Holy Moly!! I haven’t used that phrase in a while, but it seems to be making a comeback through the Math Twitterverse. I’m on board.

It perfectly describes my thoughts after my first block classes Thursday and Friday this week. We used a new app from Riley Lark (the ActiveGrade dude) called ActivePrompt (which is apparently a working title, real name TBA) and it works like this.

Here’s what we did with it:

I had absolutely no idea what to expect from this. It was one of those lessons that could be a rock star moment or a complete failure. I have awesome kids, so my hopes were high. Here’s how it went:

Both classes started by making two vertical lines. They split themselves into two groups and each group decided on a “column” (the word they used at the time). My favorite question from this part (directed at another student): “Does it matter how far down I go?” That question led them to make sure that no ones dots were on the same point (the “rows” had to be different). After they got it, I asked them how they did it. Here’s what they said:

Since that was too easy, I next told them to create two more parallel lines that were neither vertical nor horizontal. They pretty quickly realized that it was all about slope. Their idea was to assign everyone a number and then have them go up and over by that amount (1,1) (2,2) (3,3). The real genius idea (only one class did it this way) was to translate that first line up using the rule (x , y) –> (x , y+1). Yeah, ok they didn’t write it that way, but that’s what they did, we’ll talk about notation later. Also interesting was how they decided on an origin without even really discussing it. It just sort of happened. They started talking about going “up and over” and pointing at each others screens and there you go, an origin is born. One class chose the top left and the other chose the bottom left. It was actually hard to get them to realize that they had even done this task of choosing an origin. That’s why it’s last on our list:

**This video clip shows one student explaining the slope concept to another. He can’t do it without a starting point!!**

One of the classes realized that it would have been quicker to simply give each group an equation (same slope different y-intercept) and then generate the points to graph the equation by giving each group member a number. **HA!! BUSTED!! Inventing algebra to solve a problem that doesn’t necessarily require it, huh?! Who are you and what have you done with my 9th graders?! **Here’s how they explained it to me:

My favorite thing about this activity was that there were so few rules. Just make parallel lines. But they ended up having real discussions about how changing the slope or the y-intercept affects the way the graph looks. Our current learning targets are: “Graph an equation using the slope and y-intercept” and “Describe the effects of changing m or b on the graph of y=mx+b”. I could’ve had them watch me do a few examples (in class or in a video) and then practice similar problems. But it seems like that’s asking them to start with the method and end with the problem. Shouldn’t the method exist only because of the problem?? And yes, I know this one’s not “real world”, in the sense that it didn’t have a context like “the city planners need to make two parallel roads to connect blah blah blah”, but isn’t that context unnecessary here? As long as the problem comes before the solution, it shouldn’t matter how “real world” it is.

As an extension I had them try to make a star shape with their points. We only had time for this in one of the classes. They decided which points would best describe the star, then made a sketch to help everyone place their dots. It was interesting that they decided to base their dot locations relative to one of the star points instead of the origin they had previously used. Anyway, we kinda ran out of time, so it didn’t come out looking great, but their plan was good and would’ve worked. Here’s what their sketch looked like:

**How I know it worked:**

*Everyone* was getting it. Not just the genius group that came up with the idea to write the equations first, but everyone. They all were engaged to some degree in the task and could see how the location of their individual dot was decided upon and how it fit with the other dots. One student in particular had been struggling with graphing in a major way. I was kinda stuck on how to help her. But after participating in the activity, using a starting point and then going “up and over”, something clicked and she was able to see that the y-intercept was the starting point, and the slope was the “up and over”. Just telling her that wasn’t making her understand it, she had to do it. I used this Khan exercise as a quick formative assessment, and she got it right away. BAM!!

One final thought: technology was integral to this activity. Without the laptops (or some other connected device), I wouldn’t have been able to use ActivePrompt. And the Khan exercise, while not essential, did make a great formative assessment (all my kids have accounts, so I can see how they did on the exercise instantly). For some other ways to use ActivePrompt, as described by the creator, check out this post on his blog.

This is a neat start!

More and more I think we’ll stop calling this “tricking” kids into learning. Students inventing their own abstractions is the only real learning; what we think of as a regular class with lecture etc is much less directly responsible for learning, I think.

I’m excited to see more~

Agreed! I mean “tricking” in a joking way, but you’re right it sends the wrong message. I’m seeing very clearly this year that real understanding only comes when they figure it out themselves.

Hey, AWESOME activity! I’m going to steal it as an intro-activity for Precalculus next year so students start getting used to creative problem-solving and working in groups.

As another extension that might not be as difficult as making a star, you could ask them to make perpendicular lines? Or perhaps other shapes, such as a square, but it’s not allowed to be made of horizontal & vertical line segments?

Great activity–thanks for sharing it!

Thanks! Yeah I agree the star was a little tricky for them, a simpler shape probly would’ve worked better. Next time…