**Situation:
Quadratic Equations**

**PRIME at UGa**

**May 2005: Eric Tillema**

**October 2005 Revision**

**PROMPT**

** **

Mr. Sing presents problems like 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 = -3.

Then we can solve for x = 2 or x =
-4. He then turns his students loose to solve some
problems 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. He notes, however, that many students still
were not able to factor correctly.

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
are present in the material he is trying to communicate to his students? In
what ways might Mr. Sing work with his students to develop the concepts he is
trying to communicate? What knowledge might Mr. Sing need in order to develop
these concepts with his students?

**Commentary**

** **

**Mathematical Foci**

*Mathematical Focus 1: *

Introduce a quantitative situation like the following: I
have a square whose side is length *x + 1*.
Can you make a representation of this square and figure out the length of *x
*when the area of the square is 9 m^{2}?
In using this approach, a dynamic representation is helpful because it allows
students to move the values of *x *(as
seen in the GSP attachment). It leads to a number
of possible levels of solution to the problem. At a basic level, it allows
students to move the quantity (*x+1*)^{2}
in GSP to find out when the area will be 9 m^{2}. In constructing the
representation, the student may also be able to coordinate some of the
factoring issues that they seem to be experiencing in the vignette. For
instance in creating the square whose sides are *x +1*, the student can name the square algebraically
either as (*x +1)*^{2} or
through the sum of its parts. This may allow the student to have a quantitative
experience of why *(x +1) ^{2} = x^{2} + 2x + 1* and further to coordinate what the area of each
region of the function is when the total area is 9 m

*Mathematical Focus Number 2:*** **

This problem helps to demonstrate a common theme in
mathematics—finding special cases whose solution strategy is an
abbreviated form of a more general algorithm. Some of the mathematical content
involved for teachers is deciding whether to develop the general solution
strategy first and then look at special cases or to look at special cases that
might build to the general solution strategy. Given the way I treated the problem in foci 1, it makes more
sense to move from special cases to the more general solution strategy although
this technique is usually reversed in the textbooks I have read on this topic.
That is usually students learn to factor quadratic equations to find their
zeros and then are presented with special cases like how to solve the equation *(x
+ 1)*^{2} = 9. In this case, I would
probably start with specific cases and build to a generalized solution of
quadratics, using the notion of the difference of two squares and completing
the square to build up to the quadratic formula.

*Mathematical Focus Number 3:*

* *

There is an
issue in this problem also as to what *x*
represents mathematically. In the statement of the problem *x* is an unknown. However, there may be significant
confusion for what *x* stands for
if a function approach is used to solve this problem. That is in a functional
context *x *is a variable where *x
*is any but no particular value whereas when
we are solving for a particular area of square *x *becomes an unknown because we are looking for one
particular value of *x*.
Algebraically, however, there is no differentiation between the notation of
these two ideas rather it is context dependent. Therefore, it may be important
for teachers to be aware of the possible differences between thinking of *x* as a variable and *x *as an unknown and how and when they are switching
between these two conceptions in given problem situations.

*Mathematical Focus Number 4***:**

Important knowledge for addressing the issue in the vignette may be to have strategies for helping students see that general use of a formula may not always lead to the most insightful solution of a problem. Such knowledge might be in part at the heart of helping students to build a habit of mind in which they search for efficient problem solutions. Note foci 2 and 4 are mostly related to classroom culture or teacher pedagogy.