# Age-Appropriate? Not Exactly. Valuable? Absolutely.

I started something new this week. I’m not entirely sure where it will end up, but I like how things are shaping up after just one challenge.

Here’s what I presented to my students:

https://www.desmos.com/calculator/kposfnfytr

Age-appropriate for Precalculus and Calculus? Not exactly. With a slight nudge, this is something a group of Desmos-equipped 6th graders could tackle.

But… Oh. My. The blank stares. The confused looks. The surprisingly non-isolated bewilderment.

I could write a post about how students “get rusty” with things they don’t practice, et cetera, but I think something else is going on here. Many students struggled in such a way that I suspect—rather, I’m convinced—they never learned much of anything about linear functions at any depth. (I shudder to think about how deep their understanding of exponential and logarithmic functions goes, even as we’ve been working with these functions all throughout the year.) Presented with a problem in a format just slightly askew from what they’re used to, they struggled and stalled.

I could write a blog post lamenting the quality of the students passed along to me by various colleagues. But there’s more to the story, especially since I’ve taught most of these students in two, three, four, or even five other classes. An indictment on their former teachers is an indictment on myself.

So what’s my next move? How do I address the current state of graphing affairs in my own classroom and in our department as a whole? With a few resolutions:

1. Resolved, never to treat a particular mathematical topic in isolation when valuable connections are readily available.
2. Resolved, to present students with tasks which demand that they make connections between numerical, graphical, and algebraic representations.
3. Resolved, to allow key topics to spill over across the confines of individual lessons, chapters, and units.
4. In particular… Resolved, to develop a series of “match my graph” challenges to develop students’ “function sense” (or “graphing sense”?) over the course of the entire year.
5. Better yet… Resolved, to collaborate with my colleagues (in real life as well as online) to develop a series of thoughtfully sequenced/coordinated “match my graph” challenges for every course in the 7-12 sequence.

For now, I’ll create two or three challenges per week to share with my Precalculus and Calculus classes. They’ll gradually grow in difficulty, and we’ll soon shift from linear to quadratic, to power and exponential and logarithmic, to conic, to parametric, to trigonometric, and even to polar. Eventually, I hope to tag the challenges by grade level (with some challenges receiving multiple tags) so we can more easily integrate them into the rest of our courses in the department.

I’ll report back on our progress later this semester. In the meantime, if you want to create a few challenges of your own, I’d love to see them!

### Postscript

In fairness to my students, some solved the challenge rather easily (as they should have). I’ll soon provide them with a more demanding challenge, but for now, I’m interested in seeing how I can address the blank stares and confused looks that popped up on more than a few faces earlier this week.

### Update

I’ve added a lesson called Match My Graph: Linear Functions to the Lessons page. The lesson (actually, a series of mini-lessons) contains eight linear function challenges, including this one:

https://www.desmos.com/calculator/phmxercufm

### Another Update

One of my students brought a huge smile to my face with an email this weekend. Read about it here.

1. Hello Michael,
I do feel your pain with how students wipe their memories of anything remotely related to linear graphs at times. Recently I had finished working through a unit of work based on graphing quadratic, cubic and reciprocal graphs, yet when I popped a linear graph such as 3x+2y=8 (yes with fractions!), it completely through them!
This is a lovely little task which could open out into something quite deep and worthwhile exploring further and I think it could fit well into the ideas develop from the Inquiry Maths work being developed at the http://www.inquirymaths.co.uk website. I’ve been following this website, along with your blog and Dan Meyer’s blog and getting lots of useful ideas to pursue. Maybe this one could be useful to this type of starting questions too.
Best wishes,
Martin @mart_brown

2. Post
Author

Martin, thanks for sharing. I know I’m not the only one with students who struggle with what should be basic, sometimes it seems that way, so it’s nice to have a reminder that others have similar experiences.

There’s lots of great stuff over at Inquiry Maths! (And obviously on Dan’s blog as well.)

I’m looking forward to developing this idea more completely over the coming months.

Cheers!

3. What about the slider on Desmos? Does that trivialize the task? I want to use Desmos more but I was concerned about students just usu g the slider instead of thinking about the parameters of an equation.

4. Post
Author

I think adding a slider would be entirely appropriate for any of these challenges. I would caution against ONLY using sliders to solve the challenges, as I believe something else goes on in students’ brains when they have to write the equation explicitly, modify, observe, etc.

Maybe students would use sliders for the odd-numbered challenges, or for #1-4?

5. Many of my students showed up from their winter break pretty rusty with graphing linear equations – even many students who seemed proficient just before winter break. I also wanted to (self)reprimand their teacher from last semester.

I would add “verbal” to your list from resolution #2 (graphical, algebraic, and tabular).

Thanks for sharing these challenges!

6. Post
Author

Nat, thanks for the suggestion to include verbal. I try to tie in the four representations as often as possible, but I’m not sure (yet) how to launch one of these challenges with a verbal description alone. Maybe something like, “Plot a pair of perpendicular lines that intersect at (3, 4).” Or, for quadratics, “Plot a downward-facing quadratic function whose x-intercepts add up to its y-intercept.”

Is this what you had in mind? Can you suggest some tweaks (or something else entirely) that would have more potential for awesomeness in the classroom?

Also, great stuff over at http://17goldenfish.com.

Thanks for dropping by!

7. I like the ideas you suggest. In terms of building skills, your descriptions are excellent, and both of your verbal prompts would require critical thinking of your students. I might ask for two different sets of perpendicular lines from the first prompt, so you don’t just get y=4 and x=3 (…although that would demonstrate some understanding as well).

I try to contextualize graphs whenever possible as well. Here is a link to a set of openers I’ve pieced together, which ask students to move between the four representations. I used one or two each day to open class with kids who were just being introduced to linear equations.

https://www.dropbox.com/s/16qqnj01lgd4h81/GAVN%20linear-equations.doc?dl=0

Thanks for the conversation!