0, we see that h-1 gets infinitely close to -1.

E: the -1 represents the instantaneous rate of change of x^2-5x+1 at x=2. This is how fast the function is changing at that instant/point. It is graphically understood as the slope of the tangent line drawn at x=2.

I loved doing this because if a student were able to properly answer each of the questions, they really truly understand what is going on.

Switching Up Groups

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Today I switched up the groups I had in my Precalculus class and my two Calculus classes. All three classes have kids sitting in groups of 3 (and occasionally 4). They were with their original group members for the first half of the quarter. Now they are getting new group members.

I know this is a little hokey (and I tell them that), but I really want groups to get started on the right foot. I want kids to be thinking about how they interact in groups. I want kids to work at being a good group member.

So I did the following things. After they sat in their new groups, I gave them 20 seconds of silence where they were going to decide the very first thing they were going to say to their new groupmates. Then they went around and said whatever it is they wanted to say.

Then I had them write their names on a notecard. On the front, they were asked to write something they thought was a strength of theirs when working in a group. Some examples:

Good at finding new ways to explain things to people, or simpler ways

I’m pretty articulate and willing to help if someone needs it.

I’m a good listener.

I listen to your answer and compare. I do not think mine is always correct.

Not leaving people being / staying together as a group.

Then on the back, I had students write something they weren’t good at in a group setting, but wanted to work on. Some examples:

Sometimes I don’t ask questions when I might need to.

If I’m stuck I don’t ask for help enough.

If you’re moving too quickly through a problem for me, I typically won’t say anything.

I usually speed ahead.

Then I had kids go around and share what great quality they are going to be able to share with the group, and something they are going to pledge to work on with the group. What was nice is that in those few minutes when they were talking, I saw other people acknowledge and listen. One group noticed that together their strengths and weaknesses worked well together!

I know it’s hokey. I told them I knew it was hokey. But kids shared good things with each other. And maybe it’ll help a few kids be more aware of the others in their group too.

Starting Calculus with Area Functions

So I decided to try a new beginning to (non-AP) calculus this year. Instead of doing an algebra bootcamp and diving into limits, I decided to teach kids a new kind of function transformation. I’d say this is something that makes my classroom uniquely mine (this is my contribution to Mission 1 of Explore the MTBoS). I don’t think anyone else I know does something like this.

You see, I was talking with a fellow calculus teacher, and we had a big realization. Yes, calculus is hard for kids because of all the algebra. But also, calculus involves something that students have never seen before.

It involves transformations that morph one graph into another graph. And not just standard up, down, left, right, stretch, shrink, reflect transformations. Although they do transform functions, they don’t make them look too different from the original. Given a function and a basic up, left, reflect, shrink transformation of it, you’d be able to pair them up and say they were related… But in calculus, students start grappling with seriously weird and abstract transformations. For example: if you hold an f(x) graph and an f'(x) graph next to each other — they don’t look alike at all. You would never pair them up and say “oh, these are related.”

So I wanted to start out with a unit on abstract and weird function transformations. Turns out, even though the other teacher and I had brainstormed 5 different abstract function transformations, I got so much mileage out of one of them that I didn’t have to do anything else. You see: I introduced my kids to integrals, without ever saying the word integrals. Well, to be fair, I introduced them to something called the area transformation and the only difference between this and integrals is that we can’t have negative area. [1]

You can look at this geogebra page to see what I mean by area functions.

Here’s the packet I created (.docx)

That packer was just the bare backbones of what we did. There was a lot of groupwork in class, a lot of conceptual questions posed to them, and more supplemental documents that were created as I started to realize this was going to morph into a much larger unit because I was getting so much out of it. (I personally was finding so much richness in it! A perfect blend of the concrete and the abstract!)

Here are other supplemental documents:

2013-09-16 Abstract Functions 1.5

2013-09-17 Abstract Functions 1.75

2013-09-20 Area Function Concept Questions

2013-09-23 Abstract Functions 1.9375

The benefits I’m already reaping:

  • It’s conceptual, so those kids who aren’t strong with the algebraic stuff gain confidence at the start of the year
  • Kids start to understand the idea of integration as accumulation (though they don’t know that’s what they are doing!)
  • Kids understand that something can be increasing at a decreasing rate, increasing at a constant rate, or increasing at an increasing rate. They discovered those terms, and realized what that looks like graphically.
  • Kids already know why the integral of a constant function is a linear function, and why the integral of a linear function is a quadratic function.
  • Kids are talking about steepness and flatness of a function, and giving the steepness and flatness meaning… They are making statements like “because the original graph is close to the x-axis near x=2, not much area is being added as we inch forward on the original graph, so the area function will remain pretty flat, slightly increasing… but over near x=4, since the original function is far from the x-axis, a lot of area is being added as we inch forward on the original graph, so the area function shoots up, thus it is pretty steep”
  • Once we finish investigating the concept of “instantaneous rate of change” (which is soon), kids will have encountered and explored the conceptual side of both major ideas of calculus: derivatives and integrals. All without me having used the terms. I’m being a sneaky teacher… having kids do secret learning.

I mean… I worked these kids hard. Here is a copy of my assessment so you can see what was expected of them.

I love it.

Love. It.

LOVE.

IT.

I’m going to put a picture gallery below of some things from my smartboards.

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[1] To be super technical, I am having kids relate f(x) and \int_{0}^{x} |f(t)|dt