Demystifying Piecewise Functions

Piecewise functions, with their multiple rules and sudden shifts, can be a stumbling block for students. However, with engaging teaching methods and relatable real-life examples, educators can transform these functions from cryptic equations to comprehensible concepts.  It is important to break down the pieces.

Start with a clear visual representation of a piecewise function. Graphing each piece of the function separately helps students understand the different rules governing each interval. Overlaying these individual graphs creates the complete picture, showcasing the transitions between the pieces.

Bridge the gap between abstract math and the real world by using relatable examples of piecewise functions. Discuss long-distance phone call rates, where the cost per minute changes depending on the duration of the call. Cell phone plans with tiered data allowances can also be used to illustrate piecewise functions in action.

Next build understanding by breaking down the function into its component parts. Explain how to identify the different intervals by analyzing the inequalities that define them. Then, guide students through applying the appropriate rule for each interval to solve problems. Incorporate interactive activities to solidify understanding. Use online applets or physical manipulatives to allow students to explore how changes in the function's parameters affect the graph. This hands-on approach allows them to visualize the relationship between the equations and the visual representation. Don't just focus on solving for outputs; delve into real-world applications of piecewise functions. Discuss how they are used in engineering to model temperature changes during a manufacturing process,or how they can be used to calculate shipping costs based on weight and distance.

Consider going beyond the basics. Create a "piecewise function zoo" where students encounter various examples with unique applications. This could include functions modeling taxes based on income brackets or membership fees with discounts for longer commitment periods. For advanced students, explore piecewise functions with more complex rules or multiple variables. Offer opportunities for independent projects where they can research real-world applications of piecewise functions and present their findings to the class.

By employing visual aids, relatable examples, and interactive activities, educators can transform piecewise functions from daunting equations to engaging concepts. Highlighting the real-world applications of these functions further fuels student interest and demonstrates the practical value of mathematics. With the right approach, piecewise functions can become a stepping stone towards a deeper understanding of advanced mathematical concepts.

Next time, we'll look at some situations that use piecewise functions. let me know what you think, I'd love to hear.  Have a great day.

Real World Piecewise Functions

Up until recently, I never wondered when one uses piecewise funcitons in real life.  It wasn't important to me because I was only familiar with the mathematics.

After creating the hands on activity, it hit me, I didn't know when the piecewise function was used in real life.  There has to be some application because it exists since mathematics often explains the world.

So after a bit of research, I discovered several real world situations that are not contrived and make sense to me.

I assume most people reading this column have at some point worked a job where you were paid by the hour.  I had one where I worked filling the newspapers with sales flyers so I'd have days free to substitute and look for a teaching job.  The place I worked set the pay with the following parameters:  I was paid a per hour rate for the first 8 hours worked in a 24 hour period from midnight to midnight.  Anything over the 8 hours and I received time and a half for those hours.  If I worked holidays such as Christmas, I received double time.  That makes my pay a piecewise function because the amount I received depending on the number of hours I worked.

For many other jobs the piecewise is based on working a 40 hour week before a person begins to receive time and a half for overtime or double time for certain holidays or Sunday.  It all depends on the rules of the business.

Another piecewise function is based on the amount of something purchased.  The more of the one item you purchase, the less per item you pay.  I've dealt with a company where I paid full price for the first 10 items, the next 20 granted a 10 percent discount, etc.  I've seen this type of arrangement with t-shirts, water bottles, etc. 

Look at the launching of a rocket and follow its acceleration from launch to touchdown to see a piece wise function.  For the first two seconds, the rocket accelerates from zero to a certain speed.  Between two and twelve seconds, the rocket slows down as it begins to coast and due to gravity.  After 12 seconds, the parachute is released and the rocket floats back to earth. Based on the above situation, I think anyone who jumps out of a plane is going one velocity until they pull the ripcord causing the parachute to open and they are now floating down.

The next time I teach this topic, I can provide examples without students having to ask.  I feel like I've scored a win. 

Let me know what you think.  I love to hear from people.  Thanks for reading.

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.