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:
Here is another example:
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.
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 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.