Tracy Zager, in her blog Becoming the Math Teacher You’d Wish You Had shares a wonderful problem that looks simple. However, scratching below the surface a bit, the problem opens up a lot of different approaches. She aptly names the post A Problem with the Space Inside it to Learn.
Here’s the problem, which she found in a Twitter Post by Justin Lanier:
Tracy’s post discusses how her family (2 young daughters and husband) approached the problem. Comments from readers revealed even more approaches to the problem and, after initially seeing and solving the problem fairly superficially, I was drawn in. Here is my comment:
Love the problem! As you illustrated, It is accessible to people with very different levels of mathematical expertise. The question “What do you think?” turns it into a sort of 3-Act problem. What I thought was “Are they parallel or not?” I simply counted the squares to notice that the top line had a slope of 1/5 and the bottom line had a slope of 1/4, so no, they weren’t parallel.
I was done…or so I thought until I started reading the comments. Someone else may care about where they will intersect. So I guess I should care, too. Julie went about it in a way that I never would have thought by generalizing the two equations. I would have not been as clever as Julie and just used the equations y = 1/5 x + 5 and y = 1/4 x – 4. Setting 1/5 x + 5 = 1/4 x – 4 and solving for x, and then y, I would get the point of intersection at (180,41), the same answer as Julie.
What if I don’t know algebra yet and so can’t find the point of intersection by setting the y=values of the two equations and solving for x? I graphed it and noticed that at x = 0, the lines were 9 vertical units apart. At x = 20, they were 8 vertical units apart. Hmmm….could the 20 come from 4 x 5? I continued the graph to x = 40 and they were 7 vertical units apart!
Practice Standard # 8 tells me to “look for and express regularity in repeated reasoning” and # 7 tells me to “look for and makes sense of structure.” I’m pretty convinced that when x = 60 that the lines will be 6 vertical units apart. If I want to graph this, I can check out my hypotheses. At some point I will be convinced that for every increase of 20 units for x, the lines are 1 unit closer together. I fill out the following table. For the y-value of the upper line, I notice that for every increase of 20 units for x, there in an increase of 4. For the y-value of the lower line, I notice that for every increase of 20 units for x, there is an increase of 5! Wow! Another pattern! For every increase of 20 units, the y-value of the line with a slope of 1/4 increases by 5 and the y-value of the line with slope 1/5 increases by 4!
x-value of both lines distance apart y-value of upper line y-value of lower line
0 9 5 -4
20 8 9 1
40 7 13 6
60 6 17 11
80 5 21 16
100 4 25 21
120 3 29 26
140 2 33 31
160 1 37 36
180 0 41 41
So, the two lines intersect at (180, 41).
In the CCSSM, the coordinate plane is introduced in Grade 5. Standard 5.G.2 states “Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.”
I love problems that are accessible to a wide range of problem solvers. Thanks for sharing! I’m going to re-post this on my own blog www.watsonmath.com.
So, now I’m posting this on my own blog, which has lain fallow for a while. I’m just starting to get my mojo back and have another post almost ready, which focuses on the Multiplication Machine at MoMath in NYC. Stay tuned!