Points, Lines, and (Maybe Not) Inequalities

One of the things I love about Twitter is the opportunity it provides for feedback on creative work. And thinking about teaching and learning mathematics is some of the most creative work I know. I’m still amazed when folks I’ve never met in real life take time out of their day to help me improve.

Here’s a case in that beautiful point.

Points, Lines, and Inequalities

Earlier this week I created an activity designed to help students make connections between individual points and graphs of linear equations and inequalities.

Here’s the first screen:

On its own, pretty simple. But imagine a class full of students doing this. And then imagine directing their attention to a graphical overlay of every student response.

Now we’re talking. Literally. At this point, I’m asking students:

1. What do you notice?
2. What do you wonder?
3. What do these points have in common?
4. What do they not have in common?
5. Is there a way we could summarize all of these points algebraically?

And then… Turn them loose on this:

My hope is that the previous discussion laid the groundwork for the idea that these are all points where x equals 3, so x = 3 is an (obvious?) way to summarize them algebraically. That’s how it plays out in my head, anyway. More on that in a moment.

Students then move on to the next (rather similar) prompt:

(And the activity continues in a similar fashion for a total of six scenarios—four lines, two inequalities. For a closer look, check out the links at the end of the post.)

Room for Improvement

Let’s circle back to Twitter. Yesterday I shared a link to the activity, invited folks to give it a test run, and hoped a few might offer some feedback.

The response proved incredibly helpful. In particular, Bowen Kerins offered not one or two, but nearly a dozen comments about what works, what doesn’t, and how I could make this activity even better.

I have a few takeaways from that conversation about how to make this activity better. I’ll share them here in order to clarify (for myself) and share (with others) some design principles that I think may prove helpful the next time we fire up the old Activity Builder.

I’ll share the first takeaway here, and a couple more in the next few days.

Takeaway #1

The cycle in screens 1-3 (place a point, imagine all the points, summarize algebraically) is missing a crucial step. It begins simply and somewhat informally, which is helpful. And then it invites students to imagine/predict, which I also think is helpful. But then it jumps straight to the summary, without pausing long enough on the “discussion” that I described above.

My current fix. Insert this between screens 1 and 2:

By asking students to name these points (note: not just their own, but also several others), I think they’ll be more likely to see (or “hear”) the repetition that leads to our algebraic summary later on:

• “x is 3, y is mumble-mumble…”
• “x is 3, y is something-something…”
• “x is 3, y is whatever…”
• “Hey, x is always 3! So then, x = 3!”

My future fix. This activity uses an existing (and lovely) feature called copy previous, where whatever a student did on one graph screen can be pushed ahead to (or “served up” on) a later one. You’ll see this in action between the screens 1 and 2 in my original activity.

What I would love here is another feature (even more lovely?) where the graphical responses from every student are served up to kiddos on a later screen. In other words, bring the graphical overlay feature that we have in the teacher dashboard into Activity Builder itself. Apply that to my original screen 3, and I think we’re in much better shape.

And the good news? Something like this is already in development. (Side note: I continue to be amazed at the engineering skill—wizardry?—of my colleagues.)

Closing Thoughts (For Now)

I think these proposed changes will prove helpful. But I also think they stretch the activity out a bit. So maybe it’s not realistic (or even helpful) to work through vertical lines, horizontal lines, linear functions, and linear inequalities all in the same activity. Maybe this is two separate activities. Or even three.

I take some comfort knowing I’m not the only one who thinks a narrower focus here could be helpful.

With that in mind, here is my new-and-hopefully-improved activity. Give it a whirl, and let me know what you think in the comments.

Next Time

I’ll share some thoughts on making the middle section (linear functions) stronger.