Teaching m and b for meaning

It’s time to learn how to see m and b on a graph. To dip our toes into the topic, we had a nice debate in class about the rate shown in this quick video.

Is the rate 1, or 5? They couldn’t decide, because the number of marshmallows is increasing by 1, but the time keeps increasing by 5. It’s such an incredibly simple question, but their inability to decide hooked them for an entire lesson about graphing slope and y-intercept. Part of the hook was, I think, that many knew in their guts that both answers were incomplete.

I’d bet these students would have gotten the correct rate back in 7th grade. But as you know, students sometimes over-apply recent learning in a way that erodes old knowledge. So we’re here in Algebra 1, having done an entire sequence on identifying and b, and that has skewed their thinking. We’ve looked at visual patterns, like this one with m=4 and b=8:

4x+8

And we’ve looked at story problems, like this:

story problem

So far, m has always been an integer. That makes the marshmallow rate of 1/5 difficult: this lesson is the moment when we transition to m as a rate or ratio.

In this blog post, I’d like to share a Desmos lesson I made to capitalize on this moment of confusion. It turned out to be pretty engaging. You could use the lesson as-is or edit it to insert videos of yourself stuffing your face or doing something similarly silly. Here are the phases of the lesson:

1.  activate knowledge of m and b

First I shared the purpose of the lesson: how do you see m and b in a graph? We began by listing what we know about m andon the board:

  • b:
    • Beginning amount
    • stage 0 of a visual pattern
    • constant amount
    • By itself in the equation y=mx+b
  • m:
    • amount a situation grows by
    • Multiplies in the equation y=mx+b

(I’m normally a nix-the-tricks kind of guy, but m and b are arbitrary choices of letters for slope and y-intercept, so I’m okay with mnemonics here.)

2.  practice m and b

Students paired up, sharing a computer, and worked at their own pace through a bunch of card-sorts identifying m and b in this Desmos activity.

Card sorts 1

The focus is on identifying b at first. I’ve learned that when I ask students to focus on m and b simultaneously, the weakest students always mix them up and never learn the difference. If I ask only for b, they can actually learn both m and b better. Students absolutely must know the difference between m and b in story problems and visual patterns, or else they’ll never be able to comprehend representing those quantities in a graph. So keep in mind, this activity comes after 3-4 days of practicing and a little each day.

I displayed the “summary” Desmos report on the screen while student pairs did the card sorts, so they could see if they were getting correct answers by looking for the check-mark on the teacher dashboard. At this point, the Desmos pacing was set up to stop them at the end of the card-sorts.

3.  puzzle over the marshmallow video

Continuing the Desmos activity, we watched the marshmallow video as a whole class (nice laughs there) and graphed the situation, as you can see here:

video and graph

(The points can only be dragged vertically, because I didn’t want to deal with arguments about whether the marshmallow went in at 5 sec or 5.5 sec.)

The following screen asks students to identify m and b in the marshmallow-eating video. This is where we had our big debate: does m equal 1 or 5? The disagreement here drove the momentum forward — we agreed to start learning to see b on a graph, and to keep puzzling over the mystery of m. That was the end of the warmup/hook phase of the lesson.

4. practice finding b on graphs

We moved over to the next Desmos activity, which is the meat of the lesson. Its starts right off focusing students on seeing b in a graph.

Part 1 purple graph

We confirmed students’ theories by watching the video of the purple graph, and discussing m and b in the video. We were still confused about m but agreed that b=2.

Then it was time for matching graphs with their equations and values of b, starting with easy problems that require almost no knowledge transfer:

Matching with b v1

And building to more formal questions that require transfer:

Matching with b v2

5. okay, but what about m?

It was time to revisit m, but they were emotionally done being stuck on it. I needed to get them unstuck. I needed to nudge them without giving it all away. So we watched the video below with the prompt, “which half of the video shows a faster rate?”

Within a minute or so, everyone agreed that the rates were the same. So you can’t say m=1 in the bottom, because then m=2 in the top, and those are not the same. You also can’t say m=5 and m=10, because those are not the same. You need a way to say the rates that gives the same answer for both videos.

Students discussed in their pairs, and if needed, I gave another nudge: “How do you say how fast anything is happening? Like, how fast a car is driving?” Students arrived at 1/5 = 2/10. I did a live demo of the decimal rate, 0.2 marshmallows/sec, by ripping a marshmallow into 5 pieces and eating 1 piece per second. Yum. We ended by drawing the rates 1/5 and 2/10 on the graph.

Label m two ways

This led into student-paced work applying their understanding of m and b. You can see an example here:

Finally, we transitioned to drill practice on slope.  For this lesson, we did only positive slopes.

6. links to materials

  • Here are a bunch of worksheets on seeing m and b in visual patterns. You should use these before teaching the lesson in this post — they’re prerequisites.
  • Here again is the 1st Desmos activity, which prepares students by focusing on m and b in visual patterns and story problems.
  • And here is the 2nd Desmos activity, which focuses on m and b in graphs.

Maybe in the future, I’ll look back on this lesson and shake my head sadly. But the 2018 version of me thinks this lesson hits the target pretty well.

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