# 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

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.

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

• 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.

This slideshow requires JavaScript.

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

If you’re reading this post, it means that you are someone I think is amazing. Because you already are reading math teacher blogs. (Or at least one of them.) And if you’ve read stuff on my blog in the past two years, you’ll see that I’ve fallen head over heels in love with this math teacher community that I’ve found on the interwebs. They have become not only colleagues and collaborators and constant inspiration, but also they’ve become my friends.

I want as many math teachers that are out there to have as amazing an experience as I have had. I hate writing about this community because I think everyone out there thinks How hyperbolic is this guy!? Seriously?! But it’s not hyperbole. It’s real. IT’S REAL. IT’S REAL.

This past summer, four of us (Tina Cardone, Julie Reulbach, Justin Lanier, and me!) sat down and brainstormed how we could offer a way for someone to experience what we’ve experienced. We wanted to come up with a no-pressure way to help those new to the online math teacher world make their way into meeting others out there and finding access to amazing resources… and a way to let those who have been in this world all the shiny new and amazing stuff that has been generated in the past year.

And then we had it! The idea!

If you are intrigued, click on the banner and see what it’s all about. The fun starts on October 6th, 2013.

# First Day

I know I’ve been truly negligent about writing. If you know me, you know my philosophy around blogging is do it when you’re inspired, do it for yourself. And recently I just haven’t felt like it. That being said, today was finally our first day back with kids. We’ve had a week and a half of meetings without kids, but today was the first day with kids.

And I had my first class with each of them. Each class was only 30 minutes. It was designed so we could get a taste of them and they could get a taste of us. We have bonding activities for the next three days, and then academic classes will be in normal full swing the following week.

This year I’m teaching the same courses as last year (precalculus, calculus, multivariable calculus). Normally I start off the first day jumping straight into math — but it’s always been kinda boring math. Kids take notes, we have nightly work, we go to it. But this year, I wanted to mix things up a bit.

In Multivariable Calculus, I gave them an optimization problem that we do later in the year, and literally had them make a guess, we recorded the answers that their guesses gave, and then I broke them into groups and they were trying to come up with ways to solve the problem. I gave them nothing. It was interesting to listen to their conversations. Some were intuitive and geometric, others were going straight up algebraic work, and others were playing around with a mixture of both algebra and guess-and-check. I basically said they have until Monday (when we next have our class) to do whatever they can to make progress on the problem.

What’s so powerful about this is that I think I can parlay this discussion into a discussion of big ideas in multivariable calculus because they are already nudging their way towards level curves. It’s like three layers below what they were talking about today, but I think through our discussion I can dig it out. I have never really thought of multivariable calculus as a discovery based course… I don’t have enough of a command of the material (or the time) to actually do that. But heck if I myself can’t see glimpses of what that might look like. From just 30 minutes today.

The other amazing thing about that class was that I have in it a number of students I taught in my first year of teaching — they were sixth graders then, and they are seniors now. How amazing is that?

In calculus, I did something I’ve been wanting to do for years — a “3 Act” with my calculus kids. What’s been hard about this is that I always want to do them later in the year, and I always stop myself because I’m nervous I won’t do it justice, and also because I always feel like I don’t have time to spend. But today I got a taste of doing it, and I’m hooked. I did the taco cart problem which anyone from late middle school on should be able to solve. But what’s nice is that the kids seemed to get into it. And I’m not terrible at facilitating them (even though I did tell kids when they were right before we did the great reveal — which was soooooo dumb of me)! I can see this becoming a more regular part of my teaching. I already am planning on using the toothpicks 3 act in Precalculus in a couple weeks.

Precalculus was the class I was least imaginative with, because I did what I did last year. I had kids start to expand $(x+y)^n$ for various exponents, and then start to identify patterns. Kids were noticing stuff that I hoped they would, but one kid took things way further and saw Pascal’s Triangle (kinda… he saw the pattern in the coefficients) which was so lovely. The other classes I know a good number of kids from previous years, but this class is all people new to me! I’m nervous about learning their names.

Being back at school — now that kids are here — is putting me back in a more zen place, because they helped me remember what I love about teaching so much. The start-of-year-meetings are done and gone, long live the kids!

# Weekly Math Subject Chats are Live Now

The math chats for each subject area, middle school through Calculus, have all officially begun!  Here is the list of chats.

Subject Twitter # Day / Time Facilitator Middle School Math  #msmathchat Monday 9PM EDT @justinaion  @luvbcd @shlagteach Algebra 1  #alg1chat Sunday 9 PM EDT @lmhenry9  @_MattOwen_  @anthonya @kathrynfreed Algebra 2  #alg2chat

Julie Reulbach wrote a blogpost about the new Twitter chats that have formed. I've participated in two of the precalculus ones and I've gotten amazing stuff from them. Highly recommended!

# Inverse Trig Functions

At TMC13, I was in a group of people talking about precalculus. One of the exercises we did was make a list of some of the topics we found challenging to teach as teachers — and we broke out in groups to try to come up with ways to tackle those topics.

My group’s topic was inverse trig functions. (This was with April, Dan, Greg, and Andrew.)

Our initial task was to find the deep mathematical idea behind the topic… why we teach it, what we think we can get out of it conceptually… and what we sort-of converged on is that the topic really illuminates the idea of inverses and restricted domains. And that’s about it. And when push came to shove, we decided we didn’t find that restricted domain is something we really care about. We decided we didn’t really care about the inverse trig graphs, and the work we put into that side of things wasn’t really worth what we little we were able to squeeze out of it. It’s not that it is horrible, but we just didn’t couldn’t justify it.

So, honestly, we decided to just focus on inverses, and the idea of them as “backwards problems.”

Thus, we came up with two things:

1. A packet that has students secretly engage with inverse problem work before they even know what they’re doing. So the first packet is meant to be used before any unit circle trig is introduced. (A few of us, especially April, did something similar in her classes, and randomly, Greg Taylor did a my favorites on the same essential idea!)

In fact, if I were to use this in the classroom, I would not even mention the words “trigonometry.” I would focus on the idea of coordinate planes and circles, and simply leave it there.

2. A packet that students work on after they learn unit circle trig — and that more formally introduced the idea of the inverse trig functions. It tries to draw connections between the unit circle, the sine/cosine graphs, and their calculators.

There are concept-y questions for both packets. I’m including both packets below in one document. I’m posting one with a few teacher notes, and one with the teacher notes hidden. (The .docx is here if you want to edit!)

Packet with teacher notes

Packet without teacher notes

We did all this planning in pretty much an hour and a bit — from start to finish. And then I pulled together the ideas to make this document. I’m not sure I was able to capture everything we talked about, but I think I got most of the big things. Apologies to my collaborators if I totally botched the translation of our vision to reality!