## Reflection: Staircase of Complexity Breaking down the design - Section 3: Whole class discussion

I use the word 'reduce' in this lesson. I was conflicted because of the controversy over the conflict of smaller vs. equivalent. But ratios do contain this conflict and I want my students to grapple with the idea that something could be both smaller and equivalent at the same time. As the numerators and denominators change they do represent varying quantities. 2:3 represents different quantitites than 20:30. We may have 2 girls and 3 boys vs. 20 girls and 30 boys. But the relationship between the quantities remains the same: "for every 2 girls there are 3 boys". I don't want to avoid or gloss over this seemingly conflicting ideas by using less suggestive terms. I want students to confront the ideas and reconcile them.

Tell students that teachers like to use the word simplify instead of reduce and ask why that might be. It's okay to share our thinking with students and show them what's behind the curtain. Some of the decisions we make as teachers steal some of the cognitive demand from students. When we invite them to help make some of the decisions they are grappling with big ideas:

• Which describes the ratios better? Reduce or simplify? Why?
• What works about the term 'reduce'? What does it help to describe?
• What might be confusing about the word 'reduce'?
• What works about the term 'simplify'? What might be confusing?
• How can the ratios be both different and equivalent at the same time?

My students made several conclusions:

• Reducing is appropriate because we are making the numbers smaller, but the word can be confusing because smaller/bigger implies different, but equivalent ratios all represent the same pattern.
• Simpler is appropriate because the numbers are easier to work with when they are smaller, but the word can be confusing because it means easy and we don't think about making math easy and it leaves out the idea that the numbers are made smaller which my students thought was important.
• Ratios can be different and the same because its like a stamp that is used over and over again, so its the same pattern repeated multiple times and built back up from the smallest Building block. Here is the idea of multiplicative thinking!

I also struggled with whether or not to connect the points on the graph. I chose to connect them here to emphasize the equivalence by laying one line directly over another. However, in another class a student suggested that the next step was to connect the dots and I decided to address the issue. I asked students to refer to the tile context and asked what the dots represented, what the line would represent, and if the line would truly represent the tile design. Something surprising happened when a dissagreement that I hadn't anticipated ocurred:

• Some of my students concluded that we should not connect the dots because we can only 'break down' the design so far; connecting the dots would say that we could break the simplest ratio even farther.

• Other students thought about fractional pieces and concluded that in real life we might break down a ratio of 1 black tile:2 white tiles into half a black tile: 1 white tile and that connecting the dots might be appropriate.

I decided that inviting my students into the teacher's decision making process is a valuable way to surface cognitive conflict that uncovers the complexity in the content. It is very engaging for my students to be invited into the secret world of the teacher's mind and gives them a strong sense of agency at the same time.

Staircase of Complexity: 'Reduce' or 'simplify'? To connect or not to connect?

# Breaking down the design

Unit 5: Writing and comparing ratios
Lesson 3 of 14

## Big Idea: Reproducing the same pattern repeatedly results in a constant ratio.

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Standards:
Subject(s):
Math, modeling, Number Sense and Operations, cooperative learning activity, ratio, simplifying ratio, scaling, ratios, pattern, student ownership, cognitive demand
49 minutes

### Erica Burnison

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