This week, CAMI continued learning about Dan Meyer’s three-act math model by working on the Super Stairs problem. In keeping with the three-act framework, we started the meeting by watching the short video
below a few times and then posing some questions.
We came up with lots of questions, including:
- How many steps did he take on the Super Stairs?
- How many stairs did he step on more than once?
- What is the distance between turns?
- Does “super” refer to the stairs or the activity?
- Was it the same staircase both times?
- What is the difference in slope between the two staircases?
- What is the length/height of each stair?
- How many more steps does he take on the Super Stairs?
- Does he travel the same distance per minute on both staircases?
- What is the pattern for the Super Stairs?
- Which one would get his heart rate up the most?
- How many times would he have to do the regular stairs to take the same amount of steps as the Super Stairs?
- Does the landing count as a step?
- Which would take longer?
- How long would it take to complete the Super Stairs?
In the end, though, we decided to focus on the questions involving how many total steps he took on the Super Stairs and whether or not there was a “pattern” to how we could figure out the number of total steps taken. We knew that if we could figure this out, we could probably estimate how long it would take to complete the Super Stairs. These are the questions that Dan Meyer wants students to work toward, and the ones that he shows answers to in the answer video.
Before we could really dig into this and start making estimates, though, some members of the group wanted to take a minute to clarify the difference between a stair and a step so that we could use shared language when discussing the problem later. We decided that a step was an action, and that a stair was a physical thing (described in our board work as a “shelf/box thingy”). When it came time to make estimates about how long it would take the man to climb the Super Stairs, a few members wanted to see the video again so that could pay closer attention to the time. We watched the video again, and Jane determined that it took him 17 seconds to climb 5 Super Stairs. This helped everyone to make some estimates. The group also paid closer attention to how many physical stairs there were on the staircase, and we counted 21.
Estimates for how long it would take to climb the Super Stairs ranged from three minutes to ten minutes, and estimates for how many steps would be taken on the Super Stairs ranged from 80 to 480.
After discussing our estimates, everyone started working on the two questions:
- How many steps would the man take on the Super Stairs?
- How long will it take him to climb the Super Stairs?
Sharing Our Work
Our board work, including some solutions for how to find the total number of steps taken on the Super Stairs, is shown below. We had explored nonlinear sequences in other CAMI meetings leading up to this one (see Toothpick Patterns and the Pentagon Problem, and Problem-Posing with Visual Patterns in particular), and so most of the members in attendance had some background in deriving algebraic formulas that represent sequences. This is evident in the board work.
Solange created a function by using each stair as the input value of a function, and the total number of steps taken as the output. She then examined the consecutive differences in outputs to determine that the function was of the form ax² + bx + c. She also knew, though, that the value of c would be 0, because if there were no stairs, then no steps would be taken. Solange also saw that the output values were exactly twice the triangular numbers (1, 3, 6, 10, 15, 21…). If you’re unfamiliar with this term, the triangular numbers refer to Gauss’s formula for calculating the sum of the first n numbers. His formula is:
Solange multiplied that formula by 2 and came up with n(n + 1), or n² + n.
She also represented this visually, by drawing the stairs and values of n² and n. Using her formula, Solange determined that the man would take 462 steps altogether.
Maggie and Cynthia arrived at the same formula but came to it differently.
At this point, our meeting time was over, but we still wanted to see how close our predictions were to the actual time it would take the man to climb the Super Stairs.
We found that, on the whole, our guesses were much too high—the actual time was 2 minutes and 56 seconds. We attributed this to the fact that, on the first five Super Stairs, the man spends a lot more time turning around, which eats up some of the time. He climbs the stairs more quickly when he’s doing several of them in a row without turning around.
We ended the meeting by talking about how an activity like this might play out in class. How, for example, could we be sure that students would ask the questions we needed them to ask? And since it seems unlikely that many of them would be able to create formulas for calculating the number of Super Stairs, what might their work look like instead? And what would be the big takeaways for adult learners?
It also seemed important to get on the same page and define the terms we were using (the “step versus stair” discussion mentioned earlier) as a group before we started to try and solve the problem. We had a similar conversation with we looked at the Pyramid of Pennies problem defining “stack”, “pile”, and”layer”. In both cases, it allowed us to process the situation further and clarify the mathematics we were trying to represent. It would be very difficult for students (or anyone) to have a discussion without defining the terms we use in the problem-posing phase.
In the end, we all agreed that we really liked the problem and the question-posing element to it, and we agreed to check back in after we had each taught a three-act math problem in class.
Respectfully submitted by Tyler.
In attendance: Cynthia, Jane, Maggie, Solange, Tyler
Programs represented: Literacy Assistance Center, BMCC, York College, Fifth Avenue Committee
Location: BMCC Adult Learning Center, 25 Broadway, 8th Floor, Manhattan