**Situation: Quadratic
Equations**

**PRIME at UGa**

**May 2005: Erik Tillema**

**Revised November 2005**

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**Prompt**

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This
situation occurred in the classroom of a student teacher during his student
teaching. He worked very hard to create
meaningful lessons for his students and often asked his mentor teacher for
advice by asking questions similar to the oneีs found at the end of this
vignette.

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Mr. Sing presents equations of the following to his students.

He demonstrates to them that they need only take the square root of each side to get

x+1 = 3 or x +1 = -3.

Then we can solve for x = 2 or x = -4. He then turns his students loose to solve some equations like the ones he has presented and is surprised to find out that many of his students are multiplying the terms out to get

and then transforming the equation so that

and factoring this equation. Mr. Sing notes, however, that many students were making mistakes in carrying out his procedure.

He stops the class and reminds
the students that they need only take the square root of both sides to solve
these types of equations and then let's them continue working on the problems. A few days later, Mr. Sing grades the test
covering this material and finds that many of his students are still not doing
as he has suggested. At first he thinks that his students just didn't listen to
him but then he reminds himself that during the class period the students
seemed to be quite attentive.

What
hypotheses do you have for why his students are acting in this way? What
concepts might be necessary for students to understand the concept of solving a
quadratic equation? In what ways might Mr. Sing work with his students to
develop these concepts?

**Commentary**

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**Mathematical Foci**

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*Mathematical
Focus 1: *

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**Using
binomial reasoning as a basis for solving quadratic equations.**

In
many cultures (Chinese, Babylonian, Egyptian, and Greek) solving quadratic
equations arose out of solving problems dealing with lengths or areas. Perhaps the most prominent of these
types of problems were oneีs that involved rectilinear areas and
transformations performed on these areas.
To this end, it is possible to build up meaning for the equations that
Mr. Sing presented to his students through building up students binomial
reasoning in quantitative contexts.
See the attached **GSP document** to see a sequence of possible tasks that
a teacher might use to develop binomial reasoning in whole number
multiplication contexts and use this reasoning as a basis for binomial
reasoning in the context of solving quadratic equations.

**Mathematical
Focus 2:**

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**Two
methods that use a functional approach for solving a quadratic equation.**

These
types of equations often come up in the context of working on quadratic
functions. To see how to represent
(x+1)^{2} as an area of a square where the length *x* varies over the domain of all possible
lengths, see the **following GSP sketch**.
A quantitative situation appropriate for the solution of this particular
quadratic equation would be:
Suppose that you know the length of the side of a square is *x+1* where *x* represents a variable quantity, what
length would the side of the square have to be for the area of the square to be
9? In creating the dynamic
function relationship shown in GSP we have first used *x* as a variable quantity. However, when we are looking for a
specific area (the solution of a quadratic equation), we need to switch to
viewing *x* as an
unknown. In solving an equation
using functional methods, we usually switch between thinking of *x* first as the variable of a function and
then of *x* as an
unknown that represents a particular value of the function. Note that it is possible to represent *all* quadratics using areas of squares
including making a model for quadratics that have complex roots. In order to do
so involves introducing directed areas—area that has a positive or
negative orientation depending on the direction one travels around the area.

Another
possible functional method is to imagine each side of the original equation as
being a function. Then the intersection of any two functions can be thought of
as the solution of an equation. In
this case, we can think of *f(x) = (x+1) ^{2}* as representing an area function where
the graph of

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