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|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 | | | | | |
|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.
|Word/Phrase |+ |( |– |Definition/Formula |Example |
|circle | | | |The set of all points in a plane | |
| | | | |equidistant from a given fixed point | |
| | | | |called the center. | |
|center of a circle | | | |The given point from which all points| |
| | | | |on the circle are the same distance. | |
|radius | | | |a segment with one endpoint at the | |
| | | | |center of the circle and the other | |
| | | | |endpoint on the circle; one-half the | |
| | | | |diameter | |
|circumference | | | |the distance around the circle | |
|chord | | | |a segment whose endpoints lie on the | |
| | | | |circle | |
|area of a circle | | | |[pic] | |
|central angle | | | |an angle formed at the center of a | |
| | | | |circle by two radii | |
|arc | | | |a segment of a circle | |
|arc measure | | | |equal to the degree measure of the | |
| | | | |central angle; [pic] | |
|arc length | | | |the distance along the curved line | |
| | | | |making up the arc; [pic]also known as| |
| | | | |the distance around a circular arc. | |
|major arc | | | |the longest arc connecting two points| |
| | | | |on a circle; an arc having a measure | |
| | | | |greater than 180 degrees | |
|minor arc | | | |the shortest arc connecting two | |
| | | | |points on a circle; an arc having a | |
| | | | |measure less than 180 degrees | |
|semicircle | | | |an arc having a measure of 180 | |
| | | | |degrees and a length of one-half of | |
| | | | |the circumference; the diameter of a | |
| | | | |circle creates two semicircles | |
|distance around a circular arc | | | |also known as the arc length; see the| |
| | | | |definition of arc length. | |
|sector | | | |a plane figure bounded by two radii | |
| | | | |and the included arc of the circle | |
|area of a sector | | | |[pic] where N is the measure of the | |
| | | | |central angle | |
|tangent | | | |a line or segment which intersects | |
| | | | |the circle at exactly one point | |
|secant | | | |a line or segment which intersects | |
| | | | |the circle at exactly two points | |
|sphere | | | |the locus of all points, in space, | |
| | | | |that are a given distance from a | |
| | | | |given point called the center | |
|surface area of a sphere | | | |[pic] | |
|volume of a sphere | | | |[pic] | |
| | | | | | |
| | | | | | |
| | | | | | |
| | | | | | |
| | | | | | |
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.
|Date: |Topic: Circles |
|Period: | |
| | |
|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: | |
| |--[pic] |
|area of a circle |--r is the measure of the radius of the circle |
| | |
| |--[pic] |
|circumference |--r is the measure of the radius and d is the measure of the diameter |
| |--these formulas are the same because [pic]. |
|Date: |Topic: Central Angles and Arcs |
|Period: | |
| | |
|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° |
| | |
| |--a segment of a circle |
|arc |--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 | |
| |--an arc whose measure is exactly 180 degrees |
|semicircle |--created by the diameter of the circle |
| |--the arc length is one-half of the circumference of the circle |
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?
[pic]
Picture source:
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.
[pic]
Picture source:
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.
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?
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.
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
degrees at the center of the circle. [pic]
Simplify this fraction and write the decimal equivalent (round to the nearest hundredth). [pic]
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.
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.
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.
[pic]
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.
[pic]
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.
[pic]
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 |Categorical data are values which can be sorted by names or labels rather |Agree |Disagree |
| | |than numbers. | | |
|Agree |Disagree |Marginal frequencies and joint frequencies are terms that have the same |Agree |Disagree |
| | |definition. | | |
|Agree |Disagree |Relative frequencies are often stated as percentages. |Agree |Disagree |
|Agree |Disagree |Two-way tables allow us to compare two or more sets of categorical data. |Agree |Disagree |
|Agree |Disagree |Relative frequencies can be found for the whole table, just the rows, or |Agree |Disagree |
| | |just the columns. | | |
|Agree |Disagree |In order to find the conditional probability of event B given event A has |Agree |Disagree |
| | |already occurred, you must know the probability of event B and the | | |
| | |probability of event A. | | |
|Agree |Disagree |The probability of B given A has occurred, represented by P(B|A), is the |Agree |Disagree |
| | |same as the probability of A given B has occurred, or P(A|B). | | |
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 |Categorical data are values which can be sorted by names or labels rather |Agree |Disagree |
| | |than numbers. | | |
|Agree |Disagree |Marginal frequencies and joint frequencies are terms that have the same |Agree |Disagree |
| | |definition. | | |
|Agree |Disagree |Relative frequencies are often stated as percentages. |Agree |Disagree |
|Agree |Disagree |Two-way tables allow us to compare two or more sets of categorical data. |Agree |Disagree |
|Agree |Disagree |Relative frequencies can be found for the whole table, just the rows, or |Agree |Disagree |
| | |just the columns. | | |
|Agree |Disagree |In order to find the conditional probability of event B given event A has |Agree |Disagree |
| | |already occurred, you must know the probability of event B and the | | |
| | |probability of event A. | | |
|Agree |Disagree |The probability of B given A has occurred, represented by P(B|A), is the |Agree |Disagree |
| | |same as the probability of A given B has occurred, or P(A|B). | | |
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.
Two-way Frequency Tables
Below is a two-way frequency table (Table 1) with hypothetical data from 200 randomly selected students in a school.
Table 1: Hair Color versus Eye Color
| | |Hair Color |
| | |Black |
| | |Black |
| | |Black |
| | |Black |
| | |Black |
| | |Black |
| | |Black |
| |Black |Brown |Red |Blond |Total | |Eye Color |Brown |.63 |.42 |.30 |.09 |.37 | | |Blue |.16 |.29 |.30 |.70 |.36 | | |Hazel |.16 |.19 |.20 |.09 |.16 | | |Green |.05 |.10 |.20 |.13 |.11 | | |Total |1.00 |1.00 |1.00 |1.00 |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, 41.7% of the people with brown hair have brown eyes and 69.6% of the people with blond hair have blue eyes.
1. 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.
2. 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.
[pic]
3. 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.
[pic]
4. 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.
[pic]
5. 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).
6. 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) = 0.01
b. P(blond hair and blue eyes) = 0.16
c. P(green eyes) = 0.11
d. P(red hair) = 0.10
Yes, each value listed is in Table 2.
7. 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.
8. 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.
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 following formula: [pic]. In terms of our example, [pic].
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.
[pic][pic]
[pic][pic]
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 [pic]? ________________________
3. Find the perpendicular bisector of [pic] 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 [pic]. Find the measure of [pic]. ________________________
5. [pic] should have been divided into two smaller arcs—either [pic] or [pic]. Find the measure of these two smaller arcs created by [pic]. ________________________________________________
6. What is true about the two arcs measured in number five? ________________________________________________________________________
7. Using a ruler, measure the radii [pic]. What is the arc length of [pic]? ________________________
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 [pic] 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.
________________________________________________________________________
________________________________________________________________________
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 [pic] and the other chord as [pic].
Step 5. Fold the circle, laying point G onto E to bisect the chord. Open the circle and fold again to bisect [pic] (lay point T onto R). Two diameters should have been formed. Label the intersection point on [pic] as O and the intersection point on [pic] as Y.
Answer the following about Investigation 2.
1. What is the relationship between [pic]? What is the relationship between [pic]? (Hint: it may be necessary to use a protractor and ruler to help answer this).
2. Use a centimeter ruler to measure [pic]. What observation can be made?
3. Make a conjecture about the distance that two chords are from the center when the chords are congruent.
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 [pic]? A chord.
3. Find the perpendicular bisector of [pic] 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 [pic]. Find the measure of [pic]. Answers will vary.
5. [pic] should have been divided into two smaller arcs—either [pic] or [pic]. Find the measure of these two smaller arcs created by [pic]. 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 [pic]. What is the arc length of [pic]? 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 [pic] 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.
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 [pic] and the other chord as [pic].
Step 5. Fold the circle, laying point G onto E to bisect the chord. Open the circle and fold again to bisect [pic] (lay point T onto R). Two diameters should have been formed. Label the intersection point on [pic] as O and the intersection point on [pic] as Y.
Answer the following about Investigation 2.
1. What is the relationship between [pic]? What is the relationship between [pic]? (Hint: it may be necessary to use a protractor and ruler to help answer this).
[pic]are perpendicular bisectors of [pic], respectively.
2. Use a centimeter ruler to measure [pic]. What observation can be made?
[pic]
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.
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
-----------------------
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?
The probability that Jim will earn a score of 8 is 0.3 or [pic].
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?
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
................
................
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