Putting the “Use” in “Look for and Make Use of Structure.”

This is beautiful, right? Put enough straight lines in the right places and your eyes see a curve.

How many linear equations did the student use to create it? You might start counting lines and assume it required dozens. For some students, you’d be right. They typed 40 linear equations and corrected a handful of typos along the way.

But other students created it using only four linear equations and many fewer errors!

The seventh mathematical practice in the Common Core State Standards asks students to “look for and make use of structure.” The second half of that standard is a heavier lift than the first by several hundred pounds.

Because it’s easy enough for me to ask students, “What structures do you notice?” It’s much more difficult for me to put them in a situation where noticing a mathematical structure is more useful than not noticing that structure.

Enter Match My Picture, my favorite activity for illustrating my favorite feature in the entire Desmos Graphing Calculator and for helping students see the use in mathematical structures.

First, we ask students to write the linear equations for a couple of parallel lines.

Then four lines. Then nine lines.

It’s getting boring, but also easy, which are perfect conditions for this particular work. A boring, easy task gives students lots of mental room to notice structure.

Next we ask students, “If you could write them all at once — as one equation, in a form you made up — what would that look like?” Check out their mathematical invention!

Next we show students how Desmos uses lists to write those equations all at once, and then students put those lists to work, creating patterns much faster and with many fewer errors than they did before. With lists, you can create nine lines just as fast as ninety lines.

What are the four equations that created this graph? Personally, I find it almost impossible to discern by just looking at the graph. I have to write the equation of one of the lines. Then another. Then another. Then another, until that task becomes boring and easy. Only then am I able to notice and make use of the structure.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Fantastic! Can’t wait to try this with kids in a few weeks.

    I’m not sure though if I’m ready to sign up for “boring + easy” = perfect set up.” I’d always prefer “interesting + easy” to “boring + easy,” because boring can mean farting around in class or whatever. If interestingness is a function of novelty and doability, then I’d say that “novel + easy” is the perfect set up for looking at structure.

    In particular, I’m not worried that kids need to experience boredom in order to feel the need for lists. The need for lists will become clear once they see how interesting they are.

    • In particular, I’m not worried that kids need to experience boredom in order to feel the need for lists. The need for lists will become clear once they see how interesting they are.

      Kids need adequate working memory resources in order to ask themselves, “What’s the same about all these equations? What’s changing? Is the change predictable?” Novelty will subtract from those resources, so this is an area where I don’t mind seeing a student say, “Ugh. Nine lines? Seriously?

    • I see what you’re going for, but I’m having trouble making sense of it. What I’m hearing from you is that *whenever* someone has available cognitive resources to pay attention to something new they are ALSO bored. But working memory is all about cognition and boredom is an emotion — I think? I’m not sure how interestingness relates to working memory, but I don’t think I believe that you’re always bored when you’re ready for learning.

      But, anyway, isn’t your teaching philosophy that you want students to be bored so that they feel a need for the existence of lists? I didn’t think that your take was that boredom was required cognitively.

    • Perhaps “familiarity” is better than “boredom” here. If I’m trying to help you understand the abstraction we call a “chair” and I’m showing you item after item, there is a moment of productive familiarity where your brain has the resources to describe that abstraction. If those items I’m showing you are still novel, it’s much harder to describe it.

      Boredom through repetition often creates the need for efficiency. Sometimes that boredom also frees up the cognitive resources necessary for satisfying that need. I don’t think I agree with that second statement as often as the first, though.

  2. Clearly I am missing something. I am unfamiliar with this software and it may respond to my concern, which is: How do you restrict the equation lines to quadrants?

  3. Dan,

    I always enjoy and benefit from your ideas. The questions generated by your student’s graphs remind me of one of my favorite precalculus challenges. I thought folks might like this variation of the problem. I usually spend a couple of days having the students think through the problem below and create and explain their “creation” to share with the class. It often becomes their very first original proof of their own conjecture.

    Really fun question. Read it twice.

    Consider the family of lines written in implicit form ax + by = c, where the coefficients a, b, c is an arithmetic sequence. What is interesting about this family and give an argument to support your supposition.

    The goal is to connect the algebraic structure of the equation with the observed geometric structure of the graphs and to use the algebraic form to explain the geometric form.

    Modify the equation ax + by =c by changing the form of the equation or by changing the structure of the sequence; for example, consider ax + by^2 = c, and ax^2 + by = c, with a, b, c in arithmetic progression. Explain what you see. Can you describe what you will see before you create the graphs?

    What about ax^n + by^m = c, or acos(x) + by = c, with a, b, c arithmetic? Suppose a, b, c are Fibonacci?

    Now, consider a new the family of lines written in implicit form ax + by = c, but now a, b, c are in geometric progression. I dare you to explain what you see. Does your argument work with non-linear families like ax^2 + by^2 = c, with a, b, c, in geometric progression?

    Suppose ax + by = c and a*b = c and c = 24. (or a^2 + b^2 = c^2, with c = 10, or any variation you choose).

    The lines you constructed in your opening example are more complicated since c is not fixed.

    They are of the form ax + by = c, with a*b = c and a + b = 10. The “curve” that we see created by this family of lines is a boundary which separates the points which can lie on one of the lines from points that cannot fall on any line whose equation is restricted by our sequence of coefficients. With some help, students can figure out what those boundary functions are: in the case of ax + by = c, with a*b = c and a + b = 10.

    Using the methods devised by playing with the ideas above, we find that the boundary is the parabola

    x^2 – 2xy + y^2 +20x – 20y + 100 = 0

    and its reflections (this is a 45 degree rotation of the parabola, y = sqrt(2)x^2/20 + 5sqrt(2)/2).

    Thanks for a nice problem to keep me from grading this morning.


  4. This is great…and so timely! I’ve been teaching math individually to a couple of students in a special education environment. Both of them are artistic. I’ve been having them work on Desmos creating art. This post gives me some great ideas to go forward with. Thanks!

  5. Yeah. The list feature in Desmos is crazy when you use it to draw all kinds of stuffs. Recently I ran through this guide which used the list feature to embed parameters into sine waves and quadratics,. Definitely some food for teachers when it comes to activities for students.

  6. i’m excited to try this out with the students. i’ve recently done linear functions with my 8th graders; this will be a nice review and extension.

    i have my own desmos activity that is also a bit about seeing structure, modeled on the MYP criterion B (investigating patterns) rubric.

    i’m curious what you think: https://teacher.desmos.com/activitybuilder/custom/5954c8a34d6ae00b7a5b5831

    and a request: i hope the activity builder will allow for math type in regular response boxes because this all ends up looking hideous, unfortunately.

    • Dig the activity, Kate. I wonder if there are some useful extensions to ask students where they manipulate bands on the geoboard rather than just answering questions about the geoboards you provide. If that sounds interesting, copy and paste this link into any graph component in an activity and see what you can do!

  7. Thanks for sharing this! I am a pre-service teacher, and one of the things I have noticed is it is hard to have students recognize what the similarities are between certain graphs or even certain types of problems. Having them do an activity of this sort really helps them to develop those recognition skills needed to be successful in mathematics. This is definitely something I want to implement in my future classroom. Thank you for sharing this!