Download GEOM_U8_BLM_Final

Unit 8, Activity 1, Vocabulary Self-Awareness
Word/Phrase
+

–
Definition/Formula
Example
circle
center of a circle
radius
circumference
chord
area of a circle
central angle
arc
arc measure
arc length
major arc
minor arc
semicircle
distance around a circular
arc
sector
area of a sector
Blackline Masters, Geometry
Page 8-1
Unit 8, Activity 1, Vocabulary Self-Awareness
tangent
secant
sphere
surface area of a sphere
volume of a sphere
Procedure:
1. Examine the list of words/phrases in the first column.
2. Put a + next to each word/phrase you know well and for which you can write an accurate
example and definition. Your definition and example must relate to this unit of study.
3. Place a  next to any words/phrases for which you can write either a definition or an
example, but not both.
4. Put a – next to words/phrases that are new to you.
This chart will be used throughout the unit. As your understanding of the concepts listed
changes, you will revise the chart. By the end of the unit, you should have all plus signs. Because
you will be revising this chart, write in pencil.
Blackline Masters, Geometry
Page 8-2
Unit 8, Activity 1, Vocabulary Self-Awareness with Answers
Word/Phrase
+
circle
center of a circle
radius

–
Definition/Formula
Example
The set of all points in a
plane equidistant from a
given fixed point called the
center.
The given point from which
all points on the circle are
the same distance.
a segment with one
endpoint at the center of the
circle and the other
endpoint on the circle; onehalf the diameter
circumference
the distance around the
circle
chord
a segment whose endpoints
lie on the circle
area of a circle
A   r2
central angle
an angle formed at the
center of a circle by two
radii
arc
a segment of a circle
equal to the degree measure
of the central angle;
arc measure
arc measure
arc length

360
circumference
the distance along the
curved line making up the
arc;
arc length
major arc
minor arc
semicircle
Blackline Masters, Geometry
arc measure
arc length

360
circumference
also known as the distance
around a circular arc.
the longest arc connecting
two points on a circle; an
arc having a measure
greater than 180 degrees
the shortest arc connecting
two points on a circle; an
arc having a measure less
than 180 degrees
an arc having a measure of
180 degrees and a length of
one-half of the
Page 8-3
Unit 8, Activity 1, Vocabulary Self-Awareness with Answers
distance around a circular
arc
sector
circumference; the diameter
of a circle creates two
semicircles
also known as the arc
length; see the definition of
arc length.
a plane figure bounded by
two radii and the included
arc of the circle
A
area of a sector
tangent
secant
sphere
N
 r 2 where N is
360
the measure of the central
angle
a line or segment which
intersects the circle at
exactly one point
a line or segment which
intersects the circle at
exactly two points
the locus of all points, in
space, that are a given
distance from a given point
called the center
surface area of a sphere
SA  4 r 2
volume of a sphere
4
V   r3
3
Procedure:
1. Examine the list of words/phrases in the first column.
2. Put a + next to each word/phrase you know well and for which you can write an accurate
example and definition. Your definition and example must relate to this unit of study.
3. Place a  next to any words/phrases for which you can write either a definition or an
example, but not both.
4. Put a – next to words/phrases that are new to you.
Blackline Masters, Geometry
Page 8-4
Unit 8, Activity 1, Vocabulary Self-Awareness with Answers
This chart will be used throughout the unit. As your understanding of the concepts listed
changes, you will revise the chart. By the end of the unit, you should have all plus signs. Because
you will be revising this chart, write in pencil.
Blackline Masters, Geometry
Page 8-5
Unit 8, Activity 4, Sample Split-Page Notes
Date:
Period:
Topic: Circles
Parts of a circle:
radius
--one-half the diameter
--one endpoint is the center of the circle, the other is on the circle
--used when finding the area of a circle
--a segment whose endpoints are on the circle
chord
--a chord which passes through the center of the circle
diameter
Formulas:
-- A   r
--r is the measure of the radius of the circle
2
area of a circle
circumference
Blackline Masters, Geometry
-- C  2 r or C   d
--r is the measure of the radius and d is the measure of the diameter
--these formulas are the same because d  2r .
Page 8-6
Unit 8, Activity 4, Split-Page Notes Model
Date:
Period:
Topic: Central Angles and Arcs
central angle
--an angle whose vertex is the center of the circle and sides are two radii
--the sum of all central angles in a circle is 360°
arc
--a segment of a circle
--created by a central angle or an inscribed angle
--has a degree measure (called arc measure)
--has a linear measure (called arc length)
--an arc whose measure is less than 180 degrees
minor arc
--an arc whose measure is greater than 180 degrees
major arc
semicircle
Blackline Masters, Geometry
--an arc whose measure is exactly 180 degrees
--created by the diameter of the circle
--the arc length is one-half of the circumference of the circle
Page 8-7
Unit 8, Activity 5, Circular Flower Bed
Consider the diagram of the flower bed below:
What is the total area this flower bed would cover in the owner’s yard?
If the walking paths around the inner circle and the crescent shaped flower beds are to be
covered in straw, pebbles, or some other medium, how much material would be needed to cover
that area?
The owner wishes to put edging around each section of the flower bed. How much edging will be
needed?
Picture source: http://blog.oregonlive.com/homesandgardens/2007/06/plant_a_circular_vegetable_gar.html
Blackline Masters, Geometry
Page 8-8
Unit 8, Activity 5, Circular Flower Bed with Answers
Consider the diagram of the flower bed below:
What is the total area this flower bed would cover in the owner’s yard? Approx. 490.87 sq ft.
If the walking paths around the inner circle and the crescent shaped flower beds are to be
covered in straw, pebbles, or some other medium, how much material would be needed to cover
that area? Answers provided for this question and the next are samples as students will need to
make some assumptions in order to complete calculations (for example, they might approximate
the area between crescent shaped flower beds as a rectangle of dimensions 1.5 by 3.25). The
intention here is to have students explain their reasoning and persevere in solving the problem.
Teacher facilitation to assist students in solving the problem will be necessary. Sample answer:
Approx. 154.45 sq ft. However, this type of material is typically sold in cubic yards, so assuming
1 in depth (1/36th of a yard), approximately 0.48 cubic yards would be needed.
The owner wishes to put edging around each section of the flower bed. How much edging will be
needed? Sample Answer: Approximately 208.47 feet of edging would be needed.
Picture source: http://blog.oregonlive.com/homesandgardens/2007/06/plant_a_circular_vegetable_gar.html
Blackline Masters, Geometry
Page 8-9
Unit 8, Activity 5, Arc Length and Sector Area Part I
Group Members: ______________________________________________________________
1. Which can did your group receive? ______________________
2. What is the circumference of your can in centimeters (round to the nearest millimeter)?
_______________________
3. Determine the length of the diameter and the radius of the can (do not forget units).
Describe your method for determining these measures below.
Diameter __________________
Radius __________________
4. In the space provided, use the compass to draw a circle with the radius and diameter you
found in question 3. Then divide the circle into four equal parts. You may use a different
sheet of paper if necessary to draw the circle.
Blackline Masters, Geometry
Page 8-10
Unit 8, Activity 5, Arc Length and Sector Area Part I
5. What is the measure of each central angle in the circle constructed in question 4?
_________
6. Write the ratio of one central angle measure (from question 5) to the total number of
degrees at the center of the circle. _________
Simplify this fraction and write the decimal equivalent (round to the nearest hundredth).
_________
7. What is the length of the arc formed by one of the central angles mentioned in question 5
(remember to use the correct units)? ____________
Describe how you found this arc length.
8. Write a ratio that compares the arc length in question 7 to the total circumference of the
circle. _________ Simplify this fraction and write the decimal equivalent (round to the
nearest hundredth). _________
9. What is the area of the circle you drew in question 4 (remember the units)? ___________
10. What is the area of one of the four sectors formed in question 4 (remember the units)?
____________ Describe how you found this area measure.
11. Write a ratio that compares the area of one sector in question 7 to the total area of the
circle. _________ Simplify this fraction and write the decimal equivalent (round to the
nearest hundredth). _________
12. What pattern do you see in questions 6, 8, and 11? Why does this pattern occur?
Blackline Masters, Geometry
Page 8-11
Unit 8, Activity 5, Arc Length and Sector Area Part I with Answers
Group Members: ______________________________________________________________
1. Which can did your group receive? Answers will vary
2. What is the circumference of your can in centimeters (round to the nearest millimeter)?
Answers will vary
3. Determine the length of the diameter and the radius of the can (do not forget units).
Describe your method for determining these measures below.
Diameter Answers will vary
Radius Answers will vary
4. In the space provided, use the compass to draw a circle with the radius and diameter you
found in question 3. Then divide the circle into four equal parts. You may use a different
sheet of paper if necessary to draw the circle.
Blackline Masters, Geometry
Page 8-12
Unit 8, Activity 5, Arc Length and Sector Area Part I with Answers
5. What is the measure of each central angle in the circle constructed in question 4?
90 degrees
6. Write the ratio of one central angle measure (from question 5) to the total number of
90
360
Simplify this fraction and write the decimal equivalent (round to the nearest hundredth).
1
= 0.25
4
degrees at the center of the circle.
7. What is the length of the arc formed by one of the central angles mentioned in question 5
(remember to use the correct units)? Answers will vary
Describe how you found this arc length.
Answers will vary. Some possible methods may include dividing the circumference by
four or using a piece of string to measure the length then measuring the length of the
string.
8. Write a ratio that compares the arc length in question 7 to the total circumference of the
circle. Answers will vary Simplify this fraction and write the decimal equivalent (round to
the nearest hundredth). Answers will vary however the decimal approximation should be
0.25.
9. What is the area of the circle you drew in question 4 (remember the units)? Answers will
vary
10. What is the area of one of the four sectors formed in question 4 (remember the units)?
Answers will vary Describe how you found this area measure. Answers will vary. One
method will probably be to divide the total area by 4.
11. Write a ratio that compares the area of one sector in question 7 to the total area of the
circle. Answers will vary Simplify this fraction and write the decimal equivalent (round to
the nearest hundredth). Answers will vary however the decimal approximation should be
0.25.
12. What pattern do you see in questions 6, 8, and 11? Why does this pattern occur?
The pattern should be that the ratios (specifically using the decimal approximations)
should be equal to ¼ or 0.25. This happens because the total number of degrees (360)
has been divided into four equal parts. Therefore, the sector area and arc length are each
¼ of the total area and circumference.
Blackline Masters, Geometry
Page 8-13
Unit 8, Activity 5, Arc Length and Sector Area Part II
1. In the space provided, use a compass to draw a circle with a radius of 3.5 centimeters. Divide
the circle into 6 equal parts. Shade one of the six parts. The questions below with be about
the shaded region.
2. State the circumference and area of the circle. Remember to use the correct units. Round
your answers to the nearest hundredth.
3. Using the shaded sector of the circle, find the measure of the central angle, the area of the
sector, and the arc length of the sector. Justify your answers with explanations or work.
Remember to use the correct units. Round your answers to the nearest hundredth.
4. Describe a formula that might be used to find arc length. Use the appropriate vocabulary
(circumference, central angle, etc.) to explain what variables are used in the calculations.
5. Describe a formula that might be used to find the area of a sector. Again, use appropriate
terminology for the variables to be used in the calculations.
Blackline Masters, Geometry
Page 8-14
Unit 8, Activity 5, Arc Length and Sector Area Part II with Answers
1. In the space provided, use a compass to draw a circle with a radius of 3.5 centimeters. Divide
the circle into 6 equal parts. Shade one of the six parts. The questions below with be about
the shaded region.
2. State the circumference and area of the circle. Remember to use the correct units. Round
your answers to the nearest hundredth.
Circumference = 21.99 cm
Area = 38.48 cm2
3. Using the shaded sector of the circle, find the measure of the central angle, the area of the
sector, and the arc length of the sector. Justify your answers with explanations or work.
Remember to use the correct units. Round your answers to the nearest hundredth.
Central Angle = 60 degrees
Area of the sector = 6.41 cm2
Arc length = 3.67 cm
4. Describe a formula that might be used to find arc length. Use the appropriate vocabulary
(circumference, central angle, etc.) to explain what variables are used in the calculations.
N
Arc Length 
2 r
360
N = the measure of the central angle; r = radius of the circle
Students may not give this exact formula but should have some representation of the
circumference of the circle and the ratio of the measure of the central angle to 360.
5. Describe a formula that might be used to find the area of a sector. Again, use appropriate
terminology for the variables to be used in the calculations.
N
Area of a sector 
 r2
360
N = the measure of the central angle; r = radius of the circle
Students may not give this exact formula but should have some representation of the total
area of the circle and the ratio of the measure of the central angle to 360.
Blackline Masters, Geometry
Page 8-15
Unit 8, Activity 6, Concentric Circles
Blackline Masters, Geometry
Page 8-16
Unit 8, Activity 8, Anticipation Guide
Name _____________________
Date _____________________
Directions: Read each of the statements below. Circle “Agree” or “Disagree” under the
appropriate column heading (Before Lesson or After Lesson). Be prepared to explain your
reasoning for your choice.
Before Learning
Statements
After Learning
Agree
Disagree
1. Categorical data are values which can be sorted
by names or labels rather than numbers.
Agree
Disagree
Agree
Disagree
2. Marginal frequencies and joint frequencies are
terms that have the same definition.
Agree
Disagree
Agree
Disagree
3. Relative frequencies are often stated as
percentages.
Agree
Disagree
Agree
Disagree
4. Two-way tables allow us to compare two or
more sets of categorical data.
Agree
Disagree
Agree
Disagree
5. Relative frequencies can be found for the
whole table, just the rows, or just the columns.
Agree
Disagree
Disagree
6. In order to find the conditional probability of
event B given event A has already occurred,
you must know the probability of event B and
the probability of event A.
Agree
Disagree
Disagree
7. The probability of B given A has occurred,
represented by P(B|A), is the same as the
probability of A given B has occurred, or
P(A|B).
Agree
Disagree
Agree
Agree
Blackline Masters, Geometry
Page 8-17
Unit 8, Activity 8, Anticipation Guide with Answers
Name _____________________
Date _____________________
Directions: Read each of the statements below. Circle “Agree” or “Disagree” under the
appropriate column heading (Before Lesson or After Lesson). Be prepared to explain your
reasoning for your choice. “Correct” answers have been italicized. Be sure to have students
justify their reasoning. It may be possible that students have a valid reason for selecting an
opposite response “After Learning” based on a different interpretation of the statement(s).
Before Learning
Statements
After Learning
Agree
Disagree
1. Categorical data are values which can be sorted
by names or labels rather than numbers.
Agree
Disagree
Agree
Disagree
2. Marginal frequencies and joint frequencies are
terms that have the same definition.
Agree
Disagree
Agree
Disagree
3. Relative frequencies are often stated as
percentages.
Agree
Disagree
Agree
Disagree
4. Two-way tables allow us to compare two or
more sets of categorical data.
Agree
Disagree
Agree
Disagree
5. Relative frequencies can be found for the
whole table, just the rows, or just the columns.
Agree
Disagree
Disagree
6. In order to find the conditional probability of
event B given event A has already occurred,
you must know the probability of event B and
the probability of event A.
Agree
Disagree
Disagree
7. The probability of B given A has occurred,
represented by P(B|A), is the same as the
probability of A given B has occurred, or
P(A|B).
Agree
Disagree
Agree
Agree
Statement 4: Two-way tables compare only two categorical sets of data at a time.
Statement 6: For a conditional probability, the probability of B AND A, or P(B and A), must be
known, not just the probability of B.
Statement 7: This may be true for rare cases; it is not the norm.
Blackline Masters, Geometry
Page 8-18
Unit 8, Activity 8, Relative Frequency and Probability
Two-way Frequency Tables
Below is a two-way frequency table (Table 1) with hypothetical data from 200 randomly selected
students in a school.
Eye Color
Table 1: Hair Color versus Eye Color
Brown
Blue
Hazel
Green
Total
Black
24
6
6
2
38
Brown
40
28
18
10
96
Hair Color
Red
6
6
4
4
20
Blond
4
32
4
6
46
Total
74
72
32
22
200
The data displayed in the table is called categorical data because the values in the survey are
names or labels. The color of someone’s hair (e.g., black, brown, red, blond) or the color of their
eyes (e.g., brown, blue, hazel, green) are examples of categorical variables.
A two-way table is a useful tool for looking at relationships between categorical variables. A
two-way table compares data from two categorical variables. In the example above the variables
are Hair Color and Eye Color. The entries in the cells in the tables above are frequency counts,
the measure of the number of times an event occurs. The Total column and row are called
marginal frequencies while the entries in the body of the table are called joint frequencies.
1. Look at the marginal frequencies for Eye Color (Total column). Which color has the
strongest representation?
2. Look at the marginal frequencies for Hair Color (Total row). Which color has the
strongest representation?
3. Now, compare the joint frequencies. Which color combination (Hair and Eye Color) has
the largest frequency?
4. What other observations can you make about the data in the table?
Blackline Masters, Geometry
Page 8-19
Unit 8, Activity 8, Relative Frequency and Probability
We can also display the data as relative frequencies in a two-way table. Relative frequencies are
ratios of the frequency counts to the total counts. For example, the relative frequency of students
4
1
with blond hair and blue eyes is
or
. Often, relative frequencies are stated as values
200
50
between 0 and 1 or as percentages (for the example above, the relative frequencies could also be
stated as 0.02 or 2%). Two-way tables can show relative frequencies for the whole table, for
rows or for columns. The tables below show the different types of relative frequency tables.
Table 2 shows the relative frequencies of the whole table, Table 3 shows the relative frequencies
of the rows, and Table 4 shows the relative frequencies of the columns.
Eye Color
Table 2: Relative Frequencies for the Whole Table
Brown
Blue
Hazel
Green
Total
Black
.120
.030
.030
.010
.190
Brown
.200
.140
.090
.050
.480
Hair Color
Red
.300
.030
.020
.020
.100
Blond
.020
.160
.020
.030
.230
Total
.370
.360
.160
.110
1.000
Hair Color
Red
.081
.083
.125
.182
.100
Blond
.054
.444
.125
.273
.230
Total
1.000
1.000
1.000
1.000
1.000
Hair Color
Red
.300
.300
.200
.200
1.000
Blond
.087
.696
.087
.130
1.000
Total
.370
.360
.160
.110
1.000
Values may not total 1.00 due to rounding.
Eye Color
Table 3: Relative Frequencies for Rows
Brown
Blue
Hazel
Green
Total
Black
.324
.083
.188
.091
.190
Brown
.541
.389
.563
.455
.480
Values may not total 1.00 due to rounding.
Eye Color
Table 4: Relative Frequencies for Columns
Brown
Blue
Hazel
Green
Total
Black
.632
.158
.158
.053
1.000
Brown
.417
.292
.188
.104
1.000
Values may not total 1.00 due to rounding.
Each table above can give different information to help understand the relationship between hair
color and eye color. In the Relative Frequencies for Rows table (Table 3) we notice most people
with blue eyes have either brown or blond hair, with 38.9% and 44.4% representing those
respective categories. However, if you look at the Relative Frequencies for Columns table,
Blackline Masters, Geometry
Page 8-20
Unit 8, Activity 8, Relative Frequency and Probability
41.7% of the people with brown hair have brown eyes and 69.6% of the people with blond hair
have blue eyes.
5. What other observations can you make about the data?
Probability and Relative Frequency
What is probability? Remember from earlier mathematical studies that probability is the ratio of
favorable outcomes to the total possible outcomes in a given sample space. In terms of the
categorical data above, let us determine the probability of some events. Refer to Table 1 to
answer the following.
6. If we were to select one of the 200 students at random, what is the probability that the
student would have brown hair? Justify your answer.
7. If we were to select one of the 200 students at random, what is the probability that the
student would have blue eyes? Justify your answer.
8. If we were to select one of the 200 students at random, what is the probability that the
student would have red hair AND hazel eyes? Justify your answer.
9. Look at the values you just calculated and compare them to the values in the relative
frequency tables. What do you notice about each value?
10. Find the probability of the following using Table 1. Does your statement in question 9
still stand true? Explain.
a. P(black hair and green eyes) = ____________
b. P(blond hair and blue eyes) = ____________
c. P(green eyes) = ____________
d. P(red hair) = ____________
11. If the 200 people in this study represented a sample of the total school population, what is
the expected probability that a person randomly selected in the school would have brown
hair and hazel eyes? Explain your reasoning.
12. Are the events described here independent or dependent? Explain.
Blackline Masters, Geometry
Page 8-21
Unit 8, Activity 8, Relative Frequency and Probability
The joint and marginal frequencies listed in the table can be used to determine conditional
probabilities. The conditional probability of an event B in relationship to an event A is the
probability that event B occurs given that event A has already occurred. The notation for
conditional probability is P(B|A), read as the probability of B given A. For example, what is the
probability that one of students selected from those with hazel colored eyes has blond hair? This
is considered a conditional probability because we are using the given group of only those
students with hazel colored eyes as the sample space instead of the entire group of 200. This
would be written as P(blond hair|hazel eyes) read probability of blond hair given hazel eyes.
To determine the conditional probability of B given that A has occurred, we can use the
P  A and B 
following formula: P  B | A  
. In terms of our example,
P  A
P  blond hair|hazel eyes  
P  hazel eyes AND blond hair 
.
P  hazel eyes 
13. Where can we find P(hazel eyes AND blond hair)? What is P(hazel eyes AND blond
hair)?
14. What is P(hazel eyes)?
15. Calculate P(blond hair|hazel eyes).
16. Describe a different method of calculating/determining the conditional probability
P(blond hair|hazel eyes).
Find the following conditional probabilities. Be sure to justify your answers.
17. P(black hair|blue eyes)
18. P(blue eyes|black hair)
19. What is your interpretation of the probabilities you found above?
20. Approximately what percent of students with red hair have green eyes?
Based on the work you have completed here, how are two-way frequency tables helpful?
Blackline Masters, Geometry
Page 8-22
Unit 8, Activity 8, Relative Frequency and Probability with Answers
Two-way Frequency Tables
Below is a two-way frequency table (Table 1) with hypothetical data from 200 randomly
selected students in a school.
Eye Color
Table 1: Hair Color versus Eye Color
Brown
Blue
Hazel
Green
Total
Black
24
6
6
2
38
Brown
40
28
18
10
96
Hair Color
Red
6
6
4
4
20
Blond
4
32
4
6
46
Total
74
72
32
22
200
The data displayed in the table is called categorical data because the values in the survey are
names or labels. The color of someone’s hair (e.g., black, brown, red, blond) or the color of their
eyes (e.g., brown, blue, hazel, green) are examples of categorical variables.
A two-way table is a useful tool for looking at relationships between categorical variables. A
two-way table compares data from two categorical variables. In our example above the variables
are Hair Color and Eye Color. The entries in the cells in the tables above are frequency counts,
the measure of the number of times an event occurs. The Total column and row are called
marginal frequencies while the entries in the body of the table are called joint frequencies.
1. Look at the marginal frequencies for Eye Color (Total column). Which color has the
strongest representation?
The category with the highest frequency is brown eyes.
2. Look at the marginal frequencies for Hair Color (Total row). Which color has the
strongest representation?
The category with the highest frequency is brown hair.
3. Now, compare the joint frequencies. Which color combination (Hair and Eye Color) has
the largest frequency?
The color combination with the largest frequency is brown hair and brown eyes
4. What other observations can you make about the data in the table?
Answers will vary. Listen to students answers and be sure to ask for justifications for
their reasoning/thinking.
Blackline Masters, Geometry
Page 8-23
Unit 8, Activity 8, Relative Frequency and Probability with Answers
We can also display the data as relative frequencies in a two-way table. Relative frequencies are
ratios of the frequency counts to the total counts. For example, the relative frequency of students
4
1
with blond hair and blue eyes is
or
. Often, relative frequencies are stated as values
200
50
between 0 and 1 or as percentages (for the example above, the relative frequencies could also be
stated as 0.02 or 2%). Two-way tables can show relative frequencies for the whole table, for
rows or for columns. The tables below show the different types of relative frequency tables.
Table 2 shows the relative frequencies of the whole table, Table 3 shows the relative frequencies
of the rows, and Table 4 shows the relative frequencies of the columns.
Eye Color
Table 2: Relative Frequencies for the Whole Table
Brown
Blue
Hazel
Green
Total
Black
.12
.03
.03
.01
.19
Brown
.20
.14
.09
.05
.48
Hair Color
Red
.30
.03
.02
.02
.10
Blond
.02
.16
.02
.03
.23
Total
.37
.36
.16
.11
1.00
Hair Color
Red
.08
.08
.12
.18
.10
Blond
.05
.44
.12
.27
.23
Total
1.00
1.00
1.00
1.00
1.00
Hair Color
Red
.30
.30
.20
.20
1.00
Blond
.09
.70
.09
.13
1.00
Total
.37
.36
.16
.11
1.00
Values may not total 1.00 due to rounding.
Eye Color
Table 3: Relative Frequencies for Rows
Brown
Blue
Hazel
Green
Total
Black
.32
.08
.19
.09
.19
Brown
.54
.39
.56
.45
.48
Values may not total 1.00 due to rounding.
Eye Color
Table 4: Relative Frequencies for Columns
Brown
Blue
Hazel
Green
Total
Black
.63
.16
.16
.05
1.00
Brown
.42
.29
.19
.10
1.00
Values may not total 1.00 due to rounding.
Each table above can give us different information to help understand the relationship between
hair color and eye color. In the Relative Frequencies for Rows table (Table 3) we notice most
people with blue eyes have either brown or blond hair, with 38.9% and 44.4% representing those
respective categories. However, if you look at the Relative Frequencies for Columns table,
Blackline Masters, Geometry
Page 8-24
Unit 8, Activity 8, Relative Frequency and Probability with Answers
41.7% of the people with brown hair have brown eyes and 69.6% of the people with blond hair
have blue eyes.
5. What other observations can you make about the data?
Answers will vary. Listen to students answers and be sure to ask for justifications for
their reasoning/thinking.
Probability and Relative Frequency
What is probability? Remember from earlier mathematical studies that probability is the ratio of
favorable outcomes to the total possible outcomes in a given sample space. In terms of the
categorical data above, let us determine the probability of some events. Refer to Table 1 to
answer the following.
6. If we were to select one of the 200 students at random, what is the probability that the
student would have brown hair? Justify your answer.
96
P (brown hair) 
 0.48
200
7. If we were to select one of the 200 students at random, what is the probability that the
student would have blue eyes? Justify your answer.
72
P(blue eyes) 
 0.36
200
8. If we were to select one of the 200 students at random, what is the probability that the
student would have red hair AND hazel eyes? Justify your answer.
4
P(red hair AND hazel eyes) 
 0.02
200
9. Look at the values you just calculated and compare them to the values in the relative
frequency tables. What do you notice about each value?
Students should notice that the values are the same as those in the relative frequency
table for the whole table (Table 2).
10. Find the probability of the following using Table 1. Does your statement in question 9
still stand true? Explain.
e. P(black hair and green eyes) = 0.01
f. P(blond hair and blue eyes) = 0.16
g. P(green eyes) = 0.11
h. P(red hair) = 0.10
Yes, each value listed is in Table 2.
11. If the 200 people in this study represented a sample of the total school population, what is
the expected probability that a person randomly selected in the school would have brown
hair and hazel eyes? Explain your reasoning.
The expected probability that a person would have brown hair and hazel eyes is 0.09.
The cell containing the relative frequency for brown hair and hazel eyes is 0.09.
12. Are the events described here independent or dependent? Explain.
These are independent events because the color of hair or eyes does not affect the color
of the other.
Blackline Masters, Geometry
Page 8-25
Unit 8, Activity 8, Relative Frequency and Probability with Answers
The joint and marginal frequencies listed in the table can be used to determine conditional
probabilities. The conditional probability of an event B in relationship to an event A is the
probability that event B occurs given that event A has already occurred. The notation for
conditional probability is P(B|A), read as the probability of B given A. For example, what is the
probability that one of students selected from those with hazel colored eyes has blond hair? This
is considered a conditional probability because we are using the given group of only those
students with hazel colored eyes as the sample space instead of the entire group of 200. This
would be written as P(blond hair|hazel eyes) read probability of blond hair given hazel eyes.
To determine the conditional probability of B given that A has occurred, we can use the
P  A and B 
following formula: P  B | A  
. In terms of our example,
P  A
P  blond hair|hazel eyes  
P  hazel eyes AND blond hair 
.
P  hazel eyes 
12. Where can we find P(hazel eyes AND blond hair)? What is P(hazel eyes AND blond
hair)?
P(hazel eyes AND blond hair) can be found in Table 2.
P(hazel eyes AND blond hair) = 0.020.
13. What is P(hazel eyes)?
P (hazel eyes) = 0.160
14. Calculate P(blond hair|hazel eyes).
P(blond hair|hazel eyes) = 0.125
15. Describe a different method of calculating/determining the conditional probability
P(blond hair|hazel eyes).
One method is to find the frequency of students with hazel eyes and blond hair from Table
1 and divide it by the total number of students with hazel eyes. A second method is to use
the relative frequencies for rows in Table 3.
Find the following conditional probabilities. Be sure to justify your answers.
16. P(black hair|blue eyes)
P(black hair|blue eyes) = 0.083
17. P(blue eyes|black hair)
P(blue eyes|black hair)=0.158
18. What is your interpretation of the probabilities you found above?
8.3% of the students who have blue eyes have black hair while 15.8% of students with
black hair have blue eyes.
19. Approximately what percent of students with red hair have green eyes?
20% of students with red hair have green eyes.
Based on the work you have completed here, how are two-way frequency tables helpful?
Two-way frequency tables help organize data in a way that allows us to easily identify the
relative frequencies and probabilities of different events.
Blackline Masters, Geometry
Page 8-26
Unit 8, Activity 9, Conditional Geometric Probability
4
1
2
3
Jim and Susan are playing a game
using the two spinners at the right.
Points are awarded for each round by
adding the value of the two slices after
both spinners have been spun. The
highest score a player can earn is a 9.
Jim spins the first spinner and it lands
on 3. What is the probability that
when he spins the second spinner he
will earn a score of 8 this round?
3
5
2
1
Blackline Masters, Geometry
4
Page 8-27
Unit 8, Activity 9, Conditional Geometric Probability with Answers
4
1
2
3
The probability that Jim will earn
3
a score of 8 is 0.3 or
.
10
3
5
Jim and Susan are playing a game
using the two spinners at the right.
Points are awarded for each round by
adding the value of the two slices after
both spinners have been spun. The
highest score a player can earn is a 9.
Jim spins the first spinner and it lands
on 3. What is the probability that
when he spins the second spinner he
will earn a score of 8 this round?
2
1
Blackline Masters, Geometry
4
Page 8-28
Unit 8, Activity 10, Diameters and Chords
Date______________
Team Members___________________
Use the following guide to investigate the relationships that occur between the diameter
and chords of circles.
Investigation 1
1.
Using a compass, draw a circle on a piece of patty paper. Fold the circle in half twice to
locate the center of the circle. Label the center C.
2.
Pick any two points on the circle (do NOT use the endpoints of the same diameter). Label
the points G and H. Using a straightedge, draw the segment connecting G and H. What is
GH ? ________________________
3.
Find the perpendicular bisector of GH by folding the paper so that G lies on top of H.
Unfold the paper and label the endpoints of the diameter just created as J and K.
4.
Draw CG and CH . Find the measure of GH . ________________________
5.
GH should have been divided into two smaller arcs—either GK and HK or
GJ and HJ . Find the measure of these two smaller arcs created by JK .
________________________________________________
6.
What is true about the two arcs measured in number five?
________________________________________________________________________
7.
Using a ruler, measure the radii CG and CH . What is the arc length of GH ?
________________________
What are the arc lengths of the two arcs measured in number five?
_______________________________________________________________________
What is true about the lengths of the two smaller arcs compared to the larger arc?
____________________________________________________________________
8.
Using a ruler, measure GH and the two smaller segments created by the intersection of
the diameter and the chord. ______________________________________________
9.
What conjecture can be made if the diameter of a circle is perpendicular to a chord?
________________________________________________________________________
Does this conjecture apply to the radii of a circle? Explain.
________________________________________________________________________
________________________________________________________________________
Blackline Masters, Geometry
Page 8-29
Unit 8, Activity 10, Diameters and Chords
Investigation 2
Follow the steps below in order to answer the questions that follow.
Step 1.
Use a compass to draw a large circle on patty paper. Cut out the circle.
Step 2.
Fold the circle in half.
Step 3.
Without opening the circle, fold the edge of the circle so it does not intersect the
first fold.
Step 4.
Unfold the circle and label the circle. Find the center by locating the point where
the compass was placed and label the center M. Darken the diameter which should
pass through the center. Locate the two other folds and darken the chords created
by these folds. Label one chord as GE and the other chord as TR .
Step 5.
Fold the circle, laying point G onto E to bisect the chord. Open the circle and fold
again to bisect TR (lay point T onto R). Two diameters should have been formed.
Label the intersection point on GE as O and the intersection point on TR as Y.
Answer the following about Investigation 2.
1.
What is the relationship between MO and GE ? What is the relationship between
MY and TR ? (Hint: it may be necessary to use a protractor and ruler to help answer this).
2.
Use a centimeter ruler to measure GE , TR, MO, and MY . What observation can be made?
3.
Make a conjecture about the distance that two chords are from the center when the chords
are congruent.
Blackline Masters, Geometry
Page 8-30
Unit 8, Activity 10 , Diameter and Chords with Answers
Date______________
Team Members___________________
Use the following guide to investigate the relationships that occur between the diameter
and chords of circles.
Investigation 1
1.
Using a compass, draw a circle on a piece of patty paper. Fold the circle in half twice to
locate the center of the circle. Label the center C.
2.
Pick any two points on the circle (do NOT use the endpoints of the diameters). Label the
points G and H. Using a straightedge, draw the segment connecting G and H. What is
GH ? A chord.
3.
Find the perpendicular bisector of GH by folding the paper so that G lies on top of H.
Unfold the paper and label the endpoints of the diameter just created as J and K.
4.
Draw CG and CH . Find the measure of GH . Answers will vary.
5.
GH should have been divided into two smaller arcs—either GK and HK or
GJ and HJ . Find the measure of these two smaller arcs created by JK . Answers will
vary.
6.
What is true about the two arcs measured in number five? They have the same measure,
which means they are congruent.
7.
Using a ruler, measure the radii CG and CH . What is the arc length of GH ? Answers
will vary.
What are the arc lengths of the two arcs measured in number five? Answers will vary.
What is true about the lengths of the two smaller arcs compared to the larger arc? They
have the same measure, which means they are congruent.
8.
Using a ruler, measure GH and the two smaller segments created by the intersection of
the diameter and the chord. Answers will vary.
9.
What conjecture can be made if the diameter of a circle is perpendicular to a chord?
If the diameter of a circle is perpendicular to a chord, the diameter bisects the chord and
the arc.
Does this conjecture apply to the radii of a circle? Explain.
Yes, this conjecture also applies to the radii of a circle. A radius is a part of the diameter;
therefore, these properties are true for the radii.
Blackline Masters, Geometry
Page 8-31
Unit 8, Activity 10 , Diameter and Chords with Answers
Investigation 2
Follow the steps below in order to answer the questions that follow.
Step 1.
Use a compass to draw a large circle on patty paper. Cut out the circle.
Step 2.
Fold the circle in half.
Step 3.
Without opening the circle, fold the edge of the circle so it does not intersect the
first fold.
Step 4.
Unfold the circle and label the circle. Find the center by locating the point where
the compass was placed and label the center M. Darken the diameter which should
pass through the center. Locate the two other folds and darken the chords created
by these folds. Label one chord as GE and the other chord as TR .
Step 5.
Fold the circle, laying point G onto E to bisect the chord. Open the circle and fold
again to bisect TR (lay point T onto R). Two diameters should have been formed.
Label the intersection point on GE as O and the intersection point on TR as Y.
Answer the following about Investigation 2.
1.
What is the relationship between MO and GE ? What is the relationship between
MY and TR ? (Hint: it may be necessary to use a protractor and ruler to help answer this).
MO and MY are perpendicular bisectors of GE and TR , respectively.
2.
Use a centimeter ruler to measure GE , TR, MO, and MY . What observation can be made?
GE  TR and MO  MY
3.
Make a conjecture about the distance that two chords are from the center when the chords
are congruent.
When two chords are congruent, they are equidistant from the center of the circle.
Blackline Masters, Geometry
Page 8-32
Unit 8, Activity 12, Tangents and Secants
Blackline Masters, Geometry
Page 8-33
Unit 8, Activity 12, Tangents and Secants
Blackline Masters, Geometry
Page 8-34
Unit 8, Activity 12, Tangents and Secants with Answers
1
mDEB
2
1
mCDB  mDB
2
mADB 






1
m AD  mCB
2
1
mAEC  m AC  mDB
2
mAED 
mE 
Blackline Masters, Geometry
1
mBC  m AD
2
Page 8-35
Unit 8, Activity 12, Tangents and Secants with Answers
Blackline Masters, Geometry
mE 
1
(mDB  m AD )
2
mA 
1
mBDC  mBC
2


Page 8-36
Unit 8, Activity 14, Surface Area of a Sphere
Blackline Masters, Geometry
Page 8-37