At some point last semester, I remember feeling apprehensive about using balance scales as a model to introduce linear equations. I think I was worried about students’ ability to move beyond the model and consider things like negative and decimal coefficients. Would they be lost when they no longer had their crutch?
I should have trusted them.
Nat Banting has written about balance scales here. Although the metaphor of balance scales is referred to often in math classrooms – during the linear equations unit, students must hear the mantra “whatever you do to one side, you do to the other” at least 1000 times a class – the model really becomes powerful when students actually work with balance scale problems, and that they do so right from the start. Nat writes:
My Grade 9 students don’t see an equation for the first two weeks of their unit of solving linear equations. That is because I think students get all bogged down in the notation, and lose their problem solving intuition.
I decided to follow Nat’s advice. For a few days, my students worked through sequences of problems such as these, increasing in difficulty, where the goal is to find the “weight” of the square:
I grouped students randomly and set them to work on whiteboards. I didn’t mention variables or equations. Beyond a few clarifications here and there, all students were quickly able to get to work. If they wanted to use guess-and-check, great – once the problems got harder, they would move on to more efficient strategies. After a few of days, all students eventually came to use strategies that looked very much like solving equations, and soon moved fluidly to using algebra to solving the problems. When they explained their work to the class, they would say things that sounded very much like the teacher mantra, “whatever you do to one side, you do to the other.” It’s not that the metaphor changed – it’s that it came from the students, and was grounded in their own reasoning and understanding of equality.
As for my worries about negative and non-integer values? For a problem like the second one above, students noticed that (given that circles were worth 1) the squares must be “lifting” the scale up, and came to realize that they could assign them negative weights. Once we moved to solving equations without the balance scale context, students were able to solve problems such as -3x – 5 = 6, because although they had now moved beyond the model, their understanding of equality was firm enough for them to adapt. Eventually, we moved on to solving equations with variables on both sides, which warranted a brief return to the balance scale context. “Wait, can you also take away variables from both sides?” “Why not?” They got it.
It’s amazing what a good concrete or visual model does to support student thinking and to smooth the transition to more abstract problems. But when I discuss this with other teachers, almost invariably I hear: “Yes, but I know that my kids need to see examples of how to solve it first, then try it themselves.” I can’t speak for all students’ needs, but I wish that as mathematics educators, we had more faith in their intelligence and ability to reason in sophisticated ways, if they are provided the opportunity to do so. As Nat suggests,
Give them to students. Ask them to explain their thinking. All involved may be surprised by dormant algebraic thinking that just needs an intuitive trigger.
Speaking of not trusting your students: I was really apprehensive of giving balance scale problems to my Grade 10 students today, who are nearing the end of a unit on systems of equations. I wanted them to give solving systems by elimination a try, but I wasn’t sure they were ready – most have stuck faithfully to one particular algebraic strategy and have not shown interest in moving to more efficient methods. Also, I knew that they hadn’t worked extensively with balance scales the year before. I should have trusted them.
I presented the class with the following problem:
Happily, many students recognized, and shared during the whole-group discussion, that setting up a system is possible, but not necessary: if you recognize that removing 2 squares is equivalent to subtracting 6, you can easily deduce that a square is worth 3 and a circle is (consequently) worth 10. First, this signaled to me that students were developing flexibility with linear systems, recognizing both what a solution is and that there are multiple ways of finding it. In other words, they weren’t simply following an algebraic procedure blindly. Second, it made the following suggestion natural, because the students could already imagine manipulating the weights on the scales: “What if, instead, we imagine removing 2 squares and 6 circles from the first equation? How much would we have to remove from the right side?” They got it. We translated this to equations.
I set students to work on the following sequence of problems, suggesting they try a similar strategy for each. I didn’t tell them what to do when the equations didn’t have a variable with a common coefficient. Instead, I circulated, asking questions where necessary to trigger thinking.
I found the following questions helpful in getting students to generalize the elimination method to systems where there aren’t similar terms with the same coefficient (e.g., 4x + 6y = 72 and 2x + 6y = 66, as in problem A):
- “What made this strategy work for the other problems?”
- “If this same strategy were to work, what would you want to have on this scale?”
- “Am I allowed to add weight to one side? How can I make sure the balance is maintained?”
For some groups, all it took was the first question for a student to say “OOOOOOOHHHHHHH! You can just multiply this one by 3,” and they were on their way. For others, it took a bit more questioning, but they got there. (When students asked “Is this right?” – they should know better – I suggested they check to see if their answer worked for both scales. My students are still working on developing strategies to analyze the reasonableness of results.) The visual model helped immensely, and when we moved to algebraic representations, students would make references to adding or subtracting weight. Several students commented on the efficiency of this strategy in comparison to others; I’m curious to see if elimination will become their new method of choice.
If you’re looking for more balance problems, Nat has a variety in his balance scales post. They are also very easy to make on your own – I whip them up in a matter of minutes using Paintbrush (the Paint equivalent for Mac), making liberal use of cut and paste. You may wish to grab the blank balance scale image below. The problems themselves are typically pulled from the textbook.
Most importantly, trust your students. If they depend, and maybe even insist, on being shown procedures before applying them to problems, maybe it’s because they’ve had too few opportunities to try problem solving on their own, and to develop confidence in their reasoning skills. All involved may be surprised with the thinking that emerges, as I was happy to find today.