This year, I’ve been resting on all the worksheets and smartboards I’ve made in years past. Because of the time commitment spent heading the disciplinary committee and implementing SBG in calculus, there just hasn’t been time to reinvent the (unit circle) wheel. It actually kinda sucks, because I love creating new lessons, worksheets, smartboards, whathaveyou. It makes me look at whatever I’m teaching from a fresh perspective, because I’m forced to ask “how do I want to approach this topic so that my kids will get it.” I don’t know why I love it, but I do. When I’m using my existing material, I am not forced to ask that question a second time. And when I am using someone else’s material, I don’t get any say in the matter.
Anyway, for reasons beyond me, I can’t seem to find anything I made in years past on piecewise functions for my Algebra II kids. I decided to whip up a guided worksheet which would be at their level and walk them through it. My book and Ms. Cookie both approach piecewise functions in the same way. It’s actually a way I really like. But introducing it in this way, and having them make these charts, takes time I don’t have. I am not sure it will make sense to my kids if presented as an opening salvo. I think it’d make a lot more sense if they first get a basic introduction to piecewise functions, and then see it work with these numerical tables.
So for the first time in a long time, I whipped (my hair back and forth…) up a guided worksheet that I hope will go over well:
That’s all.
Continuous Everywhere but Differentiable Nowhere 
This is a pretty sweet worksheet. I can see myself adapting it to include some interval notation questions (Increasing/decreasing/constant) with my precalc kids and perhaps include some asymptotic behavior. Perhaps something like “Now create a piecewise function that includes at least one vertical asymptotes that is always increasing.” Thanks for sharing :)
As someone who is likely to win a “Sam Shah Instructional Materials Devoted-Fan-of-the-Year” Award, I am always jazzed to hear that you’ve invented some great new worksheet, activity, or unit I can steal. But I’m also excited to hear about your investigations into other areas like heading up the disciplinary committee. Every experience enriches every other experience, and it’s cool to see you deepening your teaching practice in these new ways. After all, those experiences are likely to help keep teaching interesting and worthwhile for you, which bodes well for my desire to have — and keep — wonderful colleagues like you.
Elizabeth (aka @cheesemonkeysf on Twitter)
Cool! I’ve been starting piecewise functions with my kids as well, and I was going to post my worksheets in a few days. (I think I still will, since mine look very different from yours; I only made the worksheet as a scaffolded alternative to the problems in the book… So, it’s not as comprehensive but it’s more broken down, in a way. My kids practice the basic evaluations using just problems from the textbook.)
Great worksheets! It’s always interesting to see how someone else presents the same topics.
Cool! I think each year I teach them differently. I always think I have a way so they’ll get it. It rarely works out that way. But I’ll try to post an update on how this works.
I think it went pretty awesomely. We did the first page together as a class, and then I had them do the second page on their own. They had no difficulties, really. They totally latched onto it, and I think it was the “building” aspect which made it concrete for them. They saw the graph, and they built the function.
In previous years, I recall showing the function, and explaining how it worked. The domain part really confused them, for some reason. But they totally took like fish to water.
Since we only had 15-20 minutes, I can’t say that it is solidified in their brains. But next time we meet (after midterms), I highly suspect that it will be pretty easy for them to do the problems.
Tax brackets might make a nice “real world connection” for piecewise functions. As in: “how can we write a function that describes how much someone will be taxed based on their income level?” I’m sure there are good questions that could follow.
EEP! Right before class i had that thought of at least showing them a graph of the tax bracket / income tax relationship — but i only found a crusty image online. I still showed it, but it wasn’t as effective as if I researched it and created something less… dodgy. If anyone knows of a good site that explains how the income tax thing works (and how it relates to piecewise functions), I can use it in my next class (which will happen in a week, because of midterms). If not, I think I have a nice guest blog post waiting for someone who wants to write up a little bit about it (if you don’t have a blog)!
Sam, we actually had a whole income tax policy unit, debate included! If you want, I can send you more info. There are also lots of other cost-break examples for piecewise functions… And I let my Salvadoran kiddies work on a Salvadoran mandatory-Christmas bonus break down, and they really enjoyed it. :)
Yes please! Or better yet, you should (if you can) post it on your blog so everyone can benefit! LOVES IT!
OK, gotcha.
Very well done! The introduction was just as effective as the final practice problem. I really like how you get students to describe the pieces before graphing begins–this is the key step for my students. Great job on getting students to “jump” between picture view, and equation view with the domain. If the student knows what the piece looks like, and what its domain is, then critical learning happened. I use the taxi-cabs and postal rates for examples since everyone has been in a cab, and almost everyone has mailed a package.
I am a new subscriber to the blog. I was pre-researching piecewise functions to see what my students will find when they google piecewise functions when I came across this blog.
I am currently teaching a course I recently designed for high school seniors named “Math In The Real World”. Each unit explores a different class of functions all of which must be motivated by simple things that go on around us all on a regular basis. For each function class I have my students research examples of where that particular type of function is useful in modeling real world situations. For piecewise functions I chose the concept of the outside temperature rising and falling over the course of a day. Realistically the temperature will not rise and/or fall in a completely linear fashion but we make that assumption anyway. I give the students the verbal function and have them translate it into a table then a graph and finally into algebraic form. Example: At six in the morning the temperature was 20 degrees F. The temperature then rose at a constant rate of 2 degrees per hour for 5 hours before beginning to fall at a constant rate of 3 degrees per hour. As the temperature piecewise function can be considered continuous the graph can be made by connecting all the plotted points with a ruler but if we look at other situations such as the cost of parking your car in a parking lot with rates that go up each hour you can introduce step functions in a piecewise way. Students really latch onto anything involving money so I also have them create piecewise functions that represent their bank balance over time if they go through periods of constant saving/spending.
I love this. Very real. It sounds very good for an Algebra 1 or maybe Algebra 2 class!!!
I used this as my notes to review the basics of piecewise functions in honors precalculus on Friday! They worked fantastically! Thank you for sharing!
You’re so welcome! Thanks for sharing that you used it. It literally makes me so happy when someone tells me they tried something in their class and it worked!!!
Hi Sam,
Great worksheet. I use examples of keeping track of a bank balance when you save at a constant rate for a while and then spend at a certain rate for a while. I assume no intererst compounds at first and then introduce that later as an “exponential” piece. The fun part for the students is trying to come up with the linear function algebraically that represents the second (or third etc) piece as the y intercept is not intuitive. They usually have no trouble with the tabular or graphical representations of functions representing bank balance but struggle at first with the piecewise formula. It is a great way to introduce ninth grade algebra students to the idea of domain and why it is so important.