The Brick and a Half Problem
by Brian Seitz and Inchul Jung
"If a brick weighs 3 pounds
plus half a brick, how much does a brick and a half weigh?"
How would students respond to
How would you respond?
How could technology be used
to visualize the problems and its solution?
Looking at this problem from a geometric perspective,
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
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
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
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
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).
"If a whole brick weighs
2 pounds plus one-third a brick, how much does a brick weigh?"
"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.