Michael Pershan has been writing a thought-provoking series on teaching students about equations this summer. I find bar models to be a very powerful way to nurture students' understanding of algebraic symbols and the process of solving equations.
How I Use Bar Models
Visual representations make it easier for the students to reason for themselves what steps to take and the order of the steps when solving for unknown quantities. Algebraic symbols obscure the important relationships for the unitiated, and before I used bar models, I struggled to make those relationships tangible for the students. And it seemed I was always out in front, coaxing them on. I couldn't find a way to have them be able to take the lead and really discover things for themselves. But if I put a problem like this up on the board:
How I Use Bar Models
Visual representations make it easier for the students to reason for themselves what steps to take and the order of the steps when solving for unknown quantities. Algebraic symbols obscure the important relationships for the unitiated, and before I used bar models, I struggled to make those relationships tangible for the students. And it seemed I was always out in front, coaxing them on. I couldn't find a way to have them be able to take the lead and really discover things for themselves. But if I put a problem like this up on the board:
and ask the students what the value the x's represent, almost immediately a very high percentage of the class can jump in and figure it out on their own. They can tell me why we subtract 47 from 175, and why we have to do it before we divide by 4... and why those are the operations we do. Once the students get the hang of it, they can solve some pretty complex word problems, including problems that could be represented as simultaneous equations, with insight and creativity. Some students prefer bar models to algebra as we start to make the transition, because this makes so much more sense, and contains more information.
Here is another example:
Here is another example:
Representing subtraction can be done in different ways. This is the way I do it. The students pick up on that pretty quickly. Question 2 is part of the way I build bridges between algebraic notation and bar models. And notice that there are a number of problems that this diagram facilitates. Suppose I had told you that Andrea had 51 more mangoes than Bill. Do you see where that would go in the diagram, and how you would find the number of mangoes that Andrea has? Or the number of mangoes Andrea has, or the number Bill has. Any of those could be the starting point. And notice that we aren't actually solving for x, we are asking a slightly more involved question, that might take a lot of groundwork for the students to untangle using algebra, but is almost trivial in the visual context. I could instead ask how many Bill has, or how many more Andrea has, or what the total is, etc. depending on the information we start with. This is a pretty deep understanding of how all of these quantities are related to each other, and how to get one from another.
How We Get There
We begin by translating words to bar models, learning to represent "5 times as many" or "6 more" or "shared among 6 vases" and their inverses. We also represent them algebraically, so immediately the students see a connection. I also give them algebraic expressions and talk about what bar models represent those relationships accurately. Then when we are solving equations later, I often reference a quick sketch when a student is stuck. It seldom has to be a full model of the problem, just an illustration of the part they are hung up on.
I don't use algebra tiles, so that the students don't need manipulatives when there may be none available. They can just make their own drawing (or I often encourage them to start imagining without drawing as we get further into the year). I also don't get bogged down in rigorously modeling negative quantities or general approaches to fractions (there are some situations, like 2/3 x = 28, which work very well with bar models... 3x = 5/7 not so much). For instance, I would handle something like Michael was discussing, a problem of the form b - ax = c, this way:
First, we would treat problems like -7x = 56 as a simple extension of 7x = 56. If I have 7 x's, and I want to know how much 1x is, so I divide by 7. So I ask the students what about -7x = 56? Someone suggests dividing by -7, and we check the answer, and it works. We don't worry about how to represent negatives with bar models.
How We Get There
We begin by translating words to bar models, learning to represent "5 times as many" or "6 more" or "shared among 6 vases" and their inverses. We also represent them algebraically, so immediately the students see a connection. I also give them algebraic expressions and talk about what bar models represent those relationships accurately. Then when we are solving equations later, I often reference a quick sketch when a student is stuck. It seldom has to be a full model of the problem, just an illustration of the part they are hung up on.
I don't use algebra tiles, so that the students don't need manipulatives when there may be none available. They can just make their own drawing (or I often encourage them to start imagining without drawing as we get further into the year). I also don't get bogged down in rigorously modeling negative quantities or general approaches to fractions (there are some situations, like 2/3 x = 28, which work very well with bar models... 3x = 5/7 not so much). For instance, I would handle something like Michael was discussing, a problem of the form b - ax = c, this way:
First, we would treat problems like -7x = 56 as a simple extension of 7x = 56. If I have 7 x's, and I want to know how much 1x is, so I divide by 7. So I ask the students what about -7x = 56? Someone suggests dividing by -7, and we check the answer, and it works. We don't worry about how to represent negatives with bar models.
We talk a lot about the equivalence of 4 - 6 and 4 + -6... that the two terms are being combined, and that can in practice involve either addition or subtraction. So the above situation is 4 combined with -6x to get 7. So the reasoning is the same as in 4 + 6x = 7. Subtract 4 to be able to see the value for -6x by itself. And if -6x = 3, then we have to divide by -6 to find out what 1x is.
Likewise with a problem like 3x = 5/7, the bar model would simply be:
and would simply illustrate that we need to divide 5/7 by 3.
I use the bar models when they make it easier, and work by analogy at other times.
The Benefits
The students' work with bar models gives meaning to all of those old maxims:
One extra benefit is that they see the problems more broadly and flexibly, as in the mango problem above. We could change the given values or ask for something besides just x. The relationships in the expression are the essential piece. They could use the information to write their own problems.
What Comes Next?
Algebra should serve exactly the same role - be a compact picture of a set of relationships that can be manipulated to make new relationships - so I build connections between the two representations by going back and forth. I ask the students to draw bar models from equations, and to write equations from bar models, and write both from word problems. I draw parallels between the way I want the students to show their algebraic work and the work we do with bar models. And as we do more and more work with algebraic notation, I will go back to a bar model whenever a student seems to be fishing for what to do next without really understanding why. My hope is that by the end of the year the students understand that algebraic equations don't just materialize in the homework pages of a math textbook. They represent numerical relationships that the students understand and can manipulate knowingly and effectively.
Likewise with a problem like 3x = 5/7, the bar model would simply be:
and would simply illustrate that we need to divide 5/7 by 3.
I use the bar models when they make it easier, and work by analogy at other times.
The Benefits
The students' work with bar models gives meaning to all of those old maxims:
- Do the opposite operation
- What you do to one side, you do to the other
- Simplify the equation
- Get x by itself
One extra benefit is that they see the problems more broadly and flexibly, as in the mango problem above. We could change the given values or ask for something besides just x. The relationships in the expression are the essential piece. They could use the information to write their own problems.
What Comes Next?
Algebra should serve exactly the same role - be a compact picture of a set of relationships that can be manipulated to make new relationships - so I build connections between the two representations by going back and forth. I ask the students to draw bar models from equations, and to write equations from bar models, and write both from word problems. I draw parallels between the way I want the students to show their algebraic work and the work we do with bar models. And as we do more and more work with algebraic notation, I will go back to a bar model whenever a student seems to be fishing for what to do next without really understanding why. My hope is that by the end of the year the students understand that algebraic equations don't just materialize in the homework pages of a math textbook. They represent numerical relationships that the students understand and can manipulate knowingly and effectively.
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