LESSON PLAN 2
Algebra 1 – Infusing Discrete Math (Block 1-3), Grades 9-12
Problem Solving Strategies (Linear Equations, Quadratic Equations, Sets)
Applied Problems
(Boxes)
Instructor: Dr. Michael Aristidou
Example 1: We are cutting x by x squares out of the corners of a 22 by 22 unit square and
folding up the sides. What size squares should we cut in order to have a box with a bottom
area of 64 square units?
Solution 1: (practical) The trainer, could use Legos or paper (and scissors) and guide the
participants to achieve the following construction. We describe the steps below:
X
X
X
X
8
22
8
X
X
X
22
X
STEP 1: Use Legos to make a 22x22 square.
STEP 2: Use a different color Legos (blue) to indicate a 64 sq.units central square (i.e. 8x8,
hence its side is 8 units) in the center of the 22x22 larger square built in STEP1.
STEP 3: Now that the participants identified that the side of the blue square is 8 units, they can
remove the corner squares in order to build the box upwards, and at the same practically count
how much x was “cut off” (as the 22-8 = 14 must then be divided evenly. That x = 7, and it is the
answer to the problem.
Solution 2: (algebraic) The trainer could also employ an algebraic method for the problem, as
follows:
We use the figure as guide, and realize that the side of the blue square must be 22-2x. Therefore,
its area must then be (22  2x) 2  64 . The latter, after is foiled, stands for the following quadratic
equation:
x 2  22x 105  0
which by factoring (or formula) gives x = 7 or x = 15. The solution is x = 7, as x = 15 must be
rejected (since it causes the blue square to have negative sides).
[Note that the above equation could have been also solved by the root method, and hence avoid foiling.]
The instructor, at this point, could also use the opportunity to discuss the two
different solutions above, including the advantages and disadvantages of each
one.
Also, to strengthen his arguments in favor of the algebraic method, he could
also introduce the following variations of the above problem. These variations,
could make the above problem not so intuitive and also without integer
solutions.
Example 2: We are cutting x by x squares out of the corners of a 10 by 10 unit square and
folding up the sides. What size squares should we cut in order to have a box with a bottom
area of 100 square units? [These one has solutions x = 10 and x = 0]
Example 3: We are cutting x by x squares out of the corners of a 10 by 10 unit square and
folding up the sides. What size squares should we cut in order to have a box with a bottom
area of 144 square units? [These one has NO solutions]
Example 4: We are cutting x by x squares out of the corners of a 20 by 20 unit square and
folding up the sides. What size squares should we cut in order to have a box with a bottom
area of 200 square units? [These one is the problem in the video, and has solutions
x  10  5 2 and x  10  5 2 ]