# Polygonal Crystals

In Geometry, we’ve been trying to turn the course on its head. Recently, we’ve been working on reasoning and proof. One thing students weren’t able to prove was that the sum of the interior angles of a triangle always add up to 180 degrees. (That’s because we haven’t gone over anything involving parallel lines.)

They even blew up balloons and drew triangles with two right angles on them (using protractors and rulers).

[Note to self: students cannot tie balloons. Also, they will be found scattered around the student center later in the day if you let kids keep their balloons.]

But we said: assuming that you know that on a plane the sum of the interior angles of a triangle add up to 180 degrees, can you prove that quadrilaterals have a sum of 360 degrees for their interior angles.

And each group was able to latch onto the idea of dissection without me saying anything… breaking the quadrilateral into two triangles.

But then… then… they started to say something that scared me. They said “there are two triangles, and since each triangle is 180 degrees, the quadrilateral is 360 degrees.”

To you non-geometry teachers, this might not seem problematic. But I immediately thought: “oh gosh, these kids think of triangles and quadrilaterals and the like as having some inherent property that can be added to others. They aren’t saying the sum of the interior angles of the triangle is 180 degrees… they are saying the triangle has 180 degrees.

So I gave them a follow-up question (which I’m proud of):

“The Blue Triangle is 180 degrees. The Pink Triangle has 180 degrees. So the Giant Triangle (the blue and pink triangles combined) must have 360 degrees. How is this possible? Did we just break math?”

One group had someone who figured it out right away, but the others took a good amount of time trying to figure out where this argument failed. I loved it because it really showed them a misconception they had.

It was the perfect question, because over the summer the other geometry teacher and I came up with the following (which we are in love with) involving triangle dissection:

Finally, to check if each group understood where these came from, we had them write a “triangle dissection expression” for the sum of the interior angles of this pentagon:

Fin.

# 2, 4, 8

How many of you out there have seen the Veritasium video where he goes around talking to random people on a beach, asking them to guess his rule with… heck, just check it out:

Did you watch it? No? Seriously, watch it!

Okay, good. Ever since I saw this video, I wanted to try it in my class. I wanted to have my kids guess the rule. I wanted to be the one saying “yes” and “no” to their three numbers. And today, finally, I did.

In geometry, we’re starting a short excursion into proof and reasoning. Yes, we’ve done a few proofs. And my kids are learning to justify their ideas. But we’re about to embark on a few days where they think about proof, the importance of proofs, assumptions, and other such things. And in our next class, we’re going to start talking about induction and deduction. So today, this was a perfect warm up.

I gave each group a whiteboard. They threw up three numbers. I said yes or no. After they got 3 yes/nos, they were allowed to guess the rule. Then they did it again. Some groups were looking to get the rule in the least amount of guesses. Others were guessing willy nilly. It definitely took longer than I thought. No group got it, but they were making interesting choices with their three numbers in order to figure out the rule.

After about 7-8 minutes of this, I stopped them. I started playing the Veritasium video. It was awesome. Why? Because the random beach people that Veritasium interviewed gave almost identical answers as my kids gave! They saw that the way they were approaching the problem was the same. They heard a few more sets of three numbers that worked/didn’t work, and then I paused the video. Why? Because I heard kids whispering and murmuring that they think they had it. I gave those kids an opportunity to share what they thought the rule was (I did not confirm or deny their guesses). And then I finished the video.

I loved doing this because the kids were totally engaged. And when we start talking about induction and deduction, counterexamples, and keeping an open mind when problem solving, we can use this exercise as an activity we can refer back to.

# Polygonal Numbers

I just finished up arithmetic series, and I wanted to push my Advanced Precalculus students to think hard. I usually provide them with enough scaffolding that I know they will be able to get from Point A to Point B. But today I decided I wanted to “be less helpful.”

I told the groups to mix themselves up — for a change of pace. And then I handed this out.

I told each group that they had one opportunity to call me over so I could give them A Big Hint. Then I let them go, giving them giant whiteboards to work on if they wanted.

In my two classes, I had: one group solve the problem explicitly (they had a formula that worked) and one group come up with a recursive solution that really impressed me! The other eight groups were at varying stages of understanding. Most all the groups were gung ho about working, and most all the groups started discovering all these patterns.

Only one of the ten groups asked for The Big Hint, which means my kids have perseverance! I did give varying degrees of mini-hints to kids as I saw them progress, to nudge them this way or that way.

At the end of the second class, a few kids said how they are now braindead, because they did so much thinking. They were exhausted. As a teacher, I call that winning!

I’m still at a bit of a loss as to what I am going to do tomorrow. Since kids hadn’t really finished, I thought I would have them work a bit more. I’m not good at debriefing. Also, many kids have different observations, and I don’t have time to really dwell on this. This wasn’t supposed to even take the whole day!

My plan is to give all my groups collectively A Big Hint, give each group 12 minutes to figure out how to find the nth pentagonal number, and then after those 12 minutes are up, I will give them this:

We will go over the triangular and square numbers together as a class. And then WHAM, I leave them with the pentagonal and p-gon figures to figure out on their own.

Wish me luck that it goes well tomorrow. I can see it crashing and burning, or being a good wrapup.

# A Follow Up (regarding Centers of Rotation)

It’s 10:44pm on a Saturday evening, and I have been thinking about math. Whoooo hooooo! I finally got a chance to ponder how I’m going to attack this question that I posed earlier this week. For those too lazy to click:

Imagine you’re a geometry teacher, and you want students to discover a “foolproof method” to coming up with the center of rotation (given an original figure and a rotated figure). You also want students to be able to understand (deeply) and articulate why this method gives you the center of rotation.

Although not perfect, I have whipped some stuff up in the past hour that I hope will get to the heart of this question. Of course in my head, I have class discussions, and we gently get at this. These sheets alone don’t get us there. But if you’re interested, feel free check ‘em out.

(Also, since I just whipped them up, there might be some things that need fixing/tweaking…)

There are five of them (all combined).

• Rotations of two points [after this, through discussion, get student to make the connection between the perpendicular bisector as all points equidistant from the endpoints of a segment and the radii of a circle]
• Rotation of a line segment [after this, through discussion, get students to recognize that they are really considering two perpendicular bisectors… we are looking at one perpendicular bisector to find all possible points which will rotate one end of a line segment to the new point and a second perpendicular bisector to find all possible points which will rotate the other end of a line segment to the new point… for both endpoints to simultaneously rotate to their new location, we have to look for the intersection of the perpendicular bisectors!]
• Rotation of three points [after this, make it clear that this is not the same as saying that you can trace a triangle on patty paper in two different places and find a center of rotation that will bring the first triangle to the second triangle… in fact, maybe I should have this as an exercise…]
• Center of rotation practice
• Rotations of a complex figure

(.docx)

PS. If you’re talking about multiple “center of rotation”s, do you say “centers of rotation” or “center of rotations” or “centers of rotations”? It makes me think of culs de sac, which indeed is the plural of cul de sac. Thank you Gilmore Girls.

# Mind Blown

[Cross Posted on the One Good Thing Teach blog]

Setup: We’ve been talking about perpendicular bisectors in various contexts in geometry. But they were just making observations and working on some simple proofs.

Last night in Geometry, students were tasked with the following:

It turns out that #3 is impossible, and #4 is possible with some guess and check. This sets up the background to have kids see something neat.

And now they see that for a triangle, the perpendicular bisectors of the sides all meet at a point. And that is rare and weird.  They then were asked to look at the point that the perpendicular bisectors meet at and the vertices of the triangles and make a conjecture.

Only one student “saw” it. It was fascinating for me that it was so hard for everyone else to see it! Others had conjectures that might have been true for right triangles or isosceles triangles or equilateral triangles… but not that were universally true.

For the rest of the class, to get them there, I did the following:

This was a huge setup for my “one good thing.” There were gasps, and one student said, and I quite, “MIND BLOWN.”

This weekend they are going to try to figure out what the what is up?!

PS. Yes, I am fairly certain that the setup of having students see the rarity of perpendicular bisectors meeting at a point, as well as having them look and fail to see something inside a set of seemingly random points was crucial for the big reveal. In fact, the fact that they didn’t discover it on it’s own was so powerful when they ended up seeing it.

UPDATE: The file I used is here.

# Substitution (…and Continued Fractions)

Today in Precalculus I went on a bit of a 7 minute digression, talking about continued fractions. You see, a recursive problem showed up (we’re doing sequences): Write out the first five terms of the following sequence:

$a_{n+1}=\sqrt{2+a_n}$ where $a_1=\sqrt{2}$

So obviously they go like: $a_1=\sqrt{2}$,$a_2=\sqrt{2+\sqrt{2}}$, $a_3=\sqrt{2+\sqrt{2+\sqrt{2}}}$, $a_4=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}$, and $a_5=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}$

So great. Awesome. NOT. Booooring. So I showed them the decimal expansions:

$\approx 1.414, \approx 1.848, \approx 1.961, \approx 1.990, \approx 1.998, \approx 1.999, \approx 1.9998, \approx 1.99996, \approx 1.999991, \approx 1.999997647$

WHOA! This is getting closer and closer to 2… Weiiiird…

And then I say I can show them this will continue, and we can find a way to show that $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}$ [where the pattern continues forever] will practically become 2.

To do this, I start with something else. I don’t know why, but I really wanted to show them a continued fraction first, to get the point across easier than with the square root. This was the continued fraction.

I went through a frenetic mini-lecture, and I think I had about 40% of the kids along with me for the whole ride. I’m not sure… maybe? But later a kid came by my office, and I thought of a better way to show it. Hence, this blogpost, to show you. (I have seen teachers use this method when teaching substitution when solving systems of equations… but I have never used it myself. I’m dumb! This is awesome!) This is what I did when showing the kid how to think about this in my office.

First I took a small piece of paper and I wrote the infinite fraction on it.

Then I flipped it over and on the back wrote what it equaled… Our unknown $x$ that we were trying to solve for.

I emphasized that that card itself represented the value of that fraction. The front and back are both different ways to express the same (unknown) quantity we were looking for.

Then I took a big sheet of paper and wrote $1+\frac{1}{}$ where I left the denominator blank. And then I put the small card (fraction side up) in the denominator of the fraction…

And I said… what does this whole thing equal?

And without too much thinking, the student gave me the answer…

Yup. We’ve seen that infinite fraction before. That is $x$!

Flip.

THAT FLIP IS THE COOLEST THING EVER FOR A MATH TEACHER. That flip was the single thing that made me want to blog about this.

Now you have an equation that you can solve for $x$… and $x$ is what you’re trying to find the value of. This equation can easily be turned into a quadratic, and when you solve it you get $x=\frac{1+\sqrt{5}}{2}\approx 1.618$ (yes, the Golden Ratio). And it turns out that is close to what we might have predicted…

Because in class, we (by hand) calculated the first few terms of $a_{n+1}=1+\frac{1}{a_n}$ where $a_1=1$… and we saw: $1, 2, 1.5, 1.66666666, 1.6, 1.625, ...$

And when I drew a numberline on the board, plotted 1, then 2, then 1.5, then 1.66666666, then 1.6, then 1.625, we saw that the numbers bounced back and forth… and they seemed to be getting closer and closer to a single number… And yes, that single number is about 1.618.

COOL! [1]

BACK TO OUR REGULARLY SCHEDULED PROGRAM

So after I showed them how to calculate the crazy infinite fraction, I went back to the problem at hand… What is $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}$?

Let’s say $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}}}=x$

Then we can say $\sqrt{2+x}=x$

And even simply by inspection, we can see that $x=2$ is a solution to this!

Fin.

[1] What’s neat is that yesterday I introduced the notion of a recursive sequence that relies on the previous two terms. So soon I can show them the Fibonacci sequence (1,1,2,3,5,8,13,…). What does that have to do with any of this? Well let’s look at the exact values of $a_{n+1}=1+\frac{1}{a_n}$ where $a_1=1$.

$2, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \frac{13}{8}, ...$. WHAAAA?!?!

Lovely. It’s all coming together!!!

# You Spin Me Right Round: A Challenge for Geometry Teachers

We are now delving into some interesting coordinate geometry. We’re also beginning to use patty paper. And today the other geometry teacher and I had an awesomely fun conversation that revoles around something you might find it fun to think about.

Here’s the impetus/setup.

We have just finished talking about translations, and we’re moving on to reflections and rotations. When we introduce rotations to kids, we give them a backwards problem pretty early on: here is a figure and here is the rotation of the figure. Try to find the center of rotation. Use patty paper and guess and check.

This is an awesome exercise (inmyhumbleopinion) because it has kids use patty paper, it has them kinesthetically see the rotation, and it gives them immediate feedback on whether the point they thought was the center of rotation truly is the center of rotation. Simple, sweet, forces some thought.

This is our intro. We have a few more exercises in our problem packet similar to this. But then we move on. Le sigh. This felt wholly unsatisfying to us, because at its heart we left our kids hanging. We never get at the obvious question. How do I find the center of rotation without guessing and checking.

How do we take this introduction and make it mathematics?

The Challenge

[Baby challenge: Figure out how to find the center of rotation, given any figure and a rotated figure. This is not totally simple, okay? I’m calling it a baby challenge only because the next challenge is sooooo hard!]

Imagine you’re a geometry teacher, and you want students to discover a “foolproof method” to coming up with the center of rotation (given an original figure and a rotated figure). You also want students to be able to understand (deeply) and articulate why this method gives you the center of rotation.

What do you do to achieve these twin goals?

(This is what the other teacher and I were discussing today, and having loads of fun doing it!)

You don’t need to answer this question in the comments. (Though you’re welcome to throw down any ideas.) I’m actually not looking for advice. (We’re well on our way to coming up with an answer to this.) I just thought it would be a fun thought exercise for you, if you like thinking of lesson planning/curriculum design. This is the type of stuff I love thinking about — when I have time!

This is backwards planning at it’s most fun, in my opinion. I have a deep result that is abstract and hard to grasp. I have very concrete 9th graders who I want to get from knowing almost nothing to discovering, understanding, and marveling at this great mathematical insight. How do I get from Point A to Point B?