I Noticed a Pattern in My Pattern

Introduction

“What is a pattern?” – “It is when something is repeating like ABAB.” I found at the beginning of grade 3 that my students could build repeating patterns. We’ve been working throughout the year on identifying rules in increasing and decreasing numerical and visual patterns, and I wanted to find a way for all this scattered practice to come together.

The credit for the idea goes, yet again, to Simon Gregg  who shared how he used Desmos with his students to graph the visual patterns they created. He shared his blog post with the details of the lesson and students’ work. I wanted to adapt it for my classroom and I hoped that my students would notice the connections between the model, the symbolic numerical representation and the diagram/graph.

Questions and Thoughts: Before

I was anxious about the logistics of the lesson. Thank you, Sarah, for moral support!

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Will everyone be able to build an increasing pattern?

Will all students identify which step is which when translating the pattern into the table?

If we work in math journals, will someone spend an hour creating a table to fill out?

Will the students be able to handle the technology part independently?

Preparing for the Lesson: Getting Ready to Make a Pattern

I used one of the three-act math prompts from Graham Fletcher’s website to get students thinking about the growing patterns. After watching the video, I asked my students to build steps 4 and 5 of the pattern, label the steps and the number of tiles in each step and write down the pattern rule. Then students had to figure out how many tiles would be in step 10 without building or drawing all intermediate steps. This part has gone smoothly enough to proceed to the next lesson.

Preparing for the Lesson: Getting Ready to Make a Table and a Graph

Many of my students used the table to organize the numerical information about the pattern to find the number of tiles for step 10. We created and extended the table together with the class. I decided to use teacher.desmos.com to utilize the dashboard feature that would allow me to keep track of the students work and to bring it on the interactive whiteboard for future discussions. Here is the activity screen I used to for my lesson. We graphed the pattern together and discussed the behaviour of the graph.

The Creation

The lesson went much smoother than it did in my nightmares. I made a table template to simplify the logistics. Most of the students were successful in creating an increasing pattern that follows a specific rule. Some students created patterns that were increasing, but they needed support to identify that their rule was not consistent.

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There was often a “break” in the rule between step 1 and the rest of the pattern.
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Some patterns had obvious visual rules but were tricky to represent in a table.

While most of the students worked with addition, some students tried to triple or double the number of cubes they use in the pattern.

Pattern 1: A Pyramid

Pattern 2: A Fire

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Pattern 3: Plus Four

Pattern 4: “It’s a bit up and down but still increasing.”

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Students had some great notices and wonders.

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The really exciting things started happening when students crowded around the interactive whiteboard and started analyzing their own and each others’ graphs. Someone noticed that if there is an unusual change in behaviour in a graph, there might be a mistake somewhere in the calculations. One student said, “If the number of cubes is the same on every step, I think the graph will be just the straight line, not tilted.” Someone else asked, “How can you make a circle if there will be more than one Y for one X?”

I had a moment of insight.

Making Predictions

During my lunch break I threw together my template for the afternoon reflection which included three questions and three spaces for the sketches.

-Without numbers, sketch how the graph of your increasing pattern looked like.

-Sketch how you think a graph of a decreasing pattern might look like.

-Sketch how you think a graph of a repeating pattern might look like.

I asked students to try and explain their intuition.

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There were a few interesting ideas considering the behaviour of the graph of a decreasing pattern.

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Many students tried to look at the graph differently rather than switching the direction of the slope. As if they re-located their point of view rather than changing the orientation of the graph to represent the decrease. I wonder why and I currently have no explanation.

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This reminded me of a graph of an absolute value function, and it seems to me that the student’s logic was also describing something similar.

As expected, repeating patterns had the largest variety of predictions. Most of the students went with the equally spaced bumps. A few sketched a graph of a constant.

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There was an attempt to express a repeating pattern as a periodic event.

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And I need to ask some questions about this graph: How far do you think these lines can go up and down?

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Questions and Thoughts: After

My students need more Desmos in their lives. The interface appeared to be 8 year old friendly. I did not need to explain anything about the coordinate grid. I think, if we try graphing decreasing patterns, there will be more clarity about the structure. I loved the teacher dashboard with all the functionality of tracking my students’ work and the ability to pause the lesson and to control the slides. I would love to have the function to print students’ work from the dashboard implemented because printing 50 clipped screenshots was somewhat time consuming.

I wonder if showing all three steps of the process before engaging with it pushed many of my students to work numerically rather than visually. Many of the students were just building the “rods” with the cubes instead of going for more creative shapes. Would the outcome have been different if I asked them to build the patterns first, took the pictures, and then introduced the next two steps? If anyone tries, please let me know how it goes.

I also wonder what my next step(s) should be. I am considering giving kids their graphs and pictures of their patterns and finishing their notices and wonders first. Some of the students wanted to test their predictions for decreasing and repeating patterns. Some started experimenting with creating pictures with their lines. I am also thinking about reversing the process; giving my students a graph and asking to create a table and to build a pattern. We will start with wonders and then I will make my decision.