**Modeling Algebraic
Expressions****
****
**

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**Math
Models Home Page**

**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

**1. What is the net value of
the 15th arrangement?**
**2. Which arrangement in the
sequence has a net value of 81? 225? 405?**
**3. What are some equivalent
expressions for the net value, v(n), of the ***nth
arrangement*?

**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:

**What are the net values of
the 20th, 40th, and 100th arrangements? (This is called
***evaluating the expression *for n = 20, 40, or 100*. *We
use function notation *v(n) *- read 'v of n' - to mean "the
net value of the nth arrangement". In this question we are looking
for v(20), v(40), and v(100).)
**Which ***n* gives v(n) =
90? (This is *solving the equation *for n when v(n) equals
90*.)*
**If ***v(n-1) - v(n) = 50,
*then* n* = ?

**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.**