Breaking down the design

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SWBAT express and explain a model for simplifying ratios.

Big Idea

Reproducing the same pattern repeatedly results in a constant ratio.

Intro & Rationale

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.

Warm up

15 minutes

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 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.


Whole class discussion

30 minutes

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!


4 minutes

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.