Tag Archives: experiment

Students Lead the Review

For the first time in their high school careers, my students (juniors and seniors) are taking a math final that is cumulative over the entire year. To support them in preparing for this, I took a page from my own senior-year-math-teacher’s playbook and organized a student-led review. Each student signed up for one or two carefully defined topics from the study guide for which they would lead a 10-15 minute review in the final days of class. Each was required to meet with me in advance of their presentation to discuss the details of their plan. So far, those one-on-one meetings have yielded great teaching moments. The presentations themselves have mostly been rough. (I think it’s a healthy eye-opener to the travails faced on the teacher’s end of things.)

Next year I’d like to do this kind of review at the end of first semester as well as at the end of the year. That way they’ll have a chance to learn from their first attempt at teaching and hopefully do better and feel more at ease the second time.

*On a sort of related note, I broke the study guide for the final exam into two parts. There is a list of specific skills that will be tested, but there is also a list of concepts and definitions students will need to know/understand/be able to describe. The student presentations focus on specific skills; I field questions about the concepts/definitions.

Reducing Due Dates

This year I get a trigonometry class all my own! Last year I taught trig, but under the direction of another teacher. This time it’s all me, baby.

I’ve been slowly going through the book, preparing a general outline for the course and putting together a syllabus, and I have an idea. Whenever I get a new idea, I feel certain that this is gonna be it, the idea that makes all the difference and turns my students into enthusiastic proto-mathematicians. So I might be overestimating the value of this idea, but, on the other hand, maybe this is gonna be it!

In truth, it’s not my idea so much as it’s an idea I’m adapting (mostly stealing) from my first college math professor. (Thanks, Dr. Kent!) Here’s my plan: for each section that we cover in the book, I’ll list on the syllabus every problem I want the students to solve. Then, instead of assigning a list of problems each night, I’ll assign two or three sections each week (or so). For example, on a Thursday I might assign book sections 7.3 and 7.4 to be turned in the next Thursday. On each day in between (classes at my school meet every day), the first five to ten minutes of class will be spent answering questions about problems on the assignment.

Here are my hopes: that having time in class to ask questions about the homework before it is due will lead to greater accuracy; that students will feel more energy for a task for which they have bit more autonomy (namely, they can choose when in that week to work the problems); that some will choose to get ahead by trying to figure out material we haven’t gone over in class yet; and that when those students ask questions about advanced problems, others will listen and, instead of hearing authoritative teacher-talk, will hear a peer’s request for understanding.

*May 20, 2013: This turned out to require far more executive functioning than my students were ready for. It went out the window in favor of daily homework assignments before the end of August, although we held onto the five minutes for questions at the beginning of each period.

Graphing Geometry

On Tuesday a question bubbled up in me for no apparent reason, saying, “What does happen when you vary the side lengths of a rectangle but keep the perimeter constant?” To others in the math teaching blogosphere, I’m sure this question is old hat, and even I recognize it from the days of my own high school homework. I can picture myself reaching a question about varying side lengths at the end of an assignment (you know, in the word problem section), weighing the point value of the problem against the inconvenience of figuring it out, and opting to not worry too much about leaving it half done.) But on Tuesday I flipped over the answer key in my hand, made a table of values, and graphed area vs. height.

After noting and confirming that a perfect square yields the maximum area, another question struck me and I thought, “I wonder if I can write the equation for that graph.” It looked similar to a y=–x2, so I wrote in the adjustments that would put the max of that graph in the same location as my max, (6.5, 42.25). I expected to need to make additional adjustments to fit my curve, but what do you know, it was exactly right!

So, said I, I bet I can do that for right triangles with the same give and take between height and base (for every increase in height, with a  max height of 12, the base decreases by the same amount). Looking at it now, of course it makes sense that you’d divide the rectangle equation by two, but I went through a process of fitting the curve to see how it would turn out.

With success number two in the bag, I went on to holding the area of those right triangles constant at 50 and relating the base length to the height.

Then, finally, using the same data about right triangles with area set at 50, I related perimeter to height and truly offered myself a challenge. I quickly pegged it as developing from y=1/x, but had no clue how to make it fit. After a fair bit of graphical exploration, it suddenly dawned on me: all I have to do is write the process of finding the perimeter as a function of height! Connection made!

What a triumph. This is what I want for my students, and how to “give” it to them is clearer to me now. My students know how to use area and height to solve for a right triangle’s base (b=2A/h); they know how to use base and height to find the hypotenuse (a2+b2=c2); they know how to use all three sides to find the perimeter. But they may not know that they can use these geometric pieces to explore totally cool graphical relationships.

Data Gathering Woes

Last Friday I took advantage of a teacher work day to try to formulate some cool math problems based on the law of sines. I ended up on the soccer field with a partner, a camera, a 100 ft. measuring tape, and a compass.

The compass
A somewhat helpful compass.

The value of accurate tools was revealed when I remeasured the angles multiple times with the low-quality compass and got multiple measurements. Lacking any better resources, I resorted to choosing whichever of the measurements led me to the correct calculation.

Now, I’m not fabricating the data, since the numbers I settled on were actual measurements I took. And since I know the law of sines to be reliable, using it to judge the truth of uncertain data is defensible. But… it just feels wrong somehow. Maybe I can assuage my conscience by making the judging of data part of the lesson. Or maybe I should find a compass with sighting guides.

*Interesting but irrelevant note: Despite our best intentions to create a non-right triangle, my data-gathering partner and I ended up with one angle that measured almost exactly 90° (ahem, somewhere between 88° and 95°). Subconscious determination?

Experiment Results

The experiment was a success! Students walked into class today asking to know what the inverse trig functions do (they phrased it as, “What does the negative-one mean?”), two students had reasonable theories to explain it, and one hit it right on the nose.

Before going over the worksheet, I teamed up the students for an activity where one student would write a sine or cosine value on the board, indicating whether it were a sine or a cosine; the next student would name one or several angles with that sine/cosine value and draw the angle/s on the unit circle. They were performing the inverse sine and cosine functions without knowing it. Later when a student attempted to explain his theory for the meaning of sin−1, he suddenly came out with, “I think it’s what we were doing on the board!”

An Experiment in Forethought

Today I’m conducting an experiment in forethought. For homework I gave my trig students a worksheet with problems they don’t know how to solve (arcsine, arccosine, and arctangent) mixed in with problems they do know how to solve (sine, cosine, and tangent), with the instruction that they were to complete as many of the problems as they knew how. At the top of the page is a list of twenty-six possible solutions, of which fourteen are used. (You can find the worksheet online here like I did.) I hypothesize that two results will come of this.

First, the students will be frustrated about not getting clearer instructions and therefore needing to figure out for themselves which problems to solve.

Second, after seeing sin−1 sandwiched between regular old sin, cos, and tan problems with a list of possible answers, the students will come to class tomorrow with ideas about what sin−1 means . Some may go beyond having an idea and may understand it quite well. If so, my experiment will have been a success by allowing a little forethought on the part of the students to ease the introduction of a new function.