This is from Illustrative Mathematics (the people over there do wonderful work. Plus they are lovely.) The problem I posed to my college algebra class was this:
I had them try it on their own and as I circulated the room, I noticed about 3 methods: taking the square root, putting the problem in standard form and then factoring, or putting in standard form then applying the quadratic formula. After clearing up errors and misconceptions, I was confident we understood that the answers were x = 3 and x = 9.
What now? We could gather up all of the methods they came up with and make a lovely list. Or we could take a look at the method that students almost 100% of the time ignore/forget/dismiss: graphing.
Step 1: Understand what a “solution” looks like graphically.
We separated the equation into two quadratic functions which both were equal to y. Now we had a system of equations and this group knows that systems of equations have solutions at points of intersection.
Step 2: Without a graphing calculator, sketch each of these two functions to approximate how they cross. “Expect to be wrong and give it a go anyway so that we can all learn from each other.”
Step 3: Examine some of our solutions. As I expected, only about 2 students had a solid understanding of where y = (2x-9)^2 sat on the xy-plane.
I categorized the errors into three groups:
A. The negative 9 means the graph is shifted down.
B. Our answers when we solved were x = 3 and x = 9, so this graph must cross there.
C. I don’t want to be wrong so I’m only graphing y = x^2.
Step 4: Look at the graphs of some similar quadratics like y=(x – 3)^2 and see if that thinking applies here.
*I feel like here is where understanding happened.
Step 5: Take the gridlines and axes off of the Desmos graphs and find the points of intersection.
Interesting to see them “know” that x=3 and x = 9 from solving this equation algebraically somehow applied here, but they weren’t sure how.
Eventually we arrived at (3,9) and (9,81).
We explored a method that most (if not all) students don’t think of when asked to solve an equation: Graphically. But I think it’s an important one when trying to figure out how the pieces of functions and algebra fit together. Yes, we could practice factoring, the quadratic formula, and completing the square all day long. But in the end, know those individual methods doesn’t give my students an idea of how those solutions connect with the actual functions they represent.
This problem made me think about what we tell students when we explain methods of solving equations. Any time we show a student a method, we are inexplicitly stating that this method has higher status than any other. Giving students an opportunity to solve a problem using their prior knowledge is important to the learning process. Their way of solving isn’t always going to be algebraic and building from where they are at is vital to creating a foundation of understanding. If they start with “guess and check,” help them build structure from that rather than insist that the algebraic method of solving is superior. In the case of the Problem above, any algebraic method was probably the most efficient but it isn’t always.
For example, what about this problem from Nrich. Would your students approach it algebraically first? Or is there foundation elsewhere?
I think the graphing method is interesting because while it seems “easier” to students (since Desmos or whatever does all the work for you), it’s actually making the problem a LOT harder and then finding the answer. Instead of looking at the two answers to the original question, we generalize each side of the equation to consider all INFINITE solutions to y = x^2 and similar for the other side and from among all infinite coordinate pair solutions to each, we pick out the two that satisfy both.
God, if I had a dollar for every time I said “Guess and Check is a legitimate problem-solving method, but….” I wouldn’t have to play the Powerball this week.