PIecewise Function

The idea for this column came from a question posted on Twitter about ways to help students understand piecewise functions better.  I can understand the teacher's request for help because our students have difficulty combining two or more functions into one graph.

I gave it some thought and came up with an idea I thought I'd share with everyone.  It is a physical way to see how they combine.

First draw the two individual graphs on graph paper.  I made the one shown to the left.  I included the two functions, I listed as an f(x) and g(x).

I used two different colors so the graphs are distinguishable as possible.

In the second photo, I show the individual graphs against a white background.

This prepares the student for the next step.  Since the x^2 graph is used till the value of x = 1, the students will cut the graph at x = 1.

The second graph is also cut at x = 1 because that is where the second piece begins.

For the final step, have the students tape the two graphs together at x = 1 so they match up.

With the two colors, it is easy to see where one graph ends and the other begins.

At this point, students can put in the circle indicating the < part to see how well they fit.

The last piece would be having students fill out a small questionnaire in which they use the completed graphs to determine which graph is associated with which part of the graph.
I might ask "f(3) is found on which graph?" so they have to relate values to points and lines.

Students can repeat this exercise with several different piecewise functions to see how the functions fit based on the criteria.  Some will have a smooth transition like this one while others have huge jumps between one function and the next.

I also found this game called Polygraph: piecewise functions on Desmos.  It is designed for students to improve their vocabulary regarding piecewise functions, first by playing against the computer, then against each other only after answering questions designed to help them reflect on what they are learning.

I think part of the problem may be that students are not exposed to these types of functions until they hit high school.  I don't think its covered in middle school.  I do teach it but usually not until Algebra II because I'm usually bringing my Algebra I students up to where they should be when they enter the class.

In a couple days I will discuss the use of piecewise functions in real life.  When I took math in high school and college, the teachers never discussed the situations one runs into where they might have to use these types of functions.

Let me know what you think.  In the meantime, I'm working on ideas for sketch notes and graphing activities for linear inequalities and systems of linear inequalities.  I'll share those when I get them developed.

Thanks for reading.