# Two Problems that Got Me To Think

Here are two problems that have gotten me to think a lot.

The first one came from my Precalculus co-teacher James. We had been finishing up our unit on combinatorics and also creating new groups, and he devised a great question. So here’s the two-part problem I posed to my kids:

First Problem: We have a class of 14 students, with two groups of 3 and two groups of 4. If I were to have a computer program randomly create new groups: (a) what is the total number of different configurations/outcomes we could have? (b) what is the probability that your entire group was the exact same if you were in a 4-person group?

I thought I solved it successfully and was feeling really confident. Then James told me I was wrong. Then I tried but didn’t understand his logic. So I made a simpler case, and then I thought I understood it. My brain hurt so much. I kept switching back and forth between a couple different answers. It was marvelous! Finally, I felt like I understood things and felt confident. I shared it with my class, and lo and behold, a couple students got what I got, and a couple students didn’t. But the students who didn’t convinced me with their logic. And then I shared their thinking with James, who didn’t have the same answer, and he too was convinced. And I thoroughly enjoyed being wrong and telling the kids that this problem messed with my head, and they helped me see the light!

The second problem came from a student who emailed me about wanting to become a better problem solver. And they shared this old entrance exam for this summer camp they were thinking of possibly applying for, and wanted some guidance. The problem that I got nerdsniped by and ended up spending hours working on over Thanksgiving break was as follows:

Second Problem:

This is from the 2019 entrance questions for a summer program. I think I was able to successfully solve (a) and (b). And then I think I solve (c) for n=3 and n=4 (and got an answer for n=5, but haven’t proved it is optimal). And I have no way to even start thinking about (d). But what I thought was lovely is how many different places my brain when went trying to think through this problem. And the neat geometric structure that arises out of the setup. (Even though I wasn’t able to fully exploit this structure in my thinking.)

I hope you enjoy thinking about these!

# Mathematical Habits of Mind

I haven’t been blogging for a long time. As you can imagine, the pandemic took a toll on teachers, and at least for me and my teacher friends, we were working insane amounts of time, and it was so hard. Emotionally, physically, intellectually. At the time, I just didn’t have it in me to blog about the experience.

But now we’re about to start a new school year. And I’m vaccinated. And my students are vaccinated. And we’re wearing masks. And my classes are going to be with all my kids together in a single room [1], which is such an awesome thing compared to last year.

One of the classes I’m teaching this year is Advanced Precalculus. Another teacher, my friend James, is also teaching the same course. And he’s new to my school this year, and so when talking about the course, he shared with me how he formally incorporated Mathematical Habits of Mind in his teaching in previous years. And interestingly, last year, I toyed with the idea of formally getting kids to be metacognitive about problem solving strategies — but decided to focus on something else instead. So when James shared this idea with me, I got excited.

Right now I have an inchoate idea of how this is going to unfold. Hopefully I’ll blog about it! But for now, I wanted to share with you posters I made using James’ Mathematical Habits of Mind. Most importantly, here is a link to James’ original blogpost with his habits of mind and rubric.

Photo of the posters hung up in one of my rooms:

I know, I know, the lighting is terrible. The key words are:

## Experimenter, Guesser, Conjecturer, Visualizer, Describer, Pattern Hunter, Tinkerer, Inventor

If you want these posters, the PDF file is here.

And here are all of them shared as a single sheet, and not as a poster.

Of course, if you’re a math teacher, you know there are a lot of lists of mathematical habits of mind. We agreed to use the ones James had already been using. But there are many alternative or additional things we could have included.

At the very least, I know that as we get kids to think about what strategies they’re using to solve problems, we’ll also see where there are lacuna in our curricula in terms of using those strategies. Or maybe we’ll discover it doesn’t have as much problem solving as I imagined in it. All entirely possible, since we — the kids and James and I — will all be looking through what we’re doing through our metacognitive Mathematical Habits of Mind lens.

[1] The reason I note this is because at the end of last year, I was teaching students live simultaneously in three places: they were in two different classrooms and there were a few at home on zoom. Yes, seriously. When I mention that to teachers and non-teachers alike, they asked how that was even possible. It was… a lot.

# Tiny Game Re: Euler’s Number

I’m teaching Algebra 2 this year and the other teacher and I decided that we should introduce e to our kids. The reason it’s challenging is that it’s hard to motivate in any real way. You can do compound interest, but that doesn’t do much for you in terms of highlighting how important the number is. [1] I asked on Twitter for some help, and I got a ton of amazing responses (read them all here). My mind was blown. This year, though, I didn’t have time to execute my plan that I outlined at the bottom of that post. So here’s what I did:

1. The core part of what I did to get the number to pop up was to use @lukeselfwalker’s Desmos activity. I like it for so many reasons, but I’ll list a few here. It starts by “building up” a more and more complicated polynomial of the form $(1+\frac{x}{n})^n$, but in a super concrete way so kids can see the polynomial for different n-values. It shows why the x-intercept travels more and more left as you increase n, so when you finally (in the class discussion) talk about what happens when n goes to infinity, you can have kids understand this is how to “build” a horizontal asymptote. It gets kid saying trying to articulate sentences like “this number is increasing, but slower and slower” (when talking about the value of the polynomial when $x=1$. And they see how this polynomial gets to look more and more like an exponential function as you increase the value of n. If you want to introduce e, this is one fantastic way to do it.
2. A few days later, I had everyone put their stuff down and take only a calculator with them. They paired up. (If someone didn’t have a pair, it would be fine… they just sit out the first round.) On the count of three, both people say a number between 0 and 5. (I reinforce the number doesn’t have to be an integer, so it can be 4.5 or something.)Then using their calculators, they calculate their score: they take their number and raise it to their competitor’s number. The winner has the higher number. (If it’s a tie, they go again until there is a winner.)

Then the loser is done. They “tag” along with the winner and cheer them on as they find another winner to play. This goes on. By the end, you have the class divided into two groups each cheering on one person. (I learned this game this year as an ice breaker for a large group… it’s awesome. This is the best youtube video I could find showing it.)

Finally there is a class winner.

So I then went up against them.

And when we both said our numbers, I said: e.

The class groans, realizing it was all a trick and I was going to win. We did the calculations. I obviously won.

We sit down and I show them on my laptop how this works:

The red graph is my score, for any student number chosen ($e^x$).
The blue graph is the student score, for any student number chosen ($x^e$).

Clearly I will always win, except for if my opponent picks e.

I tell kids they can win money off of their parents by playing this game for quarters, losing a few times, and then doing a triple or nothing contest where they then play 2.718. WINNER WINNER CHICKEN DINNER!

3. After this, I show kids these additionally cool things (from the blogpost), saying I just learned them and don’t know why they work (yet), but that’s what makes them so intriguing to me! And more importantly, they all seem to have nothing to do with one another, but e pops up in all of them!

I re-emphasize e is a number like $\pi$ and I showed them this to explain that it pops up in all these places in math that seem to have nothing to do with that polynomial we saw. And that even though we don’t have time to explore e in depth, that I wanted them to get a glimpse of why it was important enough to have a mathematical constant for it, and why their calculators have built in e and ln.

That is all. I honestly really just wrote this just because I was excited by the “game” I made out of one of the properties of and wanted to archive it so I would remember it. (And in case someone out there in the blogoversesphere might want to try it.)

UPDATE: Coconspirator in math teaching at my school, Tom James (blogs here) created the checkerboard experiment using some code. You can access the code/alter the code here. The darker the square, the more times the number for the square has been called by the random number generator. And with some updates, you can make more squares! In the future, we can give this to kids and have them figure out an approximation for e.

[1] And introducing it with compound interest means you have to assume 100% interest compounded continuously. Where are you going to get 100% interest?!?!

# Clothesline Math – Logarithm Style

I remember when I first heard about Clothesline Math, I was excited by all the possibilities. And in a few conference sessions with Chris Shore, I saw there was so much more than I had even imagined that one could do with it!

It’s basically a number line, that’s all. But it’s a nice public giant number line which can get kids talking. Today I came back from spring break and before break, students learned about logarithms. However I wanted to have them recall what precisely logarithms were… so I created a quick Clothesline Math activity.

I hung a string in the classroom. I highlighted it in yellow because you can’t really see it in the photo…

I then showed them this slide – explaining the string is a number line…

I then showed them this slide, which explains what they have to do if they get two of the same number. (I brought cute little clothespins, but mini binder clips or paperclips would have worked just as well):

And then I gave them the rules of play:

I handed out the cards and let kids go. It was nice to see they didn’t get tripped up as a class on too many of them, but I got to listen to debates over a few trickier ones, which we collectively resolved at the end.

Here are the cards I handed out: .DOC FORM: 2019-04-01 Clothesline Math – Logarithms

Here is a picture of some of the cards. The two on the left are average level of difficulty. The two in the middle caused my kids to pause… it took them time to think things through (they haven’t learned any log properties yet). The one on the right doesn’t belong on the number one (it is undefined) and the kid who got that card immediately knew that. Huzzah!

Here’s a picture of the numberline at the end.

And… that’s it!

I was excited to try it out as a quick review activity. And it worked perfectly for that!

(Other things of note: Mary Bourassa made a clothesline math for log properties and shares that here. The author of Give Me A Sine blog does something similar here, but has kids create the cards. I couldn’t find anything with basic log expressions — so I made ’em and am sharing them in this post. Chris Hunter has a nice tarsia puzzle that sticks with basic log expressions here, but I wanted to try out clothesline math so I didn’t use that!) But if anyone has others out there involving logs, I’d love to see them in the comments!)

# Polar. Graph. Contest.

Here’s what I hung up last week:

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:

… from something so simple …

(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:

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!

[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.

# 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.

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.

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

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!

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.

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:

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 $\frac{1}{\sqrt{5}}$.

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.

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.

Fin!

# 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.

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.

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:

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.

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…

…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.