### The Brick and a Half Problem

### by Brian Seitz and Inchul Jung

**Problem:**

**"If a brick weighs 3 pounds
plus half a brick, how much does a brick and a half weigh?"**
**How would students respond to
this question? **
**How would you respond? **
**How could technology be used
to visualize the problems and its solution? **

**Looking at this problem from a geometric perspective,
a student**

may construct a visual, technological representation of the problem as such.
A whole brick is represented below.
**The second visual representation is of a whole
brick, **

split into two equal parts/halves.
**The brick weighs 3 pounds plus half a brick**

Thus, one way to think of the problem is that,

1 brick = 3 + 1/2(brick)

Assume x=1 brick, thus x = 3 + x/2

So. the brick weighs 6 pounds total.

**That is one way to think of the half-brick problem.**
**There are many others. We will look at how students
could possibly **
**construct their understanding of this problem
and also, how**
**they may misinterpret the question asked.**

**Using Algebra, a student can allow x to be the
weight of the brick and a half. Therefore, one brick would weigh **
**Half of this brick would weigh**
**Now the equation for the entire brick and a
half would be as such**
**Using simplification rules**
**This logic accounts for a students representation
of a brick as an entire brick and a half. A student who is comfortable is
their use of algebraic problem solving may never have to visually construct
a representation of the **

**Another algebraic representation of the problem
is **

allowing one brick to be x.

Thus, if we allowed the one brick to be x, then the equation

would be as such
**x = 3 + x/2**
**thus, x/2 = 3**
**and x = 6.**
**This algebraic representaion is the equivalent
geometric **

relation of our first problem above.

Although, the geometric representaion gives a visual construction for students
to work with.

**The wording of the problem is what is at first intmidating.**

One may think that the problem is asking for one brick

as the answer, which is in fact only 6 pounds. Also, the statement

"3 pounds plus half a brick" is an ambiguous one.
**Students may have difficulty in this problem. **

I initially was confused with the question, but portraying it

visually (through technology) I was able to clearly see the solution.

Extensions of the problem can be found in many facets.

a) Altering the values of the half-a-brick from 3 to another value (2
, 8). Changing the fraction from 1/2 to another fraction (1/3 or 1/4).

For example:

**"If a whole brick weighs
2 pounds plus one-third a brick, how much does a brick weigh?"**
or

**"If a whole brick weighs
8 pounds plus one-fourth a brick, how much does a brick weigh?"**
As well, these extensions allow a student to see
the construction of a notion of one-variable algebraic equations.

**Brian's homepage**
**Inchul's homepage**