Trying to find math inside everything else

Archive for the ‘math’ Category

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!

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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.

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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.

Whole Class Test

I teach an SAT Math Prep this year, which has been an interesting challenge. We basically started off with lessons on all the different content in the exam, then had a long section on tactics (which can be framed as test-taking tactics but I noticed are often just tactics for solving problems in general, which was nice). But we reached the end of those, and the (in-school) SAT is a month away. The obvious thing to do is to just keep doing practice exams, but that can get a bit boring, for both me and the students. Plus, the class that meets Tues/Thurs hasn’t had very many graded assessments this marking period, so I needed to give them something.

I had decided that grading them on correctness in a practice SAT is not appropriate. I had told them this before, and they knew their grades on their assignments were more for things like how they applied the tactic we were learning. But last class they walked in and I gave them a Part 3 exam (the non-calculator part) and told them it would be graded – but there would be a plot twist. For right now, just take it individually, except this half of the room should start from the back and go forward. Oh, and you get 5 fewer minutes than normal.

While they were working, I went around on my whiteboards and put up the numbers 1 through 20 well spread out, and an ABCD for 1-15. (I wish I had taken pictures!) This started to get them suspicious. When time was up, I told them my grading scheme: it was out of 5 pts, and they lost a point for every question they got wrong. So if you got 15 right, that’s a 0. But! They had the remaining 20 minutes of class to work together and figure out what the right answers should be. And if anyone got less than 15, the whole class lost a point – forcing them all to work together. (With limits, of course – they won’t be penalized for that kid who went to the bathroom for 15 minutes during this, for example.)

A suggestion I made to them was to go around and make votes for their answer for each question. A clear consensus might mean that that is the right answer. However! Don’t be afraid to put your answer down even if everyone else’s is different. I’ve seen questions where only one person got it right. I told them they need to convince each other of what the right answer is.

Let me tell you, I heard so many great conversations as they and I went around the room. Because it’s the SAT, no one gets them all right, so everyone is being pushed to make a convincing argument that their answer is right. Students who weren’t sure got explanations from others. It was delightful!

About halfway, I noticed a clear consensus for about 15 of the 20 questions, but the middle 5 were really quite split. So I lead the class in sharing out their reasoning for some of those questions – never saying what the right answer was, but again letting them convince each other.

It was a nice collaborative effort – I highly recommend it.

Integration First

Last year I went to a PD at Math for America that was about approaching calculus from a geometric point of view. The presenter mentioned during it that, historically, the idea of the integral was developed first, followed by the derivative, and then the limit. Yet in many calculus courses, they are taught in the exact reverse order. I decided that, should I teach calc again in the fall, I’d do integration first.

Well, school is rapidly approaching, and so I’ve been thinking about it again. I did so searching and found this intense forum discussion (oh, old Internet), which pointed me in the direction of the Apostol’s Calculus 1 textbook, which starts off with integrals. The post also had a bunch of arguments about why I shouldn’t do it. One of the notable arguments was that in order to fully teach integration (including u-substitution and integration by parts), you need differentiation. But I actually view that as a benefit, not a downside, because it forces a more spiraled approach. I can start with integrals, then go to differentiation, and then tie them together.

In general, I feel like area is a much more approachable subject than slope. My years of teaching Algebra I to 9th graders certainly seems to support that claim. But I also think it’s easier to understand the linearity of integration than the linearity of slope. “If you add together two functions, the area under the new function is the sum of the areas under the old functions” seems much more evidently true than “If you add together two functions, the slope of the tangent line for each point of the new function is equal to the sum of the slopes of the tangent lines at the same points on the old functions.”

Of course, Jonathan has already worked to restructure his calculus course, and I plan on taking a number of cues from his more spiraled sequence – but still with integrals first.

Here’s what I’m thinking:

Q1 (Intro to Integrals) – (Sam’s Abstract Functions, Area Under Stepwise Functions/Definite Integrals, Properties of Integrals, Riemann Sums, Area Under a Curve, Power Rule for Integrals, Trig Integrals, some applications)

Q2 (Intro to Derivatives) – (Average vs Instantaneous RoC, Tangent/Secant Lines, Power Rule, Trig Derivatives, some applications)

Q3 (Fundamental Theorem) – FTC, Chain Rule/u-substitution, Product Rule/Quotient Rule/Integration by Parts, Curve Sketching/Shape of a Graph)

Q4 (More Applications) – Related Rates, Optimization, Volume, etc

How does that sound?

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)

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.

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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.