Monday, June 3, 2013

How One Student Stopped Me From Dropping the Ball

Every year, June review time is when I drop the ball. I usually give out all kinds of review packages and old exams, together with answer keys, and say to my students "Here you go, my part's done, now it's up to you - good luck!" I also drone on and on during the last few classes about all the bazillions of things we did all year, during which time I literally put myself to sleep. I do this because my brain swirls with these kinds of thoughts: "Well I can't possibly go over everything anyway...If they don't know it now they never will....They need to do the work now not me....and anyway this is SO BORING!"  Basically, I drop the ball on the field and say, here it is, if you want it, come get it.

But that's not the same thing as passing the ball, so that someone else benefits from your momentum. This year, I tried to pass the ball. What made that happen? One student's words.

A couple of weeks ago, I asked my students for some input on how they wanted to review for their June exams. They gave me their ideas via a google form. Here are a few:
The best way for me to review is to gather all my notes and start rewriting down the rules to remember them but mostly...practice , practice , practice!
Review packages, those are the best thing for me to review a whole year's worth of stuff. 
My ideal review would be to actually review everything we did since the beginning of the year and not only do practice..... I think we would just need to refresh everything in our minds. 
do some problems from every chapter but i know you know how to do the best review ever :) 
That last one is the one that did it. "I know you know how to do the best review ever." That came from a student who has put in an effort of 200% all year, and more importantly, has always taken the time to tell me how much he appreciates my efforts, so he had my ear. When I read that, all those swirling, bored, tired, I'm-now-dropping-the-ball thoughts thudded to the floor. I did NOT want to let this student, or any of my students, down.

So I put together some overviews that actually did make it interesting, at least for me, who has seen it all before a zillion times. Based on their reactions, it didn't work perfectly, and I already see lots of places to improve. But I feel better about this than anything I've ever done at this time of year in my entire career.

First, I must mention, that all year we've used a graphic organizer for each of the functions we've studied. I call it the Wheel of Function (get it? Wheel of Fortune?....sigh....). Every time we finish a new function, we go around the wheel of function and summarize it using these headings, so this is not the first time I've tried to give them a Big Picture of some sort:

But this time, it became the Wheel of FunctionS:

I used the wheel to make activities that would:
  • cover a heck of a lot of material in a very short time
  • meet them halfway in terms of content - start with activities at the lower end of Bloom, then work our way up
  • get them up to speed so they can do the practice they need/want
  • deepen understanding for some
  • cause understanding for others
  • at the very least make them do some review during class if nowhere else
Here are some that we did, and some that I came up with afterwards, with the instructions I gave as captions. By the way, these were all given to them in powerpoint format, so that they could move the tiles around:

Graphs: Start with simple recall:

Match the groups of graphs to the function rule

Then get into more detail:
Move each tile into the appropriate column

After more matching, look at the properties from a comparative point of view, leading to some Higher-Order-Thinking: There are similarities between log and square root function domains, so why is the only difference the inclusive/exclusive brackets? Or - The exponential and log graphs look very similar - how can you tell the difference between them?

Definitions of basic functions:

For this one I had them all writing on the board, just for variety. It was interesting to see how little everyone remembered. And frightening. It was a great opportunity to go over the algebra behind how we simplified them, and there was an aha moment for at least one student who said she hadn't noticed that we never did simplify the trig functions. Next year, maybe we will by using identities (AT LAST SOMETHING USEFUL HAS COME FROM IDENTITIES!!!)

Fill in the simplified form for each function

Solving equations and finding intercepts:

Before getting into the nitty-gritty algebra, an overview of What to Expect When You're Solving an Equation. (I think next time, I'll have them fill in the orange part first.) The idea behind the purple part was that for some equations, it's obvious when there's no solution - for example, if a quadratic equation has no solution, some kids find out only when they try to take the square root of a negative number, and their calculator lets them know. But for some equations, it's easy to miss when there is no solution, because the algebra doesn't always alert you, for example, in this equation,        it's very tempting to square both sides, but the fact is that there is no solution to it. I wanted them to be on the alert for those situations, as well as to realize, perhaps for the first time in their math lives, that linear equations ALWAYS have a solution, as do logarithmic ones:

1. Fill in the number of intercepts that the given function can have: none, 1, 2, or infinitely many.
2. Indicate whether it's obvious when the given type of equation has no solution, and what it looks like when that is the case.

......that last one needs work. Not clear.

That was last week. Today I showed them this series of slides, in which I tried to summarize probably way too much about solving inequations and finding the rule for a function, again from a visual and comparative point of view:

I think I may have exploded a few heads......including my own. But I will definitely use it next year, and if you see a way to make it better, please feel free to comment.

Bottom line, I am so grateful to all my students, especially Mr. 200%, whom I shall call Albert today, and he'll know why.

Thanks for reminding me, Albert, that even though my job description is "Teacher", you and your peers truly deserve that title.

And oh yes: Catch!


  1. No wonder that "Albert" has given 200% all year long - he knows you are listening to him.

  2. Wheel of functions is an excellent idea!! I'm sharing on Facebook. Thank you so much!

    1. I'm so glad you liked it, and thank you for sharing it! Please feel free to use and modify as you see fit, and if you get a chance, let me know how it goes over, I'm always happy to hear feedback, good or bad. Tks again! (Wish I knew your first name...)

  3. I'm interested in using this in my Precal class as a year long graphic/mental organizer. Could you tell us a bit more about what information you would put in categories like "operations," "composite," "Solving equation," or "solving inequation"? Or if you have an example of what a filled out wheel of function would look like that would be really helpful.

  4. I am also interested in using this to help my students organize their knowledge about functions. Could you also send me more information about the other categories as well?