This lesson provides data about different tile designs to students in a variety of ways, in a table, in a graph, and in statements. Students need to determine which of the designs used the same proportions of black and white tiles. This is a good way to get students to choose an appropriate mathematical tool, either the graph or simplified ratios or both. They will also have to write an argument with supporting evidence to explain their choices. The questions "how do you know?" and "what makes you say that?" and "what could you do with this information/data to help you decide?" are the teachers most useful tool in this lesson.
Before we start the warm up I have students check in with each other to see what they did with their homework Which show a proportional relationship.docx. Some may have simplified ratios to show which designs were equivalent to the three proportional graphs matching proportion to graph.docx and some may have plotted points on the graph and extended the lines of the proportional graphs. I choose student work to display under the document camera and have someone explain what it shows.
The warm up Show all proportional to Chloe.docx shows a single point on a graph showing the ratio of black to white tiles used by Chloe. Students are asked to show all the other floor designs that would use the same proportion of black and white tiles as Chloe's design. I expect several students to begin drawing a straight line through zero and Chloe's point (6,9). As soon as one or two students have gotten up to get rulers I think the secret will be out and others will get the idea. Some students may tell each other what to do when they pass their desks, which is fine with me.
When we go over what students did, I have one come up and draw the line. Then I would ask what ratio Chloe used and how we could use this information to test our line. By simplifying 6:9 to get a ratio of 2:3 some students may draw the "stairs" going up 3 (white tiles) and over 2 (black tiles) to show that each point following the constant ratio lies on the line. Or they may explain scaling up the ratio 2:3 and show that those points also fall on the line.
This warm up reinforces the idea of how to use the graph as a tool to determine proportionality.
Each student gets a graph with four points labeled "Sam, Alfonso, Angel, and Julianna" showing the ratio of black to white tiles each of these people used in their floor designs. Students are told these were the four people from their homework that did not use the proportions shown in the graphs they were given. They are asked to figure out if any of them used the same proportions as each other.
I expect students to use a variety of strategies:
I circulate to make sure students are sharing their ideas and looking at each others papers. If an entire math family group appears to be stuck I might suggest they get up and see what others are trying.
I ask some students to come use the document camera to show and explain what they decided. If one of the ideas above is not shared I would ask either:
Next I give students more design data in different forms (table and statements) and tell them to continue figuring out Who used the same proportions as whom. As I circulate I share ideas with the class that their peers are coming up with.
For homework homework proportion argumentation.docx students need to write an argument to support each of their decisions about who used the same proportions of black and white tiles. They are given helpful vocabulary and sentence frames to help them write their arguments. They are told to explain how the ratios show which people used the same proportions and also explain how this shows on the graph.
I suspect some students will just try to fill in the blanks in the text box that gives them sentence starters. I look for this as I circulate and have them reread the directions that say to use a separate piece of paper. I clarify that they will have to write multiple arguments and that the sentence frames are simply examples to help them get started.
Many of my students will not do the written part of this assignment, which is why they have a group assignment based on it in the next lesson (Writing arguments). This makes sure they engage in the writing process at some level.