probabilities when outcomes are equally likely 622 1/25/11 3:36 pm links to the future adjusting the...

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Interactive Teacher’s

Lesson Guide

CurriculumFocal Points

Family Letters

EM Facts Workshop Game™

Algorithms Practice

eToolkitePresentations AssessmentManagement

Common Core State Standards

620 Unit 7 Probability and Discrete Mathematics

Advance PreparationFor the optional Readiness activity in Part 3, assemble the Probability Meter Poster. Write events on large stick-on notes. See Part 3 for suggestions. Be certain to display only the English version of the poster or both the English and Spanish versions simultaneously. For the optional Readiness activity in Part 3, obtain the book G Is for Googol: A Math Alphabet Book by David M. Schwartz (Tricycle Press, 1998).

Teacher’s Reference Manual, Grades 4–6 pp. 153–157, 172

Key Concepts and Skills• Add fractions with like denominators.

[Operations and Computation Goal 3]

• Identify all possible outcomes of an experiment. [Data and Chance Goal 3]

• Calculate probabilities when outcomes are equally likely. [Data and Chance Goal 3]

• Compare predictions based on theoretical probabilities with experimental results. [Data and Chance Goal 3]

Key ActivitiesStudents calculate the probabilities for various experiments with equally likely outcomes.

Ongoing Assessment: Recognizing Student Achievement Use journal page 248. [Data and Chance Goal 3]

Key Vocabularyoutcomes � equally likely � event � probability

� favorable outcomes � possible outcomes

MaterialsMath Journal 2, pp. 247 and 248Student Reference Book, p. 376transparency of Probability Meter (Math Journal 2, Reference Page 2; optional) �

1 set of double-6 dominoes

Playing Solution SearchStudent Reference Book, p. 332Math Masters, pp. 404 and 473per group: complete deck of number cards (the Everything Math Deck, if available)Students practice finding solutions to inequalities.

Math Boxes 7�1Math Journal 2, p. 249 Students practice and maintain skillsthrough Math Box problems.

Study Link 7�1Math Masters, p. 217 Students practice and maintain skillsthrough Study Link activities.

READINESS

Reviewing Vocabulary and Describing EventsProbability Meter Poster � stick-on notesStudents express the probabilities of different events using words, fractions, and percents.

READINESS

Reading about ProbabilityG Is for Googol: A Math Alphabet BookStudents read about the language and everyday uses of probability.

ENRICHMENTPlaying Carnival GamesMath Masters, pp. 218 and 219per partnership: 1 six-sided die, 1 coinStudents apply their knowledge of probability to determine strategies for maximizing winnings for games of chance.

EXTRA PRACTICE Playing Grab BagMath Masters, pp. 449–4523 six-sided dice � scissorsStudents practice substituting values for variables and finding probabilities.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

Probabilities When OutcomesAre Equally Likely

Objectives To review the basic concepts of probability; and toprovide experiences finding probabilities for events when alloutcomes are equally likely.

��

620_EMCS_T_TLG2_G6_U07_L01_576922.indd 620 3/24/11 1:27 PM

Probability ConceptsLESSON

7�1

Date Time

Math Message

The spinner at the right has 5 equal sections. Two sections are blue. If you spin itmany times, the spinner is likely to land on blue about �

25� of the time. Therefore, the

probability of landing on blue is �25�, or 40%.

Using the spinners shown below, write the letter(s) of the spinner next to thestatement that describes it. A spinner may be matched with more than one statement.

A B C D E F

Example:

This spinner will land on blue about 2 out of 3 times.

1. There is about a �14� chance that the spinner will land on blue.

2. This spinner will land on blue 100% of the time.

3. There is about a 50-50 chance that this spinner will land on white.

4. This spinner will never land on white.

5. The probability that this spinner will land on blue is �35�.

6. This spinner will land on white about twice as often as on blue.

7. This spinner will land on white a little less than half the time.

8. The probability that this spinner will land on white is 75%.

9. Suppose you spin Spinner A 4 times and it lands on white every time. What is the probability that the spinner will land on white on the fifth spin?

10. If you spin Spinner A 90 times, how many times would you expect the spinner to land on blue? 60

D

FC

EB

ED

A

�13�

A and C

Math Journal 2, p. 247

Student Page

Adjusting the Activity

Lesson 7�1 621

Getting Started

9 _ 10 9 _ 10 ; 90%

9 _ 12 3 _ 4 ; 75%

Math MessageComplete the problems on journal page 247.

Mental Math and Reflexes Students rename dictated fractions in simplest form and as percents. Suggestions:

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 2, p. 247)

Ask students to use a variety of probability phrases to rephrase some of the statements in the Math Message. Record these statements on the board. For example, the statement “There is about a 1 _ 4 chance that this spinner will land on blue” might also be stated as follows:

� If I spin the spinner many times, I expect it to land on blue about 1 out of 4 times.

� The spinner will land on blue about 25% of the time.

� The spinner is unlikely to land on blue.

� The probability that the spinner will land on blue is 0.25.

� The ratio of the number of times the spinner will land on blue to the total number of spins is about 1 in 4.

Have students explain the following:

� The spinner will land on blue about 1 _ 3 as often as it will land on white. The spinner will land on white 75 times out of 100 and will land on blue 25 times out of 100. 25 _ 75 = 1 _ 3

Make a transparency of the Probability Meter and refer to it while discussing probability phrases. (Students have a copy on page 376 of the Student Reference Book and on Reference of Math Journal 2.)

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

ELL

27 _ 75 9 _ 25 ; 36%

9 _ 20 9 _ 20 ; 45%

56 _ 64 7 _ 8 ; 87.5%

156 _ 234 2 _ 3 ; 66. ⎯ 6 %

Interactive whiteboard-ready ePresentations are available at www.everydaymathonline.com to help you teach the lesson.

Mathematical PracticesSMP1, SMP2, SMP6Content Standards6.RP.1, 6.NS.6c, 6.EE.8

621-626_EMCS_T_TLG2_G6_U07_L01_576922.indd 621 3/20/12 9:37 AM

Domino ProbabilitiesLESSON

7�1

Date Time

A set of double-6 dominoes is shown below. 148–153

Suppose all the dominoes are turned facedown and mixed thoroughly. You select one domino and turn it faceup.

1. How many possible outcomes are there? possible outcomes

2. Are the outcomes equally likely?

When the possible outcomes are equally likely, the following formula is used to find the probabilities:

A favorable outcome is the outcome that makes an event happen.

Probability of an event �

3. What is the probability of selecting each domino?

Find the probability of selecting each domino described below.

4. A double 5. Exactly one blank side

6. No blank sides 7. The sum of the dots is 7.

8. The sum of the dots is greater than 7.

9. Exactly one side is a 3. 10. Both sides are odd numbers.

Yes

�278� �2

68�

�2218� �2

38�

�298�

�268� �2

68�

number of favorable outcomesnumber of possible outcomes

�

28

�218�


Student Page


▶ Deciding Whether Outcomes PARTNER ACTIVITY

Are Equally Likely(Math Journal 2, p. 248)

On the board, list some chance experiments like those suggested below. Identify the possible outcomes for each. Ask students to decide if the outcomes are equally likely or not.

Assign journal page 248. Make sure students understand the following:

� Selecting a domino is a chance experiment with equally likely outcomes.

� There are 28 dominoes. Each domino has the same 1 _ 28 chance of being selected.

Experiment Possible Outcomes

Outcomes Equally Likely?

Flip a coin. HEADS, TAILS Yes

Roll a fair die. 1, 2, 3, 4, 5, 6 Yes

Roll a loaded die(a die with one face weightedheavier than the other faces).

1, 2, 3, 4, 5, 6 No. The face opposite the weighted face will come up more than 1 _ 6 of the time.

Spin the spinner. Black, white Yes. The black and white areas are equal.

Spin the spinner. Black, white Yes. The black and white areas are equal.

Spin the spinner. Black, white No. The black area is larger than the white area.

Spin the spinner.1

67

2345

1, 2, 3, 4, 5, 6, 7

Yes. The numbered areas are equal.

Drop a thumbtack. Lands point-up, lands point-down

Can’t tell; must collect data by experimenting

Send a letter. Delivered, returned, lost

No. The chance of the letter being returned and the chance of it getting lost are both very small.

Students write their names on cards and put them in a bag. A volunteer picks one card without looking.

Name of each student who wrote his or her name on a card

Yes (if all cards are the same size); No (if cards are different sizes)

621-626_EMCS_T_TLG2_U07_L01_576922.indd 622 1/25/11 3:36 PM

Links to the Future

Adjusting the Activity

Lesson 7�1 623

▶ Finding the Probability

WHOLE-CLASS ACTIVITY

of an Event(Math Journal 2, p. 248)

An event is a specific set or group of possible outcomes. For each event listed in Problems 4–10, students must identify and count the number of dominoes satisfying that event. Once students have counted those dominoes, they may think in different ways to name the probability. For example:

Problem 4: The event is select a double.

� The favorable outcomes are the dominoes that are doubles. There are 7 favorable outcomes: 6-6, 5-5, 4-4, 3-3, 2-2, 1-1, and blank-blank.

� The number of possible outcomes is 28 because there are 28 dominoes in the entire set.

� The fraction number of favorable outcomes ___ number of possible outcomes = 7 _ 28 , or 1 _ 4 is the probability of the event.

Problem 7: The event is the sum of the dots is 7. Possible strategies:

� There are three dominoes having a sum of 7: 6-1, 5-2, and 4-3 dominoes. The probability of picking a domino with a sum of 7 is 3 _ 28 .

� Three dominoes have a sum of 7. Each of these dominoes hasa probability of 1 _ 28 of being picked. The total probability of picking a domino with a sum of 7 is 1 _ 28 + 1 _ 28 + 1 _ 28 = 3 _ 28 .

Have a set of double-6 dominoes available for students who might benefit from actually gathering dominoes rather than just looking at pictures.

Students will use the second strategy described in the Problem 7 discussion (above) to find the final probabilities for tree-diagram problems in Lesson 7-5.

Ongoing Assessment: Journal Page 248 �Recognizing Student Achievement

Use journal page 248 to assess students’ abilities to identify outcomes and to calculate probabilities. Students are making adequate progress if they are able to solve Problems 1–10. For Problems 4–10, students’ answers should agree with probabilities found using the formula. [Data and Chance Goal 3]

PROBLEMBBBBBBBBBBOOOOOOOOOOOBBBBBBBBBBBBBBBBBBBBBBBBBB MMMMMEBLEBLLBLEBLELLLBLEBLEBLEBLEBLEBLEBLEBLEEEMMMMMMMMMMMMMMOOOOOOOOOOOOBBBBLBLBBLBLBLLLLLLPROPROPROPROPROPROPROPROPROPROPROPROPPRPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPROROROROROROROOPPPPPPP MMMMMMMMMMMMMMMMMMMMEEEEEEEEEEEELLELEEEEEEEEELLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRRRRRRRPROBLEMSOLVING

BBBBBBBBBBBBBBBBBBBBB EEELEMMMMMMMMOOOOOOOOOBBBLBLBLBBLBBROOOROROROROROROROROROROO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGGGGLLLLLLLLLLLLLVINVINVINVINNNVINVINVINVINVINVINVINVINV GGGGGGGGGGGGOLOLOOLOLOLOLOOLOO VINVINVLLLLLLLLLLVINVINVINVINVINNVINVINVINVINVINVINNGGGGGGGGGGOLOOLOLOLOLOLOLOOO VVVVLLLLLLLLLLVVVVVVVVVOSOSOSOOSOSOSOSOSOSOOSOSOOOOOOSOSOSOSOSOSOSOSOSOSOOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVLLLLLLVVVVVVVVVLLLLVVVVVVVVLLLLLLLVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIISOLVING

To help students differentiate between favorable and possible outcomes, draw the following table on the board. Write numbers in the appropriate columns as you discuss Problems 4 and 7.

Number ofFavorable Outcomes

Number ofPossible Outcomes

Probabilityof Event

7 28 7 _ 28

3 28 3 _ 28

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

NOTE To practice finding the probability of independent and dependent events, see www.everydaymathonline.com.

621-626_EMCS_T_TLG2_U07_L01_576922.indd 623 1/25/11 3:36 PM

Math Boxes LESSON

7�1

Date Time

5. Complete.

a. m = 368 mm

b. cm = 0.245 m

c. 32 mm = m

d. 45.2 cm = mm

e. 0.25 mm = cm

6. Solve. Solution

a. w _ 8 = 16

b. 60 _ p = 5

c. 3 _ 7 = t _ 28

d. d _ 18 = 4 _ 6

1. Fill in the missing equivalents. Write fractions in simplest form.

210 72 73 113

2. Solve the following equations. Check each solution by substituting it for the variable in the original equation.

a. 8x - 1 = 11

Solution: x =

b. 2 _ 5 y + 6 = 10

Solution: y =

c. -20m + 20 = -20

Solution: m =

3. Indicate whether each inequality is true or false.

a. 2 _ 3 ∗ 9 > 8

b. -4 ≤ -3 - 1

c. 48 - (6 ∗ 4) > 20

d. 8 - 10 ≠ 13 - 15

4. Graph the solution set for k > -2 on the number line below.

0 1-1-2-3 2 3

55 59 60 251 252

244241

Fraction Decimal Percent

9 _ 10

0.98

60%

7 _ 25

12.5%

1 1 _ 2

2

10

w = 128

t = 12p = 12

d = 12

0.368

false

false

24.5

truetrue

0.0324520.025

1 _ 8

3 _ 5

49

_ 50

0.9

0.60.28

0.125

90%98%

28%

EM3MJ2_G6_U07_247_277.indd 249 3/17/10 11:18 AM


Student Page

Study Link Master

STUDY LINK

7�1 Outcomes and Probabilities

Name Date Time

Complete the table.

Use the problems from the table to answer the following questions. Express each probability as a percent.

3. What is the probability of selecting a quarter from the coins in Problem 1?

4. What is the probability of choosing a factor of 20 from the cards in Problem 2?

5. Suppose you spin the spinner from the Example in the table. Complete the number sentence below to determine the probability of the spinner landing on A or C.

� �

Probability of A Probability of C Probability of A or C75%50%25%

37.5%100%

Simplify the expression using the order of operations.

6. 3.8 � 6.4 � 0.2 � 1.8 º 2.6 � 3.2 � 0.8 27.12

Possible OutcomesExperiment Outcomes Equally Likely?

Example: Spin the spinner.

1. Choose a coin.

2. Choose a factor of 20.

1 20 10

2 4 5

DD D

DNQQ

Q

A

C

BNo. The area for C istwice as large as eachof the other 2 areas.

No. There is anunequal numberof each type ofcoin.

Yes. Eachnumber card isa factor of 20.

A, B, C

‰,Â,Í

1, 2, 4, 5, 10, 20

Practice

150–153

Math Masters, p. 217


2 Ongoing Learning & Practice

▶ Playing Solution Search

SMALL-GROUP ACTIVITY

(Student Reference Book, p. 332; Math Masters, pp. 404 and 473)

Algebraic Thinking Students first played Solution Search in Lesson 6-12. They may create sets of Solution Search cards to use in place of the ones provided.

After students have finished playing the game, have each student draw one Solution Search card. Tell students to draw a number line on an Exit Slip (Math Masters, page 404) and graph the solution set for the inequality shown on their card. Remind students to use a solution to check the graph.

▶ Math Boxes 7�1

INDEPENDENT ACTIVITY

(Math Journal 2, p. 249)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 7-3. The skill in Problem 6 previews Unit 8 content.

Writing/Reasoning Have students write a response to the following: Explain what happens to the solution of Problem 2a when you multiply each term of the equation

by 2. Sample answer: Since the equations are equivalent, the solution is the same.

▶ Study Link 7�1

INDEPENDENT ACTIVITY

(Math Masters, p. 217)

Home Connection Students identify outcomes, decide whether outcomes are equally likely, and find probabilities.

621-626_EMCS_T_TLG2_U07_L01_576922.indd 624 1/28/11 10:18 AM

Lesson 7�1 625

3 Differentiation Options

READINESS

SMALL-GROUP ACTIVITY

▶ Reviewing Vocabulary and 5–15 Min

Describing EventsTo provide experience with ways in which students can describe the probability of an event, have them position stick-on notes of events on the Probability Meter Poster according to each event’s likelihood. Ask students to then use basic word phrases, fractions, and percents to describe the likelihood of each event.

Suggestions:

� You will see a living triceratops in your lifetime. Impossible; 0 _ 100 ; 0%

� The sun will rise tomorrow. Certain; 100 _ 100 ; 100%

� You will have homework tonight. Sample answer: Very likely; 4 _ 5 ; 80%

� A flipped coin will land on HEADS. 50-50 chance; 1 _ 2 ; 50%

� A rolled six-sided die will land on a number less than 6. Very likely; 5 _ 6 ; 83. ⎯ 3 %

Write additional events as appropriate for your class.

READINESS PARTNER ACTIVITY

▶ Reading about Probability 5–15 Min

Literature Link To introduce probability concepts, have students read “P is for Probability” from the book G Is for

Googol: A Math Alphabet Book by David M. Schwartz (Tricycle Press, 1998). Use the illustrations and examples from the excerpt as a springboard for discussion about the language and everyday uses of probability.

0 . 0 1

1

910

45

7—10

23

58

56

78

3—10

15

110

0

16

18

13

38

1—20

110 0

1920

810

,

25

4—10,

14

28

,

50——100

34

68

,

35

610

,

2—4, , 3—6

, 4—8, 5—10

, 10—20,

-

0 . 5 0

45% 0 . 4 5

40% 0 . 4 0

35% 0 . 3 5

30% 0 . 3 0

0 . 2 5

20% 0 . 2 0

15% 0 . 1 5

10% 0 . 1 0

5% 0 . 0 5

0. 0 0

50%

25%

0%

0 . 3 7 5

0 . 3 3

0 . 1 6

0 . 1 2 5

95% 0 . 9 5

90% 0 . 9 0

85% 0 . 8 5

80% 0 . 8 0

0 . 7 5

70% 0 . 7 0

65% 0 . 6 5

60% 0 . 6 0

55% 0 . 5 5

1 . 0 0100%

0 . 8 7 5

0 . 8 3

0 . 6 6

0 . 6 2 5

75%

-

-

-

9 9——1000 . 9 9

50–50

CERTAIN

IMPOSSIBLE

1—2

VERY I

KELY

UNL

VERY

LIKELY

L

K

I

L

Y

E

EXTREMELY

UNLIKELY

EXTREMELY

UNLIKELY

50–50CHANCE

LIKELY

Probability Meter

621-626_EMCS_T_TLG2_G6_U07_L01_576922.indd 625 10/11/11 2:11 PM

Booth 1

Two in a Row

Flip a coin twice. If the coin lands on the same side both times, you win a prize coupon.

Booth 3

Roll It Up

Roll a die twice. If the second roll is a greater number than the first, you win a prize coupon.

Booth 5

Make the Call

Predict the roll of a die. If that numbercomes up, you win a prize coupon.

Booth 2

Odd Tail Toss

Flip a coin once and roll a die once. If you get TAILS and an odd number, you win a prize coupon.

Booth 4

10 or More

Roll a die twice. If you get 5 or greater both times, you win a prize coupon.

Booth 6

7 or More

Roll a die twice. If the total of the rolls is 7 or greater, you win a prize coupon.

LESSON

7�1

Name Date Time

Carnival Games

At the carnival, you will play 10 games and will try to win as many prize coupons as possible. You must visit at least three different booths.


Teaching Master

LESSON

7�1

Name Date Time

Carnival Games Records

Below, record the number of each booth you visit. Make a tally mark for each prize coupon you win during your 10 games.

1. Describe a strategy for winning the greatest number of prize coupons in 10 gamesif you must visit at least 3 different booths.Answers vary.

2. At which booths does it seem easy to win?Sample answer: Booths 1, 3, and 6 provide the greatest chances of winning.

3. Describe how you would change the rules of one game to make it easier to win.Sample answer: At Booth 2, flip the coin and roll the die once. If you get TAILS and a number greater than 2, you win.

Booth Number Number of Prize Coupons Won

Total Number ofPrize Coupons Won


Teaching Master


ENRICHMENT PARTNER ACTIVITY

▶ Playing Carnival Games 15–30 Min

(Math Masters, pp. 218 and 219)

To further explore probabilities and outcomes, students roll dice and flip coins for a set of carnival games. They find the best possible combination of booths to visit to maximize winnings over 10 games.

Share the following summary of outcomes with students or write the headings of the table on the board and work with them to complete it.

Booth Number ofFavorable Outcomes

Number of Possible Outcomes Favorable Outcomes

__ Possible Outcomes Probability of Winning

1 2 4 2 _ 4 , or 1 _ 2 50.0%

2 3 12 3 _ 12 , or 1 _ 4 25.0%

3 15 36 15 _ 36 , or 5 _ 12 41. ⎯ 6 %

4 4 36 4 _ 36 , or 1 _ 9 11. ⎯ 1 %

5 1 6 1 _ 6 16. ⎯ 6 %

6 21 36 21 _ 36 , or 7 _ 12 58. ⎯ 3 %

Have students suggest winning strategies based on the probabilities. The best theoretical combination would be visiting Booth 6 eight times and Booths 1 and 3 each one time.

Have partners compare their individual tallies to the probabilities. They will likely notice that their individual results often differ. Ask why this might be. Their samples are too small to reliably reflect the probabilities.

Consider having students design their own set of fair games that are easy or difficult to win.

EXTRA PRACTICE SMALL-GROUP

ACTIVITY

▶ Playing Grab Bag 15–30 Min

(Math Masters, pp. 449–452)

To provide extra practice with calculating probabilities, have students play Grab Bag. Provide each team with a set of Grab Bag playing cards (Math Masters, pp. 450 and 451), game directions, and record sheets (Math Masters, pp. 449 and 452).

Students roll 3 dice to determine the numbers of objects in a set from which one object will be drawn at random. They calculate the probability of drawing that particular object.

621-626_EMCS_T_TLG2_G6_U07_L01_576922.indd 626 10/11/11 2:11 PM

probabilities when outcomes are equally likely 622 1/25/11 3:36 pm links to the future adjusting the...

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