The Desmos overlay function turns out to be a really nice way to introduce 2-variable inequalities. Today we did an activity that centered around the screen shown below. A link to the activity is at the bottom of this post. For the prices shown, each student tried to create some data points that totaled more than $40, some points that totaled exactly $40, and some points that were less than $40.

It’s pretty simple work, and it’s low-stress because students can basically try anything. We’re making use of all answers whether they total to more than $40, less than $40, or exactly $40, so they can just play with numbers without worrying that they’re doing it wrong. When you overlay their graphs, you get something like this:

This is a really nice intro to linear inequalities. The green points should all lie on a line, the blue points below the line, and the red points above. Someone else in the MTBoS blogged about a similar lesson several years ago. I don’t remember who it was, but if anyone knows, I’d be happy to credit her. (I think it was an 8th-grade teacher in Massachusetts).

## Nitty-gritty teaching moves

Here are discussion notes to wring the most out of the screen shown above. Students may become confused if they (correctly) find some green points that are actually higher than some of the red points, or blue points that are the same height as red points. I reassure them that if they know their calculations are correct, the whole-class discussion will solve the mystery of the graph. Once enough data has been gathered, I show a few students’ responses, and highlight what students found interesting or confusing. For example, “Is it weird to you that the point (2, 10) is blue, but the point (8, 10) is red, and yet they’re at the same height?”

If students don’t express puzzlement, then I just ask if students see patterns for why the different colored dots ended up where they did. Prompts to use:

- Do you see any green points in the wrong place? How do you know?
- If you have a SMARTboard, you can circle where the green points should be, to highlight that they’re on a line. Draw the line in green marker. Cross out any green points that are in the wrong place for the sake of clarity.

- Why is the slope negative?
- Students will have a legit hard time with this. That’s because they’re still thinking in the logic of Slope-Intercept Form, not of Standard Form. That is, as you move to the right on the graph, you’re increasing the number of pizza slices. If
*y*represented the total cost, it would increase as you move right. But here,*y*represents the corresponding number of candies you can buy to get the same $40 total cost. Buying more pizza means you can afford fewer candies. It’s important not to skip this discussion, because it’s the key conceptual transition from Slope-Intercept Form (where*y*is the output) to Standard Form (where*y*is another input that adjusts to keep the total amount constant, and is only an output implicitly).

- Students will have a legit hard time with this. That’s because they’re still thinking in the logic of Slope-Intercept Form, not of Standard Form. That is, as you move to the right on the graph, you’re increasing the number of pizza slices. If
- Do you see any red or blue points that ended up in the wrong place? How do you know?
- If any of red/blue points did end up in the wrong place, asking this question is a great opportunity to draw out the pattern. A student can circle a wrong point and say, “That point is incorrect because all the blue points are supposed to be below the green line, and the red points are supposed to be above.”

At the end of the discussion, I guide the class into translating their work into a Standard Form equation. To do this, I choose some green points and show how to prove on *one line* of the calculator that they make $40. For example, (5, 10) works because 4(5)+2(10)=40. And (20,0) works because 4(20)+2(0)=40. The equation for this pattern is 4`x`+2`y`=40.

Here I throw in the vocabulary of “Standard Form”.

Finally, I ask students to type the equation 4`x`+2`y`=40 into their expression list. Some will be impressed when they notice that the graphed line goes through all their green points.

The real kicker is when you ask them change the = to a >, and the graph shades above the line. For some reason, that gets audible gasps, and a few comments like, “Okay, that’s actually kind of cool.”

*Admission of guilt: yes, most of this blog post is cribbed from the teacher notes I wrote for this screen of the Desmos activity. Hey, recycling, right?

## Am I crazy, or do we lump Algebra topics together all wrong?

The longer I teach, the more confused I am by the topics we choose to lump together into units. This lesson is a great example. Learning to think in Standard Form, when you’ve only ever thought in Slope-Intercept Form, is a legit conceptual shift, as I described above. It’s essentially the shift from thinking of a function defined explicitly to one defined implicitly. It’s conceptually huge! And yet we lump those topics together because they both graph as lines. From our expert perspective, they belong in the same topic. But it will take novices a week to start seeing them as they same topic!

Granted, I do teach Standard Form equations a bit in my Slope-Intercept unit, but all we do is transform them into Slope-Intercept and then graph them. They’re just puzzles… hmm, someone wrote this equation in a weird form, and we have to fix it. That’s all. We don’t discuss the logic of Standard Form at that time, because it’s almost the polar opposite of the logic we’re studying, and student brains so often mush opposite ideas together in a confusing muddle.

Instead, and as far as I know I’m the only person who does this, I teach Standard Form as my intro to 2-Variable Inequalities. Because before students really see that *y* (the number of candies) is really determined by *x* (the number of pizza slices) in a clear way that can be expressed in an explicit form, their natural thinking really is guess-and-check. Some guesses are too high, some are too low. Their natural thinking is really in terms of inequalities. So now, we’ll study graphing inequalities, both in Standard Form and in Slope-Intercept Form. This will include drill practice and application scenarios.

After that, and only after that, will we start graphing Standard Form by finding the *x*-intercept and *y*-intercept. The goal is for students to see the line generated by Standard Form as a solution set — a bunch of points, all of which satisfy *Ax*+*By*=*C*. If I introduce Standard Form and show graphing-by-intercepts too early, students will see the line as a shape, not a solution set. They’ll see it as the line that *connects the intercepts*, but they won’t see that the points between and beyond the intercepts are also solutions.

Someday, I’ll write a post explaining/ranting at other topics that I think are paired weirdly in Algebra 1. Here’s a taste of that…I’m throwing it out there in the hopes that someone in the comments or on Twitter will either agree with me or convince me that I’m wrong: Isn’t solving quadratics by taking square roots much more closely connected to the order of operations than to solving by factoring? Because the work involved is basically unwinding the equation by reversing GEMDAS. Shouldn’t it be taught early in the year, when you’ve just finished order of operations and grouping symbols, and you’re moving into solving equations?

## Link to materials

Here is the full Desmos activity.

“If I introduce Standard Form and show graphing-by-intercepts too early, students will see the line as a shape, not a solution set. They’ll see it as the line that connects the intercepts, but they won’t see that the points between and beyond the intercepts are also solutions.” Thanks for letting me where I’ve gone these 16 years of teaching!! Haha! Is there another way, more of an animation or gif, that can show the negative slope as a relationship between the two quantities?? Meaning, how can I create a graph (maybe with sliders) that as I increase x(pizza here I believe), they can SEE the y (candy) decreasing?? Good work Kevin!!

Ooh, I like that idea of an animation or a gif. That’s a really good idea. I’ll keep an eye out. Thank you for the encouragement — means a lot.

*know; *wrong

Typos, sorry!