I begin this lesson with the height vs shoe size equation from yesterday's lesson on the board. An example would look like "y = 0.4x - 14". I ask my students to pair-share what they think it would look like as a graph and why? After a few minutes I randomly select students to share what they talked about and sketch their ideas on the front board. (MP4) Since they also need to explain why they think the graph will look like that I get a quick assessment of my students' understanding of graphing.
I leave the sketches posted, give each student a sheet of graph paper, and ask them to graph the equation. I remind them that x represents height in inches and y represents shoe size. I stay with x and y for now, but may be able to switch to f(x) and x when more of my students are familiarized with those terms in earlier classes. I also tell them that their graphs should have the axes and scales labeled. (MP6) When everyone is finished I ask them to compare what they've graphed to the sketches and congratulate themselves if their sketches were close to being correct. I then ask them to use their graph to predict the shoe size of someone who is 58 inches tall. Some students will say that it's not possible because they didn't make their graph big enough. I commiserate and suggest that our work yesterday included people as tall as 76 inches and as short as 28 inches, and that because the data is for human heights and shoe sizes that should have given them an idea of how to set up the scales for their graph. I then repeat my question about predicting the shoe size of someone 58 inches tall let volunteers give their best estimates. I finish by saying that scientists did similar calculations and used paper and pencil graphs to build the Eiffel Tower and to put men on the moon.
You will need copies of the Graphing Problems for this section of the lesson. I tell my students that now that we've worked through the toy story example as a group they will be working independently to solve two-variable problems by graphing them manually. I say that labels are important and that precision will make their work much easier. I remind them that they are welcome to use the rulers and graph paper I have and ask that they not crowd their graphs but also don't spread them to fill the page unless necessary. I distribute the Graphing Problems and ask if there are any questions. (MP1, MP2, MP4) While my students are working I walk around offering encouragement and redirection as needed. For example, I a student is struggling to come up with a model I might suggest that they estimate a few data points to get an idea of the shape of the graph.
When everyone is done I tell them that they will be self-checking their papers so that they have a reference for graphing when they need it. I project each problem on the board using my document camera and projector so my students can see the graphs as well as the points I estimated.
To close this lesson I ask my students to compare, in writing, making predictions from a graph as they did today and using an equation like they did yesterday. I explain that I don't just want to know which one they think is easier or faster, I want to know the strengths and weaknesses of each method. If you'd like sample questions to guide this reflection you might try these: