Graphing linear functions may be the most important topic in Algebra 1. While proportional reasoning is a prelude to functions, this is the first formally identified function presented to them. The graphing leads to slope and intercepts, beyond the entry point for graphical representations to functions. This post presents an activity that can serve as the entry point for linear functions.
The activity is presented on a Google Jamboard, which provides manipulatives. It begins with a relevant context for students, money and being paid for a job. This allows them to engage the function using money. Before using numbers, they engage the work context through images. They are presented the table and then graph representations of the function before getting into the equation. Here is a FB Reel showing the movement of the images.
There are 3 categories of slides. Here is a description of each.
Table and graph clocks for hours and dollar bill for the money.
They graph the whole hours first, then fractional hours (1/2 and 1/4) to see that there are points “squeezed in between each other. This leads to the idea of infinite number of points. In turn, this leads to the idea of the line are a visual means to present all the points. The points can be presented as solutions. Hence, the graph presents all the solutions for the function.
Table and graph with numbers on sticky notes that can be moved from the table to ordered pairs to positions on the coordinate plane.
The equation, with sticky notes to show numbers substituted in for the variables and then moved to ordered pairs with parentheses.
When our 3rd child was born, we decided to buy a used Honda Odyssey as 3 young kids were not fitting into a sedan. Being the stats geek I am (master’s in statistics at the University of South Carolina – total geek) I collected mileage and price data for all the used Odysseys for sale on dealer sites throughout South Carolina. I then created a the scatterplot shown below. I went to a dealer, showed an agent my graph, and he immediately exclaimed “Where did you get that? We create graphs like that every week!”
It was this experience that led me to the idea of using used car data to introduce linear functions. Shopping for a used car has proven to be a relevant, real life activity the students enjoy.
Used car shopping to collect data on 10 used cars of a single make and model.
Creating a scatterplot for price vs mileage of the used car of choice.
Creating a line of best fit (regression line) to model the data.
Creating a linear bi-variate equation (regression equation) to model the data.
The activity is presented on a WORD document (feel free to revise). It shows screenshots to walk student through the Carmax website (subject to Carmax revising their website). The screenshots make it easy for the student to navigate, which increases independence. (NOTE: there is an ample number of Youtube videos on using Google Sheets for this activity.)
The end product looks like this. Note the importance of using 1000s of miles as the slope is more meaningful, -$140.64 per thousand miles, as opposed to 14 cents per mile. I would start with the scatterplot alone to unpack the variables, the relationship between the variables, and the ordered pairs. Then the line and equation can be introduced to show a meaningful use of the line and the equation. The y-intercept has meaning with “0 miles” equating to a new car (I do not explain that new cars have miles already accumulated until we unpack the math).
Graphing linear functions and the underlying concept are challenging for many students. The video below shows a scaffolded approach to teaching how to graph. This approach also addresses the concept of the graph as a visual representation of all possible solutions (see photo above). Students often do not realize that the line is actually comprised of an infinite set of points which represent all the solutions. Here is a link to the document used in the video.