##
* *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, but equivalent ratios all represent the same pattern.*different* - Simpler is appropriate because the numbers are
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.*easier* - Ratios can be different and the same because its like a stamp that is used over and over again, so its the same pattern
and built back up from the smallest Building block. Here is the idea of multiplicative thinking!*repeated multiple times*

**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:

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

- 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

Lesson 3 of 14

## Objective: SWBAT express and explain a model for simplifying ratios.

*49 minutes*

This lesson breaks down what students built in last night's homework in which they designed a floor pattern by repeating the same pattern piece over and over. Today they are doing this in reverse. They are looking at the designs created by others and trying to determine what smallest piece they started with. This activity helps to reinforce the constant nature of the changes when scaling a ratio up and down. It also helps them make sense of the mathematics of simplifying. The visual and kinesthetic patterns of repetition are particularly useful to ELL students.

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#### Warm up

*15 min*

Students are asked to look at floors designed by their math family group for homework Floor design template.docx and pattern block tiles.docx last night and figure out which of the four patterns they used.

Since three of the patterns have the same ratio of black to white tiles I expect some disagreement. A student might create a design using one pattern, but their partner argues that they used a different one. I listen for the "yes you did....no I didn't" argument and encourage them to show each other why they think that. I may refer back to that original sentence frame in the warmup again and ask them to use this. This argument helps to strengthen their idea of the equivalence of simplified ratios. The natural peer instruction that happens here is helpful for differentiation.

I expect students to catch each other's mistakes as well. Many students will have tried to repeat the pattern, but may have made some mistakes. I may suggest that they outline the repeated pieces in order to show the pattern or find the mistake. A few may have missed the whole point of repeating the pattern, but I don't expect many to have done this since it likely would have been caught in class when it was started yesterday.

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#### Whole class discussion

*30 min*

**Ask students to describe how they went about figuring out which pattern everyone used.** I expect they will try to "build up" from the smaller pieces and replicate the building process rather than simplify. Showing students the process of breaking down the pattern helps them visualize the simplifying process. At this point I really want to zero in on the math of simplifying, so I make a clear connection between reducing the design to smaller and smaller parts and dividing by common factors.

Once it's been reduced down to the simplest ratio I like to use the sentence frame **"for every 2 black tiles there is one white tile"** and show that this is still true as we build the floor back up. This emphasizes that the simplest ratio still represents the pattern no matter how large it gets.

I like to ask them also what makes it hard for them to figure out which pattern was used. They may say that its harder when the numbers are bigger which can emphasize the benefit of simplifying. I expect them to notice that they can't tell the difference between the patterns whose ratio of black to white tiles is the same, which can be used to reinforce equivalent ratios.

I like to bring up 3 student samples for the class to look at under the document camera to figure out the ratio of black to white tiles used in each. **I specifically choose two that used the same ratio and one with a different ratio. **

Next we see how the designs compare on a graph. I have students come place points on the graph to show each ratio. Someone may or may not notice they form a straight line. Either way I connect the points. This is something we will take more time on in a later lesson. We do the same for the second sample. After completing the table for the third one which is equivalent to the first I ask them to predict where this line will be. Many of them will realize it will go right over the top of the one that used the same ratio. This really helps to reinforce the idea of equivalence!

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#### Homework

*4 min*

The homework who built it.docx is similar to the activity students did in class. They are given four different black and white tile patterns. This time all have unique black to white tile ratios and none can be simplified. Each problem tells about a person who tiled their floor using one of the designs. It tells the number of white tiles and the number of black tiles used by each person and asks students to figure out which pattern they used. Students need to simplify the ratio in order to match each floor to the pattern used to build it.

I like to use student names in my assignments because it makes the kids feel more included and important. I think it improves their attitude and engagement in class.

#### Resources

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- UNIT 1: Order of operations & Number properties
- UNIT 2: Writing expressions
- UNIT 3: Equivalent Expressions
- UNIT 4: Operations with Integers
- UNIT 5: Writing and comparing ratios
- UNIT 6: Proportionality on a graph
- UNIT 7: Percent proportions
- UNIT 8: Exploring Rational Numbers
- UNIT 9: Exploring Surface Area
- UNIT 10: Exploring Area & Perimeter

- LESSON 1: Which is the blackest?
- LESSON 2: Designing the floor pattern
- LESSON 3: Breaking down the design
- LESSON 4: Part to whole ratio
- LESSON 5: The secret side of ratios
- LESSON 6: Comparing ratios
- LESSON 7: Ratio soup assessment day
- LESSON 8: Scaling up ratios
- LESSON 9: Terminology for scaling ratios
- LESSON 10: There's an ap for that!
- LESSON 11: Let's get organized!
- LESSON 12: Navigating a data table
- LESSON 13: Mistakes & Peer Instruction
- LESSON 14: Mickey Mouse Proportions