Monday, December 3, 2012

7 Best, 5 Worst

It's time for a quarterly review. Here's the good and the awful from this year's teaching.

7 Best 

1. Paul Salomon's Introduction to Proving Stuff about Exponents - The idea is to use function notation to prove things about exponents without being distracted by the repeated-multiplication model that we (rightly) inculcate our kids with in the early years. Maybe the best thing about this problem is that it really does force everybody in the room to use proof. There's no mushiness, no intuition to fall back on, just cold reason. The second best thing might be getting student work that looks like this:

2. Exponents for Functions - And all it took was a little bit of nudging to get kids to understand why the hell f^-1 should refer to the inverse of f. It was beautiful. Then we drew the analogy farther, figuring out what a rational "power" of composition would have to mean.

3. Encryption and Inverse Functions - Not huge, but it gave me a language for talking about invertibility. Plus, it was a ton of fun. ("Can you give us, like, enough time to actually figure the code out?")

4. Swap and Solve with Equations - My kids were struggling with equations. They could handle anything that you could undo the steps on, but that thing don't work if you've got variables on both side of an equation. I wanted to share with them the "you've got equal weights on a balanced scale" thing, but I couldn't make it snappy. 

This was a blast. I gave everyone an index card with a number on it, and they had to write an equation that had that number as its solution. Then, they gave their equation to a pal and asked them to solve it.

Why did this work? Because if you want to stump your friend you need to write a hard equation. And once some jerk reveals what makes x + 30 - 20 + 4 - 7 + 1 = 10 a pretty easy problem ("Oh, come on Mr. P, you gave it away!") you have to up your game. To use fancy man language, there was a load of intellectual need in that room.

5. 100m Dash/Stratos Space Jump - We used the 100m dash to talk about linear regression, and the Space Jump to break it. Both of these problems fundamentally worked as contexts for using the line of best fit to make predictions.

6. Height v. Shoe Size - I love making graphs on the white board. This was a particularly fun way to introduce two-variable data to my Algebra students. They put the post-its at their height and shoe size. Hey, look, there's a trend there. And we can talk about outliers too. The next day I took this picture and abstracted everything but the datapoints, leaving a scatterplot. (Explicitly imitating this guy.)

7. Constructing Number Tricks - This was pretty similar to my swap and solve activity with equations, and it worked in a similar way. Kids like coming up with their own things.

5 Worst 

1. Guess-Check-Generalize - This was a boatload of frustration for me. Guess-Check was an easy sell for me; I'm still looking for buyers on Generalize. I tried lots of problems, drawn from CME and Park Math, and they did hook kids in, but every time that I brought in any abstractions I lost the crowd. My one minor success was with this pretty on-the-nose worksheet. Next time I teach this I'm going to try that sort of on-the-nose stuff earlier, and I might also wait until all my kids are extremely comfortable solving equations to attempt teaching this strategy.

2. Life Expectancy - I blogged about this guy already, but it bears repeating: this was a huge disaster lesson for me.

3. Graphs of Inverse Functions - No idea how to teach this. I'm, like, 1 for 6 in attempts to teach this thing, and I'm pretty sure that the one win was a fluke. Maybe the issue is that I just find it really cool that the graphs of a function and its inverse reflect across y = x, and I expect kids to find it as cool as I do. That very well might be the problem, since I tend to teach this by asking kids to graph and bunch of functions and their inverses and keep an eye out for something cool.

Or maybe the issue is that they're not comfortable with technology and graphing interesting functions is cumbersome? Whatever it is, I don't know how to make what really should be a cool idea pop for students.

4. Defining New Symbols - So promising! I love the problems, some of my kids love the problems, and it seems like a great way to practice evaluating expressions while also ramping-up the sophistication for the stronger kids.

It was way too hard for the kids just getting used to variables and expressions, and my attempts at explaining this stuff were just met with blank stares. (We lost a day to me trying, like, three different ways of explaining this to a eerily quiet room.) I love this idea, but I'm not yet sure how to make it work.

5. Percentage/Fractions - Don't know how to teach 'em, especially quickly, especially to Algebra students who have never quite gotten them and need to know them for more advanced topics. I tried a bunch of stuff, and it all kind of failed. The one thing that I'm feeling better about is division by a fraction, which I'm pretty sure that I know how to teach now.* The issue is everything else.

* Next year you can be sure that I'm going to draw out the distinction between two different division models very early. Is 10/2 = 5 because 10 split up into 2 even groups would have 5 members each, or because there are 5 groups of 2 in 10? Only one of these models really works well for 10/0.5.

Bonus: Solving Equations, in General - I don't know how long it takes most teachers to get kids up to speed on solving linear equations, but holy cow it took me a while. We've got to speed things up, I think.


I wouldn't mind seeing your "X Best and Y Worst" post. I think that would be fun.


  1. That was clever, tying the multiplicative inverse into inverse functions. I've always felt the notation to be more confusing than anything, particularly once it's associated with trigonometry. There it seems to work. Very nice. Hopefully I remember to (or rather, find the time to) do something similar when I'm redoing things next semester.

    In terms of graphing inverse functions, what was the so-called "fluke"? Maybe if you approach it from the perspective of what's similar about functions which act as their own inverse as being "cool"? (Any line with slope -1 for instance.) Not that I've done that; I tend to build the graphs off of the fact that domain and range get swapped, so what does that imply... which seems more specific as compared to how you do things. (There's far too much material to get through in the Gr 11 U-course...)

    As far as the soapbox goes, I haven't posted up individual good and bad lessons, though last month I did a few generics in terms of a positive roundup:
    And negative roundup:

    There WAS a pretty good Exponents lesson as part of Day 1 of my DITLife, followed by some screwups in another class on Day 2, same blog channel, different blog times. Everything else is random...

  2. Though I'm not a math teacher, just a wanna be mathematician who does some tutoring on the side, I thoroughly enjoy this blog. However, this post has me scratching my head on one point. On #1 of the good, I see the student calculating 3^-3 and obtaining 1/9. If I'm reading things right, the answer was marked as correct. However, unless I'm just too fuzzy on a Saturday morning after a long week, 3^-3 = 1/27. The student's process was certainly correct, but 3^3 = 27, not 9. Am I missing something or reading it wrong?

    1. Damn. Teacher mistake. That's embarrassing.

    2. Welcome to the human race, eh?

  3. For Worst #4, I think it can be done. I'll give it a go early next year with less text and more practice problems with numbers only. Day 2, they'll make their own "function".

    Well done, discussing mistakes and areas for improvement. A post in that direction is coming soon.