## Sunday, October 12, 2008

### Graph Savvy

I've had former students come back to visit lately and they talked about the expectations of their college professors. One student who's in calculus says that the professor expects them to have a solid, quick-recall grasp of basic graphs (what does the sine graph look like? the square root function? the tangent graph?, etc). Yeesh, note to self on not just teaching that for one week but reinforcing and recalling it periodically.

Then that reminded me of a dream I had the other night. I like memory tools, and I dreamt that one way for the kids to remember the square root of x graphs is that, "SEE, the graph LOOKS like the square root symbol". Nerd alert. I'm going to try that this year.

Then I also started thinking of how to make all these graphs become second nature to the kids. Maybe periodically, and consistently, I can pull out "cards" with (say) pink cards graphs and yellow cards the functions, and maybe they can have a running contest with themselves and keep track of the time it takes them to match them up, and eventually (months? weeks?) we can have a "beat the clock or teacher" contest of a match up game.

I also want them to have the quickness of thought as to, "okay, I don't remember this shape, but hey! let me quickly make a table and plot the points to help myself (without the teacher prompting me or just sitting there and not doing anything)". Hmmmm, how to teach this skill. ... Maybe also a periodic and consistent "game" of "here's a weird function on the board, quick like a bunny, find 5 points on the graph or plot 5 points or something".

1. It's more than recognizing different graphs (although that is part of it).

What I want them to understand is that square root is the inverse of x^2 so the graph looks like half a parabola reflected through y=x and they should understand why it is only half of a parabola.

It is understanding that if you know the graph of y=f(x) then
y-h=f(x-k) is the same graph translated and that idea works whether you are graphing y=mx+b or (x-a)^2+(x-b)^2=r^2 or y=a(x-k)^2+h.

2. Anonymous6:40 PM

Thanks for your feedback. I'm guessing my students know the inverse relationship between x^2 and sqrt(x) enough to say which are inverse operations of each other. Also if prompted they can do the "reflecting thing" .... I believe that in their minds these are all different "topics" of math and they may not see the connection of identifying the graphs, and then also keeping in mind the inverse graph relationships and at the same time the transformations.

I'll keep that in mind to keep making the connections.

3. I've found that "graphing simon says" helps a bit too. I'll call out "absolute value" and students have to show the shape of the graph with their arms. Or "square root of x" or "x cubed" or ... you get the idea. Admittedly the trig graphs are kinda hard to do this way.

Then we start including transformations. "Absolute value of the quantity x + 3" - so they need to make the "vee" and move 3 steps to the left.

It helps some of them. For some of them the moving around for a bit is just a nice change of pace.

4. A few years ago, I was at a Pre-AP workshop where the presenter does Row Races in her class whenever there's a few minutes left at the end. She would say either parent functions or transformations of them and the first person at the board to have drawn it correctly wins a point for their row.

Think back to grade school and multiplication races to give you some ideas... Around the World maybe? Flashcards?

5. Wow, I love all these ideas. As an intern, I am just in the sink or swim stage, lol. But I have included similar "game" strategies with my grade 10 math. I have the class (very small class) divided into two teams. These teams will be kept for the entire semester for the purpose of quizing or just playing around. Winning points are added to each teams totals. The other day, we had 5 minutes left, so I put 8 equations on the board where the students must put in the form y=mx+b. I wanted each student to also find the slop and the y-intercept. The class is made up of 10 students, therefore 8 had to find the solutions and 2 had to check their groups work for errors.
I love reading your blog....keep up the great work. Oh, if you have a math website for your students, would you be able to put up a link on your blog.
Thanks,
Michelle Munro

6. Anonymous12:06 PM

I'm loving these extra ideas, too. They'll be great to reinforce and teach various topics.

I do have a math website for my students, but I want to keep my real name out of the mix here.If you want the link, just send me e-mail, and I'll give it to you. (math_mambo@yahoo.com)

7. sam shah11:10 PM

I might be a bit old school, but I show my students in Alg II seven "basic functions" (y=x, y=|x|, y=x^2, y=x^3, y=sqrt(x), y=cubert(x), y=1/x) and we come up with the graphs together.

We talk about important points to get correct when graphing (e.g. you need to get (1,1) and (-1,-1) for y=1/x).

And how to come up with the graphs in case they blank.

And then I say: "You have no homework today except to come to class tomorrow knowing these basic graphs. You will be quizzed. In fact, here's a copy of the quiz. Seriously." [Hand out blank copy of quiz, which is just seven blank grids with the seven functions written above them]

And I tell you, it's cold, but it works. From that point forward, most of my students know those basic graphs and their key points for the rest of the year. (We tend to do this right before we do function transformations. And them knowing these key points pay off a million fold.)

(I also do this when we hit trig and we learn the sine, cosine, tangent, cosecant, secant, and cotangent graph.)

Another teacher at my school does "function arobics" in her precalc class. And then when she hits conics, she pairs students together and they do "conic yoga."

Sam Shah

8. sam shah11:10 PM

I might be a bit old school, but I show my students in Alg II seven "basic functions" (y=x, y=|x|, y=x^2, y=x^3, y=sqrt(x), y=cubert(x), y=1/x) and we come up with the graphs together.

We talk about important points to get correct when graphing (e.g. you need to get (1,1) and (-1,-1) for y=1/x).

And how to come up with the graphs in case they blank.

And then I say: "You have no homework today except to come to class tomorrow knowing these basic graphs. You will be quizzed. In fact, here's a copy of the quiz. Seriously." [Hand out blank copy of quiz, which is just seven blank grids with the seven functions written above them]

And I tell you, it's cold, but it works. From that point forward, most of my students know those basic graphs and their key points for the rest of the year. (We tend to do this right before we do function transformations. And them knowing these key points pay off a million fold.)

(I also do this when we hit trig and we learn the sine, cosine, tangent, cosecant, secant, and cotangent graph.)

Another teacher at my school does "function arobics" in her precalc class. And then when she hits conics, she pairs students together and they do "conic yoga."

Sam Shah

9. Have you tried the online Math Games at Hooda Math yet?
hoodamath.com

10. Anonymous9:19 AM

Ooh, no I haven't seen the "hooda" site yet. Thanks for the tip.