Inventing Equation Solving Methods
patterns and grouping, patterns and predicting, predicting and generalizing, grouping for counting, writing formulas, grouping for writing equations, grouping for generalizing, describing patterns with formulas and models , integer lessons: counting piece model, sum and difference models, dimensions and net values, area models
In this lesson, students use algebra piece models to represent algebraic expressions, and to solve equations in which two algebraic expressions are set equal to each other. We start with these two pattern sequences:
What observations about the patterns can you make? How would you describe them with an algebraic expression (formula)?
Suppose that we see Sequence A as having three columns of the arrangement number plus four 3(n + 4) or three columns of the arrangement number plus twelve (3n + 12), and the B sequence as having five columns of the arrangement number (5n). We could create algebra piece models like these:
We can use the algebra pieces to determine' for which 'n' the two patterns have the same net value for the same arrangement number. Symbolically,
For which 'n' is v(A) = v(B)What would you try to do with the pieces?
Here's a strategy that students have come up with...
Let's look at another example. Suppose we're given the equation n2 - 2n = 120. We can model each side of the equation as an algebra piece collection...
If n - 1 equals 11, then n must equal 12. In both of these examples, we reason from the model to find out more information. As in the last lesson, students begin rcording their 'moves' with some kind of symbolic notation as they become ready, developing the symbolic equation solving skills necessary for algebraic fluency. We explore lots of patterns, each with its own twist or new idea.