This
lesson begins with a personalized version of the somewhat familiar
"Surfer and Spotter" problem (Bennett, 2002, p. 82) and extends to
regular polygons with an even number of sides.
Group students in pairs. Display the Triangular Island overhead on the projector.
Then, tell the following story:
Suppose that you and a friend have become stranded on a desert island,
which is in the shape of a regular triangle. Gilligan is a lazy,
incessant bore, so you’d like to build a hut far away from him. In
addition, you need to place your hut on the island so that the sum of
the distances to each of the three shores will be as small as possible.
Students may wonder why each person would want the sum of the
distances to be as small as possible. You can explain that minimizing
the sum of the distances makes getting to each side of the island as
easy as possible. For realism, state that the island is covered with
lots of vegetation; if the sum of the distances is minimized, you will
have to clear less vegetation.
In addition, you might also want to tell students that
Gilligan is very selfish. Without discussion, he plans to place his hut
at the exact center of the island, which means that students will have
to place their huts elsewhere. (This is not completely necessary if
students work in pairs, as suggested. But if you would prefer that
students work individually, this will ensure that not all students
place their huts at the center. Later in the lesson, students can
compare their locations to Gilligan's location at the center to find
out that the sum of the distances is the same.)
Divide the class into pairs, and give each pair a copy of the Triangular Island
activity sheet. Tell students they each are to decide where to locate a
hut and find the distances to each side. (Eventually, you will want to
give each student his or her own copy of the activity sheet. For the
moment, however, require students to work together on the same sheet.
This will ensure that they choose different locations for their huts,
which will make it more likely that they notice that the location does
not affect the sum of the three distances.) Note that students may need
guidance about how to measure distance to each side. That is, inform
them that they should measure the perpendicular distance.
Select several student papers that show different locations for
the two huts, and have these students share their ideas with the entire
class. Ask, "Which locations seem to be best for the huts? Why?"
[Students should notice that the sum of the distances for every
location will be the same.]
Follow up by asking, "Is this sum related to any particular
characteristic of the triangle?" [This question is difficult and
students may just randomly suggest various characteristics—side length,
perimeter, area, or height.] Record the possibilities for future
reference. This part of the lesson is to initiate thought about the
problem and possible characteristics of the triangle. This is not a
time to press for exact answers; that will come later. For now, have
students put aside their paper-and-pencil work, but they might return
to it when they reconsider the list of possibilities.
Hand out the Beyond Triangular Island activity sheet. With the Triangle Island applet, students can explore the distance to the sides for equilateral triangles and squares. With the Hexagon Island
applet, students can explore hexagons. To explore octagons, students
will have to draw the figures themselves and find the measurements
using a ruler or other measuring device; or, geometry software such as
Geometer's SketchPad^{®} can be used.
You may need to provide directions to students about how to
locate the applets and how to use them. Project the URL of the applets
on the overhead projector, write them on the chalkboard, or tell
students to do the following:
- Go to the home page of the Illuminations web site (http://illuminations.nctm.org).
- Go to the Activities section.
- Click on "View All Activities," or press the Search button.
- From the list of activities, scroll to the link for Triangle Island, Hexagon Island, or Octagon Island.
Once the applets have been found, students can click on the plus
sign in front of Instructions to learn how to use the applet. The
applets may be used in a whole-group setting or by pairs of small
groups of students in a lab setting, depending on the availability of
technology.
As students gather the first set of measures for a triangle,
verify that they are able to correctly manipulate the applet. The
applet allows students to explore triangles of different sizes,
something that would be difficult if students continued to use only
paper triangles. (Note that students were never told how big the island
was, so exploring multiple sizes is important.) To complete Question 1
on the activity sheet, allow students to gather data for triangles of
other sizes.
In whole-group discussion, students should formulate
conjectures based on the measures they accumulated. Ask, "What
conjecture can you make about the sum of the distances to each side of
the triangle for any location of the hut?" [For different locations of
the interior point, the sum of the distances to each side of the
triangle is constant.] Then ask, "Is this sum related to any particular
characteristic of the triangle? Explain your reasoning." [The sum
appears to be equal to the height of the triangle. Students may reason
through this observation by considering what happens when the hut is at
a vertex of the triangle. Dragging the interior point to each of the
vertices of the triangle, students may note that the distances to two
of the sides become zero while the distance to the third side becomes
the height of the triangle.]
Return to the paper triangles. Ask, "How do your results from
the applet match your earlier results with the paper triangles?"
[Students may note that the two sets of results are similar, so long as
their measurements for the paper triangle were accurate.]
Point out to students that the island could have other shapes.
It could be square, or it could have more that four sides. Ask students
to consider a few different cases. In order to have a class set of data
from which students can generalize a relationship for all regular
polygons with an even number of sides, assign one of the remaining
three polygons (square, hexagon, octagon) to each pair of students.
Each pair starts with its assigned polygon and then moves through the
other two polygons, time permitting. For each polygon, the pair must
complete the table and respond to the questions on the activity sheet.
[For the square, the sum of the distances appears to equal twice the
height of the square; for the hexagon, the sum appears to equal
three times the height of the hexagon; and for the octagon, the sum
appears to equal four times the height of the octagon.]
After students have had an opportunity to consider at least one
of the square, hexagon, or octagon, and depending on time, the students
return to a whole-group setting. A student from a pair originally
assigned to each polygon shares the pair’s results. Other students are
responsible for questioning the pair about their reasoning.
When the results for all of the polygons are public, pose the
following question: "How might you generalize your results to islands
of other shapes?" If this question is too general, students’ thinking
may be supported by asking them to think about a concrete case
involving a regular polygon with a large even number of sides. For
example, what would be the relationship between the sum of the
distances to the sides of a 50‑gon and the height of the 50‑gon? [The
sum will be equal to 25 times the height of the 50‑gon.] In general,
how can this observation be stated, that is, in terms of an n‑gon? [The sum will be equal to n/2 times the height of the n‑gon. Students should see that this generalization applies only to n‑gons with an even number of sides. Different relationships exist for polygons with an odd number of sides.]