Investigation: Perimeter and Area
Pose the following problem to the students:
Tanya Teen started her own summer business - putting on birthday parties for
small children. Her neighbors agreed to loan her square card tables to seat the
children for refreshments. However, when some of the neighbors were away on
vacation, Tanya couldn't use their tables, and she really hated hauling the
tables back and forth. Therefore, using as few tables as possible was important
to her. Because all the children wanted to sit together, she had to place the
card tables together into rectangles. Only one child could sit on each side of a
card table. Her first party had eighteen children. How many tables did Tanya
need to borrow?
This problem is also found on the Tables at a Birthday Party activity sheet.
Students may use tiles to display a 1 × 2 banquet table on the
overhead projector.
Ask, How many people can be seated? [6]
Alternatively, the problem can be posed as, What is the area of a
rectangle with a perimeter of 18 units?
Allow small groups of students to explore the problem with
the square tiles.
Ask, Which rectangles used the fewest square tiles?
[1 × 9 or 9 × 1]
The most square tiles? [4 × 5 or 5 × 4]
How many tables did Tanya need?
The language of tables and people sitting at tables is an appropriate story
to link the world of students to the world of mathematics. In this example,
tables represent area, and people sitting at tables represent perimeter.
Translate the language of unit tables and people into area and perimeter. In
describing the dimensions of a rectangle, use bottom and side edge in place of
length and width. This allows students to include rectangles 4 × 5 and 5 × 4 as
different rectangles. Later the terms width and length can be
introduced. Arrange all the rectangles with a perimeter of 18 in a table, such as the one shown below.
| Length (units) |
Width (units) |
Perimeter (units) |
Area (square units) |
| 1 | 8 | 18 | 8 |
| 2 | 7 | 18 | 14 |
| 3 | 6 | 18 | 18 |
| 4 | 5 | 18 | 20 |
| 5 | 4 | 18 | 20 |
| 6 | 3 | 18 | 18 |
| 7 | 2 | 18 | 14 |
| 8 | 1 | 18 | 8 |
A blank table is found on the Tables at a Birthday Party activity sheet.
Ask the students to observe patterns in the chart. As the length
increases, the width decreases, and as the length increases, the area increases
to a certain point and then starts to decrease. The later pattern is a verbal
description of the graph called a parabola, as shown below. This question builds an
understanding of the effect of changing one variable, say length, on another
variable, such as width or area.
If we allow lengths to be real numbers, is there a rectangle with a
perimeter of 18 that has a larger area? Students will usually try a
rectangle with a side of length 4.5 units. The area is 20.25 square units.
Trying sides with lengths larger and smaller than 4.5, say 4.4 or 4.6 or 4.55,
will yield areas that are smaller than 20.25. The shape of the rectangle with
largest area for a perimeter of 18 is the square rectangle that is closest in
shape to a square, 4 × 5 (if we restrict our lengths to whole numbers).
Several graphs can be drawn to represent the data in the table above.
The graph of the area versus the length (or width) is a
parabola, which is the graph of a quadratic function.
Area vs. Length
The graph of the length versus the width is a linear graph.
Length vs. Width
Using the graph of the parabola, ask students, What happens between the points (4, 5) and
(5, 4)?
This is where the maximum point will occur if we extend
our dimensions to include real numbers.
Ask, What are the dimensions of the rectangle whose area is 16 square
units? [6.5 × 2.5]
What is the area of the rectangle whose length is 5.5 units?
[19.25]
The graph visually describes the effects of increasing the
length on the area.
Can a length be 9? [No, because the width would then be 0.]
Ask, What happens if the graph is allowed to cross the horizontal
axis?
Investigation: Predicting the Maximum Area
Have students predict which rectangle has the maximum area for a fixed perimeter.
Discussion. Ask the class to predict when the area seems to "turn around." It reaches its maximum area when the shape of the rectangle is as close
to a square as possible. If we have real numbers, then the rectangle with
largest (maximum) area is a square. Try out this conjecture with perimeters of
20 units and 24 units. Have students guess when the area turns around and then
find all the rectangles with integer lengths that have the given perimeter.
For a perimeter of 20 units, students may need to list all pairs of whole numbers
whose sum is 20/2, or 10, and note that the pair (5,5) is the turn around. Thus
the square with sides of length 5 and area of 25 is the rectangle with the
largest area for a fixed perimeter of 20 units.
For a perimeter of 24 units, the rectangle with the largest area is also a square, with sides of length 6 units and area of 36 square units.
For a perimeter of 34, which rectangle has the
maximum area? If the dimensions are whole numbers, then the rectangle with
dimensions 8 × 9 has the maximum area. If the dimensions are real numbers, then
the square whose sides have length 8.5 has the maximum area.
Investigation: Generalizing the Process for Finding the Maximum Area
Have students generalize the pattern for finding the area of a rectangle given a fixed
perimeter.
Discussion. Ask the students to describe how they found all the
rectangles. Allow the students time to talk about the process and then to
translate the process into symbols. Find all pairs of whole numbers whose sum is
half the perimeter. Use these pairs of numbers as the lengths and widths of the
rectangles and then compute the areas. Or find the pair of numbers that are
almost equal; this rectangle has the maximum area. For real numbers, since we
know that the rectangle with the maximum area for a fixed perimeter is a square,
use the number that is one-fourth the perimeter for the length of the side of
the square. The
following sets of equations could also be used:
Basic Equations
2(length + width) = Perimeter, or
length + width = Perimeter ÷ 2
length × width = area
Advanced Equations
Looking toward Algebra: The equation that describes the area of a
rectangle with fixed perimeter is a quadratic equation, which is studied in more
detail in algebra. Quadratic equations always have the shape of a parabola (as shown previously), except some open down and have a maximum point and
some open up and have a minimum point. The maximum or minimum pint can also be
found algebraically.
Have students observe the symmetry of the parabola and how
this relates to the pairs of rectangles; for example, the rectangles 4 × 5 and 5 × 4 have the same area. They lie on the same horizontal line. The perpendicular
line drawn from the maximum point to the horizontal axis is the line of
symmetry. This problem provides intuition and understanding for some very
important concepts of algebra and calculus. Maximum and minimum points of more
general functions are studied in calculus. Geometrically, the maximum or minimum
points occur when the tangent line to the graph is a horizontal line or has
slope zero. As the students generalize the patterns they have observed and
discussed, they are "sneaking" up on the notion of variable in a very natural
way.
Investigation: Factor Pairs
Have students find all the factor pairs for 36.
Discussion. Ask students to describe all the rectangles that have an
area of 36. Let them use square tiles or grid paper. Some students will
recognize that this task is equivalent to finding all the factor pairs of 36. To
find the rectangle with the least perimeter, students need to find the rectangle
with a shape closest to a square or the factor pair where the factor pairs begin
to repeat. For 36, these are the factor pairs:
1 × 36
2 × 18
3 × 12
4 × 9
6 × 6
9 × 4
12 × 3
18 × 2
36 × 1
The factor pairs repeat after the pair of factors (6, 6). The factors in this
pair are equal, thus 36 is a square.
Let students explore an area of 12, 28, or 64.
Ask them to
conjecture about when the factor pairs will begin to repeat.
Ask, If
you were going to find all the factor pairs of a number, how many numbers would
you have to check? [1 through the number that when squared is closest to the
number; for 12 it is 3; for 28 it is 4; for 64 it is 8.]
Ask students
if they can predict which numbers are perfect squares (square of a whole
number). The number is a square number if it has a factor pair with both factors
equal or with an odd number of factors.
