## Trying to find math inside everything else

### The Secret of the Chormagons

(Doesn’t that title sound like it should be a YA fantasy book?)

I shared this Desmos Activity Builder I made on Twitter, but never wrote a post about it. Oops! Let’s do that now.

First, here’s the activity.

This activity is basically adapted from Sam’s Blermions post that I had helped him think about but he did all the hard work of creating questions and sequencing them. I’ve used the blermion lesson in the past and it went well, but that last part lacked the punch I was hoping for, having to work with analog points and compiling them together myself. But then I thought – wait, Desmos AB does overlays that will do this beautifully, if I can just figure out how to get the Computation Layer to do what I want. So I set to it.

The pacing helped pause things to conjecture about what chormagons are – I stopped them at slide 5 so they can make those conjectures. I then advanced the pacing to 6 for the “surprise,” and 7-8 to make further conjectures. Look at some of theirs below. Even showing them everyone’s, I tended to get conjectures about the shape their points make – a trapezoid, a pentagon, a heptagon, etc. (One person seemed to guess the truth, but that was uncommon.)

Then I showed them the overlay. I saw at least one jaw literally drop, which totally made my day. We talked about why it might be a circle, and what that means for these shapes, then I introduce the proper terminology “cyclic quadrilaterals.” (I taught this lesson as an interstitial between my Quads unit and my Circles unit.)

Speaking of proper terminology, you may notice I called them Chormagons here, as opposed to Blermions. That’s important – Blermions is very google-able, it leads right to Sam’s post! But Chormagons led them nowhere. (And they were so mad about that!)

Now Chormagons will lead right here. So this is an important tip if you use this DAB: make sure you change the name of the shape!

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### One Problem, Eight Ways

I had a pretty good lesson recently that I wanted to share. It was at the end of my quadrilaterals unit, and so we were working on coordinate proofs. I love coordinate proofs because you can get so much information from just a pair of coordinates, which lends itself to lots of different ways of solving the same problem. Add to that how many different ways there are to prove something is a square, and we have the start of something good.

I gave the students the above sheet, starting off with some noticing/wondering about the graphed figure. Then I assigned each table a different method to prove that the quadrilateral is a square. Each group was off to their whiteboards to get started. It was really great to see each group discussing the problem so intently, and it reminded me how easy it is to facilitate discussion when up at the vertical whiteboards. Afterwards, the students went around in a gallery walk to compare their proofs to the other methods. They analyzed how they were similar, how they were different, and thought about which method they might prefer in the future. (Some comments included things like preferring method 2 because it only involved slopes, even though it involves more lines.)

The whole lesson went so smoothly and had tons of intra- and inter-group discussion. Need to use the structure again.

### The Great Geometry Review

Since Kate asked us to post more unsexy things, I thought I would throw up this review book I made for geometry, which basically covers all the things students should “know” (not necessarily be able to do, or deeper understandings) for the course, especially for the NY Regents (Common Core). The students can fill in the blanks and are then left with a nice study guide. So far my students seem to like it! (Although one student said they wouldn’t do it if they didn’t get a grade for it – so frustrating!

Great Geometry Book (doc)

Great Geometry Book (pdf)

### Natural Circle Measures

Yesterday I introduce radians to my students for the first time. I started out by asking why they thought a circle had 360°. There were a few good answers – four right angles makes a circle, so 4*90 is 360; a degree is some object they measured in ancient Greece, and so a circle was made of 360 of them; something to do with the number of days in the year. All good answered, but I told them it was completely arbitrary based on the Babylonian number system.

Once we decided that it was arbitrary, I asked them to come up with their own method of measuring a circle. I would classify their responses into three categories

1. Divide the circle up into 200 “degrees” (most common)
2. Divide the circle up into 100 “degrees”
3. Divide the circle up into 2 “degrees” (least common)

I was expecting 100 “degrees” to be the most common, so I was very surprised to see that most of the students want to split the triangle into two sections, each with 100 parts.

I have been a proponent of tau for a while, as I thought it was natural to think of radians as pieces of a whole circle, but my students were clearly thinking of the circle as two semicircles right off the bat.

I pushed the students who came up with the third way in a whole class discussion. If this whole semicircle is one student-name-degree, what would you call this section? And so we got to using fractions of those degrees. That made a pretty easy transition into radians. I went a little into the history; instead of using a degree, some mathematicians decided to use names based on the arc length – and so that semicircle’s angle was 1 π radians, instead of 1 student-name-degree. And the fractions we used were the same.

This almost made me doubt my tau ways – maybe π was more natural. But then, as we started converting angles from degrees to radians, some students kept complaining that, for example, 90° was 1/2 π instead of 1/4, since it was clearly a quarter-circle – so maybe I can stay a tau-ist.

### Angle Chasing

On Friday our school was supposed to have a Quality Review, but it was canceled at the last minute. (That’s a whole ‘nother story.) But that pushed me to do a lesson that I probably wouldn’t’ve done otherwise, so that’s good. I actually think it went pretty well.

I noticed in our last exam that I should probably explicitly teach angle chasing as a problem solving strategy, so I asked the MTBoS for some good problems. Justin Lanier came through in the most wonderful way. So I picked out some problems into a nice sequence that would use a bunch of the theorems we’ve already done.

I wanted the students to work as a group up on the whiteboards, so I gave each person in each group a different color marker. I then had the students write a key in the corner. Each student’s color represented 1-3 of the theorems that they would have to use to solve the problems. Then they would draw up the diagram of the problem. As they went through, each person was only allowed to write when their theorem was used to deduce the measure of the angle. That way, with the colors, I could actually trace through the thought processes they used to solve the problem, which was really nice. (I wonder if I can use that as an assessment some how, having students trace through the same process. Maybe as a warm-up, once I get my smartboard working again.)

Here’s some pics of their great work.

### Quadrilateral Congruence

Stressful as it is, I am loving teaching new courses. When I first start teaching, I felt like I was learning new stuff all the time, stuff about algebra (and how it connects to other courses) that I didn’t know I didn’t know, and now it keeps happening with geometry, especially with the more transformational tinge CC geometry has.

One of the things that struck me was, last week, when I used this Illustrative Mathematics task as a follow-up to my lesson about the diagonals of quadrilaterals. I feel like the understanding I had internalized that you can prove triangles congruent with less information because they are rigid structures, but quadrilaterals are not, so there are no quadrilateral congruence theorems. But I realized that’s not true.

Last time, we constructed all of the special quadrilaterals by taking a triangle and applying a rigid motion transformation. That meant that every special quadrilateral can be split into two congruent triangles. Therefore, if you had enough information to prove one pair of triangles is congruent, you could prove the whole quadrilaterals are congruent. So if we’re looking at SSSS in terms of the triangles, we really only know two sides of the triangles. Since that’s not information to prove the triangles congruent, then it’s not enough for the parallelograms. But SAS is enough for the triangles, so it’s enough for the parallelograms. Here’s a non-parallelogram example. Here are two isosceles trapezoids with the same diagonals, same legs, and the same angle between the diagonals and one of the bases, but the trapezoids are not congruent. But that’s because, when you look at the triangles, we have Angle-Side-Side, which we all know is not a congruence theorem. If, instead, we had had SSS (a leg, a base, and a diagonal), then they would be congruent.

### Building Quadrilaterals and Their Diagonals

I wanted a lesson to explore the properties of the diagonals of different types of quadrilaterals, but the curriculum map I was following just lead to Khan Academy, and that’s not really my speed. And some scanning through MTBoS resources didn’t find me what I wanted, but chatting out my half-formed ideas with Jasmine in the morning focused the idea into what I did in class today.

I started by having the students draw 6 triangles: 3 scalene, non-right triangles; 1 isosceles non-right triangle; 1 scalene right triangle; and 1 isosceles right triangle. Then we used each of those figures to create a quadrilateral by making some sort of diagonal. Each time, I asked them to identify the quadrilateral and what they noticed about the diagonals. First, take one of the scalene triangles and reflect it over one of its sides. Thus we created a kite – which we know because the reflection creates the congruent adjacent sides. Then we can use the properties of isosceles triangles – we know the line of reflection is the median of the isosceles triangles because of the reflection, so it is also the altitude, meaning the diagonals are perpendicular.  Then, take another scalene triangle and reflect is over the perpendicular bisector of one of the sides. This makes an isosceles trapezoid – we know the top base is parallel to the bottom base because they are both perpendicular to the same line, and it’s isosceles because of the reflected side of the triangle. Then we notice the diagonals are also made of a reflected side of the triangle – and so we can conclude that the diagonals of an isosceles trapezoid are congruent.

For the third one, I asked them to draw a median and then rotate the triangle 180°. The trickiest bit here is to prove that this is a parallelogram – previously we had classified the quadrilaterals by their symmetries, so using the symmetry definition we could say any quad with 180° rotational symmetry is a parallelogram. Or we can use the congruent angles to prove the sides are parallel. Once we did that, we saw that, because we used the median, that the intersection of the diagonals is the midpoint of both – and thus the diagonals bisect each other.

I then tasked them to figure out how to make a rhombus, rectangle, and square out of the remaining triangles using the triangles. Because we proved the facts about the diagonals of the parent figures, we could then determine the properties of the diagonals of the child figures.

I think it went pretty well – the students performed the transformations and easily saw the connections between the diagonals. Tomorrow I think we’ll do something about whether or not those diagonal properties are reversible – if every quad with perpendicular diagonals is a kite, for example.