Class Participation

I’ve always had class participation as 10% of the students’ grade.  The main reason is that I want the class to be as interactive as possible; I don’t think many of them will “get it” by just listening to me and taking notes.  And if they don’t get it, they won’t be able to practice at home; they’ll just bring back incomplete homework.  So I try to engage the students deliberately; when I call on them, I call on the ones who don’t volunteer answers.  If someone is struggling with an answer, I might call on another student to bail them out.

And one of my favorite things to do is to ask for a volunteer to work/answer a problem.  If they get it right, they get to pick the classmate to answer the next one.  And so on around the room.  They like this a lot; I see a lot of paybacks where A called on B yesterday so B calls on A today, but it’s kept friendly.  So I like the class participation part of the grade, but I don’t generally leave it up to the students; I’ll engage them and make them earn that 10%.  With the way the electronic gradebook works, I don’t enter the class participation grade until the end of the quarter, so a borderline grade will often get a small boost at the end, which is nice.

Precalculus Stew

Precalculus does seem to be a hodge-podge of different ideas, doesn’t it?  In talking to students, I’ve explained the reason for this as this is the last chance a math student has of picking up certain skills before moving on fully to calculus.  I taught 6th grade math one year, and I noticed that the book (Saxon Math 76) was all over the place in the subject matter.  But I realized that the reason for this was that the students would be starting prealgebra next, and this was the last chance to hone the skills they’d learned before moving on.  I always pitch the hodge-podge nature of precalculus as a positive to my students: “Hey, if you don’t like what we’re doing, there’s always a chance you’ll like what we do next.”  And this has been true of a number of students.  A few of them who don’t like the insight required for algebra absolutely love the rigor and plodding nature of matrices and/or statistics.

But there are some major themes running through precalculus which give us a few categories to classify the disparate subjects into.  I talk about precalculus as being a little bit Algebra III (into which I group matrices), a little bit Trigonometry (which I also call Geometry’s big brother), a little bit Complex Numbers (into which I group polar graphing and vectors), and a little bit Statistics/Probability.  So those are the four major themes I’d identify.

BFT

Below is something that’s been unique to my classes (as far as I know).  If you’ve taught trigonometry, you’ve seen the unit circle with the points marked for the special angles.  I have another tool that I use, which we call the BFT (Big Freakin’ Table).  The columns are “deg”, “rad”, and the six trig functions.  There are 17 rows, all blank, for the special angles from 0 to 360 degrees.  We fill it in as we learn the special ratios from the 45-45-90 triangle and the 60-60-60 triangle.  Then we expand the knowledge of trig to the ray passing through (x,y).  But presenting all of the data in this form allows me to point out some patterns and relationships that I think are key for good for understanding trig:

1. Relationship between degrees and radians – the students will know how to convert from one to another, but they’ll also be able to see that pi/6 = 30 deg and 7pi/6 = 210 deg.  That kind of proportionality reinforces the understanding, in my opinion.
2. Cofunctions – in the first quadrant, sin and cos are the same numbers but reversed order.  Also, they get that sin increases but cos decreases in the first quadrant.  The order of the columns reinforces the cofunction relationship; they’re all grouped together.
3. Symmetry – they can see that sin repeats the same numbers coming back down in the second quadrant as going up in the first.
4. Reference angles – they can see how even the negative numbers are repeats of the positive, just following the same pattern as sin pushes into the third quadrant.
5. Identities – we compute tan and cot from sin and cos, and sec and csc by reciprocating cos and sin.  Done 17 times or so, this helps stick the relationship.  In fact, by the time we get to csc 210 degrees, there are several ways of easily figuring it out from the data on the table, and all of them work.  The order of the columns reinforces the reciprocal relationship; the reciprocals are the middle two (tan and cot), the next two out (cos and sec), and then the next two out (sin and csc).

When we’re done, I stress to them that from the quadrant 1 sin values we can get the quadrant 1 cos values.  From there we can get the rest of the quadrant 1 trig values.  Knowing how the reference angles work and the positive/negative pattern, we can get the entire rest of the table.  So all we need are 5 little values to get the entire table.  And I discovered a few years ago there’s even a pattern there:

• sin 0 deg = sqrt(0)/2
• sin 30 deg = sqrt(1)/2
• sin 45 deg = sqrt(2)/2
• sin 60 deg = sqrt(3)/2
• sin 90 deg = sqrt(4)/2

deg	rad	sin	cos	tan	cot	sec	csc

Analytical, Numerical, or Graphical?

Despite using technology all of the time, I am a late-adopter when it comes to teaching upper school math.  I was convinced that an analytical approach to problem-solving was best, since it requires critical reasoning skills.  What’s an analytical approach?  It’s the kind you probably learned in school as well: for the equation x^2 = 3x+4, you move everything to the left-hand side, factor, and solve.

And this is great for strong math students, but some people think differently.  For limits on rational expressions of x as x approaches infinity, the best way to convey what happens may be to just use really, really big values of x and plug the expression into a calculator.  This type of approach is called numerical.  For the example above, you can show how the x^2 value gets smaller and then larger again, and how 3x+4 just gets larger.

A graphical solution would involve graphing the left-hand and right-hand expressions and seeing where they cross.  There are downsides, of course; you need to know that your graphing window includes all of the solutions you’re looking for, and while that point looks like (4,16), is it exactly (4,16)?

But while I used to look at the graphing calculators as a crutch to critical thinking skills, I now see it as a useful tool in allowing the student to use numerical and graphical methods to solve problems.  And I feel this is beneficial because each method can strengthen the comprehension of the other methods.  For a student who thinks visually, the use of the graphing calculator may aid in his understanding of why the analytical method works, and so on.  So while I’m a little late in incorporating the graphing calculator into the precalculus curriculum, we’re doing it this year.

Greetings!

This is the blog of a part-time precalculus teacher at a private school.  I’ve started this as part of a project to get math teachers blogging in order to share ideas, but I’m happy to have anyone follow along.  I’ll try not to write for any one audience.