Instead of beginning this lesson with a story problem on the board, I post a table that has the headings "Height" and "Shoe Size". I post my height in inches and my shoe size and invite my students to post their own data. When everyone has posted, I tell them we are going to use the data to predict the shoe size of my son, who is 6'4" tall. I use him because he represents a height that is different from anyone in class. Someone usually raises the question about mens shoe sizes being different from womens shoe sizes. I ask for suggestions and allow the class to reach consensus about what to do. Since my class is small this year, I may suggest that we keep all the data together and simply adjust mens sizes down one as an easy fix and good approximation. If you have more data, you can separate the tables by gender. When this is done I ask my students which is the independent variable and which is dependent. Once we've decided to put height on the x-axis and shoe size on the y-axis, I walk them through the process of finding a linear equation or equations to model the data. (MP4) These equations vary a bit each year depending on the class but usually look something like "y = .4x - 14". Some students will question the x-intercept being a negative number which provides an excellent opportunity to discuss what reasonable values would be for shoe size and height. This example is nice because students can easily see that humans can not have a height or shoe size of zero. In fact, even toddlers are generally over 24 inches tall. Now that we have the equation, I ask my students to answer the original questions, the shoe sizes of my granddaughter and son. (MP1) While they're working I walk around offering encouragement and assistance as needed. When everyone is done, I ask for volunteers to share their predictions. (Which should be approximately 16 for my son and 2 for my granddaughter)
For this section of the lesson you will need copies of the Data Challenge, rulers, and small (5cm) pieces of string for each team. I tell my students that they will be working with their front-partner for this part of the lesson. I say that they will be collecting data and developing equations to model that data and make predictions. I distribute the Data Challenge and ask the teams to review the directions. I ask if there are any questions, then walk around offering encouragement and redirection as needed. This activity engages and challenges most students because they collect real data, model it and use their models! (MP1, MP2, MP4)
After about 25 minutes or when all the teams are done, I have my teams number off by fours. I randomly choose from the "#1" teams to share their work to problem one using the document camera and projector. I invite the class to critique their work and encourage them to respond, then call up another "#1" team to present. (MP3) When all of the "#1" teams have presented I randomly call the #2, #3, and #4 teams to present in turn.
I close this lesson by giving my students a notecard and asking them to write one new thing they learned about mathematics today and one thing they believe they need to understand better. Some students will state that they understand it all perfectly well and to those students I suggest that they select something they would like to understand even better than they already do. This is their ticket out the door today and it gives me some insight into areas I may need to re-teach or at least discuss a bit more with my class.