Geometry, Teacher\'s Edition

Let the Radius be One –The simplification of only considering polygons inscribed in a unit circle by letting the
radius be equal to one, may seem a bit odd to students. Let them know that this is being done in preparation for more
advanced trigonometry. In trigonometry ratios of side lengths of similar triangles are considered, and the size of the
triangles in not important. Letting them know that this simplification becomes useful in the future will reassure them
that the course of their mathematics education is well designed.

Additional Exercises:

1. The area of a regular hexagon is 24

√

3 cm^2. What is the length of each side?

Answer:

24

√

3 =

1

2

∗s∗

1

2

s

√

3 ∗ 6 Each side has a length of 4 cm.

s= 4

Geometric Probability

Count Carefully –Counting is the most challenging aspect of probability for students. It is easy to make an error
when thinking of all the possible outcomes and determining how many of them are favorable. The best way to guard
against errors is to make logical, orderly lists. The goal is for the students to see a pattern so that eventually they
will be able to get the count without listing all of the possibilities.

TABLE2.14:

Outcome Favorable? Outcome Favorable?

(N 1 ,N 2 ) No (D,N 1 ) No

(N 1 ,D) No (D,N 2 ) No

(N 1 ,Q) Yes (D,Q) Yes

(N 2 ,N 1 ) No (Q,N 1 ) Yes

(N 2 ,D) No (Q,N 2 ) Yes

(N 2 ,Q) Yes (Q,D) Yes

Does Order Matter? –One of the hardest decisions for a new probability student to make when analyzing a situation
is to determine if different permutations should be counted separately or not. Take Example Two from this section.
Is there a first coin and a second coin? Does it matter if Charmane sees the quarter of the dime first? In this case,
it does not because the two coins are taken at the same time. To compare have the students consider the situation
where first one coin is drawn and then a second. In this case there are more possible outcomes. The probability
remains the same, 50%, but now it is calculated as 126 instead of^36. It is not always the case that considering order
results in the same probability as when order is not considered. Note that this example can be reduced to the simpler
question of whether a quarter is one of the two coins drawn.

Additional Exercises:

1. A man throws a dart at a circular target with radius 6 inches. He is equally likely to hit anywhere in the target.
   What is the probability that he is within 2 inches of the center of the target?

Answer:

π 22

π 62

=

1

9

≈. 11 There is an approximately 11% chance that he will hit within 2 inches of the target.

2.10. Perimeter and Area