Trying to find math inside everything else

Archive for the ‘lessons’ Category

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.

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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.

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?

The Alien Bazaar

Michael blogged and tweeted about exponentiation being numbers instead of operations, which made me think of a lesson I attempted last year. It didn’t go quite as planned, but I liked the idea, so I said I’d write about it.

At a Math for America meeting, we were working on problems with different bases. I was talking with the facilitator about how glad I was that my 4th grade teacher taught us alternate bases, as I’m sure it was pivotal in greatly improving my number sense. Later on in the session, we were talking about “real world math,” and I brought up how it’s not the real world that is important, but that the setting of the problem is authentic and internally consistent. The presenter remembered a lesson she had seen that was totally fictional, but still felt authentic: if aliens with different numbers of fingers (and thus, presumably, differently-based number systems) were at a galactic bazaar, how would they sell things to one another?

I loved the idea, and so wanted to extended it beyond the simple worksheet-based problems that it was. I wanted to have an alien bazaar right in my classroom. So I developed 6 species of aliens, each with a different number of fingers. Each species of alien brought a different set of items to the bazaar, to sell to the others. Each species also had a shopping list of items it needed to purchase before it could head back home. (I set up groups of four to be all the different races.)

Each species’s inventory was given to them in a folder containing all the items. (Each species had a monopoly on one specific item, but was in competition with the other items, and no one race had enough to satisfy all customers.) The first task was, given the prices of each item in decimal, to convert them to the number system of the group’s race and set up a shop with a display and the items spread out.

After that was complete, each group was given blank-check currency, color-coded by race. Two members of each race were to act as shoppers, going around to the other races and 1) figuring out how much their item cost, 2) figuring how much money that would be in their own number system, and 3) making sure the shopkeeper agreed with their payment.

The other two students stay behind to run the shops, checking the work of each shopper to make sure they were not ripped off. This made is more like a bazaar – both the buyer and the seller had to agree on the price being paid, which can be tough when the buyer and the seller use different number systems!

I was really excited by this idea, but it didn’t pan out quite the way I wanted. I think the main problem was that there wasn’t enough time – these are big, unfamiliar ideas and the whole process needed more time than I was willing to give it. I taught the lesson in the hopes of strengthening their ideas about exponents and scientific notation, but since those are such a small part of the 9th grade Integrated Algebra curriculum, I couldn’t really devote a lot of time to it. I hear they are big parts of the 8th grade common core standards, so this might work well in an 9th grade classroom or even earlier.

What I did do, though, to salvage the situation, is to segue the lesson into an exploration of polynomials, and this is related even more so to what Michael talked about with exponents being numbers.  When we have the number 132, that number is really 1 \cdot 10^2 + 3 \cdot 10^1 + 2 \cdot 10^0. But that’s only for us humans. If that number were written by the 5-fingered aliens, it would be 1 \cdot 5^2 + 3 \cdot 5^1 + 2 \cdot 5^0. And, in general, if we wanted to figure out what that number means for any number-fingered alien, we would use 1 \cdot x^2 + 3 \cdot x^1 + 2 \cdot x^0. So we looked at that in Desmos.

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We start with x=4 because 132 doesn’t make sense as a number for a base smaller than 4. And now we can see all the different values of 132 in the different bases. As we go into numbers with more digits, we start to get more interesting polynomials, which was also fun to explore.

I’d love to hear feedback.

Estimation180 and Absolute Value Graphs

So as I was getting ready to teach absolute value graphs a little while ago, I came across this post from Kate Nowak about a lesson she did with it. I liked the idea but…I didn’t like the idea of having to “get my butt into overdrive” to collect data from staff and students about such a thing. I wanted a lesson for the next day, so I didn’t really have time for that.Screen shot 2014-03-26 at 6.52.24 PM

But then I thought, well, my students have been doing Estimation180 all year long. Maybe there’s a way I can use that? I even tweeted Kate about it, but was left to my own devices. (Though I suppose this is finally the write up I promised.)

 

I thought about what was different between what we’ve done with Estimation180 and Kate’s task, and then it hit me – Kate’s lesson is all about one guessing event, but we have loads of different ones. At that point we have done ~30 estimations. What if we could do some comparisons?

 

My premise was this – Mr. Stadel, who runs the Estimation180 site, wants to implement a ranking system where all the estimations are listed as “easy” “medium” “hard” etc. But how can he tell when one is hard or not? He knows all the answers, so he can’t used himself to judge. So I told Mr. Stadel that we have lots of data from our class and we could probably use it to come up with a system.

[Aside – this was the 3rd or 4th day of the new semester, and to complete the task I asked students to use the estimation sheets from the previous semester. They revolted, because they claimed I had told them they could throw those out! Which I vowed I didn’t…though, to be honest, it’s possible I did, since I hadn’t thought of this lesson yet. Luckily enough students had not thrown them out so that it could still work.]

So I reviewed what the estimations we did were and I told each group that they have to pick one estimation that they wanted to evaluate. Then they had to collect data from their classmates (and from the binders of other classes, through me) – the estimate each person made and what their error was. Once they have collected enough data, they have to make an Error vs Estimate graph and see what happens. Then I had them make some analysis on whether this counted as a difficult task or not. I didn’t have them compare graphs at the time, but I totally should have.

I think it worked pretty well and many of the students understood why it should be a V-shaped graph. They were at first surprised about where the vertex was, but then it made sense, especially comparing many different error graphs.

Estimation Difficulty Rating (Word format)