Implementing Routines – Visual Patterns

I’ve written a bit about instructional routines and my excitement about implementing several of them in my classes. While I’m working to take routines that other smart folks have designed and make them work for my class, I’m also taking great ideas from those routines and wrapping them into things I already do.

Here are some questions I’m thinking about:

  • How can I set up class so that students listen to and learn from each other?
  • How can I set up class so students are accountable for doing mathematical thinking?
  • How can I set up class so that students have supports to engage in meaningful mathematical discourse?

Grace Kelemanik, Amy Lucenta and Susan Janssen Creighton have lots of great ideas for how to meet these goals in the routines they share in Routines for Reasoningand many of their ideas fit into structures I already use. I’ve integrated a numbero of their ideas into my warmup using visual patterns. This is how I’ve changed my routine the last few weeks.

What I Used To Do

Once a week, my warmup is a visual pattern It’s worth noting that I teach 90 minute blocks, so spending 10-12 minutes on a routine to start class takes less time from instruction than it would from a shorter period. I would put up the pattern with the same five questions, like this:


  1. Draw the next step.
  2. How many squares will be in the sixth step?
  3. How many squares will be in the nth step?
  4. What would the 0th step look like?
  5. How many steps will it be until there are 200 squares?

I have my students at tables, and students would work for 6-8 minutes, either alone or working with others at their table. I would circulate and pick out a few students to share different strategies for their expression for #3. They would come to the board, write their equation, and explain how they came up with it, occasionally drawing additional images to support their reasoning. Then I would open up the floor to any different strategies that weren’t shared. Sometimes we would talk about#4 or 5. Other times we would just move on.

Many of my students really enjoy working with visual patterns, and I’ve found this routine effective for practicing algebra skills and seeing problems from multiple perspectives. However, many of my goals with this routine are left to chance. It’s easy to be a freerider and avoid engaging, or tune out during the discussion, or copy the work of someone else.

What I’m Doing Now 

I don’t think this is perfect, but after a bunch of experimenting with different structures I’ve found some small tweaks that I think make a big difference.

I start with the same questions, and ask students to work on their own for the first 3-4 minutes. Then, each student shares their expression for #3 with a partner, or if they haven’t come to an expression they discuss how they see the pattern growing and work together to try to come up with one. I circulate and pick out two different strategies. The student whose strategy is being shared goes to a whiteboard to write the expression and explain where it came from, while their partner goes to the projector and points to the different parts of the pattern to illustrate the thinking that led to the expression. After two pairs share, I open the floor to different strategies, and additional pairs can share in the same format. Then I ask students to verify that all of the expressions on the board are equivalent, and to sketch what they think the 0th step would look like, working with their partner. If there is disagreement I may highlight two ideas and have students discuss in partners which they agree with, otherwise I validate the consensus and we move on. I make explicit that the focus of this routine is on thinking about algebraic expressions — #1 and 2 are useful places to start that thinking, and #4 and 5 are extensions to dive deeper into the pattern.

These aren’t profound changes; they’re focused on clarifying the goals of the routine, increasing accountability of both individual and partner work, and supporting high-quality discourse at several points in the routine. I do think that these changes are particularly impactful for my lowest-performing students. While they aren’t massive changes, they increase engagement and the quality of discourse for students who were most likely to be disengaged and inarticulate.

I’ve just finished the experimentation phase and I’m sure I will continue to develop structures for thinking about visual patterns. But already students are more engaged in partner work and more likely to be able to follow along with another student’s explanation. And part of the purpose of a routine is that, as students become comfortable with the routine itself, they are better able to focus on the mathematics as they spend less time thinking about what they’re supposed to do next and more time listening to their peers.

5 thoughts on “Implementing Routines – Visual Patterns

  1. howardat58

    Some comments:
    1. The top line of the squares diagram is better with six squares, and the bottom line with five squares. I think more consistent,
    2. Are you familiar with the differences of the quadratic terms, as in
    … 5 10 17 26 … a quadratic
    ……5 . 7 . 9 … first differences
    …….. 2 . 2 ….. second differences (all the same)

    3. “And part of the purpose of a routine is that, as students become comfortable with the routine itself, they are better able to focus on the mathematics as they spend less time thinking about what they’re supposed to do next and more time listening to their peers.”
    Supposed to do next ?????????

    1. dkane47 Post author

      I am familiar with the differences of quadratic terms, though I think those goals may be ambitious for this type of warmup. With respect to your third question — many students spend significant energy figuring out if they will be asked a certain question, who they should pay attention to, which questions are important, etc — routines are meant to mitigate some of the resulting cognitive load.

    1. dkane47 Post author

      I also like that question — but it’s hard to know exactly what it should look like for students. That’s the least accountable part of this routine.

  2. Pingback: I Have A Real Hard Time With Number Talks And I Need Help | La Vie Mathématique

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