Developing the Idea of "Solution"
How can we get students to COME TO UNDERSTAND without just telling them? In particular, some questions I ask myself include: How can we get kids to really internalize what “solve” means, what “solution” means? What can we ask them so they don’t lose site of that while they go about the business of “solving”?
Some thoughts I had in pursuit of an answer: Two classrooms from 2 different districts have inspired this post—the students in an integrated math 1 class at the high school level, and the students in an 8th grade class at the middle school level. It’s a challenge to get students to think about the larger picture sometimes. When we can get kids to see how their common sense has a place in math, they can see much more. They seem to think math isn't a place for common sense.
Both classes had been working with graphing equations like 2x - y = 3, and were launching into systems of equations. The teachers were focused on how their new texts were approaching three methods of solving systems (starting with the graphing method.) In one classroom, the teacher let me ask the class a couple of questions.
Equation on the board: 2 x - y = 3.
I wanted to find out: 1. Do they recognize the difference between this and solving an equation with one variable? 2. Do they realize that when they solve algebraically, they are finding out what values can be substituted to make the equation true—and that is called a solution, or that a graph is a picture showing all solutions?
First question: “How many people think this equation has a solution?” Many hands go up. Interesting, this will be fun. “How many people can think of an x-value and a y-value that you could plug-in to make this equation true?” They thought for a second or two, only a few hands were raised, and then I asked them to talk at tables. Of the 7 or 8 table groups in the classroom, only 2 of the table groups started coming up with more than one solution. The others paused after they had one. So I brought them out of talking with their groups, and got some kids to report out what they had. This moved the rest of the groups to consider there were more solutions. As they gave their x-y pairs, I listed in an x-y table on the board.
Second Task: I used a second equation from the system (which had not been shown on the board yet) and asked them if they thought this equation had a solution, an x and y value that you could plug-in to make it true. [Say the second equation was 2 x + y = 21. I don’t remember the exact equations now.] They thought it would, and I asked them to take a minute at their table groups to find a few. They were doing this by guessing and checking. No one had thought to graph to find all the solutions. Which was fine, I was trying to set them up to see that the graph could save them all of this trouble guessing and checking. I didn't let this go too long, just long enough to see if they made progress on the idea that there was more than one solution.
3rd Task: To the class, “Okay, I have a new question for you. Do you suppose there might be an x value and a y value that can make BOTH equations true, that is a solution to this equation AND to this second one?” I pause for a couple seconds of think time, (which should probably have been a longer pause because I could improve my wait time.) “How many people think there might be an x value and y value that works in this first equation and also in this second equation?” Some raised hands. “How many people think there might be more than one x value and y value to make them both true?” Some different raised hands. “How many people think there won’t be a solution to both?” Some different raised hands. “How many people aren’t sure?” Some raised hands. “How many people think we are going to find out?” Most hands were raised. “Okay, work with your group to figure it out.”
That set the hook—now they are searching for the answer to this puzzle, and they are clear they are searching for something to make both equations true, which is the definition of a “solution”, so we are grounded in the conceptual definition, and I can start working in the word “solution” as we go here.
They did find the solution for both while they were in groups. And then as the teacher and I were checking with groups, we asked groups if they thought there might be another solution to make both true (which of course there wasn’t, but we wanted them to figure that out.)
Question sequence to move from groups to whole class discussion: “How many of you found some solutions?” Most kids raised a hand. “Hmm, perhaps as a class, we found a bunch of solutions. How many of you think we might have different solutions for the x-y pair that makes both equations true?” Not many raised hands. “How many of you think we might have the same solution?” Most raised hands.
Please click on the link below the picture to listen to me describing and showing how to facilitate the students looking for a common solution, and how to set them up to see there can only be one solution.
Then asked, “How many people are pretty sure this is the only solution to both? How can we be sure there won’t be another solution that works in both way out in the table or something?” This was asked about the table with the intention of giving them a chance to own the idea about using the graph as a way to think about this, a chance for them to feel clever which will provide confidence and help them persevere later. A couple of kids noticed that values were getting farther apart in their tables which meant no other solution was possible even if you keep increasing or decreasing the x values (which was cool because after the next part, we can ask students to look for how they can see that in the graph.) But that wasn’t most kids, which is okay because they were starting to recognize they would have to check many pairs of x-y values; this would allow them to see the power of using a line to see all the solutions at once.
While the kids worked in groups during the last task, the teacher and I had a brief discussion, and had decided to ask them to graph the equations next. We told them we would take a break from this guessing and checking, and go back to something we had learned before. We asked them to graph a couple of equations not mentioning they were the same ones and to use the same axes (they had graph paper):
1. 2x - y = 3
2. 2x + y = 21
They noticed the equations were the same ones from before. Most of them changed to slope intercept form to graph, which is telling of them not stepping out to see a larger picture of what they had been doing, and realizing they had some “points” already. So we asked, “what do you suppose would happen if you graphed the x and y values from your table that you made before?” Some were starting to make the connection, and knew those points would be on each line, and other checked and saw that it did (which helped them come to understand that the graph is a picture of the solutions.) Then they noticed where the common solution to the 2 equations was, at the intersection of the two lines!
Potential Next Questions: Now we were off and running, and could ask, “What is another solution to the first equation that we don't have in the table? (Seeing if we can get them to find more solutions by looking at the points on the line.) “Do you think there could be another solution that works in both equations that we haven’t found yet?” and “Why or why not?” (Giving them a chance to notice that unless the lines will bend and intersect again, there won’t be another place where they have the same solution.”)
We didn’t get all of that discussion done on this day, but the teacher continued the next day.