Piecewise Functions

Piecewise functions have been something my students have always struggled with. They don't always get they are taking parts of the function and graphing them on the same graph. They'd like to graph the whole graph of all functions on the same graph or they don't have an idea where to get the points to graph. So, I am changing up piecewise functions this year based on some suggestions from the Math Twitterblogosphere.

The first thing I did was introduce piecewise functions via Mathalicious' Domino Effect lesson. (shameless ad - I've met Karim and had wonderful conversation with him. He and his company are doing some great things to help teachers. It is worth the money to subscribe and have access to the lessons. Go check it out. I'll wait. :-) ) I had used this at Hedge's suggestion - she had used it in her Algebra 2 classes to introduce piecewise functions.

On the second day, I adapted what Maggie has done with her PreCalculus students when introducing piecewise functions. She created an investigation where the students graphed the individual functions and cut out the pieces needed and put them together on a graph. What a great idea! Granted, right now we are working with piecewise linear functions, but the idea is fantastic. So, I took her idea and modified it to fit what we were doing with piecewise functions. Here is what I came up with:


I should add here that the graph is larger and has the printed axes on it - for some reason, I can't get it to show correctly when uploading to either box or scribd. If you want the file, try the updated post or send me an email at lmhenry9 at gmail dot com.

We only had time to work through the first two functions, We did the first one (mostly) together as a class, so that students would understand how to put them together. Once the first one was complete, we talked about what a piecewise function is and how it is created. We discussed the domain restrictions and why we cut out the individual pieces. On the second function, we talked about how to determine which x-values to use in the table and they walked through the process. When it came time to cut out the functions, students asked how to determine where to cut the graphs and we discussed the domain restrictions again. Here are some samples of what my students did:

When the kids returned on Monday, we went over the last question and discussed how to deal with the more open-ended domain. I then went over 2 piecewise examples with them without doing the formal cut-and-paste. With the remaining 20 minutes, I had students work on 5 problems in class. Most students got through all 5 problems and I was surprised that not only did they have very few questions on how to start, but that they were working successfully on the problems they attempted.

What a difference compared to previous times I have taught this concept! In the past, I have not even broached this topic with the regular Algebra 2 students, only the Advanced students, and even then, they have struggled with it. What a huge difference - students actually seemed to understand what parts to graph and to put it together on one graph. I am really pleased with how this turned out. Thanks Hedge, Mathalicious, and Maggie for all the great parts to this lesson!

Follow up on the Trip slide

I put the anyqs slide I had created in front of my freshman math students today. As I put the slide up, I told them that my family and I were planning to drive to Houston, Texas this summer to see my sister-in-law. I explained to them that I usually use Google Maps to prepare for a trip but this time I looked at four websites. They saw this slide (well the first page of it):



My 3rd period class wondered pretty quickly why Bing was so different than the other ones. They zeroed in rather quickly on the idea that the speeds the websites assumed we were driving were different. They also came up with the formula to figure out the speed quickly (distance / time). As we were getting ready to do the calculations, they talked about rounding the times to make it easier to calculate. We talked about what to round to and I did steer them on Mapquest and Yahoo to round to part-hours rather than to the nearest whole hour. We talked about why that would make more sense. Throughout the discussion we talked about factoring in time for bathroom stops and stops to eat. We also talked about the possibilities of getting slowed down in traffic and/or construction. It was a very good discussion, although it wasn't a whole class discussion. About 1/2 to 2/3rds of the class was engaged in it. We talked about there being different speed limits in different states, so we went to the internet to find a current list of state speed limits. We labelled each state with its speed limit and talked about the different terms in the chart. Once we looked at the map, they decided which one they thought would make the most sense. This class choose Google Maps because they felt it would best reflect the amount of stops needed (not counting the over night one). We had a little time in this class to actually look at the routes and compare two of them. I think if I were to do this again, I'd have them look at the directions in pairs and compare and constrast them (not just the routes, but the information given).

3rd Period Slide


In  my second class, the discussion went basically the same. With this class, I had closer to 2/3rds of the class involved in the discussion. I had a couple of students who haven't been actively interested in math all year take an active part in the discussion today. One of these students made some excellent observations, including one where he said that "Bing can't be right because we have driven down to Florida straight through and it took us 19 hours. This trip is farther than that, so it can't be a 20 hour trip." This class needed more prompting - they did come up with the question pretty quick and they were able to come up with speed = distance / time, but I had to prompt them a little more with why there were differences. This class wanted to know which one was "right" and that brought out some good things. They wanted to rely on a GPS - they felt that would be the most accurate. We did not have time to look at the actual routes, but we talked briefly about the differences in routing (and the differences in mileage). In this class, I did poll them on which one they would pick (like I did with 3rd period - but I didn't write down the numbers), but unlike 3rd period, I asked them to share why they picked the website they did. I jotted down some of the reasons on the slide. We ended up with Mapquest more here - they felt that Google was too slow, but the kids who picked Bing also made a more convincing argument here.

5th period slides (there are 2)


I was pleased with the discussion. I think if I had done this earlier in the year, the discussions may have been a little more productive, but generally they went well. My students were engaged - which for there being 12 days left of school, was excellent. These are lower ability freshmen (some discipline issues as well) so I felt it was a win. I think they got something out of it. I think I may develop this into more of an activity at a later point, similar to what Colin suggested and possibly what Max suggested. Right now, I just don't have time to get it there. I thought their suggestions were good and are worth considering.

All in all, I think it went well. We had good discussion and talked about math in a real world context, which for these kids, is important to see. It cemented even more for me why it is important to incorporate this kind of stuff into the mathematics classroom. Students need to see the relevance of mathematics in their world, and for these students, that relevance provides motivation to engage with the material. They won't just do it because they're in school and that's what the teacher says to do like my top students and even most of my average students. Maybe I need to start with this course and see what I can find to make the math more accessible to them. The curriculum mirrors somewhat an Algebra curriculum - maybe finding those connections to the material will help them to be more successful with it. Definitely some things to think about. Doing this does make me even more excited to head to the NCTM Reasoning and Sense Making Workshop in Orlando - to learn about the possibilities that I have not considered in depth until now. My class will be a more exciting place next year!

Trip - class plan version 1

I am trying to set this up as an in-class deal with my lower-level freshmen. I am trying to follow what Dan Meyer set up as a framework here.

Here is the setup:
 (note, page 2 is blank)


Ask students for what they wonder about this slide. I am looking for them to question why the trips look basically the same and have different times.

Here is where I am new to this and looking for guidance...

I think I would provide to them the routes the websites came up with. They have previous experience with d = rt, so I don't think I will need to provide time to instruct them on it (although they may need some coaxing to come up with it). I think I pretty much let them have at it. I am debating whether to break them down into smaller groups and each group has one or two of the four routes to look at or have each small group have all four routes to look at.

They would resolve the question and we would discuss why the results they came up with are significant. I hope that it would lead to a discussion of how do you plan how much time a longer trip would take and the things you need to take into account. We have 50 minute class periods and I am expecting this to take most/all of the class period.

I feel like I don't have a lot to go on here. I have repeated said that I have very little (if any) background on this, so I know this is a learning experience for me. Please offer any and all thoughts on this - I am looking to try this next week (year is winding down and I figure I have nothing to lose - and hopefully engage my students for the period).  Thanks.