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Archive for the ‘lessons’ Category

Vimes’ Theory of Socioeconomic Injustice

As I was planning my linear functions unit, I noticed a problem about someone choosing between two electric companies, with the standard idea of one having a larger start-up cost but a lower monthly rate. I realized these problems are very common, and they reminded me of of a quotation from my favorite author, Terry Pratchett:

The reason that the rich were so rich, Vimes reasoned, was because they managed to spend less money.

Take boots, for example. He earned thirty-eight dollars a month plus allowances. A really good pair of leather boots cost fifty dollars. But an affordable pair of boots, which were sort of OK for a season or two and then leaked like hell when the cardboard gave out, cost about ten dollars. Those were the kind of boots Vimes always bought, and wore until the soles were so thin that he could tell where he was in Ankh-Morpork on a foggy night by the feel of the cobbles.

But the thing was that good boots lasted for years and years. A man who could afford fifty dollars had a pair of boots that’d still be keeping his feet dry in ten years’ time, while the poor man who could only afford cheap boots would have spent a hundred dollars on boots in the same time and would still have wet feet.

This was the Captain Samuel Vimes ‘Boots’ theory of socioeconomic unfairness.

– Men at Arms

I wanted to share this concept with my class, so I looked for problems. One thing I realized, though, was that many of these problems involved one person choosing between two things. This makes it so there is one clearly correct answer.

But oftentimes, in the real world, people don’t have a choice. One person can afford the upfront cost to pay less in the long run, but another person can’t, and winds up paying more overall, as in the Pratchett quote above. So I decided to reframe the problems as comparisons between two people, to highlight that injustice.

The lesson started with a model problem, then I gave each group a different problem from the set below.

(I went through this page of unisex names to make all of the problems gender-neutral.)

Each table worked on a different problem (with some differentiation on which group worked on which), then they jigsawed and, in their new groups, they shared out their problems and how they solved them. Then, most importantly, they looked for similarities and differences between the problems.

We then read the Pratchett quote and discussed its meaning, and students had to agree or disagree (making a claim and warrant for each). We had a good discussion on whether it really applied to today’s world or not.


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.

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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.

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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!

Circles, Lines, and Angles

My math coach gave me this idea as we were planning my Circles unit. I think it went fairly well, so I’ll share it here. The idea is that we have, essentially, three basic objects that we’ve combined in different ways in geometry: circles, lines (including segments), and angles. So, as an opening activity to the unit, the task was this:

“Think of as many ways as possible to combine those three objects.”

First they brainstormed individually, as I reminded them that they can use multiple lines or angles or circles if they wanted. Then they went up to groups and made a master list per pair or group, eliminating ones that were “pretty  much” the same. I gave them some vocabulary based on what I saw they drew, and they had to use that vocabulary to describe what each drawing had. Finally, they chose one example and created one neat, fully correct example, in color that we combined into class posters. (I approved what they chose, to ensure a variety of possible layouts.)

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Between my two classes, they came up with almost every scenario I could think of that we would learn in the unit, with the exception of Tangent Line & Radius, which I drew and put in myself. Now they are hanging in the classroom, acting as a guide for our journey into circles.

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.

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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.

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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.


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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.

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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.

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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.

The Cold War

In my first year teaching I came up with this activity for working with quadratic-linear systems, based in the Cold War and missile defense. It didn’t work as well as I hoped, mostly because it was too complicated, but I like the core of the idea. Maybe now, with more experience and the brainstorming power of the MTBoS, we can think of a way to make it work. But first, I’ll describe what .i actually did.

Students entered the room to find the desks rearranged – four big group tables, and the room split down the middle by a wall of desks, representing the “Iron Curtain.” Each student was then randomly assigned to one of four groups: US Missile Command, US Missile Defense, USSR Missile Command, and USSR Missile Defense. (Only one student, the son of the Georgian consulate, demanded to be switched from the USSR group to the US side.)

Each student then had two roles – one of the roles was their job on the team. Treasurer, secretary, chief engineer, etc. These roles were public. Their other roles were secret – they were things like Double Agent, Handler, FBI Agent, Innocent.

The idea was that each missile team was trying to build a missile that could hit the other country, while bypassing their missile defense. And the missile defense teams were trying to shoot down the missiles. The missiles were represented by quadratic equations and the missile defense by linear functions. But the best way to find out what the other side was planning is through espionage.

Of course, the thing they’ll probably learn is that the missile defense fails and everyone dies – we all lose the cold war.

Below are the files I made way back when. What are your ideas to make this workable?