Describing Patterns With Formulas and Models
patterns and grouping, patterns and predicting, predicting and generalizing, grouping for counting, writing formulas, grouping for writing equations, grouping for generalizing, integer lessons: counting piece model, sum and difference models, dimensions and net values, area models
Students can begin to formalize their algebra skills by building on the patterning and grouping skills and concepts developed in earlier lessons (see red topic links above). We can use Algebra Pieces to create models for formulas and describe the patterns they describe. Here is a set of Algebra Pieces:
Students use their knowledge of dimension and area models, grouping strategies and formula generating when working with Algebra Pieces (see red topic links above).
Here's an example of a tile pattern; What algebraic expression describes the pattern?
The pattern represented by the expression 4n + 1 can be generalized as a linear or area model using the Algebra Pieces.
We can reason from the model to answer questions like
This is a simple example of how the algebra pieces can be used to make algebraic expressions and formulas meaningful for students.
Here's the first four figures in another pattern; what algebraic expression could you use to describe the net value, v(n), of the nth arrangement?
Based on their observations, students often build this model for the nth arrangement:
Try to use the model to help you answer these questions:
As students work through problems and patterns such as these, they are building the foundation for formal symbolic algebra, and we discuss standard form and notation alongside the exploration and discussion. In the next lesson, we'll look at solving equations in more detail.