18Jun04
Perimeters and Areas and Islands and Such
1. What do we mean by area
of a figure?
2. On the grid at the right,
each square represents 1
mile by 1 mile (thus, 1
square mile). What is the
approximate area of the
island whose map is on this
grid? (Do you recognize the
island? Which island is it?
Hint: it is a Caribbean island
where distances are
measured in miles instead of
kilometers.)
3. What is the perimeter of this 5. Using squared paper, draw at least 3 different shaped
page? (You could use a
rectangles each with area 24 square units.
ruler.)
6. Find the perimeter of each rectangle you drew in #5.
4. How could you figure out
7. A question to think about: How many different rectangles
the perimeter of the island
could be drawn with an area of 24 square units?
on the grid at the right?
Rosalie A. Dance and James T. Sandefur, 2004
This material is based upon work supported by the National Science Foundation under Grant No. 0087068.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and
do not necessarily reflect the views of the National Science Foundation.
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Teaching Guide for
Perimeters and Areas and Islands and Such
Introduction: The geometric concepts of perimeter and area are critical concepts for
understanding our world. To help students clarify the concepts of area and perimeter before
beginning this small project, you might have the whole class find the area and the perimeter of
your classroom. This is easy to do if you are blessed with classroom floor tiling that is 1 square
foot per tile, but it is worthwhile in any case. Help students to take in the notion that “area” is the
amount of space inside a plane figure and so requires a “square unit” measure while perimeter
is the length of its boundary, and so is measured in linear units. If your room is not tiled
conveniently, your students will need to measure. Spend some time discussing the concept of 1
square unit (a square foot, a square yard, a square meter; whatever unit you choose to use).
Distinguish that idea from a unit of length.
When students have a basic understanding of the language (area, perimeter, and unit
measures), they are ready to begin the project. Note that the students themselves will need to
come up with language to explain area in question 1. This should not have been completely
done for them during the discussion of area of the room. Students should work in groups of 3 or
4. Encourage them to engage in discussion, staying focused on the tasks at hand. Respect for
one another’s thinking is key to a successful group.
Answers and teaching suggestions:
1. An appropriate answer might be, for example, “the amount of space inside a figure in 2dimensions.” We are not looking for rigorous definition here. Students need to understand
that area refers to the amount of space inside the boundaries of a plane figure. They should
get past the notion that area is “length x width”, useful though that may be in the right
circumstances.
2. Any answer from perhaps 22 to 32 square miles is acceptable. The island is St. Thomas; its
area is about 26 or 27 square miles.
3. Assuming this has been printed on 8.5 x 11 inch paper, the perimeter would be 39 inches. Or
99 cm.
4. One could surround the island with string, or thread, and then stretch the string out and
measure it. Or, one could estimate the distance by estimating the distance needed to
traverse each individual square. You might mention the fact that the actual perimeter of the
island is much larger than the estimates you would get using this map because there is much
detail along the boundary that cannot be shown on this small map. An island’s boundary is
fractal in nature. Furthermore, the actual perimeter of the island is modified daily by the
seas.
5. Students will probably draw rectangles of size 1 by 24, 2 by 12, 3 by 8, and 6 by 4. Of
course, there are many others.
6. The perimeters of the rectangles listed in answer 5 are, respectively, 50, 28, 22, and 20. One
point to make in this lesson is what the words perimeter and area mean. The St Thomas
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map problem is intended to broaden the student’s understanding beyond rectangles.
Problems 5 and 6, however, permit them to apply that understanding in the context of the
familiar rectangles.
7. Students might at first think that the 1 x 24, 2 x 12, 3 x 8, and 4 x 6 are the only four; others
might argue that there are eight, thinking that the 3 x 8 is different from an 8x3. If someone
thinks of using examples like ½ x 48 or 8/3 x 9 or 0.1 by 240, you might have an interesting
discussion about “many”, or even “infinitely many.” This discussion should only go as far as
your class seems to want to take it. It is very valuable to take this opportunity to help
students see how the use of fractions to get another 24 square unit rectangle works. For
example, if they want a rectangle with a side of ½ , they can divide 24 by ½. To see how to
do this, they can ask themselves how many halves are in 24. If the want a rectangle with a
side of ¼ , they can divide 24 by ¼, asking themselves how many fourths are in 24. Many
students will know the rule for dividing by fractions; few will have thought about why it works.
This discussion can help them to better understand fractions and division by a fraction. It is
not at all necessary that students understand the concept of infinity, but they will enjoy
mentioning it.
This lesson might alert you to some students’ lack of understanding of the meaning of fractions or of
operations on fractions. If so, assign those students to do Lab Activities 1 – 4 with a tutor.