As you might notice, I haven’t been so great about follow through on this blog. Lots of ideas to blog, but I never get around to them. But Tina Cardone requested posts about just what happened in a single class, and I told myself I can blog this. I’m going to focus on my 6th period 10th grade math class from yesterday where I used one idea I had just gotten from some colleagues last weekend, and another idea I came up with on a plane last weekend.
10th grade at my school has aways been about functions. We used to use a text where we started (see chapter 3) with the definition of a function, what is it, what do we mean by it’s graph, how can we do transformations to the graph, what does it mean to compose functions, and finally what it meant by the inverse of a function. All of this was the first unit of the year, after which we proceeded to look at polynomial and rational functions, then exponential and inverse functions, and finally trigonometric functions and their inverses. A few years ago I tried changing the order so that in the first semester we did all of that except we didn’t look at the concept of inverse, logarithms, inverse trig, or solving polynomial equations, exponential equations, trig equations, or solving triangles. The theme of inverse and how it could be used to “solve” was done in the 2nd semester. I did like this better, but this year we changed up the order again. Now we start with looking at sequences, difference equations, and modeling without worrying about formal definitions of functions. Our idea in doing so was to really get to know and connect with functions in specific contexts before worrying about the concept more abstractly toward the end of the year. I should write much more about this approach later, but for now what’s relevant to Monday’s lesson is that I was about to introduce the inverse trig functions, but our class doesn’t yet have an abstract concept of function or the inverse of a function. We do, however, have experience with logarithms and using them to find when an exponential growth model reaches a given point, and we have experience using square roots to help find when a quadratic growth model reaches a given point. In terms of trigonometry, we’ve defined trig functions in terms of modeling the sun’s apparent motion across the sky (and back around under the flat Earth until it rises the next day). We’ve also explored how that understanding can help us distances from knowledge of angles. My goal for Monday was to draw on these experiences to lay the ground work for defining the inverse trig functions with the details presented in an assigned reading Monday night to be followed up on the next day (today).
So this past weekend I was working with two colleagues to do some PCMI planning and they told me about this great idea they had picked up at a recent conference (NCTM?) in Orlando. So I don’t know exactly whom to credit for this, but the way I understood the idea which they called “the bagel” was as follows. A circle–I guess that’s the bagel–is drawn on paper with about four spokes coming out from it. A question is given, perhaps as homework, for which there is a short answer. At a table each student writes their answer to the question in one of the regions formed by the dividing spokes. Then the table can discuss and debate their answers and try to convince others of the correctness of one answer. If the entire table becomes convinced of one answer, they can place it in the center. I often throw out problems or questions where I want each student to give the question some individual thought first before then discussing it with the table. The bagel seemed like a great way to try to make sure some more individual though was given before table discussions commenced, and for me to be able to quickly see all the individual responses and quickly gauge where there was agreement and where there was disagreement.
On Monday I started class with four short problems (no calculators) with the instructions “Do by yourself first in your notes..NO DISCUSSION YET”. The problems (1) simplify (if possible) (2) solve (if possible) (3) simplify (if possible) (4) solve (if possible) . I also had a fifth problem for those who did the first four quickly (5) . While the students were working individually on these problems I placed a sheet divided into 4 rectangles labeled 1-4 and in each was a bagel described above. I then gave the students the bagel instructions. The idea in starting with these problems was that I wanted to students to do some refresher work with logarithms (we hadn’t really used them in over a month), I wanted them to think about the idea of an inverse function and how it can be used to solve an equation, and, most importantly, I wanted them to think about the subtlety involved in square roots where the radix refers to one particular root, but when solving we want to list all solutions unless there is context involved.
Almost all were able to do problems (1) and (2) and the student or two who wasn’t was quickly convinced. They just needed a refresher. Problems (3) and (4) though led to some interesting discussions. Problem (3) some wanted the answer to be 5 or 19 and others just wanted 5. Three tables became convinced of 5, two became convinced of 5 or 19, and 1 remained divided when I turned to a whole class discussion. For problem (4) five tables became convinced of although at most tables at least one student originally had just . In fact, at one of the tables three of the four had that answer, but the fourth person convinced the other three of her answer. The sixth table also became convinced of just . All together this was about 10 minutes.
Whole Class Discussion
On the board I pointed out the two consensus answers for problem (3) pointing out that each was able to gain the confidence of an entire table. I asked for a volunteer to try to convince other tables of his or her table’s answer, and then for a volunteer in the other direction. In hindsight, I think instead of volunteers I should have called upon specific people who had already been convinced to change their own answer. The discussion continued for a little with no input from me except calling on hands, and then I said, “Let’s come back to this and look at problem (4).” Again I shared the answers and called upon volunteers. This time one student defended the but then noted that he wanted to change his answer on problem (3). He reasoned that if could mean 7 or -7, then there would be no need for the in front of the . This argument seemed to convince most of the class, but I chimed in at this point also with the question if was 7 or -7 what would . Nobody in that class seemed to like that it would have to be 14 or 0 or -14. (Interestingly, the other section of this class required some more convincing and more prodding from me.) In this section this took about another 15 minutes.
Always-Sometimes-Never meets Rock-Paper-Scissors
In the second part of this lesson I wanted students to think about how in the context of a right triangle a given trig ratio should determine a unique angle even if we didn’t have a name for that angle. I like the idea of always-sometimes-never questions but hadn’t yet done any this year. I thought this would be a good place to try some, but as I was planning this lesson I considered how to collect their responses. I don’t do poll everywhere because students are supposed to have cell phones in school (although of course they all do). I thought about individual whiteboards (which I also haven’t used yet but should) but decided that was unnecessary. Instead I thought about having students raise their hand with different hand signals. It occurred to me that rock-paper-scissors provided some clearly recognizable hand signals and the idea of everyone needing to throw out their signals at once without waiting to see what someone else does was perfect. So I explained the “game” to the students and said use rock (fist) for always, paper (palm) for sometimes, and scissors (peace sign) for never.
After also taking a moment to define what it is meant by “knowing” a triangle we played always-sometimes-never with the following statements. (Statements were given one at a time with discussion and debate in-between, but in this post I’ll present all of them together now).
(1) If the area of a square is known, then the length of its sides is known.
(2) If the area of a rectangle is known, then the length of its sides is known.
(3) If a right angle and second angle are known, the triangle is known.
(4) If a right angle, a second angle, and a side are known, the triangle is known.
(5) If a right angle and one side is known, the triangle is known.
(6) If a right angle and two sides are known, the triangle is known.
The idea of the first question was to set the stage for the usefulness of giving a name to a number we know exists. So in the discussion that followed it I pointed out that if the area of the square was 5, we agreed there was a determined side length but if we thought about it that symbol didn’t tell us how long the side was but rather gave the name to a number we knew was a little more than 2.
This post is already too long and I need to get to sleep so I won’t go into detail on all of the discussions that followed from each statement, but I will note some interesting things that came up. The game worked great. Every student was engaged and we had some great discussions stem from it. I always gave a few moments to think and then said “1-2-3 throw”. If a student raised their hand immediately or didn’t raise a hand at all I compared it to what would happen if actually playing rock-paper-scissors. Most statements had just two responses and a little debate between students reached agreement. Sometimes often came up because of ambiguity in my statement and I said that was great. A sometimes response would often lead me to refine my statement and revote. For example, for statement (2) one said sometimes because if the rectangle were a square we’d know (in the other section the issue came up if the area of the rectangle was prime would we know the side lengths and other students pointed out we didn’t require integers). For (6) one student pointed out that we needed to know which sides we had (that reasoning could have also applied to (4). In all this was about 20 minutes and I finished the class with introducing the homework which was to do the reading including any exercises the encountered in the reading.