Modeling problems

This week, I attended the High School Math team’s common planning time.  We were trying to select some common assessment tasks for this standard:


We started by looking at the tasks from Illustrative Math.  The first task we chose was:


As soon as this problem came on the screen,  I started whispering excitedly to my colleague Robyn.

“Robyn! It is a visual pattern!  This is great!”

Robyn smiled. Robyn and I are in a K-12 learning group. Recently, we have been discussing the value of using visual representations in math class.

Robyn and I both went to work solving the Illustrated Math problem.  I could “see” the first expression right away.

Here is a rendering of what was drawn on my paper:

I thought, “So n must be the number of dots on the bottom row which also corresponds with the step number.”

Then I started thinking of the second expression:



I couldn’t “see” this expression in the first image, but I could see it in the subsequent steps.


I checked in with Robyn. “Is the square in the middle of the image?”

She said, “I think so.”

I went back to thinking of the first step. I realized the first step was a little trippy because the square I was seeing in the subsequent steps was actually made out of circles.  That is why I can’t see it in the first step. It isn’t there. In the first step, n is the number 1 so 1 dot squared is still one dot. Weird. I was about to check in with Robyn again, but I missed my chance because it was time to discuss the problem as a whole group.

Somebody said, “Time’s up. What do we think?”

One teacher said, “I don’t see it. I haven’t done any dot problems.”

Another one said, “I am voting down this problem.”

And finally, “This is over the top. I would have to spend a lot of time to teach this and it would take away from what we have to do.”

I couldn’t say what I wanted to so I wrote it on my paper:


Not enough time?  Over the top?

In my head, I was thinking “this is what we have to teach.  This is where we have to spend our time.”

As a learner, I was feeling really frustrated inside.  When I took Algebra I in High School, it was all procedures.  I never understood one bit of it because procedural recall isn’t my strength.   If I don’t understand something, I won’t remember it.  I hated Algebra, but I loved Geometry.  Geometry made sense to me. I could see it.  I remember thinking “Why can’t Algebra be more like Geometry?”  Back then, I thought Algebra and Geometry were two completely different subjects that had nothing to do with each other.  Now, I realize that Geometry made more sense to me because I could “see” it. I wonder if Geometry and Algebra are more intertwined than I ever realized. I would love to take these classes again, but with an integrated approach.

So…. I chose not to say anything.

One of the teachers said, “I think I can see it, but I don’t know how a student would explain this. How would you answer this question?”

Robyn spoke up.  She asked, “How would you explain it?”

He started  to explain where he saw n squared in the image.

Robyn kept asking questions to draw out his thinking.

She asked, “Why?”, “Can you explain where +2 is?”,  “What does the (n+2) squared represent?”

Robyn’s questions helped me understand my colleagues thinking.

Finally, someone said, “I don’t know how we would expect students to write all that.”

I said, “I didn’t write it. I drew it.”  I held up my rough sketch.


I think Robyn said, “If you do these types of problems with kids on a regular basis, they get good at seeing the expressions and explaining their thinking.”


“I would not use this or teach this because we have so much to do. I am not going to waste time teaching this when I barely have time to teach everything else. This is “over the top”. It is nice to have over the top, but I don’t have time for it.”

I didn’t say anything.

“Well,”  someone else said, “It is time to vote. We have other problems to look at.”

We lost.  3 to 2.

We analyzed the other Illustrative Math problems and we chose to use  The Physics Professor and Mixing Fertilizer.  These are both great tasks. Why do I still feel like we are jipping our kids because we left out the Dots tasks?  It is one task.  Leaving out one task can’t have that much of an impact on our Algebra curriculum?

Or can it?

Maybe it isn’t about the task.  Maybe it is about what the task represents.  The reason we are choosing common assessment tasks is to deepen and calibrate our understanding of the standards.  Unfortunately, I think calibrate has become somewhat of a loaded word in our district. When I say “calibrate our instruction”, I think some teachers hear “stifle, homogenize, anesthetize” our instruction.  I think, sometimes, my definition of calibrate gets lost in translation.

To me, calibration is an ongoing process. You can read more about my definition of calibration here, but I think the dual purpose of calibration is professional growth and equitable math experiences for students. At the heart of calibration is transparent and collaborative reflection.   Calibration means continuously and collaboratively asking, what do we want our students to learn?  Why do we want them to learn it?  How do we want them to learn it?

I guess the reason I can’t seem to “let go” of the Dots task is because it represents a crucial answer to the questions I just asked.  It represents the integration of visually representing Algebra.  I feel pretty strongly that all students should be able to “see” Algebra.  Earlier, I refered to myself as a “visual learner”.  I often wonder about the value of this term. Should there be a certain kind of learner who sees things visually or should visual learning be an expectation for all students?  Is there value in being able to move from the abstract to the visual and back to the abstract?  As a learner,  I realize now that it is really important for me to able to move fluidly from the visual image  to the expression and back to the visual image. Isn’t this the essence of modeling with mathematics?

Dan Meyer describes modeling  as “the process of turning the world into math and then turning math back into the world.”

I think I can see how the other tasks we chose would offer opportunities for modeling with mathematics, but I still want the Dots task.  I can’t seem to let go of it. Maybe it is personal.










5 thoughts on “Modeling problems

  1. Don’t let go of the dot tasks Coach! I spent 3 class periods last year with my 6th graders during equations and expressions and I’ll say with considerable confidence that at least one third of them, probably half, would be able to give you at least one expression that your pattern represents within 5 minutes. I found it to be extremely valuable to spend the time to teach this visual pattern recognition to my kids AND they loved doing the work. At first they were skeptical, but as they solved the easier patterns and saw they have success, they took off. The task offers a great path to differentiation as it allows me to assign a batch of more difficult patterns to quicker studies and assign batches of easier patterns to those learners struggling to grasp the concept. As I head into this unit 3 with expressions and equations, I plan to spend at least the 3 class periods on patterns this year with the idea that it makes accessing the standards of the unit EASIER to grasp for the students.

    I don’t think it’s personal. Trust your instincts.


    1. I agree. This is not the cream on top. This is it. There are two things I think really help with understanding the anatomy of an expression, one is to graph it, and the other is to see how it can be represented by a visual pattern.

      Lots can be done with these in elementary, like this, with my Grade 4s:
      (When I’ve done this kind of thing since, I’ve added graphing the pattern to the mix.)


    1. I can try. I saw “n” as the number of dots in the bottom row. It is also the step number. I can fill in the dots to “complete” the bottom row and then square it. I have to subtract the corner dots that I added to make the square. I saw (n+2)^2-4 as completing the “outside” square and then adjusting the total versus n^2+4n is using the “inside” square and then adjusting by adding the 4 groups of dots that surround the inside square. Does that make sense? It helped me to draw it out.


  2. Simon I love your post! I think it would be wonderful to share your students videos with our students. Bill (6th grade teacher and teammate of mine who posted above) and I are starting to work with visual patterns with his students. It would be fun for us to share some of the videos with his students and ask them the questions you asked in your blog. Bill and I have been talking about slowing down the process in order to get students to really own the relationships – taking more time with the visual and word descriptions of the patterns so the students really own the relationships inherent in the problem. I am hoping to blog about our thinking because we had a truly inspirational conversation around the first pattern problem we used with his class. Hopefully, I will make time. I will keep you posted.


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