It was a cover letter that gave a link to a really simple website, and on that website was an educational philosophy, a few sample tests, and some student work. Although it was pretty basic, what I liked was that on that simple site I got a *much* better sense of who this candidate was. I loved the idea. And I decided then and there that I would create my own teaching portfolio online that would capture who I was as a teacher.

This past summer, I did it.

To be clear: this isn’t a *reflective* teacher portfolio. It’s a *descriptive* teacher portfolio. It is something that I put together — a mishmash of snippets — that together hopefully gives a solid sense of who I am, what I do, what I believe in. I think calling it a **visual teaching resume **or a** wunderkammer** best describes it. (Click on the image to go to the site.)

There are a few missing things that I would like to add to this site at some oint:

- I would like to add everyday samples of student work. Not projects. Just everyday stuffs.
- I would like to add a section about the two week history of science course I designed and implemented with another teacher this year. (See Days 80-87 on my 180 blog for more.)
- I would like to add a section about the “Explore Math” project (more info here and here) I did in Precalculus this year.
- I would like to finish the student quotation page. I actually have quotations typed for a number of previous years, but I do not have more recent years ready.

It was pretty simple to make (I used the free website creator weebly) and I hope if I ever were to go on the job market, it would catch the eyes of whoever had the giant stack of cover letters and resumes in front of them. I wasn’t really going to make a post about my visual resume, or share it with anyone, because I thought: *who would care?*

But heck: maybe someone out there is going on the job market and thinks the idea is worth replicating? So I decided to post.

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**The question in the survey:**

The “Explore Math” project is something I’ve never done before. I explained my reasoning behind it — which is I wanted to encourage you to see that there is so much more than our curriculum covers, and let you just have fun looking at math stuff outside of our curriculum… and get some easy credit for it (almost everyone is getting full credit for the first batch of things I’ve seen). However, as a teacher, I know something like this could easily be seen as busy work, and that was my big concern — that it would feel like a chore rather than something you actually want to do.

This is me laying my cards on the table. If I came to you in the student center and told you this and asked you for your thoughts, what would you say?

**Every Student Response In Entirety:**

I really liked the Explore Math project and I definitely would say it was an overall success. I loved how many options we were given for what we could do, and the fact that you gave us the options was great because otherwise it can feel like you are just trying to desperately research and find a topic to write about. My Explore Math topics I thought were extremely interesting, and it was cool to even connect some to the stuff we were learning in class. It was a lot of writing, which is something foreign for math classes, and also made it kind of difficult to grasp exactly how to format what we were writing (five page essays for each topic?). One other thing that was a little stress-inducing was the deadline and I know it was for a problem for most people that it often happens that when there are multiple assignments due on one day, students leave them all and do them in bulk. Because of this, having the deadline of the first three due in February was definitely helpful. Overall, I really loved the assignment.

I really liked this project! I found a lot of things about math that I would have never known about if we weren’t assigned this project. I learned new formulas, new (very addictive games), great youtube channels and informative popular articles. I found an entirely new community online that I did not know existed.

At first I expected it to feel like a bit of a chore but when I actually sat down and did it, it was pretty fun. I think it was great that there were multiple ways you were allowed to “explore math.” I also thought it was amazing I could play around with the project a little bit to find areas of math that are aligned with my personal interests. Being able to think about how math affects our society, in a math class, was an amazing interdisciplinary activity. I think it’s good that not every option was a math puzzle — that would have felt constrictive.

I would say as long as the students are innovative, interested and patient people the project sounds wonderful. The student, if very interest in math, should be encouraged to further their mathematical understanding, and find means in which math is even more interesting to them as it was prior. Emphasizing the point that one (the student) does not need to seek the more difficult problem or most tedious theorem is also very helpful, as the student will be encouraged to explore areas of math in which really interests them.

I would say that I absolutely love the explore math project. I have always been a person who enjoyed math that connected with the world. Being in a classroom memorizing formulas was never my interest and I was psyched when you announced the project. I think that this project can be very helpful in putting math on the global scale for students who only see it as a class in a school. This opens their eyes to new heights math can taken and how much math actually helps outside of the classroom.

I agree it felt like busy work some. I find it weird that something that’s supposed to be us having fun exploring math had a grade and time constraint attached to it. That’s one thing I didn’t like.

All I have to say is that this was not busy work; in fact it was productive and learning work. I found this to be incredibly intensive and interesting, and it broadened my horizons of the understandings of applied mathematics and sciences, and introduced me to things that I had previously trembled [at] before, like string theory, for instance. I thought this was a great project and a simple and easy way to get us thinking in a mathematical mindset, and I am definitely reaping the benefits from it, because I have come away with much more knowledge about certain aspects of math that I had previously not known. I really wouldn’t know what to change because I liked these individual explorations so much and they intrigued me so much. Thank you for giving a projected that I was thoroughly interested in, seriously!

For someone who is very interested in math in and out of the classroom, I am generally engaged with math concepts that are not a part of our curriculum. Thus, this was a good experience for me in that I was able to get credit for simply enjoying and exploring math; it also perhaps pushed me a little bit to go further than I normally would in exploring mathematical concepts online. However, for students who don’t love math outside of the classroom, I could definitely see how this might have seemed like busy-work. If you don’t genuinely enjoy math, then writing a lot about it and research about it is going to be cumbersome, but if you do, it’s enjoyable.

I really liked doing the explore math assignment. I liked that you were giving us an outlet for us to not just do the math that needs to be done in order to complete the class. This assignment allowed me, personally, to dive deeper into how math can be applied to the world and that math is actually occurring all the time. Also, I remember not really understand[ing] infinite series and then I did an explore math with infinite series that really helped me because it was a visual representation that really clicked with me.

I think that initially I thought the project might just be busy work and I didn’t really understand what we were expected to be doing. Once I read over the assignment and saw the scope of the projects we were allowed to do, I was much more interested and saw the project completely differently. I think that it is important to highlight, when giving the assignment, how broad a range of options you have when doing this, and that there are so many math projects that relate to everyday life that could be interesting if you just *think* about it, rather than relying on the assignment sheet completely to guide you.

Personally, I have enjoyed what I have done so far. Just recently, I voiced my concerns about the state of math in America and was able to comprehensive research about the bitcoin that I would not have done on my own. That being said, some of this has seemed like busy work and stuff “I just have to do for credit.” Since it seems like you genuinely want us to enjoy the project, it might be made better by making it extra credit. That way, we could be able to explore as much as we want without worrying about our grade.

I had a really awesome time doing my Explore Math assignments, but the one thing you could do to make it less busy work is make it 3 different assignments, rather than 5, and make them a little more in depth, and more interesting in that regard. I think that if the students only had to do 3, they could expand more on what they were interested in.

I really like the idea, but for me personally, it turned into busy work. Not because I find it boring but because I have so much other work that it gets pushed back towards the end of my load. I would like to spend more time on them, so possibly have it on top of the nightly work for math, designate a night specifically to explore math.

This is practically the farthest thing from busywork we can do! Repetitive problems often seem like busywork. Practice is always good, but once you have something down, it can be quite annoying to practice it over and over again. Sometimes i feel that way about homework, but with this project we’re choosing any math-y thing that interests us! We have a lot of freedom, and hopefully it piques an interest in math outside of the curriculum. This project is great, personally, I wish I had taken more time with it. As long as you don’t procrastinate too badly with it, I don’t see how this project could be a chore, unless you claim to hate math.

I LOVED this project, and I wish we got to do more things like this throughout the year. (I know we can do things like this whenever we want, but it’s really nice to get some recognition and the chance to formally share your math ideas with others.) As a side note, this project was also interesting to be doing while looking at colleges for the first time. I know that sounds like a really strange thing to say, but getting to enjoy math in new contexts, such as music theory, has given me new ideas of things I would like to pursue and take classes [on] while I am at college because we don’t always get to learn about things like this on a daily basis in high school.

I do admit that I wasn’t very enthusiastic at the start of the project, but as soon as I started I completely changed my mind. Most of the work that I did was stuff I had never done before and might never do again. I was genuinely interested in what I was doing, and it was great to be able to choose what I focused on instead of being told what to look at.

I understand why you assigned this project, and I think it is very important to see the relevance math has in the world. This breathes life into the abstract “why are we learning this” type that doesn’t appear to have anything to do with life outside the classroom. However the problems with this assignment are that I didn’t know what I was searching for. When I found the Sloane’s Gap video and paper I felt like I struck gold after seemingly endless mining. However the mining part is very un-exciting. Not un-exciting enough to undo the excitement of finding the cool stuff, but it’s not very encouraging either. I wouldn’t want this assignment to turn into a chose 5 of these pre-determined projects because that wouldn’t make anyone feel like anyone feel like they’re venturing outside the classroom. I’m not really sure what I would do to change this assignment, but I think it really is a good idea that with some refinement could become a really dynamic way to get into math. I think keeping it low pressure and “easy credit” is the way to go because stress + ambiguity about an assignment is a terrible combination that would only end in resentment from your students, and students not enjoying their work.

Honestly, I had quite a bit of fun with the “Explore Math” project as I saw many cool analogies of real-world applications of math. For example, one of my five “research topics” was the probability and randomly guessing on every SAT multiple choice question. I learned that the probability is horrifyingly low — I already knew this, but not to such an extent. Furthermore, I saw some very cool analogies in this SAT topic; for instance, if a computer were to take the SAT 1 million times a day, for five billion years, the chance of any of the SATs resulting in a perfect score on just the math section would be about 0.0001%. Crazy, I know!

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At the end, he said something powerful. The first thing one needs to do to when leading a purposeful life is to say what it is that you want to do. Articulate it aloud. And that is scary. Making it public so you can hear yourself say it, but also so someone else can hear you say it. So it becomes *real* instead of this thing that bounces around in your head but never gets out. And so at the end, he told everyone to be quiet, and he was going to say something he wanted to do, and then afterwards there should be silence… and when anyone else wanted to say something they wanted to do — something they would declare out loud — they should stand up and say it, and then remain standing. This was an open invitation to the students in these honors societies, but also to the parents and teachers there as well.

The speaker said: “I want to change the world.”

Silence.

A little more silence where everyone looked around and felt uncomfortable.

Then a student — one courageous student — got up and said something. And remained standing.

And then another. And another.

The head of the upper school said something. Then more students. Then a parent. Then me. Then another math teacher. Then more students.

At the end, every student made a declaration, and a few adults too. *It is scary*. But it also showed me how much courage our kids have. Their declarations ranged from showing others that girls can do math and science *to* spreading love *to* making people laugh *to* promoting peace *to* inventing something *to* becoming a biochemist *to* making a mark on the world. Big things and small things, lofty things and concrete things, but all things that share with the room a sense of self and a sense of purpose.

I loved watching this.

I also loved and hated how hard it was for me to come up with my thing. My purpose in life. I said:

*I want to make it so that kids see math as an artistic and creative **endeavor*.

And I meant it. Because you know what has been bouncing around in *my* head that I have been having trouble articulating? I am now pretty good at coming up with deep and conceptual approaches to mathematical ideas. And I’m okay at promoting mathematical communication. And I’m transitioning to having kids do groupwork all the time, to learn from each other — so I am not the sole mathematical authority in the room.

But all of that said: **I don’t think I teach math in a way to shows how it is an art form, how deeply creativity and mathematics are intertwined.** And I know that this is one of my charges as a teacher moving forward. It’s going to be an uphill challenge, and one that will likely take me many years to wrap my head around. The hurdles are significant. Having a set non-problem-solving-based curriculum which doesn’t allow time for much mathematical “play,” nor for the inclusion of rich problems with multiple entry points, is the largest hurdle. But there must be ways — activities or units here and there — that can illuminate the artistry and creativity of doing and discovering mathematics. And I want to be involved in finding ways for this to happen. Yes, this happens at math circles. Yes, this happens at math clubs. Yes, this happens at summer math programs. That’s where the love and excitement and understanding of the *beauty* of mathematics unfolds for many students. But I want to find a way for this to happen in a normal classroom, with normal students, with the normal constraints. That (one of) my purposes.

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Tonight I wanted to see if I could re-derive it like I did before — and lo and behold I did. I’m curious if any of you have done it the way I did it, or if there are other ways you’ve learned to approach this problem. (There is a student who I had last year who created this amazing 3-d version of this using the edges of a cube and some string. I love the idea of asking — for this 3-d figure — what *surface* is generated by the intersections of these strings.)

We start out by having these lines which form a family of curves. But of course we’re not graphing *all *the lines. If we were, we’d get something more dense like this.

The main idea of what I’m going to do to find that curve… I’m going to pick two of those lines which are *infinitely close to each other *and find their point of intersection. That point of intersection will lie on the curve. (That’s the big insight in this solution.) But I’m not going to pick two *specific* lines — but instead keep things as general as possible. Thus when I find that point of intersection for those two lines, it will give me *all* the points of intersection for *all* the lines.

Watch.

First we pick two arbitrary lines.

We’ll have one line move down on the y-axis units (and thus over on the x-axis units). And the second line will be moved down on the y-axis just a tiny bit more (down an additional units). Yes, we are going to have that tiny bit, that , eventually go to zero.

The two lines we have are:

A little bit of algebra is needed to find the point of intersection. Setting the y-values equal:

And then doing some basic algebra:

Now solving for we get:

So the point of intersection is:

Here’s the kicker… Remember we wanted the two lines to be *infinitely *close together, right? So that means that we want to go to zero. Thus, our point of intersection of these *infinitely* close lines will be:

or .

Beautiful! And recall that we picked the lines arbitrarily. By varying and plotting , we can get any two lines on our doodle.

But I want an equation.

Simple. We know that . Thus .*

Since , we have

Let’s graph it to check.

Huzzah!!! And we’re done!

I wonder if I can do something similar with this cardioid:

I think I must (for funsies) do some investigation of “envelopes” this summer. I mean, Tina at *Drawing on Math* even introduces conics with these envelopes!

**An extension for you. Do something with this 3d string-art.**

*Of course you might be wondering why I don’t say . Since is between 0 and 1, we know that must be positive.

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Earlier this year, I was struck by this post by Fawn Nguyen. It’s rare that I read something and it just keeps rattling around in my brain, and won’t let me forget it. (Thanks Fawn, for being an annoying bee attacking my brain!) If you’re too lazy to click the link, the TL;DR version:

Fawn has her kids go to Math Munch and explore and play with mathematics it based on what interests them. She has her kids keep track of what they do with this sheet:

What I loved about this? **It gave kids the freedom to explore mathematics that interested them. The assignment was fairly low-pressure. **

I wanted to do something similar. I knew I wanted it to be low-pressure to do, fairly easy to grade, and really focus on what the *kids* want to do. Thus, **Explore Mathematics! **was born.

[.docx]

Students are asked to engage with mathematical things that they are interested in during the third quarter. There are two deadlines, so they are working on them continuously and not rushing at the end to finish them. (Also to make marking them easier for me.) There is a low-pressure grading structure, which reinforces the notion that this is more about just engaging and less about “doing the right thing.” In total, I’m making it worth about half a normal test.

I don’t know exactly how this is going to turn out. But I’ve already had a student present a piece of mathematical artwork he’s made, and I’ve had a couple fun conversation with kids about things they’re thinking of doing/looking at. I hope this fosters a lot of fun mathematical conversations between me and the kids about the things they’re finding (and of course, among the kids themselves).

The biggest concern is making this assignment not *seem like* or *become* busywork for the kids. I don’t want it to seem like added work just for the sake of extra work! That’s the fine line I am trying to navigate — sort of “forcing” kids to carve out some time here and there in their busy schedules to get exposed to the cool things out there. I have to figure out how I can create this feeling in the kids. Maybe that means I will give up some classtime for them to work on this every-so-often, to show them I value this sort of exploration. Wish me luck on this.

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and derive

and

simply by dividing both sides of the original equation by or .

I did this same this year.

Except later on, a few weeks ago, I saw a post on twitter talking about introducing trigonometric identities through graphs on the unit circle — and having kids come up with their own identities. I loved this idea and planned to make a whole thing about of it. So far I’ve given students one thing I’ve made as a result of this idea (and that worked out super well).

From this, students were able to come up with the three Pythagorean Trig Identities we saw above, but also a fourth one that was totally unexpected.

I had them all pick a different angle and substitute it into the left hand side and the right hand side of the last equation. KABAM! Whoa! SAMESIES! (Note to self: Next year make a dynamic visualization of this triangle on Geogebra, like this but better/cleaner.)

Instead of doing a whole unit on Trigonometric Identities, the other teacher and I are slowly giving students a problem here and a problem there to practice with and find new strategies, over a couple weeks. I hope that works! And maybe if I have time, I’ll make a follow up activity. Maybe giving the drawing below but without anything labeled but the radius of the circle, and having kids fill in each of the lengths and find various identities? They can use the ratios of similar sides… but also if triangles are similar, they can also use the ratios of the perimeters! Or knowing that the ratio of the areas of two similar figures is simply the square of the ratio of two corresponding sides? Also, maybe just maybe kids could generate inequalities — like the area of this one triangle will always be less than the area of this other triangle?

I don’t quite know as things aren’t fully formed in my head yet. If anyone has any ideas, or existing resources, pass ‘em along!

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***

How can we create a realistic pathway for underserved students to become scientists, mathematicians, engineers, and programmers?

That question launched the Summer Program in Mathematical Problem Solving. You see, my view of mathematics was shaped by the incredible experiences I had outside of school: summer programs, math clubs, math contests and more that gave me access to abstract mathematics. I felt that creating a way for all kids to get this experience and then shepherding through more programs for advanced study through middle school and high school could finally make it possible for them to succeed at the highest levels.

SPMPS has been hugely successful, and many of our kids have gone on to great high schools and summer programs. We just finished our third year, and next year we’re opening a second site. That means we’re looking to double our pool of instructors. Sam invited me to do a guest post to invite you all to come to our program as faculty, and I hope you will!

The program is an incredible place to work at. You are invited to create your own classes, on pure topics such as number theory, combinatorics, or logic; or applied topics such as circuit design, astrophysics, and digital communications. You can also teach a problem solving course. The students are amazing, and despite the many challenges they face they have a great abilities for abstract reasoning. They also have a huge hunger for learning: they do seven hours of mathematics per day and they love it! Some of the results last year include proving the infinitude of primes, solving a challenge problem to determine if it’s possible to put + and – signs between the numbers 1 2 3 … 50 to get 0, constructing a binary adder using AND/OR/NOT gates, and sending an image from one computer to another using sound.

If you want to learn more, take a look at our website and then contact us to get a lot more information and the application itself. If you’re not interested in coming yourself, please pass it on to someone who might be. I’d love to tell you more, and hopefully to work with you this summer!

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I think I now have a way that might help students to get conceptually understand what’s going on. I only had the insight 10 minutes ago so I’m going to use this blogpost to see if I can’t get the ideas straight in my head… The point of this post is *not* to share a way I’ve made the chain rule understandable. It’s for me to work through some unformed ideas. I am not yet sure if I have a way to turn this into something that my kids will understand.

So here’s where I’m starting from. Every “nice” function (and those are the functions we’re dealing with) is basically like an infinite number of little line segments connected together. Thus, when we take a derivative, we’re pretty much just asking “what’s the slope of the little line segment at ?” for example.

Now here’s the magic. In my class, we’ve learned that **whatever transformations a function undergoes, the tangent line undergoes the same transformations**! If you want to see that, you can check it out here.

For a quick example, let’s look at and .

We see that is secretly which has undergone a vertical stretch of 2, a horizontal shrink of 1/5, and has been moved up 1.

Let’s look at the tangent line to at . It is approximately .

Now let’s put that tangent line through the transformations:

Vertical Stretch of 2:

Horizontal shrink of 1/5:

Shift up 1:

Now let’s plot and our transmogrified tangent line:

Yay! It worked! (But of course we knew that would happen.)

The whole point of this is to show **that tangent lines undergo the same transformations as the functions** — because the functions themselves are pretty much just a bunch of these infinitely tiny tangent line segments all connected together! So it would actually be *weird* if the tangent lines didn’t behave like the functions.

So why not look at function composition in the same way?

**We can look at a composition of functions at a point as simply a composition of these little line segments. **

Let’s see if I can’t clear this up by making it concrete with an example.

Let’s look at .

And so we can be super concrete, let’s try to find , which is simply the slope of the tangent line of at .

I’m going to argue that just as and are composed to get our final function, we can compose the tangent lines to these two functions to get the final tangent line at .

Let’s start with the . At , the tangent line is (I’m not showing the work, but you can trust me that it’s true, or work it out yourself.)

Now let’s start with the square root function. We have to be thoughtful about this. We are dealing with which really means that we’re taking the square root of 9. We we want the tangent line to at . That turns out to be (again, trust me?): .

So now we have our two line segments.

We have to compose them.

This simplifies to:

Let’s look at a graph of and our tangent line:

Yup!

Where did we ultimately get the slope of 2 from? When we composed to two lines together, we multiplied the slope of the inner function (12) by the slope of the outer function (1/6). And that became our new line’s slope.

Chain rule!

For any composition of functions, we are going to have an inner and an outer function. Let’s write where we can clearly remember which one is the inner and which one is the outer functions. Let’s pick a point where we want to find the derivative.

We are going to have to find the little line segment of the inner function and compose that with the little line segment of the outer function, both at . That will approximate the function at .

The line segment of the inner function is going to be

The line segment of the outer function is going to be

I am going to keep those terms blah1 and blah2 only because we won’t really need them. Let’s remember we *only want the derivative* (the slope of the tangent line), not the tangent line itself. So our task becomes easier.

Let’s compose them:

This simplifies to

And since we only want the *slope* of this line (the derivative is the slope of the tangent line, remember), we have:

.

Of course we chose an arbitrary point to take the derivative at. So we really have:

Which is the chain rule.

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Everything I do in my course aims for this. Sometimes I succeed. Sometimes I fail. But I don’t lose sight of my goal.

Each year, I have parts of the calculus curriculum I rethink, or have insights on. In the past few years, I’ve done a lot of thinking about ** limits **and where they fit in the big picture of things. Each year, they lose more and more value in my mind. I used to spend a quarter of a year on them. In more recent years, I spent maybe a sixth of a year on them.

*Okay, not really. But kinda. I’ll explain.

First I’ll explain my reasoning behind this decision. Then I’ll explain how I did it.

For me, calculus has two major parts: the idea of the derivative, and the idea of the integral.

Limits show up in both [1]. But where do they show up in derivatives?

- when you use the formal definition of the derivative

and… that’s pretty much it. And where do they show up in integrals?

- when you say you are taking the sum of an infinite sum of infinitely thin rectangles

and… that’s pretty much it. I figure if that’s all I need limits for, I can target how I introduce and use limits to really focus on those things. Do I really need them to understand limits at infinity of rational functions? Or limits of piecewise functions? Or limits of things like as ?

Nope. And this way I’m not wasting a whole quarter (or even half a quarter) with such a simple idea. All I really need — at least for derivatives — is how to find the limit as one single variable goes to 0. C’est tout!

This was our trajectory:

(1) Students talked about average rate of change.

(2) Students talked about the *idea* of instantaneous rate of change. They saw it was problematic, because how can something be changing *at an instant*? If you say you’re travelling “58 mph at 2:03pm,” what exactly does that mean? There is no time interval for this 58mph to pop out of, since we’re talking about an instant, a single moment in time (of 2:03pm). So we problematized the idea of instantaneous rate of change. But we also recognized that we understand that instantaneous rates of change do exist, because we believe our speedometers in our car which say 60mph. So we have something that feels philosophically impossible but in our guts and everyday experience feels right. Good. We have a problem we need to resolve. What might an instantaneous rate of change mean? Is it an oxymoron to have a *rate of change* at a *instant*?

(3) Students came to understand that we could *approximate *the instantaneous rate of change by taking the slope of two points *really really really* close to each other on a function. And the closer that we got, the better our approximation was. (Understanding why we got a better and better approximation was quite hard conceptual work.) Similarly students began to recognize graphically that the slope of two points really close to each other is actually almost the slope of the tangent line to the function.

(4) Now we wanted to know if we could make things exact. We knew we could make things exact if we could bring the two points *infinitely *close to each other. But each time we tried that, we got either got two points pretty close to each other or the two points lay directly on top of each other (and you can’t find the slope between a point and itself). So still we have a problem.

And this is where I introduced the idea of introducing a new variable, and eventually, limits.

We encountered the question: “what is the exact instantaneous rate of change for at ?

We started by picking two points close to each other: and

This was the hardest thing for students to understand. Why would we introduce this extra variable . But we talked about how wasn’t a good second point, and how also wasn’t a good second point. But if they trusted me on using this variable thingie, they will see how our problems would be resolved.

We then found the average rate of change between the two points, recognizing that the second point could be really faraway from the first point if were a large positive or negative number… or close to the first point if were close to 0.

Yes, students had to first understand that could be *any* number. And they had to come to the understanding that represented where the second point was in relation to the first point (more specifically: how far horizontally the second point was from the first point).

And so we found the average rate of change between the two points to be:

We then said: how can we make this exact? How can we bring the two points infinitely close to each other? Ahhh, yes, by letting get infinitely close to 0.

And so I introduce the idea of the limit as such:

If I have ,** it means what blah gets infinitely close to if gets infinitely close to 0 but is not equal to 0. That last part is key.** And honestly, that’s pretty much the entirety of my explanation about limits. So that’s the 5 minutes I spend talking about limits.

So to find the instantaneous rate of change, we simply have:

This is simply the slope between two points which have been brought *infinitely* close together. Yes, that’s what limits do for you.

And then we simplify:

Now because we know that is *close *to 0, but *not equal to 0*, we can say with confidence that . Thus we can say:

And now as goes to 0, we see that gets infinitely close to 6.

Done. (Here’s a do now I did in class.)

We did this again and again to find the instantaneous rate of change of various functions at a points. For examples, functions like:

at

at

at

For these, the algebra got more gross, but the idea and the reasoning was the same in every problem. Notice to do all of these, you don’t need any more knowledge of limits than what I outlined above with that single example. You need to know why you can “remove” the (why it is allowed to be “cancelled” out), and then what happens as goes to 0. **That’s all. **

Yup, again, notice I only needed to rely on this very basic understanding of limits to solve these three problems algebraically: **means what blah gets infinitely close to if gets infinitely close to 0 but is not equal to 0. **

(5) Eventually we generalize to find the instantaneous rate of change *at any point, *using the exact same process and understanding. At this point, the only difference is that the algebra gets slightly more challenging to keep track of. But not really that much more challenging.

(6) Finally, waaaay at the end, I say: “Surprise! The instantaneous rate of change has a fancy calculus word — *derivative.*“

Apologies in advance if any of this was unclear. I feel I didn’t explain thing as well as I could have. I also want to point out that **I understand if you don’t agree with this approach.** We all have different thoughts about what we find important and why. I can (and in fact, in the past, I have) made the case that going into depth into limits is of critical importance. I personally just don’t see things the same way anymore.

Now I should also say that there have been a few downsides to this approach, but on the whole it’s been working well for me so far. I would elaborate on the downsides but right now I’m just too exhausted. Night night!

[1] Okay, I should also note that limits show up in the definition for continuity. But since in my course I don’t really focus on “ugly” functions, I haven’t seen the need to really spend time on the idea of continuity except in the conceptual sense. Yes, I can ask my kids to draw the derivative of and they will be able to. They will see there is a jump at . I don’t need more than that.

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What is 1 Radian?” Try it. Dare ya. They’ll do a little better with: “What is 1 Degree?”

I really loved the question, and I did it last year with my precalculus kids, and then again this year. In fact, today I had a mini-assessment in precalculus which had the question:

What, conceptually, is 3 radians? Don’t convert to degrees — rather, I want you to explain radians on their own terms as if you don’t know about degrees. You may (and are encouraged to) draw pictures to help your explanation.

My kids did pretty well. They still were struggling with a bit of the writing aspect, but for the most part, they had the concept down. Why? It’s because my colleague and geogebra-amaze-face math teacher friend made this applet which I used in my class. Since this blog can’t embed geogebra fiels, I entreat you to go to the geogebratube page to check it out.

Although very simple, I dare anyone to leave the applet not understanding: “a radian is the angle subtended by the bit of a circumference of ~~the circle~~ ~~that has 1 radius~~ a circle that has a length of a single radius.” What makes it so powerful is that it *shows *radii being pulled out of the center of the circle, like a clown pulls colorful a neverending set of handkerchiefs out of his pocket.

If you want to see the applet work but are too lazy to go to the page, I have made a short video showing it work.

PS. Again, I did not make this applet. My awesome colleague did. And although there are other radian applets out there, there is something that is just *perfect** *about this one.

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It’s amazing. You’re amazing. You joined in the Explore the MathTwitterBlogosphere set of missions, and you’ve made it to the eighth week. It’s Sam Shah here, and whether you only did one or two missions, or you were able to carve out the time and energy to do all seven so…]]>

Here I’m reblogging our last mission from the Explore the #MTBoS!

Originally posted on Exploring the MathTwitterBlogosphere:

It’s amazing. You’re amazing. You joined in the *Explore the MathTwitterBlogosphere* set of missions, and you’ve made it to the eighth week. It’s Sam Shah here, and whether you only did one or two missions, or you were able to carve out the time and energy to do all seven so far, I am proud of you.

I’ve seen so many of you find things you didn’t know were out there, and you tried them out. Not all of them worked for you. Maybe the twitter chats fell flat, or maybe the whole twitter thing wasn’t your *thang*. But I think I can be pretty confident in saying that you very likely found at least one thing that you found useful, interesting, and usable.

With that in mind, we have our last mission, and it is (in my opinion) the *best* mission. Why? Because you get to do something…

View original 501 more words

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Kate Nowak played “log war” with her classes. I stole it and LOVED it. Her post is here. It really gets them thinking in the *best kind of way. *Last year I wanted to do “inverse trig war” with my precalculus class because Jonathan C. had the idea. His post is here. I didn’t end up having time so I couldn’t play it with my kids, sadly.

This year, I am teaching precalculus, and I’m having kids figure out trig on the unit circle (in both radians and degrees). So what do I make? The obvious: “trig war.”

The way it works…

I have a bunch of cards with trig expressions (just sine, cosine, and tangent for now) and special values on the unit circle — in both radians and degrees.

You can see all the cards below, and can download the document here (doc).

They played it like a regular game of *war*:

I let kids use their unit circle for the first 7 minutes, and then they had to put it away for the next 10 minutes.

And that was it!

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Today, I feel a slight glimmer of motivation. And so here I am.

Here’s what I want to talk about.

In calculus, we all have our own ways of introducing *the power rule for derivatives*. Graphically. Algebraically. Whatever. But then, armed with this knowledge…

that if , then

…we tend to drive forward quickly. We immediately jump to problems like:

take the derivative of

and we hurdle on, racing to the product and quotient rules… We get so algebraic, and we go very quickly, that we lose sight of something beautiful and elegant. This year I decided to take an extra few days *after the power rule* but *before problems like the one listed above *to illustrate the graphical side of things.

Here’s what I did. We first got to the point where we comfortably proved the power rule for derivatives (for n being a counting number). Actually, before I move on and talk about the crux of this post, I should show you what we did…

Okay. Now I started the next class with kids getting Geogebra out and plotting on two graphics windows the following:

At this point, we saw the transformations. On the left hand graph, we saw that the function merely shifted up one unit. On the right hand graphs, we saw a vertical stretch for one function, and a vertical shrink for the other.

Here’s what I’m about to try to illustrate for the kids.

**Whatever transformation a function undergoes, the tangent lines to the function also undergoes the exact same transformation**.

What this means is that if a function is shifted up one unit, then all tangent lines are shifted up one unit (like in the left hand graph). And if a function undergoes vertical stretching or shrinking, all tangent lines undergo the same vertical stretching or shrinking.

I want them to *see* this idea come alive both graphically and algebraically.

So I have them plot all the points on the functions where . And all the tangent lines.

For the graph with the vertical shift, they see:

The original tangent line (to ) was . When the function moved up one unit, we see the tangent line simply moved up one unit too.

Our conclusion?

Yup. The tangent line changed. But the slope did not. (Thus, the derivative is not affected by simply shifting a function up or down. Because even though the tangent lines are different, the slopes are the same.)

Then we went to the second graphics view — the vertical stretching and shrinking. We drew the points at and their tangent lines…

…and we see that the tangent lines are similar, but not the same. How are they similar? Well the original function’s tangent line is the red one, and has the equation . Now the green function has undergone a vertical shrink of 1/4. And lo and behold, the tangent line has also!

To show that clearly, we did the following. The original tangent line has equation . So to apply a vertical shrink of 1/4 to this, you are going to see (because you are multiplying all y-coordinates by 1/4. And that simplifies to . Yup, that’s what Geogebra said the equation of the tangent line was!

Similarly, for the blue function with a vertical stretch of 3, we get . And yup, that’s what Geogebra said the equation of the tangent line was.

What do we conclude?

And in this case, with the vertical stretching and shrinking of the functions, we get a vertical stretching and shrinking of the tangent lines. And unlike moving the function up or down, this transformation *does* affect the slope!

I repeat the big conclusion:

**Whatever transformation a function undergoes, the tangent lines to the function also undergoes the exact same transformation**.

I didn’t actually tell this to my kids. I had them sort of see and articulate this.

Now they see that if a function gets shifted up or down, they can see that the derivative stays the same. And if there is a vertical stretch/shrink, the derivative is also vertically stretched/shrunk.

The next day, I started with the following “do now.” We haven’t learned the derivative of , so I show them what Wolfram Alpha gives them.

For (a), I expect them to give the answer and for (b), .

The good thing here is now I get to go for depth. **WHY?**

And I hear conversations like: “Well, g(x) is a transformation of the sine function which gives a vertical stretch of 3, and then shifts the function up 4. Well since the function undergoes those transformations, so does the tangent lines. So each tangent line is going to be vertically stretched by 3 and moved up 4 units. Since the derivative is only the *slope* of the tangent line, we have to see what transformations affect the slope. Only the vertical stretch affects the slope. So if the original slope of the sine function was , then we know that the slope of the transformed function is .

That’s beautiful depth. Beautiful.

For (b), I heard talk about how the negative sign is a reflection over the x-axis, so the tangent lines are reflected over the x-axis also. Thus, the slopes are the opposite sign… If the original sine functions slope of the tangent lines was , then the new slopes are going to be .

This isn’t easy for my kids, so when I saw them struggling with the conceptual part of things, I whipped up this sheet (.docx).

And here are the solutions

And here is a Geogebra sheet which shows the transformations, and the new tangent line (and equation), for this worksheet.

Now to be fair, I don’t think I did a killer job with this. It was my first time doing it. I think some kids didn’t come out the stronger for this. But I do feel that the kids who do get it have a much more intuitive understanding of what’s going on.

**I am much happier to know that if I ask kids what the derivative of is, they immediately think (or at least can understand) that we get , because…**

** our base function is which has derivative (aka slope of the tangent line) … Thus the transformed function is going to be a vertical stretch, so all the tangent lines are going to be stretched vertically by a factor of 9 too… thus the derivative of this (aka the slope of the tangent line) is .**

To me, that sort of explanation for something super simple brings so much graphical depth to things. And that makes me feel happy.

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At the start of the lesson, I gave each group one colored piece of paper. (I got this idea last year from my friend Bowen Kerins on Facebook! He is not only a math genius but he’s also a 5 time world pinball player champion. Seriously.) I don’t know why but it was nice to give each group a different color piece of paper. Then I had them designate one person to be the “paper master” and two people to be the friends of the paper master. Any group with a fourth person simply had to have the fourth person be the observer.

I did not document this, so I have made photographs to illustrate ex post facto.

I started, “Paper master, you have a *whole sheet of paper! One whole sheet of paper!* And you have two friends. You feel like being kind, sharing is caring, so why don’t you give them each a third of your paper.”

The paper master divided the paper in thirds, tore it, and shared their paper.

Then I said: “Your friends *loveeeed* their paper gift. They want just a little bit more. Why don’t you give them each some more… Maybe divide what you have left into thirds so you can keep some too.”

And the paper master took what they had, divided it into thirds, and shared it.

To the friends, I said: “Hey, friends, how many of you LOOOOOVE all these presents you’re getting? WHO WANTS MORE?” and the friends replied “MEEEEEEEEEEEEEEE!”

“Paper master, your friends are getting greedy. And they demand more paper. They said you must give them more or they won’t be your friends. And you are peer pressured into giving them more. So divide what little you have left and hand it to them.”

They do.

“Now do it again. Because your greedy friends are greedy and evil, but they’re still your friends.”

“Again.”

“Again.”

Here we stop. The friends have a lot of slips of paper of varying sizes. The paper master has a tiny speck.

I ask the class: “If we continue this, how much paper is the paper master going to eventually end up with?”

(Discussion ensues about whether the answer is 0 or super duper super close to 0.)

I ask the class: “If we continue this, how much paper are each of the friends going to have?”

(A more lively short discussion ensues… Eventually they agree… each friend will have about 1/2 the paper, since there was a whole piece of paper to start, each friend gets the same amount, and the paper master has essentially no paper left.)

I then go to the board.

I write

and then I say: “How much paper did you get in your initial gift, friends?”

I write

and then we continue, until I have:

Ooohs and aahs.

Next year I am going to task each student to do this with two friends or people from their family, and have them write down their friends/family member’s reactions…

I love this.

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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 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 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 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

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

**E:** the -1 represents the instantaneous rate of change of at . 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 .

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.

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