Functions [linear, quadratic, exponential, trigonometric] and Graphing [linear, quadratic, sinusoidal; domain and range]
Interpreting Graphs [distance vs. time]
Solving Quadratic Equations
Exponents and Logarithms
Angles and Trigonometric Values [sine and cosine; deriving the values for angles in quadrant one, providing the values for other standard angles]
I find I’m a big supporter of the cumulative final, even and especially for students who struggle with long-term retention. How else will they train their minds to hold on to things? I’m an equal proponent of intelligently designed cumulative finals. My final this year was not a test designed to congratulate those with natural retention and punish those without it. We spent time throughout the year, plus a good chunk there at the end, building student retention of important skills and information, making my final an opportunity for students to take pride in having actually learned things.
Not that my final was a cake walk. The expectations in the test were high to match the value I intended it to have.
I’m cleaning out my email inbox and found this in an old, unread blog post:
(This is the blog post, by Dan Meyer. This is the original source of the image, by Rachel Kernodle.)
My efforts at teaching inverse functions this year were fun, but, alas, not very effective. I started with the “function box,” then added the “inverse function box” with the full range of appropriate sound effects, but the two “boxes” were just too similar. There wasn’t a strong enough visual signal of the opposite-ness of the two kinds of functions. Ms. Kernodle’s stapler/remover analogy could be the key to finally getting the message across.
The trig students gushed unforeseen excitement about seeing me derive the Law of Sines today. They loved it! My plain old plan for it had been this:
Go through derivation clearly but quickly so students see a connection between SOHCAHTOA and sinA/a=sinB/b=sinC/c
Demonstrate using the LoS to solve a triangle
Have students practice solving a triangle using LoS (make sure they do the practice problem)
Little did I know that step one would be so energizing and fascinating to them! Although I told them they would not have to reproduce the derivation, students who normally forget the proper function of notebook paper were furiously copying down the steps, asking questions (“In the area equations, what represents the height?” “What’s the pattern of the letters in the different equations?” “Why is it that way?”), and answering other students’ questions.
My favorite moment was when I asked the class, “Does everyone understand how I went from this to this?” A pedagogically poor question, it has traditionally been answered by silence, a dull or uncertain “Yeah,” or even a perturbed “We get it.” Today in my class it was answered by, “Yeah, you broke off that part and everything cancelled out except the a on the bottom.” The language is not terribly specific or mathematical, but it’s far more so than any of the traditional replies.
I wonder whether all derivations would strike these students as forcefully, or whether this one was especially suited to the task. Either way, I slapped a sticky note down on my lesson plan to remind myself to hit it out of the park again next year.
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 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?
Students at my school are preparing for parent-teacher conferences, for which they (the students) prepare a slide presentation about how they’re doing, where they can improve, and what techniques they’ll use. In view of that, one student came this morning to ask how he’s doing in my trig class. As I collected my thoughts I asked him to sit down and tell me how he thinks he’s doing in the class. He then delivered this unexpected simile: “It’s like we were thrown out of an airplane,” he said, “and into an ocean.”
“Before it was like we were in a boat and we’d go scuba diving, but yesterday it was like we were thrown out of an airplane into the ocean.”
The before that he mentions I assume refers to our most recent topic, trig functions on the unit circle. The thrown-from-an-airplane-and-into-an-ocean feeling comes from moving into trig functions based on triangle side ratios (SOHCAHTOA).
I’m always glad to hear students describe their understanding but, to date, this is my favorite description. The sentiment was so unexpected and the image so expressive! It is also strangely satisfying—not because I want my students to feel lost at sea, but because he could have said something far less heartening. If he’d given me a tired idiom (“It’s like we jumped from the frying pan into the fire”) or something just plain tired (“I’m lost”) the message I would have received would be that he’s not confident with the material. That’s a valuable message, but what this student communicated to me is that although he is not yet confident with the material, he’s engaging with it enough to come up with an original and vivid analogy for his struggle, and to compare it with an original and vivid analogy for a previous unit.
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!”
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.