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Polar. Graph. Contest.

Here’s what I hung up last week:

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Here’s a closeup of some of them…

 

 

These are polar graphs that students designed using Desmos. Then I printed them out on photopaper and hung them up.

This was something I wanted to do after introducing polar graphing. Why? Because one day during the polar unit, I started playing around with desmos and accidentally created:

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… from something so simple …

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(Now to be fair, desmos isn’t great with creating great complicated polar graphs… and it’s better to write them parametrically to get a bit more accuracy… so this is a bit of a lie of a graph in that it isn’t totally accurate… but it’s oh so pretty.) [1]

So after our unit on polar graphing, I took 10 minutes at the start of class to introduce this idea of a Polar. Graph. Contest. First I threw this image up:

Polar.Graph.Contest..png

I then pulled up desmos and asked my kids to shout out some polar function. I graphed it. Then I put in a slider or two. So for example, if they said r=\cos(\theta), I might have added the slider r=\cos(a\theta)+b. And then I started changing the sliders. Then I might have altered the function a bit more, like r=\cos(a\theta)+b\theta and we saw what happened. Then I gave everyone 7 minutes to just come up with something pretty.

It was magical.

Kids just started playing. They dug into old functions they had learned about. They got excited by what they were seeing. They gasped and turned their screens to show their friends. Some who were getting boring graphs saw the cool graphs their classmates were getting and were inspired to mix things up since they knew they could make neat things. #mathjoy in the house.

My heart was singing.

Then I showed kids a google doc which had all the info for the contest — and the link to the google form to submit their entries. There were initially two contests. Students needed to create the coolest polar graph with one equation. And students needed to create the coolest polar art using multiple equations. However, some students were animating the sliders and coming up with fun animations (like this or this… watch both for a while). So I added an optional third animated polar graph category.

I haven’t yet told my kids who the winners are. I want to just let them appreciate the work of their classmates for now.

After creating the bulletin board, I’ve seen kids look at the artwork. Kids from my class, but also kids from other grades. And what I’ve found fascinating is that so far, very few kids pick the same polar art pieces as their favorites. I expected everyone to love the same ones I do. But it just isn’t the case. I think when I announce the winners, I’ll have the class go to the board, have everyone point out a few that they like, and then I’ll make my grand pronouncement.

Student Feedback:

I asked my kids, when submitting their artwork, “This is something new I came up with this year. I want to know if you enjoyed doing it or not. No judgments if you didn’t. Y’all tend to be honest when I ask for feedback, and I appreciate it! I genuinely want to know. I also am a bit curious if you had any mathematical thought as you were playing on Desmos? You don’t have to say what thoughts you had (if any) — just if you had any.”

Every student responded positively. Some responses included:

  • I think this was so awesome! I love art and this felt like art to me. It is so fun when art and math intersect, I loved it!!!!!
  • Messing around with the graphs was actually more entertaining than I thought it would be. I spent a lot more time on this than I thought I would, and I feel like I’ll probably spend more time on this trying to find a really cool design I like (and possibly gaining a better understanding why the graphs look the way they do…).
  • I had so much more fun doing this than I thought I would, honestly. Once I finished my multi-equation graph, I looked at the clock on my computer and realized I had been working on it for nearly 20 minutes; it had seemed like maybe 5.
  • I really enjoyed doing this assignment. I felt that I learned a lot about polar through it. I didn’t think too much about math while making my graphs, however I thought about math a lot in order to observe and think about patterns I found in my graphs.

Now I want to be frank: there isn’t much “learning” that happens when kids are doing this assignment. This isn’t a way to teach polar. But it is a way to get kids to appreciate the power of polar when they are done working with polar, and what sorts of different kinds of graphs compared to the boring ol’ rectangular coordinate system. I just wanted kids to play, like I played, and get excited, like I got excited. It’s a slightly different way to appreciate the power of math, and I am good for that. Especially since it only took 10 minutes of classtime!

 

As an aside, I love that when I tweeted this out, a tweep said he was going to be doing this in his class after his kids learn about circles. Um, hell yeah!

Circle.PNG

 

[1] So there are two ways to graph polar in desmos. First is the straight up polar way, and the second is the parametric way. It turns out that the polar way is solid for most things, but it loses refinement at times. Let me show an example. If we graphed r=\cos(57\theta), we should get a flower with 57 petals.

And happily, if we graphed it in both polar and parametric, we get the same looking graph:

 

However if we zoom in a bunch, we can see that the red graph (the polar equation) is interesting and stunning, but just isn’t correct. While the zoomed in blue graph (the parametric equation) is more boring, but is technically correct.

 

It turns out desmos samples more points using parametrics than polar.

As a result, a few of the polar artworks my kids made aren’t “true.” Their pieces are a desmos quirk, like the red graph is above. But what a lovely desmos quirk.

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Part II: Transition from Anti-Derivatives to Integrals

In my last post, I outlined the road trip scenario I used to prime kids to think about areas under curves. It had nothing to do with anti-derivatives. And that’s important to keep in mind. This post is going to try to outline how we made the intellectual leap from areas under curves to anti-derivatives. [Update: I wrote this during 40 minutes of free time I had in school today where I didn’t want to do other things I was supposed to be doing. So I didn’t get to writing the part of the lesson where we make The Leap. This post ends where we’re literally all primed to make the leap. But indeed! I will make the leap with you! But in Part III. Which will be written. Soon. I hope.]

The road trip introduced this idea that kids can approximate how far someone traveled using a left-right-midpoint Riemann Sum approximation (we did not give it that name…). It arose naturally from the roadtrip scenario.

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We also made the conclusion that if we had more data, we could get a better approximation for how far someone traveled. To remind you, we started with this data:

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and then we got more data:

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That’s going to be our transition. We are now going to give infinite velocity data! [1]

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Wonderfully, kids had no problem with doing this. The reason I highlighted question 1c is because I was very intentional about including that question. When students graphed 1e, they often would draw:

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I didn’t correct anyone while they were working. And it was nice to hear a few groups have the requisite conversation about why we needed to connect the points. Afterwards, when we debriefed as a whole, this was something I highlighted. We knew the position at every moment in time, including at t=2.31 (as asked in 1c).

Kids continued on with Questions 2, 3, and 4. They flew through these, actually.

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These were golden. Let me say that again: these were golden. [2] It was amazing to watch kids:

  • Parse the connection between velocity graphs and position graphs
  • Understand the idea of negative velocity
  • Think about the fact that we have to specify an initial position in order to create a position graph
  • Draw a connection between motion on a number line and a graph of position v. time
  • Understand what distance and displacement are, and see the difference between the two

Seriously, just watching kids work through these problems was… well, I’ll just say this feels like something I’m super proud of creating. It didn’t take much time to do but gave us so much fodder.

We didn’t need a lot of time to debrief these four questions. I had students highlight a few things, and I made sure we brought up the fact that we were drawing line segments for the position graphs, and not something curvy. Because constant velocity means position is changing at a constant rate, it’s linear. So for example, the position graph for 3c would look like what I have below. It isn’t a parabola.

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But there was one huge thing we had to go over. With the roadtrip, we drew a connection between area and how far someone went. Most kids, as they were doing these problems, didn’t think about area. I wanted kids to think about area. So in our debrief, I explicitly asked them what our huge insight from the roadtrip was (area as distance travelled!), and if we could apply it to one of the problems.

So first I went back to question 2c. And I asked students how they calculated their answer:

pic6.pngKids said she went backwards a total of 24 units. So they did 100-24. And then I explicitly had them draw the connection to what we did with the roadtrip. This is when we talked about it being area, but “signed” to represent the backwardsness.

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To be clear, some students had already been thinking in this way (about area/signed area) when working on these problems. But most hadn’t been, so we had to bring that idea to the forefront.

Then I had a student talk through Question 3 with areas in mind:

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And finally, I asked groups to discuss how we could understand distance and displacement from the velocity graph.

 

 

There is one more part to this packet I had my kids work on that I will outline in my next blogpost in the series! But here’s an editable .docx of the file I made [2018-05-02 Velocity Graphs]. And here’s the document to view here:

Stay tuned for Part III.

 

[1] Both @calcdave and I stumbled upon the same approach for this!

[2] The only note is that a few students didn’t realize the time interval was 1/2 hour for Question 4. And it involves a fair amount of calculation.

 

Part I: Transition from Anti-Derivatives to Integrals

An Important Prelude: On one of the early days in my calculus class, I have groups imagine a roadtrip. It’s one of the very best things I do, because it problematizes the idea of how something can be going at a particular speed at a moment in time. Like a speed involves a rate of change of position in a time interval, but we don’t have a time interval at a moment in time. So saying something like “we were going at 58mph at 2:03pm” suddenly goes from a statement kids accept to a problematic statement. And at that moment, we’re ready for calculus.

Setting the Stage Now: In calculus, I had gotten my kids to take tons of derivatives, and then taught them about antiderivatives and how to take them. But of course the whole time they were doing antiderivatives, they were asking “but why are we doing this?” I got this question a few times, but I just said “you’ll see very soon… probably next week… I just want to get some of this algebraic thinking out of the way so we can focus on concepts.” [1]

The Problem: I looked back at what I’ve done in the past about how to draw the connection between the antiderivative and signed areas together, and had a bunch of pretty terrible methods. I didn’t like any of them. So I put out a call on twitter while brainstorming myself. [2] I got a ton of responses, three of which stood out to me. The first one is something I kinda did at the start of the year already (before we did any calculus), so I’m excited for when my kids see the connection…

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The second came was from the appropriately named @calcdave:

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What’s incredible is that when I tweeted out asking for help, I had already started brainstorming … and it was so amazing that we had such similar ideas that I had to take a photo of my scratch work to send David:

 


I also got a DM from Brett Parker (@parkermathed) with a screenshot of how he digs into the idea. And what was his idea?

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Yup. Driving in a car. Which of course got me thinking of the start of the year when I basically transitioned to derivatives using a road trip.

What I did: Part I

So I told kids we were going to put a pin in anti-derivatives for a short while to go on a short detour. And I used Brett’s setup, but changed it to be a followup from the roadtrip we had talked about earlier this year. We read the setup together, and then I gave each group a few minutes to talk about part (a).

sheet1Surprisingly, this was not an easy question for kids. Many didn’t instantly think distance=(rate)(time). Additionally they didn’t know what assumption to make about the speed for the minute that passed between 7:10 and 7:11. I emphasized the approximate part of the question, and really told kids they would need to make an assumption.

When we debriefed, most kids suggested Alex was probably driving 69mph for the minute, so they did 69\cdot\frac{1}{60}.

Some suggested he was maybe going 70mph for most of the minute, so they suggested 70\cdot\frac{1}{60}. And with that, I suggested that maybe he was going 68mph for most of the minute… We did the calculations for all three and saw they were pretty similar.

notes1.png(We also had discussed that in that minute, perhaps Alex started at 70mph, went to 100mph, and then slowed down to 68mph… We just didn’t know. So we were making and using an assumption, but one that is pretty reasonable.)

Then kids in their groups went to the next three parts. And each group was assigned an assumption: average speed, left hand speed, or right hand speed.

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The only errors I saw kids make in part (b) was not taking the different time intervals into account. Since we did one example which had a time interval of \frac{1}{60} hour, some groups were using that time interval for everything. But pretty rapidly most kids got there.

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In our debrief, I wrote out the calculation kids did for the left hand assumption, and then asked students what I would have to change to do the calculation for the right hand assumption. (That was written in yellow.) This question was a key question to ensure kids understood the difference.

Then we talked why this was an approximation? (We had to make assumptions about the speed of the car at all the times inbetween the times we were given.) And then kids said that to get a more accurate distance for Alex, we’d need more data.

And that’s precisely what they got when they flipped the sheet over.

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I assigned some groups the left hand assumption, and some groups the right hand assumption. I thought with so many data points, kids would be like “argh! I have to do all these calculations!” But no, there were no audible groans (I don’t think…) And they plotted the points and then did the calculations. I told kids to write out all their calculations instead of just saying how far Alex drove in the first minute, the second minute, the third minute, etc. No everyone listened. Shame. They lost out. Because when we debriefed, we saw:

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Yup. The fact we could factor out the 1/60th is key. It made this all go so much faster. Only a few kids noticed that when doing the calculations. And then we compared that with the right hand assumption:

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The same answer! Why? Kids saw that the sum inside the brakets (after factoring out the 1/60) would be the same because the starting speed and ending speed were both the same.

So we’re done, right? NO. We had two more moves to make.

First, we looked at the graph.

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And I asked: “if we were doing the left hand assumption, what would our velocity graph look like?” And we concluded that for each 1 minute interval, Alex was driving at a constant rate. So it would look like this:

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Second, I asked: Between 7:05 and 7:06, how much are we assuming Alex went? (Kids answered 67\cdot\frac{1}{60}) Here’s where I did some talking. I could probably have asked kids to think geometrically and had them come up with this, but I was running out of time.

I said: “where do we see 67 represented on this graph?” (Kids said the height of the velocity graph from 7:05 to 7:06.) I then said: “where do we see the \frac{1}{60} on this graph?” (Kids said the 1 minute length.) So I drew a rectangle. Slowly. And then shaded it in. Slowly. And turned around. Slowly.

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Yes. Audible gasps from many. I then said: “What about 7:04 to 7:05? What was our calculation for the distance Alex traveled?” And after a few more, we saw:

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Kids saw. Heck yeah. They got that the approximate distance that Alex traveled was algebraically calculated in one way. And then they saw that number had a graphical representation. It was awesome.

I left by showing the original velocity points graphed. Reminded them of our left hand assumption, and that it was just an assumption, though a pretty good one. And then I drew the curve below. And dropped the microphonic device and left.

 

All in all, this was a pretty detailed blow-by-blow. And since I did it in two different classes, and things unfolded slightly differently in each, this is an amalgam of what happened. But it’s a pretty solid recap of the story I wanted to tell and how it was told. (And it’s a testament to the help of my (twitter) frands.) Most of the time, students were working. But a lot of great conversations happened as a whole class. It is a long post, but the question we worked on was only one page [editable version to download: 2018-04-30 A Road Trip Reprise] And it probably only took 20 minutes total from start to finish. A really exciting 20 minutes for me.

Stay tuned for how I used the idea that @calcdave and I both stumbled upon to make the connection between area and anti-derivatives. Right now kids have seen that there is a connection between area under a velocity curve and the distance someone has traveled. There is still no connection to anti-derivatives. That’s coming up.

[1] To be honest, I made the decision years ago to do tons of antiderivatives before introducing the integral. I wanted all the algebraic work done and solid before we introduced the concept of the integral. I didn’t want kids messing with the idea of signed area, new notation, and tough antiderivatives all at the same time. I still kind of think this is the right decision, but some doubts have crept in. I just really hate when I have to say “you’ll see why… promise…”

[2] Apparently I asked this on my blog ages ago in 2009.

A simple vector problem with a rich set of approaches

In precalculus, we do a little bit with vectors. And last year and this year, I gave a basic problem to my kids. (I think I found it in some standard textbook.) What I love about this problem is that there are so many ways kids approached it. All essentially the same, but all different enough. Because my kids weren’t all doing it the same way, it has shown me that we are teaching them well. And also, it reminds me that a super basic problem can be a super rich problem.

vector problem

Approach 1: Since we were in vector land, a few kids solved it like this. They found the vector from P to Q, and then set the magnitude equal to 5 and did the algebra.

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Approach 2: Similarly, some students just used the distance formula that they had memorized.

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I like that this kid expanded out (-3-x)^2 and then later eventually factored. Because that was slightly different than how the student in approach 1 solved it (leaving (x+3)^2 as is, and then taking the square root of both sides.

Approach 3: One student found the equation of a circle of radius 5 around the point (-3,1). Then they realized that they were looking for the solution to a system of equations for the circle and the line y=4. So they substituted y=4 into the equation of the circle and solved!

approach3

Approach 4: Most students took a geometric/visual approach. They drew the point (-3,1) and the line y=4. Then they drew these two triangles (seeing that the vertical distance from the point to the line was 3 and the diagonal distance from the point to the line was 5, since we want a distance of 5 away from (-3,1). Then they used 3-4-5 right triangles to get the horizontal distance.

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All of these were lovely. I enjoyed seeing them all together and drawing some connections among them. Most kids were awed when they saw Approach 3. And since so many students didn’t take a straight up algebraic approach, they were like “ooooh” when they saw Approach 1/2. I supposed what I most like about this is that it really highlights how circles, distance, and vectors are all essentially tied together. I mean kids should know that circles and distance are fundamentally related (but of course they don’t always remember that). But this problem connects those two concepts with something new: vectors. And that the magnitude of vectors simply being an equation involving a circle, secretly.

***

Actually, while I’m writing this, I might as well share this other problem that had a couple of approaches. This was the basic question:

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Approach 1: Most students took this approach. They drew the vector, and then drew a smaller vector with unit length, and then used similarity to find this new vector (with a scale factor of \frac{1}{\sqrt{5}}.

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Related to this were students who simply saw the scalar that was multiplied by the vector <5,12> to be a “scale factor” that stretches/shrinks the vector by a particular factor. But why this works is because of this similar triangles argument.

Approach 2: A bunch of students used trig. They first found “the angle” and then realized that angle put on a unit circle would work! The fact that so many students saw the problem this way made me happy. I then asked what if we wanted the vector to have length 2 or 3 (instead of unit length), and they were able to answer it. We also talked about one huge deficit of this approach: you lose exactitude since they approximated the angle with their calculator. Even if they didn’t round, they wouldn’t “see” the square root of 5 pop out, when they would with the similarity argument.

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Approach 3: Okay, so strangely this year, none of my students used this approach. But it is related to the similar triangles approach, and in years past, I’ve had students come up with it. So I showed it to them so they could see another approach. It’s an algebraic approach to find the scale factor.

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Fin!

For a sphere, why is the derivative of the volume the equation for the surface area?

Yeah, so the title of the post says it all. I am teaching a standard calculus course, and I wanted my kids to see why this beautiful thing holds true.

sphere

It’s not a coincidence. And in fact, a circle also has a nice property: the derivative of the area of a circle (A(r)=\pi r^2) gives you the circumference of the circle (C(r)=2\pi r). So yesterday I decided I wanted to come up with a short investigation that at least exposes my kids to this idea.

After working for around 90 minutes this morning, I ended up with a packet, and these things on my desk which I’m going to use for illustrations (blocks, dumdums, and tape):

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UPDATE: I was shopping yesterday and found these gems. YAAAS!

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I’ll post the packet I whipped up below. It goes through the standard argument, so in that way it’s nothing special. But in the past when I taught the course, I used to just kinda stand up at the board and give a 5 minute explanation. But I wasn’t sure who was really grokking it, and I was doing too much handwaving.

The big picture trajectory:

*At the start of class, but way before doing this activity, I’m going to have kids recall what a derivative is graphically (the slope of a tangent line), and then how we approximated it before we used limits (the slope between two points close to each other). And from that, I’m going to remind kids of the formal definition of the derivative:

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* I may also start class with this problem, suggested to me on twitter by Joey Kelly (@joeykelly89). It’s a classic problem that was featured on xkcd, but oh so unintuitive and surprising!:

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*Way later in class, I will transition to this activity. The first idea is to get kids to see the connection between the volume of a sphere and the surface area of a sphere. And then again for the area of a circle and the circumference of a circle.

*Then I try to get kids to understand what’s going on with the sphere first… followed by the circle.

*Then I show kids the “better explained” explanation. Why? Because at this point, kids are spending a lot of time thinking about the algebra, and I’m afraid they might have lost the bigger picture. The algebra focuses on one “shell” of the sphere, or one “ring” of the circle. But how does it all fit together? [@calcdave sent me this video, which I’d seen before but forgot about, which has the same argument… this is where the licorice wheels above come into play.]

*Finally, I problematize what they’ve learned. I have them mistakenly make a conjecture that the derivative of the volume of a cube is going to be the surface area of the cube, and the derivative of the area of a square is going to be the perimeter of the square. But quickly kids will see that isn’t quite true. So they have to tease out what’s happening.

My document/investigation [docx version to download/edit]:

My solutions:

I haven’t taught this yet. So it could be a complete disaster. I don’t have a sense of timing. I don’t know how much of this is me and how much will be them. I am just hoping tomorrow isn’t a disaster! Fingers crossed!

More Things I Want To Highlight From Twitter

As I mentioned before, I often see neat stuff on Twitter and want to remember it, but I just “favorite it” and then forget it. Sometimes I remember to go back and look for it, but it’s arduous. So I decided if I get the time, I’ll blog about some of my favorite things which will help me remember them better.

At the very end, you’ll see a problem that nerdsniped me for a good while. It hit a sweet spot for me. 

***

First off, though, something about me! Kara Newhouse contacted me and Joel Bezaire about how we use reading in our math classes, and then she wrote an article about it for KQED Mind/Shift. I didn’t quite know what it was, but I figured why not answer a few questions!

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Even though I hadn’t heard of Mind/Shift before, it apparently is something that gets around. Because my friend Julie told me someone sent it to her, and my sister saw it in the NAIS newsletter. And even Steve Strogatz tweeted it out (with some links to my blog). So that’s random and awesome. If you want more information about how I organized the book club, here is my post-mortem after the first year of doing it. Now I am not teaching the course I had the book club in, but what’s awesome is that I have a few kids who wanted to read math books with me… so informally I meet with them for 40 minutes each 7 school days, and over donuts, we talk. First we read Flatland, and now we’re reading The Man Who Knew Infinity. And I’m loving it! So don’t think you need to do it in a class. I’ve read books and discussed them with kids one-on-one!

Okay, now that’s over! Onto the other less self-aggrandizing things!

***

I love Richard Feynman. I don’t think it would be too strong a statement to say that I would be a different person today if I hadn’t had been introduced to him when I was young, when my father gave me a copy of What Do You Care What Other People Think? And I saw this, and I immediately wanted to get it printed on a business card to hand to kids at the start of the year.

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***

Now that we’re talking about Feynman, James Propp wrote a powerful piece about “genius” which problematizes Feynman. I already knew Feynman was self-fashioning himself in the way he presented himself to the public-at-large (and his contemporaries). But this article goes further, in a reflection connecting to a powerful piece by Moon Duchin about the sexual politics of genius. He notes:

But Duchin makes me ask, for whom does Feynman’s advice work well? Who in our culture is forgiven for putting aside personal relationships in the name of single-minded pursuit of truth? Who is permitted to be a joker? And who in our culture is steered, from an early age, toward an excruciating attunement to what other people think?

I highly recommend reading James Propp’s piece.

***

Steve Phelps (@giohio) shared this desmos applet which plots lines normal to a curve.  You can change the curve!

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It reminds me of my family of curves project (post 1, post 2, post 3). I wonder if I couldn’t have kids come up with a way to get any perpedicular line to a curve in calculus, and then have them play around with this applet to generate beautiful designs!

***

Nanette Johnson tweeted out a powerful slide from a talk she was at given by @danluevanos. I don’t need the rest of the talk. I get it.

pic4.pngI often feel like a crappy teacher. Right now, I’m on day 3 of Spring Break, and since it started, I’ve been contemplating how crappy I feel about my teaching. I know I’m not a bad teacher, but … maybe I am? I don’t know. But yes, these two questions screamed at me. Because they are part of something I need to recommit myself to: focus on the positive and take the positive and multiply it. Because I always focus on questions #3 and #4, and rarely give myself time to think about #1 and #2.

***

Mark Kaercher is using Desmos to do warmups.

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Here’s the first link in his tweet: https://www.desmos.com/calculator/xxxmahtp91
Here’s the second link in his tweet: https://teacher.desmos.com/activitybuilder/custom/5a770219c1b9e208ca83895c
I love this idea of doing warmups using Activity Builder. Must remember for next year!

***

Patrick Honner wrote a great article in Quanta magazine: “How Math (And Vaccines) Keep You Safe From The Flu”. I’m just mad I didn’t send this article to my calculus kids after we did our point of inf(l)ection activity (adapted lightly from Bowman Dickson @bowmanimal).

***

This tweet from Kara Imm just made me so happy. I always believe that formalism and stuff should come after something has been explored (whenever possibe). And second graders were absolutely doing that! Sixogon! Navada! So awesome!

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***

So Anna Blinstein asked about higher level mathematics and this happened. I learned that arithmetic with complex numbers is akin to arithmetic with polynomials mod x^2+1. WHAAA?!

pic7pic8pic9pic10

Of course I took out pencil and paper and had some fun with this. Blows my mind.

***

Steve Strogatz tweeted out this interesting article by Maria A. Vitulli about Writing Women in Mathematics in Wikipedia. The abstract is here:

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***

When it comes to polynomial division, I’ve seen the connection to standard division (where x=10). But I think I need to exploit this more in my teaching, especially to make the polynomial remainder theorem seem obvious to kids (and not like magic, which they often feel, even when we’ve figured it out). Erick Lee tweeted a perfect reminder:

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***

Joe Cossette tweeted out a neat idea — a stop motion photography race between two figures. (Click here for original tweet so you can watch the video.) I wonder if this can be adapted to calculus when we talk about rectilinear motion. Regardless, I could see it be interesting to give a physical understanding to various functions. Especially when comparing them (like exponential versus quadratic). “Which will eventually win?” Plotting x^2 versus e^x is one thing, but seeing them in a race is another.

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***

David Butler read my recent post about The Law of Cosines and shared his own post which talks about how the Law of Cosines doesn’t actually need cosine in it. Worth seeing! Trig without trig!

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***

I leave you with a problem from Abram Jopp that nerdsniped me!

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More constraints: You can’t use domain restrictions. You can use compound inequalities, but desmos only allows simple ones (e.g. 2<x<3 or x<y<x+2) and not complicated ones. A good number of tweeps got obsessed. There were many different proposed solutions, but I am proud of mine. I don’t think anyone else’s was quite like it.

In the process of working on this problem, David Butler and Suzanne Von Oy reminded me of this beautiful relationship: min(A,B)=1/2(A+B)-1/2|A-B| and max(A,B)=1/2(A+B)+1/2(A-B). And so I wanted to illustrate that with this Desmos graph. I definitely want to remember this gem when I teach Advanced Algebra II at some point when we’re exploring the power of absolute value. It’s so awesome.

 

A nice proof for the Law of Cosines

We’ve been working on the Law of Cosines in my precalculus classes. And I am having them prove it by scaffolding up from specific triangles to more general triangles. And then with the most general triangles, students consider acute, obtuse, and right triangles.

Kids tend to struggle a bit on the first triangle, but as soon as they realize they need to draw an altitude, they see all that opens up with right triangles and are good. After that, for the rest of the concrete ones, they tend to breeze through. The place where they first stumble again is when they get to the fifth triangle, the one with the angle \beta. They get to L^2=(5\sin\beta)^2+(4-5\cos\beta)^2 but then don’t go any further. But since I know I want them to get to the law of cosines, I tell them to expand and look for something nice. Sometimes I’ll give them the answer (L^2=41-40\cos\beta) and then say: work your work until it looks like this, with one trig function in it. From that point on, kids are in the zone.

For years, I used to teach this by giving kids waaaay too much information.

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And I kinda told them what to do… Meh. I was jumping way ahead to get to the formula. We weren’t savoring the thinking to get to the formula. Now we are.

That being said, I ran across something quite beautiful. A stunning proof of the Law of Cosines (at least for acute triangles) on the site trigonography.

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I love it because it looks like a proof for the Pythagorean theorem.  Which is nice because  the Law of Cosines is essentially a more generalized version of the Pythagoren theorem.

The area of the bottom square (the green one) is clearly the area of the top two squares (the red and blue ones) minus two green areas. Ummm…. c^2=a^2+b^2-2ab\cos(C) anyone? You see the c^2 and the a^2 and b^2, but what you also see is that you’re taking out some area (the green bits). [1]

When introducing it to my class, I showed them this image:

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and said it was a proof of c^2=a^2+b^2-2ab\cos(C).

And I just said to observe. To make statements based on what they see visually…Anything and everything. And if students could, see if they could make connections to the equation (but without writing anything down). After a short while of observations, I opened this geogebra applet and played around. I showed them what happened when we made angle C a right angle.

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They saw the green rectangles disappear, and how this would be a proof of the Pythagorean theorem if the blue areas and the pink areas were equal to each other. And then I squiggled and smushed the triangle about and eventually kids conjectured that the blue rectangle areas were equal, the pink rectangle areas were equal, and the green rectangle areas were equal. I told them that was true, and they were going to prove that. But before doing that, I asked them: if this was true, do you see a connection between this diagram and the law of cosines?

And kids eventually got there. They saw this argument, essentially…

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…and they realized that the green rectangles were probably the thing that was being subtracted out in the law of cosines!

At this point, I gave my kids a blank paper copy of the diagram, and groups work on proving that the blue rectangle areas were equal, the pink rectangle areas were equal, and the green rectangle areas were equal. They had seen all these right triangles before,  when they were looking at the diagram and making observations, so this went pretty quickly for most of them.

I love this proof of the law of cosines! Of course when I went online, I saw so many other beautiful proofs (look here, and the links at the bottom, for some). Troll the internet and be amazed! They are so elegant! This “scaling up” one might be my favorite. And here’s one that David Butler sent me (that is on the site I linked to above). And I remember proving the Pythagorean Theorem in geometry using the crossed chord theorem, and now the same argument here can be used for the law of cosines.

 

[1] To be clear, this diagram only works for acute triangles. I haven’t yet modified the argument to work for obtuse triangles.