What twelve year old wouldn’t want to spend a few days bicycling along the ocean front, spending their days amid sunshine and ocean breezes and their nights under star-filled skies? Along the way, they get to swing through Cape May, a lovely ocean-side town filled with beautiful Victorian buildings, visit Chincoteague Island to see the annual auction of wild ponies who swim to the island from Assateague Island, and swim in the ocean. This is the context for my students’ exploration of different ways to represent and analyze data (tables, graphs, and eventually equations). The series of (Connected Math) lessons center around a set of college students who are setting up a summer bicycle tour business to earn money for school. In the series of lessons, they explore the question of how long each day’s ride should be, whether the planned route is feasible (they test out the route and collect data), where they should rent bicycles for the tour, finding the perfect price point to maximize their income, how long the drive back from the final destination will take at various different driving speeds, and the cost of taking the tour participants on a side outing to an amusement park.
Yesterday, I started the unit by introducing the problem context. I began by showing a short video clip of someone on a bicycle tour through Great Britain. I chose to begin with a video clip in order to support my English Language Learners and students from lower socio-economic households, in order to bridge language and economic divides that might make the problem context difficult to grasp. By seeing a bit of a bicycle tour, they would have better access to the problem context.
After students watched the video clip, I introduced the problem – five college students setting up a summer bicycle tour business. The first question the college students were considering was how long each day’s ride should be. I asked my students what they thought would be reasonable. This led into a nice discussion of some of the factors that might influence the answer to that question – the relative incline (uphill, downhill, flat), the terrain (pavement vs sand or gravel), the weather (riding into a wind, no wind, wind at one’s back). I then asked them if they thought it was reasonable to expect the riders to maintain a constant rate for the entire day. Some students thought not, but some students thought the riders could pace themselves. This laid the groundwork for the first part of the lesson.
In order to explore the idea of pacing, I had students conduct a jumping jack experiment. At each table group, one member of the group would perform jumping jacks for 2 minutes. Another member of the group would be the timekeeper, marking the time in 10 second intervals. Another member of the group would be the counter, counting the jumping jacks. The final member of the group would be the recorder. (I assigned tasks to group members by their seat position within the group. If a group had only three members, I had the jumper also do the counting.) In order to speed the process, I provided the recorder with a pre-made table so that he or she would spend less time copying out a table and could instead just fill in the table entries.
At the conclusion of the experiment, I had groups collect data from their recorder and complete their own data tables. Then, I asked students to describe what happened to the rate of jumping jacks as time progressed. There were a few groups throughout the day that maintained a fairly steady pace, but most groups experienced a steady decline as time passed. Some groups had a jumper who stopped completely part way through the experiment and then resumed their jumping after a short break. As the class discussed this, I asked them how they saw these changes in the tables that they had created. This gave them the chance to explore the idea of how a change in the jumping jacks compared to a constant change in time. (I had not yet introduced dependent/independent variables.) At this point, my goal was to begin to tie this lesson back to work they had done with ratio tables in a previous unit. I wanted them to see that this was not in fact a ratio table because the rate was not constant.
I told students that I wanted them to look at the trends of the jumping jack data in a graph. In order to do that, they needed to learn how to correctly make a graph. I began by introducing the concept of independent and dependent variables in a table and talking to them about the mathematical conventions. Then, I introduced the process for translating data into a line graph using a Flow Map (this is a Thinking Map used for sequential processes). I provided students with the advance organizer to complete their notes on creating a graph. I had a regular version of the organizer and a modified version of the organizer (that is more of cloze activity) to support students with learning disabilities.
After students completed the Flow Map, I had them use the data from the jumping jack experiment to make a graph. As they worked, I circulated among them, taking anecdotal records on their work. As they finished, I selected graphs to share with the class. I was very intentional in selecting graphs with errors. I then asked the class to examine the graph to see if there were any errors. This forced the students to think more deeply about the work they had been doing. When someone found an error, I gave a piece of candy to both the student who allowed us to look at his or her work and to the student who found the error. I explained that the person who let us see the mistake did as much to help us grow mathematically as the person who found the error did. (This is a common practice in my class). I made sure to share multiple graphs with errors in each period to ensure that no one student felt like he or she was the only one still learning how to do this.
After analyzing several graphs, I asked the students to explain how they saw the rate of jumping jacks changing in the graph. Here, I was laying the ground work for upcoming lessons in which they will be analyzing data in graphs.
After summarizing the lesson, I had students complete an exit card in which they had to find the error in a graph. After some thought, some of the students were able to see that the independent and dependent variables were on the wrong axis. For those students who were having difficulty, I told them to go back to their Flow Map and work through each step to see if they could find the error. Eventually, everyone successfully completed the exit card.
Following the lesson, I posted an anchor chart on the wall of the classroom that corresponds to the Flow Map that they used in their notes.