Last year I introduced systems of equations with Trashket-Ball. It was a big hit and the students were definitely engaged. I wanted to do it again this year, but I also wanted to spiral in some older material. So we played trashketball Day 1 to wrap up the linear unit and after the flight lab we’ll go back and I’ll ask them to try Day 2 where they have to try and tie to continue on the idea of systems).
I ran into a lesson by Mr Ward over at Prime Factors (the link includes his version of the task which is great, thanks!) which has students gathering data to answer the question: When is it cheaper to drive? I liked that systems could be used to solve, but it also brought back scatter plots, lines of fit, writing equations, interpreting points and more. I used his idea and adjusted it to use Desmos and include some reflections back to the original predictions. You can download my version here: Drive or Fly Lesson. (Sorry about the crazy graph color, I can’t print in color so I had to make the map very different from the circle for a black and white print).
The attached lesson is more guided than I actually used, but I wanted to provide a complete document that might be able to be used without reading my mind. Here is what I actually did:
Kids walk in and see the prompt: Is it cheaper to fly or drive? Choose and be ready to defend your answer. I wouldn’t answer any questions or give more information at that point. We took a poll and had students from each side lay out their case. We came to the super insightful conclusion: “It depends.” On what? We listed out the possibilities and factors we might need to consider and distance was by far the most common response. That led to page one in the document with the map and the three distance choices. Again, kids had to choose a picture and be ready to explain as well as come up with all the other things they’d want/need to know to actually mathematically decide.
Student Requests: Cost of the flight. Gas. Food. Time. Type of Car. Destination.
Pretty much the info I asked them to fill out in the chart. I asked them to pick a date a month or more out for airplane tickets as well as a type of car. Then they started gathering data. They were in pairs or small groups to make the data gathering take less time.
**Extension/Change I had all the students use the same flight date and car. it might have been interesting to have each group choose a different date or a different car to see the differences in final answers. I didn’t so that we could combine all the different city data to make a class set and compare their lines/intersections with the class set of lines/intersections. I liked the final compilations, but it would be interesting to have them explore how much gas mileage or how much notice for the plane ticket would change. **
Desmos was great for creating the scatter plot. I had them print the graph so they had to come up with the line of fit themselves, but we also went back to compare theirs with the actual line of best fit via Desmos regression would be to see how close they were and whether or not that error was acceptable.
Finally, after the reflections we focused on the last questions about tying back to the original guess and checking cost for more cities in and out of the “final” range. Most found they were good predictions, but some did not. We had a quick, but I feel important discussion on why that might be. (Distance is not the the only/major consideration with flight costs). We brought up strength of correlation and how that fir into the two lines. Most students came to the conclusion that the model was good, but that they might want to double check if flying into a really small town.
On the way out, I posed the question… should time to get there have another cost. (We factored in food for both and hotels, but not the value of time itself). We had a quick debate on what our time might be worth and when it would matter more. That might also be a great extension place.