Experience with multiple representations fosters students’ robust understanding of what functions are, how they behave, and how they can be composed. [from the article]

This page provides access to the activities described in our November 2012 Mathematics Teacher article “Using Multiple Representations to Teach Composition of Functions.” To download an activity, click its link below. We strongly recommend that you read the notes for each activity, as they contain suggestions for introducing the activity, monitoring student progress, and conducting class discussions. Several of the activities are also supported by short videos that you and/or your students can use to view the construction process.

(These activities are based on The Geometer’s Sketchpad Version 5, but can be adapted to other dynamic mathematics software.)

Objectives

As a result of this sequence of activities, students will

• Create a variety of functions using different representations.
• Vary the variables.
• Attend to and describe the behavior of the variables, including relative rate of change.
• Use the dependent variable of the first function as the input to the second, in a way that’s appropriate for each representation.
• Show, and later hide, the details of the mechanism connecting the second function’s input to the first function’s output.
• Identify similarities and differences between the representations, and use the similarities to create their own definition while using the differences to make sure the definition is appropriately abstract.

Geometric Representation

This is the recommended starting point, because it is easy for students to construct from a blank sketch, the variables are easy to vary, and function behavior is easy to observe and capture. The zip file includes teacher notes, short and long forms of the worksheet, and a sample sketch showing the result at different stages of completion.

Download geometric_composition.zip  |  View video

Dynagraph Representation

Dynagraphs provide students the opportunity to drag the independent variable and observe the dependent variable on two parallel number lines, an input axis and an output axis. This activity provides an excellent transition from the geometric to the numeric.

The zip file includes the prepared sketch, the student worksheet, and teacher notes. In the full activity students use transformations to create the functions, analyze them, compose them, and analyze the composed result. The activity can also be shortened to eliminate the initial creation of the functions. (revised 11/15/12)

Download dynagraph_composition.zip  |  View video

Numeric Representation

In this two-part activity, students create a composed function by using calculations. They start with an independent variable, perform a calculation, and then perform another calculation on the result of the first. By varying the independent variable, they construct a table of values and use the table to explore relative rate of change and unit rate. In part 2, they plot the results of the calculations on parallel axes to create a dynagraph and explore the connections between the behavior of the numeric variables and the shapes of the traces of the dynagraph.

Download calculation_composition.zip

Symbolic Representation

In this activity students create two functions, calculate composed values and plot the values on parallel axes, and vary the input to examine and explain the resulting patterns. Students create and plot the symbolic composition of the two functions, relate this representation to the others they've worked with, and are asked to make sense of the notation g(f(x)).

Download symbolic_composition.zip

Cartesian Representation

In this two-part activity students create two independent points on the x-axis and then use Cartesian graphs of functions f and g to evaluate both functions, resulting in dependent points on the y-axis. They transfer the output of function f to the x-axis in order to use it as the input to function g. Students use their results to construct the graph of the composed function, and then animate a point along a graphical pathway to show the roles played by the different parts of the sketch in composing the functions.

Download cartesian_composition.zip  |  View video

Update History:

03 November 2015: Updated videos to play on the web page.
01 January 2014: Created this page.