Lesson 1.2.2: Defining Functions Graphically (Day 15)

Since today is August 30th, let's look at the thirtieth topic on Shelli's list:

30. How do you support struggling students?  What intervention strategies have you used?

I responded to this same Blaugust prompt three years ago. That year, I mentioned two specific examples of times when I helped struggling students. One was when my eighth graders were having trouble with rotations (isometries), and I helped them by making them more concrete by rotating graph paper. The other involved a seventh grade computer lesson where the students needed to remember mean, median, and mode (measures of center). I jogged their memory with a "Measures of Center" song.

Well, it's still possible for me to use these interventions this year. After all, I do still sing songs in class of course, and rotations will appear later this year in the Integrated Math I curriculum. (Once again, recall that much of Math 8 and Integrated Math I overlap.) But that's not what my  kids need today.

And yes, many of my Math I students are definitely struggling. Chapter 1 of the CPM text is mostly about functions, but all Math I teachers agree that the students need review on solving equations. This is why a week of solving equations was included before we officially started Lesson 1.1.1. But as we can easily see, many students are still having trouble solving the simplest of equations.

Last week at the CPM training, we had discussed turning yesterday's Lesson 1.2.1 into a game of Math Baseball, similar to the old 1980's computer lab game (like Oregon Trail). But Lesson 1.2.1, on the official pacing guide, fell on Monday -- and it's foolish to think we'd have time for Math Baseball on a non-block Monday (or Friday). So I pushed Math Baseball back to today -- and then, instead of functions from Chapter 1, change it to equations instead. (After all, students must solve equations to, for example, derive the input from a given output.)

But still, it didn't help enough students. Too many students go for a single (1B) in Math Baseball and struggle to solve a one-step equation. And of course, many of our previous interventions (the "Don't Call Me After Midnight" mnemonic and the verses of my "Solve 'Em" song) presume that the students are proficient with one-step equations and are having trouble remembering which step in a multi-step equation goes first.

Other Math I teachers are taking even more drastic measures. Some teachers are spending even more time on solving equations and are already behind the pacing guide -- this week is supposed to be the third quiz, but some are giving the second or even the first quiz this week. And one teacher is going yet more slowly -- spending time on PEMDAS and integer operations before even solving equations.

I think back to an old book written by science fiction author Isaac Asimov -- a math book. Here he introduces algebraic concepts. And one line from that book sticks out in my mind -- "there are rules for solving equations." The only reason we solve equations the way we do is because the alternative -- solving them by trial-and-error (plugging in numbers and simplifying) -- takes too long. We are riding our bike instead of walking. But too many of our students (and this applies to many concepts, not just this one) believe that this steps constitute some sort of magic -- that the rules of solving equations are arbitrary, rather than sense-making.

Links to other Blaugust participants: Sue Jones used to be one of my Blaugust go-to sites, and yet I haven't linked to her blog until today. Here's the link to her most recent post:

https://resourceroomblog.wordpress.com/2022/08/28/the-main-point/

Jones writes about the most important things for students to know and use: "Arithmetic, ratio, proportions, expressions and simple equations, people." And that's simple equations, as in the equations that my students are struggling to solve write now.

The song for today is Square One TV's "That's Math," performed by the late Gregory Hines, a world-famous tap dancer:

THAT'S MATH by Gregory Hines

First Verse: 

Hey, thank you,
Hey, come on in here.
Let me tell you something about math,
Let's look at this bill. What have we got here?
I had two plain pies at six bucks apiece,
Add one with extra cheese, that's seven.
Add eight cream sodas, a buck a pop,
Adds up to $27.
We know the tax is five percent,
In this great state we live in.
Multiply by three, 15 percent,
That's the tip that I'll be givin'.

Refrain: That's math! That's math! That's math! That's how you figure it out.
That's math! That's math! That's what it's all about.

Second Verse:

Bottom of the ninth, we're up by three,
Full count, two outs, and three men on.
This guy comes up hitting .405,
Six homers last month alone.
Gotta think, last year he hit my fastball,
Six times out of seven,
He was three for five on my curveball,
But only one for four on my slider.
After going over the numbers,
I wind up and let one fly,
I strike him out with my slider,
And I'm voted Most Valuable Guy.

Refrain: That's math! That's math! That's math! That's what it's all about.
That's math! That's math! That's how you strike 'em out.

Third Verse:

Imagine an island three feet wide,
A thousand feet below,
I gotta drop a survival box,
To save a guy named Joe.
So I calculate my altitude,
Direction and wind speed,
And I factor in the box's weight,
And the angle I will need.
Now I'm not ashamed to say I hit,
A perfect bulls-eye and move on.
Joe can eat until he gets rescued,
I just hope he likes croutons.

Refrain: That's math! That's math! That's math! That's how you figure it out.
That's math! That's math! Bing! Beng!

Fourth Verse:

Now the only thing as great as math I see,
As far as I'm concerned,
Is music I can dance to,
Here's something I recently learned:
Music is a kind of math,
And intervals and beats.
Every time I sing a note,
Every time I move my feet.
And it's not just school,
It begins when you close your book.
Tread any path and you'll find math,
Everywhere you look.

Refrain: That's math! That's math! That's math! Bing! Beng!.
That's math! That's math! That's what you can do with math.