Functions are arguably the most important concept students study in high school mathematics, but it’s also one of the most challenging, resulting in a wide range of misunderstandings and misconceptions.
Richard Feynman famously said “What I cannot create, I do not understand.” The focus of today’s workshop is to enable students to create their own functions, with variables that students actually vary, with relative rate of change that they observe, analyze, and control, functions whose behavior creates patterns and pictures, all with the goal of developing students’ deep and broad understanding of function concepts.
▹︎ Trajectory (5 min)
Investigate Transformations as Functions: Students investigate reflection, rotation, dilation, and translation. They drag independent points, define function rules to construct dependent points, observe relative rate of change, use relative rates to distinguish between function families, restrict the independent point to lines or polygons, use meaningful function notation, and play function games.
Connect Geometry and Algebra Through Functions: In this unit students transfer their understanding of geometric functions, with variables that are points in the plane (Flatland), to linear functions, with variables that are numbers on the number line (Lineland). Students restrict their point variables to number lines (“Lineland”), measure the values of the variables, compose geometric functions on the number line, and represent the composed functions in both Dynagraph and Cartesian forms, and discover a new way of looking at linear functions.
▹︎ The Dilate Function (15 min)
Download the student worksheet. Play the Dilate Games. Step 2 Step 7 Alice Shrinks and Grows
In this activity you'll create an independent variable, a rule, and a dependent variable. You'll vary the variables, make some patterns, analyze the relative rate of change, and solve interesting challenges.
▹︎ Functions of Points → Functions of Numbers (5 min)
Download the student worksheet. Step 2 Step 3 Question 6
What happens when students restrict the domain of the Dilate and Translate function families to a number line, and then compose these two functions?
▹︎ Dynagraphs (40 min)
Create your own composed Dilate-and-Translate function restricted to a number line, modified to put the dependent variable on a second number line. Analyze how the dilation scale factor and translation vector affect the motion of the points, and how they relate the values of the variables. Finish by playing the Dynagraph game, trying to match the behavior of a mystery function.
▹︎ Discussion (15 min)
How is this approach in keeping with Feynman’s observation of creating in order to understand?
How might it work in your own classroom?
What problems would you anticipate?
How might it encourage your students to notice, and to wonder?
What other questions and comments do you have?
▹︎ Cartesian Connection (5 min)
Download the student worksheet. Step 2 Step 3 Step 6 Step 7 Play the Cartesian Games.
In the capstone of this series of activities, students create the Cartesian graph of a linear function using only dilation, translation, and rotation functions, and understand the shape of the resulting graph in terms of the dilation scale factor and the translation vector.
▹︎︎ The Fine Print