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.

 


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Inchul's homepage