The other day in my Algebraic Geometry class, we were doing this problem:

An airplane is flying 36,000 feet directly above Lincoln, Nebraska. A little later a plane is flying at 28,000 feet directly above Des Moines, Iowa, which is 160 miles from Lincoln. Assuming a constant rate of descent, predict how far from Des Moines the airplane will be when it lands.

This is one of the original problems from the PEA materials that we use in our PBL curriculum and I love using it for many reasons. This problem is on a page in the book where we are discussing slope and points that are collinear. So many students’ first idea is to think of the rate of change of the plane as it descends – at least that how I expect them to think about it. However, the student in my class who presented this problem, I’ll call her Robin, had a similar algebraic perspective. Robin realized that since the plane dropped 8,000 miles of altitude for every 160 miles across, she could just see how many times she needed to subtract 8,000 from 36,000 in order to get to the ground, then multiply that by 160. This was crystal clear to Robin, but other students were a bit confused.

So Sandy chimed in. Sandy drew a picture where the airplane was at a height of 36,000 feet and proceeded to subtract 8,000 a number of times drawing triangles as she did this. She did this until she got down to 4,000 (which was 4 times of course), and then realized she only needed another half of 8,000, so realized it was a total of 4.5 triangles that would go 8,000 down and 160 across to get down to the ground. So she multiplied 4.5 x 160 which of course was the total distance across the ground or 720 miles. However, this was not the answer that other students got.

So then Noa, who really likes algebra, says, “Isn’t 8,000/160 just the slope of the line?” Many of the other students agree with her and nod their heads. “So I just wrote the equation of the line as y=36,000-50x and graphed it on GeoGebra. Then I just found the x-intercept. But I knew that we were only looking for the distance from Des Moines to the landing point, so I subtracted 160 from 720, so the answer is 560.” This then inspired Sandy and Robin to check if their answers agreed with Noa and it did.

Just then, Anna said, “Can’t you just plug in zero for y in Noa’s equation? Why do you have to find the x-intercept on the graph? I just plugged in zero and solved for x.” Noa replies,” That’s the same thing…” which created a debate about finding x-intercepts of lines. Which then inspired another student to say that she saw it a completely different way and compared to triangles that had the same slope and set up a proportion giving her an equation that said 28,000/x=8,000/160, which of course set off a bunch of students writing other proportions that were also true.

After this discussion died down, and it seemed we had exhausted that problem, Sandy looked thoughtfully at the board and all of the different methods. She said, “That’s really cool. I can’t believe we all looked at it in so many different ways and we were all right.” And just having a student say that in a spontaneous way made the whole discussion worthwhile for me. It was such an amazing moment, that I sat and paused and let them all accept the pride in their own creativity and ability to use their own knowledge to solve the problem the way they saw fit. I was so proud of them.