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:

- What do you notice?
- What do you wonder?
- What do these points have in common?
- What do they
*not*have in common? - 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.

I made a new Desmos activity: “Points, Lines, and Inequalities” Can you help me by giving it a try? https://t.co/ogMZydVViA #MTBoS #mathchat

— Michael Fenton (@mjfenton) March 9, 2016

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.

Like the way eq of vertical & horizontal lines are addressed here – I can see making an activity solely about that. https://t.co/MJiK15att4

— Cathy Yenca (@mathycathy) March 10, 2016

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.

#### Links

I almost forgot! You can get the original activity here. Or try it out as a student here.