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# How Likely Is It by fjzhangxiaoquan

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```									How Likely Is It?, Investigation 1, Problem 1.1                                                                Completed
Choosing Cereal
Mathematical Goal                                                       National Standards         State Standards
NAEP                         6NJ 4.4.B.1,
• Develop an intuitive sense of probability through
D4a, D4c, D4g             6NJ 4.4.C.3,
a coin-tossing experiment.                                            CAT6                         6NJ 4.4.B.5
LV16.15
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,
CTBS
Student Activity CD-ROM, www.PHSchool.com
LV16.53
Materials:   Student notebooks, Overhead projector, Coins, Paper        ITBS
cups (optional)                                               LV12.PS
Pacing:      45 minutes                                                 S10
Int2.DSP, TV
LV16.15

1. LAUNCH (10 minutes)                                                                  Targeted Resources

Discuss the questions posed in Getting Ready. Introduce Problem 1.1 by                Transparency 1.1A Getting
telling Kalvin’s story about how he decided to toss a coin to decide his              Ready
breakfast. Have students predict, before they conduct the experiment,                 Transparency 1.1B Coin–
how many days in June Kalvin will have Cocoa Blast for breakfast.                     Toss Results
The following questions can help in a discussion of bias:                             Labsheet 1.1 Coin–Toss
Table
• What kinds of things could happen to affect the data you gather in
this problem?
• How should you toss a coin to be sure you have a fair trial?
• What if you always start with tails facing up when tossing a coin?
Do you think this introduces bias?
Let students know that collecting data in a random way can help to better
predict what to expect when a coin is tossed. In this case, random means
that the coin’s behavior is not affected in a predictable way by how it is
tossed.
Distribute Labsheet 1.1 to each pair or group.
• Your pair or group is going to toss a penny once for each day in the
month of June and record the result, H or T, in the Result of Toss
column.
Have students work in pairs or small groups.

2. EXPLORE (20 minutes)                                                                 Targeted Resources

As students work, make sure they are not introducing bias and are
recording their data correctly.
You may want to illustrate how to fill the table in for some sample data for
the first few days noting the cumulative nature of the table columns
marked “so far”. Students can compute the fraction of heads after any
number of trials by dividing the number of heads that occurred to that
point by the number of trial days.
When students are done with Question A, bring the entire class together
to work on Question B of the problem.

3. SUMMARIZE (15 minutes)                                                               Targeted Resources
Combine the class’s data. Ask one pair or group at a time to report how
many heads they tossed in 30 trials, and record the results in a table. After
you have collected all the data, combine it by making another table
showing a running total of the number of trials and number of heads.
Recompute the fraction and percent of heads each time.
• As the number of trials increases, what is happening to the percent
Have students do Question C as a class, in groups, or as homework

4. ASSIGNMENT GUIDE                                                                      Targeted Resources
Core 1,3–5, 19, 20                                                                     1ACE Exercise 2 Adapted
Version
Other Applications 2; Extensions 31
exercises, see the CMP Special Needs Handbook.
Connecting to Prior Units 19, 20: Bits and Pieces I

How Likely Is It?, Investigation 1, Problem 1.2                                                                 Completed
Tossing Paper Cups
Mathematical Goals                                                       National Standards         State Standards
NAEP                         6NJ 4.4.B.1,
• Continue to develop an intuitive sense of probability through
D4a, D4c, D4g             6NJ 4.4.C.1,
a cup tossing experiment.                                              CAT6                         6NJ 4.4.B.5
• Understand that probabilities are useful for predicting what              LV16.15
will happen over the long run.                                         CTBS
• Toss cups to find an experimental probability where the                   LV16.53
outcomes are not equally likely.                                       ITBS
LV12.PS
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,                      S10
Student Activity CD-ROM, www.PHSchool.com                      Int2.DSP, TV
Materials:   Student notebooks, Paper cups                                  LV16.15
Pacing:      45 minutes

1. LAUNCH (10 minutes)                                                                   Targeted Resources

Tell the story of Kalvin’s idea for eating more Cocoa Blast. Ask students to
predict if a paper cup is better than a coin for Kalvin to use. Talk with your
students about the differences between using coins and using paper cups
to generate random events.
• Will paper cups behave like coins and land on an end or a side
about the same number of times? Why or why not?
• Do you think the paper cup is more likely to land on its side or on
one of its ends?
• What are some ways that you can toss a cup?
Encourage students to handle the paper cup with care, so that it does not
become misshapen. Changing the shape of the cup can affect the
outcomes of the experiment. Help students develop a method for keeping
track of their results. You want students, over time, to decide on record-
keeping schemes for themselves. However, at this stage, they may need
a few models.
Have students work in pairs or small groups on the problem.

2. EXPLORE (20 minutes)                                                                 Targeted Resources

Encourage students to be careful about gathering their data and
• Do you think your results and other groups’ results will be the
same?
Through your questions, help them to realize that there will be a great deal
of variation among individual sets of 50 tosses, but less variation in the
class’s combined results.
When you notice that students have completed Questions A and B, bring
the entire class together to talk about the rest of the problem

3. SUMMARIZE (15 minutes)                                                               Targeted Resources

After pairs or groups have each tossed their paper cup 50 times, discuss
their findings.
• Did you all arrive at the same conclusion about which outcome (the
paper cup landing on its side or the cup landing on one of its ends)
is more likely?
Combine the data from all the groups, and find the fraction (or percent) of
times the paper cup landed on its side or an end as in Problem 1.1. This
can help students see where the relative frequency of landing on an end
students to think about Question D.

4. ASSIGNMENT GUIDE                                                                     Targeted Resources
Core 6–8                                                                               Answers to ACE and
Mathematical Reflections
Other Connections 21–23; Extensions 32; unassigned choices from
previous problems
Special Needs Handbook.
Connecting to Prior Units 21, 23: Bits and Pieces III

How Likely Is It?, Investigation 1, Problem 1.3                                                                Completed
One More Try
Mathematical Goals       .                                               National Standards        State Standards
NAEP                         6NJ 4.4.B.2,
• Develop strategies for finding experimental probabilities for a
D4a, D4c, D4g              6NJ 4.4.B.5
situation that involves tossing two coins.                             CAT6
• Begin to explore the notion of fair and unfair.                          LV16.15
CTBS
Vocabulary: probability, experimental probability                          LV16.53
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,                      ITBS
Student Activity CD-ROM, www.PHSchool.com                     LV12.PS
Materials:   Student notebooks, Coins                                    S10
Pacing:      45 minutes                                                    Int2.DSP, TV
LV16.15
1. LAUNCH (10 minutes)                                                              Targeted Resources

After discussing the vocabulary and notation, you will want to talk to
students about how Kalvin uses two coins to determine his breakfast. If
the coins match, he gets to eat Cocoa Blast and if they do not match, he
eats Health Nut Flakes.
• If Kalvin uses this method for the month of June, how many days in
June do you think Kalvin will eat Cocoa Blast?
Save these conjectures on the board or a transparency to revisit in the
summary. Demonstrate how to record the data for tossing two coins. Have
students experiment in pairs or groups for Question A.

2. EXPLORE (20 minutes)                                                             Targeted Resources

After the students have collected their data in Question A, bring the class
back together to compile the data in order to discuss Question B. Because
the two coins are the same, students may not see heads-tails as being
different from tails-heads. It is probably better to wait until the summary to
address this concern, when students will be seeking an explanation for
why match and no-match are equally likely.
Give the class time to complete Questions C and D on their own and save
the discussion for the summary.

3. SUMMARIZE (15 minutes)                                                           Targeted Resources

Question C encourages students to think about theoretical probability.
This term will be defined in Investigation 2. For now, try to encourage
students to seek explanations for the probabilities in this problem.
• So what did you find out about the probability of a match and the
probability of a no-match?
• Why is the probability of a match the same as a no-match?
If students are struggling with the realization that there are two ways to get
a no-match, consider the suggestions in the Mathematics Background on
pages 4 and 5 on strategies for finding outcomes. Question D revisits the
Law of Large Numbers with a new and unfamiliar experiment. Discuss this
as a class.
• What does the numerator in each probability tell us?
• What does the denominator in each probability tell us?
• Which probability is greater?

4. ASSIGNMENT GUIDE                                                                 Targeted Resources
Core 9, 10, 24, 25                                                                  Answers to ACE and
Other Connections 26; unassigned choices from previous problems                     Mathematical Reflections
Special Needs Handbook.
Connecting to Prior Units 24: Prime Time; 25: Bits and Pieces I

How Likely Is It?, Investigation 1, Problem 1.4                                                            Completed
Analyzing Events
Mathematical Goals .                                                       National Standards         State Standards
• Understand the concepts of equally likely and not equally                NAEP                         6NJ 4.4.B.1,
likely.                                                                     D4a, D4c, D4g             6NJ 4.4.C.3,
CAT6                         6NJ 4.4.B.5
Vocabulary: equally likely                                                    LV16.15
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,                        CTBS
Student Activity CD-ROM, www.PHSchool.com                        LV16.53
Materials:   Student notebooks, Overhead projector                         ITBS
Pacing:      45 minutes                                                       LV12.PS
S10
Int2.DSP, TV
LV16.15

1. LAUNCH (10 minutes)                                                                     Targeted Resources

Tell the story of Kalvin’s coin and his mother’s suspicion that it is not a fair         Transparency 1.4 Table
• Why do you think Kalvin’s mother is suspicious of the coin?
• What do you think it means for a coin to be “fair”?
Review his mother’s example (names chosen from a hat) to illustrate the
difference between equally likely and not equally likely. Students might
need help understanding how the table in the problem is structured.
• The “Action” column explains the situation; the “Possible results”
column describes the things, or outcomes, that can happen.
You may want to read the first entry in the table together.
Have students work on the problem individually and then gather in pairs or
small groups to discuss their answers.

2. EXPLORE (20 minutes)                                                                    Targeted Resources

Make sure groups understand that they should try to reach consensus
about each situation. Listen to how students defend their answers, and be
on the lookout for inventive ways of arguing for a particular answer.
In parts (5)–(7) of Question A, first determine the set of possible results for
each action, and then determine if the results are equally likely.
Note the results each pair or group generates. Encourage them to
convince themselves that they have found all the possible results for each
action.

3. SUMMARIZE (15 minutes)                                                                  Targeted Resources

Discuss the groups’ answers as a class. There may be some
disagreement. Encourage students to explain their reasoning as they
Record some of the actions that students make up to illustrate equally
likely events.

4. ASSIGNMENT GUIDE                                                                        Targeted Resources
Core 11–17                                                                               Answers to ACE and
Other Applications 18,Connections 27–30; unassigned choices from                         Mathematical Reflections
previous problems
Special Needs Handbook.

Connecting to Prior Units 27–30: Data About Us

How Likely Is It?, Investigation 1                                                                     Completed
Completing the Investigation

Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, Student Activity CD-ROM, www.PHSchool.com
Pacing:     65 minutes

1. MATHEMATICAL REFLECTIONS (25 minutes)                                        Targeted Resources

Students reflect on and summarize their learning at the end of the              Answers to ACE and
investigation.                                                                  Mathematical Reflections

2. ASSESSMENT (40 minutes)                                                      Targeted Resources

Investigation 1
Use these resources to assess students’ understanding of the
mathematics taught in this investigation, to create additional worksheets,      Multiple Choice Items
or to provide additional practice problems.                                     Question Bank
These questions can also be found on the ExamView CD-ROM.
ExamView CD–ROM
Questions
ExamView CD–ROM

3. SPANISH ASSESSMENT                                                           Targeted Resources

Assessment and Practice                                                         Spanish Additional Practice:
Investigation 1
Use these resources, in Spanish, to assess students’ understanding of the
mathematics taught in this investigation, to create additional worksheets,      Spanish Multiple Choice
or to provide additional practice problems.                                     Items
These questions can also be found on the ExamView CD-ROM.                       Spanish Question Bank
Spanish Assessment
Spanish ExamView CD–ROM
Questions

How Likely Is It?, Investigation 2, Problem 2.1                                                        Completed
Predicting to Win
Mathematical Goals                                                      National Standards         State Standards
NAEP                         6NJ 4.4.B.2,
• Find the theoretical probability by analyzing the possible
D4d, D4e, D4f             6NJ 4.4.B.4
equally likely outcomes involved in a game of guessing the            CAT6
color of a block.                                                        LV16.15
• Compare the experimental and theoretical probabilities.               CTBS
LV16.53
Vocabulary: outcomes, theoretical probability                           ITBS
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,                        LV12.PS
Student Activity CD-ROM, www.PHSchool.com                  S10
Materials:   Student notebooks, Opaque container with 9 red, 6             Int2.DSP, TV
yellow, and 3 blue block (you may substitute other            LV16.15
objects)
Pacing:      45 minutes

1. LAUNCH (10 minutes)                                                                  Targeted Resources

The answers for Problem 2.1 are based on a bucket containing 9 red, 6
yellow and 3 blue blocks.
Tell the story about the Gee Whiz Everyone Wins! game show. Then ask
the following questions:
• What do you think random means?
• Suppose you are a member of the audience. Would you rather be
called to the stage first or last? Why?
• We are going to play this game as a class. I will be the host, and
you will be the contestants. You will take turns choosing from the
container. We need to keep careful track of the outcome of each
choice. After each turn, we will return the container to its original
state by replacing the block and then mixing the blocks. Why do we
need to do this?
Do Question A as a whole-class activity. Ask students one at a time to
predict the color, then let them choose a block. After finding the color of
the block, have them replace the block. Have one student keep track of
the colors chosen with a chart with tally marks at the board.
• Are the data that are accumulating on the board useful for making a
prediction for the block color? Why?
• What is the experimental probability of choosing each color?
Have students complete Questions B, C, and D as a whole class or in
pairs.

2. EXPLORE (20 minutes)                                                                 Targeted Resources

If parts of the problem are done in groups, be sure that students are
analyzing the data correctly. Since there are 9 red blocks, 6 yellow blocks
and 3 blue blocks, there is a total of 18 blocks. So the probability of
9   1
choosing a red is      or .
18   2

3. SUMMARIZE (15 minutes)                                                               Targeted Resources

Explain that the probabilities found in Question B are called theoretical
probabilities. Talk about why this is a sensible name. Specifically, they are
probabilities derived from theory (what we think ought to happen) without
having performed any experiment.
Discuss parts (1) and (2) of Question C:
• Are the probabilities of choosing a blue block, a red block, or a
yellow block equally likely?
Discuss Question D. Ask students once again:
• Is there an advantage to going first or going last?
Students should recognize that there is an advantage to going last
based on good experimental probabilities can be more predictive of what
will happen over time.

4. ASSIGNMENT GUIDE                                                                       Targeted Resources
Core 1–2, 14–16                                                                         Answers to ACE and
Mathematical Reflections
Other Connections 13; Extensions 34
Special Needs Handbook.
Connecting to Prior Units 13, 16: Bits and Pieces I; 14, 15: Bits and
Pieces II

How Likely Is It?, Investigation 2, Problem 2.2                                                                  Completed
Exploring Probabilities
Mathematical Goal                                                         National Standards         State Standards
• Observe some properties of probability.                                 NAEP                         6NJ 4.4.B.1,
D4d, D4e, D4f             6NJ 4.4.B.2,
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,                       CAT6                         6NJ 4.4.B.3
Student Activity CD-ROM, www.PHSchool.com                     LV16.15
Materials:     Student notebooks, Opaque containers, Colored              CTBS
blocks                                                        LV16.53
ITBS
Pacing:        45 minutes
LV12.PS
S10
Int2.DSP, TV
LV16.15

1. LAUNCH (10 minutes)                                                                    Targeted Resources

Briefly review the ideas in this Investigation, including theoretical
probability, notation, and the context from Problem 2.1. You might want to
have a brief discussion about the use of the words “or” and “and”. An “or’
statement is true if one or both of the phases (or probabilities) occur. So,
6
in Problem 2.1, P(red) or P(yellow) occurring is P(red) + P(yellow) =        +
12
2   8
=   . An “and” statement is true if both events (probabilities) occur.
12 12
Then let students work on the problem in pairs.

2. EXPLORE (20 minutes)                                                                   Targeted Resources
As students work, pay attention to which ideas seem to be difficult for
them. If students struggle with the probability of something not happening,
you may want to ask them about what could happen, rather than pointing
them to a formula. After students have found that choosing red or
choosing yellow is the same as not choosing a blue, you might ask:
• What happens if we add P(blue) and P(not blue)?

3. SUMMARIZE (15 minutes)                                                              Targeted Resources

Since there are lots of small questions in this problem, you should focus
your summary around one or two of the big ideas your students have
found interesting or difficult. You might ask questions like the following:
• What does it mean to find the probability of something not
happening?
• How can you compute the probability of something not happening?
• How can you find P(red or yellow)?
• Can a probability ever be a value greater than 1?
• Can a probability ever be 0? Give an example from this problem.
• Can a probability ever be 1? Give an example from this problem.
Questions C and D provide you with an opportunity to assess students’
understanding of equivalent fractions and probability.
After discussing these and any other ideas from your class, you may want
to finish by having students individually work on ACE Exercise 3.

4. ASSIGNMENT GUIDE                                                                    Targeted Resources
Core 3–6                                                                              Answers to ACE and
Other Connections 17–24; unassigned choices from previous problems                    Mathematical Reflections
Special Needs Handbook.

How Likely Is It?, Investigation 2, Problem 2.3                                                               Completed
Winning the Bonus Prize
Mathematical Goals                                                      National Standards        State Standards
NAEP                         6NJ 4.4.B.4,
• Use organized lists and tree diagrams to find theoretical
D4d, D4e, D4f              6NJ 4.4.B.2
probabilities.                                                        CAT6
• Understand that experimental probabilities are better                   LV16.15
estimates of theoretical probabilities when based on larger           CTBS
numbers of trials.                                                      LV16.53
ITBS
Vocabulary: counting tree                                                 LV12.PS
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,                     S10
Student Activity CD-ROM, www.PHSchool.com                    Int2.DSP, TV
Materials:   Student notebooks, Overhead projector, Two opaque            LV16.15
containers filled with 1 red, 1yellow, and 1 blue block,
Chart paper or blank transparencies (optional)
Pacing:      45 minutes

1. LAUNCH (10 minutes)                                                                 Targeted Resources
Use the opening paragraphs of Problem 2.3 to introduce tree diagrams.              Transparency 2.3 Tree
Demonstrate how to make a tree diagram and ask students questions                  Diagrams
modeling those they will need to ask themselves:
If I toss two coins,
• What are the possible outcomes for the first coin?
• Are these outcomes equally likely?
• If you get heads with the first coin, what are the possible outcomes
for the second coin? Are these outcomes equally likely?
• If you get tails with the first coin, what are the possible outcomes
for the second coin?
• If you toss two coins, what is the probability that the coins will
match? What is the probability that they won’t match?
Discuss the bonus game with students. Make sure that students realize
that contestants must make a prediction before choosing a block from
each bag. So they must say, for example, “blue from Bag 1 and red from
bag 2.”
Ask students to make a prediction:
• What are the contestant’s chances of winning this game?
This activity works well either as a whole-class experiment or in groups,
each group having two containers of blocks.

2. EXPLORE (20 minutes)                                                            Targeted Resources

Allow students to experiment and decide when they have enough data to
make a good estimate of a contestant’s chances of winning. If students do
the activity in groups, pool class’s data before students answer Question
B. Some may need help making a tree diagram. Have them ask
themselves:
• What are the possible outcomes for the first choice? For the
second choice?
• How do you know when you have all possible outcomes?
Students may need support in determining which of the outcomes
represent a win for the contestant.
Remind students that contestants are given one guess to correctly predict
the color of block chosen from both bags.
Give chart paper or transparencies to each group to record their strategies
for finding the theoretical probabilities

3. SUMMARIZE (15 minutes)                                                          Targeted Resources
•What are the experimental probabilities for predicting each pair of
colors?
• How did you determine the theoretical probabilities?
Compare the theoretical probabilities to the experimental probabilities.
Have a class discussion about the kind of outcomes in Problem 2.1 and
the kind of outcomes in this problem.

4. ASSIGNMENT GUIDE                                                                Targeted Resources
Core 7–9                                                                             2ACE Exercise 7 Adapted
Other Connections 25–31; Extensions 35, 36; unassigned choices from                  Version
previous problems                                                                    Answers to ACE and
exercises, see the CMP Special Needs Handbook.
Connecting to Prior Units 25–28: Bits and Pieces I; 29: Data About Us;
30, 31: Bits and Pieces II

How Likely Is It?, Investigation 2, Problem 2.4                                                               Completed
Pondering Possible and Probable
Mathematical Goals                                                     National Standards         State Standards
NAEP                         6NJ 4.4.C.1,
• Deepen understanding of equally likely and not equally likely.
D4d, D4e, D4f             6NJ 4.4.B.2,
• Understand that a game of chance is fair only if each player         CAT6                         6NJ 4.4.B.4
has the same chance of winning, not just a possible chance              LV16.15
of winning.                                                          CTBS
LV16.53
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,                    ITBS
Student Activity CD-ROM, www.PHSchool.com                    LV12.PS
Materials:   Student notebooks, Coins                                  S10
Pacing:      45 minutes                                                   Int2.DSP, TV
LV16.15

1. LAUNCH (10 minutes)                                                                 Targeted Resources

A good launching strategy is to play the coin-tossing game with the class
so students understand how to score and take turns.
Initiate a discussion about what it means for a game to be fair.
• What do you think it means to say that a game is a fair game?
Make sure students understand that you are referring to games of chance
(such as bingo) rather than games of skill (such as tennis).
The coin-tossing game is a two-person game, so students need to think in
terms of two players. In fair games of chance, each player has an equal
chance, or probability, of winning. You may want to collect experimental
data as a whole class. If you do this, record the probabilities on the
Let students know that they are to examine the coin-tossing game for
fairness.
Let them work on the problem in pairs.

2. EXPLORE (20 minutes)                                                                Targeted Resources

As you listen to students discuss the game, remind them that they must
be prepared to explain why they think the game is fair or unfair.

3. SUMMARIZE (15 minutes)                                                              Targeted Resources

If you collected experimental data, then you can use this to help students
see the variability in the results of all the games that were played. From
this data, you and the class can determine relative frequencies of getting
all heads, all tails, or a matching pair. To determine the theoretical
probabilities, ask the class to share their strategies. Be sure that both an
organized list and tree diagram are discussed. The outcomes are: TTT,
TTH, THT, HTT, THH, HTH, HHT and HHH. Each of the eight possibilities
has the same chance of occurring. However, in this game we are
1
interested in three of a kind or a pair. P(three heads or three tails) = 2(     )
8
2 1                                   1    6  3
=    =  and P(two heads or two tails) = 6( ) =   = . This means that
8 4                                   8    8  4
one player has a much better chance of scoring and therefore of winning.
• How can you make the game fair?
For example,
• What is the probability of getting exactly 0 heads? Exactly 1 head?

4. ASSIGNMENT GUIDE                                                                    Targeted Resources

Core 10–12                                                                             Answers to ACE and
Mathematical Reflections
Other Connections 32, 33; Extensions 37; unassigned choices from
previous problems
Special Needs Handbook.

How Likely Is It?, Investigation 2                                                                            Completed
Completing the Investigation

Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, Student Activity CD-ROM, www.PHSchool.com
Pacing:     65 minutes

1. MATHEMATICAL REFLECTIONS (25 minutes)                                               Targeted Resources

Students reflect on and summarize their learning at the end of the                     Answers to ACE and
investigation.                                                                         Mathematical Reflections

2. ASSESSMENT (40 minutes)                                                             Targeted Resources

Investigation 2
Use these resources to assess students’ understanding of the
mathematics taught in this investigation, to create additional worksheets,             Skill Practice: Probability
or to provide additional practice problems.                                            Check–Up
Multiple Choice Items
These questions can also be found on the ExamView CD-ROM.
Question Bank
ExamView CD–ROM
Questions
ExamView CD–ROM

3. SPANISH ASSESSMENT                                                                 Targeted Resources

Assessment and Practice                                                             Spanish Additional Practice:
Investigation 2
Use these resources, in Spanish, to assess students’ understanding of the
mathematics taught in this investigation, to create additional worksheets,          Spanish Skill Practice:
or to provide additional practice problems.                                         Probability
These questions can also be found on the ExamView CD-ROM.                           Spanish Check–Up
Spanish Multiple Choice
Items
Spanish Question Bank
Spanish Assessment
Spanish ExamView CD–ROM
Questions

How Likely Is It?, Investigation 3, Problem 3.1                                                              Completed
Designing a Spinner
Mathematical Goal                                                     National Standards         State Standards
• Develop strategies for finding experimental and theoretical         NAEP                         6NJ 4.4.B.4,
probabilities in situations involving spinners.                        D4b, D4c, D4d, D4j        6NJ 4.4.B.1
CAT6
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,                      LV16.15
Student Activity CD-ROM, www.PHSchool.com                CTBS
Materials:   Student notebooks, Overhead projector, Bobby pins or        LV16.53
ITBS
paper clips, Angle rulers
LV12.PS
Pacing:      45 minutes
S10
Int2.DSP, TV
LV16.15

1. LAUNCH (10 minutes)                                                                Targeted Resources

Discuss the three spinners in the Getting Ready. Discuss Kalvin’s idea              Transparency 3.1A Getting
• Which spinner will give Kalvin the best chance of going to bed at             Transparency 3.1B Kalvin's
Labsheet 3.1 Kalvin's Spinner
Discuss the need for students to make sure that the way they conduct the
experiment won’t affect, or bias, the outcome. Ask the class:
• What kinds of things can happen to affect the data you gather in
this problem?
• How should you spin the pointer to be sure you have a fair trial?
Explain to students that Kalvin decided to make his own spinner to use in
determining his bedtime. If possible, demonstrate a couple of spins on
Transparency 3.1B. Record your data, demonstrating how to keep a tally
for each spin. After spinning two or three times, ask whether you have a
large enough sample. Distribute Labsheet 3.1 to each pair or group of
students.
2. EXPLORE (20 minutes)                                                                   Targeted Resources
• Is there a tendency for the pointer to land in the same area each
time?
Students should find that Kalvin’s chances of going to bed at 11:00 are
about (37.5%), the same as for 9:00. Ask them to compare their
experimental and theoretical probabilities.
Focus on how students are finding the theoretical probabilities. Many
students will want to compare the number of sections rather than the
relative sizes of the sections. If students are struggling with thinking about
the sizes of the angles of the sections, ask them the following question:
• Look at one of these 9:00 sections. Is the pointer as likely to land
there as in the 11:00 section?
• What is the angle size associated with each section?
Encourage students to be specific about why the 11:00 section is more
likely than the 9:00 section.

3. SUMMARIZE (15 minutes)                                                                 Targeted Resources

Discuss the data that groups collected and their answers to the questions.              Transparency 3.1C Another
Combine the data for all the groups, recalculating the fractions or percents            Spinner
after each groups’ data is added. Students should conclude that Kalvin’s
chances of going to bed at 11:00 are about 38%. To check understanding
of determining theoretical probabilities, put up Transparency 3.1C.
• If Kalvin uses this spinner, what is the probability that he will go to
bed at 10:00? At 11:00? Why?

4. ASSIGNMENT GUIDE                                                                       Targeted Resources

Core 1, 3, 4, 11–17                                                                     Labsheet 3ACE Exercise 1
Other Applications 2, 5, 6; Connections 18–21; Extensions 31, 32                        Version
Labsheet 3ACE Exercise 1 is provided if Exercise 1 is assigned.
exercises, see the CMP Special Needs Handbook.
Connecting to Prior Units 11–16, 19–21: Bits and Pieces I

How Likely Is It?, Investigation 3, Problem 3.2                                                                  Completed
Making Decisions
Mathematical Goals                                                        National Standards         State Standards
NAEP                         6NJ 4.4.B.5,
• Analyze probability situations.
D4b, D4c, D4d, D4j        6NJ 4.4.D.1
• Use probability to make decisions.                                      CAT6
• Decide whether the probability situations are fair or unfair.              LV16.15
CTBS
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,                          LV16.53
Student Activity CD-ROM, www.PHSchool.com                   ITBS
Materials:    Student notebooks, Overhead projector, Paper clips or          LV12.PS
bobby pins, number cubes, coins, colored blocks, and        S10
bags (optional)                                                Int2.DSP, TV
Pacing:       45 minutes                                                     LV16.15
1. LAUNCH (10 minutes)                                                              Targeted Resources

Discuss the Getting Ready question with students. They will quickly see             Transparency 3.2 Getting
that tossing a coin once to decide which of three people should go to the           Ready
office makes no sense. Students may have suggestions for how to use a
number cube (e.g., 1 and 2 represent Billie, 3 and 4 represent Evo, and 5
and 6 represent Carla), colored blocks in a bag or a spinner. In each case,
it is important that there are three equally likely outcomes. Help students
to see this and to express precisely what those outcomes are.
Suggested Questions When students make suggestions, ask them:
• What are the outcomes?
• Are these outcomes equally likely?
Make sure students understand that they are to evaluate each student’s
suggestion in each part of the problem. Allow them to work in pairs or
small groups. You may want to have blank spinners (Labsheet 3.2),
number cubes, coins, colored blocks, and bags available for students who
want to test their ideas.

2. EXPLORE (20 minutes)                                                             Targeted Resources

As students work, listen for whether they are being precise about the               Labsheet 3.2 Blank Spinner
outcomes in each suggestion. For example, in Questions B, Tony’s
suggestion is that an outcome of 3 or 6 corresponds to his team while an
outcome of 1, 2, 4, or 5 corresponds to Meda’s team. These are two
outcomes and the latter is more likely.
Pay attention to whether students are able to fill in the missing gap in Sal’s
suggestion in Question C. If the number from the first bag is the tens digit
and the number from the second bag is the units digit, then the suggestion
is a fair way to choose the winner. This is a different simulation from what
has been done in the unit so far. Students may need help thinking about
this. If a student has the number 0, then zero must be chosen from both
bags. So 00 represents the number 0 and 05 (choosing 0 from the first
bag and 5 from the second bag) represents the number 5.

3. SUMMARIZE (15 minutes)                                                           Targeted Resources

Base your summary on what students did and did not understand in the
problem. Make sure that students can specify the outcomes in each
suggestion and determine whether each is a fair way to make a decision.
If your students struggled, you will want to spend time carefully discussing
the various outcomes and ways to determine whether they are fair.
Suggested Questions You might ask a series of questions like the
following:
• What are the possible outcomes of when you roll a number cube?
(1, 2, 3, 4, 5, or 6)
• What are the possible outcomes of Ava’s suggestion? (There are 5:
kickball, soccer, baseball, dodge ball, or roll again.)
• Are all five of these outcomes equally likely? (No, roll again is most
2
likely. The probability is     .)
6
• Are all four of the games equally likely? (Yes, on any roll, the
1
probability of each game is     .)
6
• Is this a fair way to make the decision? (Yes, because the
probability for each game is equally likely.)
Have students talk about Huey’s suggestion in Question C. Here, the
outcomes are not equally likely. The easiest way to see this is to note that
the smallest possible sum is 10. Therefore, students 1 through 9 would
not even have a chance.
• How does Student 60 win? (If all ten number cubes came up with a
6.)
• How does Student 50 win? (Possible answers: All ten number
cubes could come up 5 or eight of them could come up 5, one
comes up 6, and one comes up 4.)
• How does Student 10 win? (If all the number cubes come up with a
1.)
• Does every student have an equal chance of winning? (No; Student
50 has lots of ways to win while Student 60 has only one and
Student 1 has no chance.)
• What is the probability of Student 1 winning? [P(Student 1 wins) =
0]
Have students discuss Sal’s method. You may want to demonstrate how
to choose and determine the numbers.
• Is it possible to choose each number from 0 to 59? [Yes; to choose
0 you must choose zero from both bags. So 00 represents the
number 0 and 05 (choosing 0 from the first bag and 5 from the
second bag) represents the number 5. You may want to make an
organized list to show that choosing each number is equally likely.]
Check for Understanding
Ask students to pick one question from Questions A–C. Determine one
more way to make the decision fairly.

4. ASSIGNMENT GUIDE                                                               Targeted Resources
Core 8, 9                                                                         Answers to ACE and
Mathematical Reflections
Other Applications 7; Connections 22–26; unassigned choices from
previous problems
exercises, see the CMP Special Needs Handbook.
Connecting to Prior Units 22 –25; Bits and Pieces I

How Likely Is It?, Investigation 3, Problem 3.3                                                          Completed
Scratching Spots
Mathematical Goals                                                      National Standards         State Standards
• Develop strategies for finding both experimental (using a             NAEP                         6NJ 4.4.B.4,
simulation of a scratch-off prize card) and theoretical                  D4b, D4c, D4d, D4j        6NJ 4.4.B.2
CAT6
probabilities (analyzing the scratch-off prize card).
LV16.15
Vocabulary: simulation                                                  CTBS
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,                        LV16.53
ITBS
Student Activity CD-ROM, www.PHSchool.com
LV12.PS
Materials:   Student notebooks, Paper clips or bobby pins, number
S10
cubes, coins, colored blocks, and bags (optional)             Int2.DSP, TV
Pacing:      45 minutes                                                    LV16.15

1. LAUNCH (10 minutes)                                                                  Targeted Resources

Describe the situation to the students and discuss possible ways of finding
the experimental probability of winning a prize with one scratch-off prize
card. This will involve doing a simulation.
Collect various simulation suggestions. Analyze their suggestions with
them. NOTE: You may want the class to try different simulations, but it
might be easier if the class uses the same simulation. One way to
simulate the game is to put two red blocks, one white block, one yellow
block, and one blue block in an opaque container, then choose two blocks
at a time. When you choose the two red blocks, you simulate scratching
off the two matching spots. Another possibility is to have five students act
as the “spots” in the front of the room. They hold tags that are hidden from
the rest of the class. Two students hold tags with matching symbols, and
three hold tags with three other non-matching symbols. A seated student
chooses two of the five students to reveal their tags. After each guess,
students mix up their tags.
Let the class explore the problem in groups of two to four students. (The
way you decide to arrange it depends on whether you do the same
simulation as a class or if students all do different simulations.)

2. EXPLORE (20 minutes)                                                                 Targeted Resources

If you let the groups decide on their own simulation, then each group must            Labsheet 3.2 Blank Spinner
decide for themselves how to collect the data and when they have
collected enough data to find a reasonable experimental probability. Be
sure that each method is mathematically equivalent to the scratch-off prize
card. Allow the groups to plan and carry out their simulations, collect their
data, and find the experimental probability.
When they have completed their experiment, students should move on to
Question B in which they analyze the outcomes and find theoretical
probabilities and then do the rest of the problem.

3. SUMMARIZE (15 minutes)                                                               Targeted Resources

Have the groups share their solutions and strategies with the class. If
groups chose their own simulation, discuss the different methods.
• When you were finding the theoretical probability, what strategy did
you use to find all the outcomes?
A list shows that there are ten ways to scratch off the spots. One of these
1
ten combinations is the matching pair, so the probability of winning is      .
10
Notice that the order in which the spots in a pair are chosen does not
matter. Students may want to list each pair twice with the letters reversed
(to represent two cards). There would be 20 pairs (2 of which are winners)
2     1
which also gives a probability of      or    .
20    10

4. ASSIGNMENT GUIDE                                                                 Targeted Resources
Core 10, 27                                                                         Answers to ACE and
Mathematical Reflections
Other Connections 28–30; Extensions 33–35; unassigned choices from
previous problems
Special Needs Handbook.
Connecting to Prior Units 29: Covering and Surrounding

How Likely Is It?, Investigation 3                                                                         Completed
Completing the Investigation

Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, Student Activity CD-ROM, www.PHSchool.com
Pacing:     65 minutes

1. MATHEMATICAL REFLECTIONS (25 minutes)                                            Targeted Resources

Students reflect on and summarize their learning at the end of the                  Answers to ACE and
investigation.                                                                      Mathematical Reflections

2. ASSESSMENT (40 minutes)                                                          Targeted Resources

Investigation 3
Use these resources to assess students’ understanding of the
mathematics taught in this investigation, to create additional worksheets,          Partner Quiz
or to provide additional practice problems.                                         Multiple Choice Items
Question Bank
These questions can also be found on the ExamView CD-ROM.
ExamView CD–ROM
Questions
ExamView CD–ROM

3. SPANISH ASSESSMENT                                                               Targeted Resources

Assessment and Practice                                                             Spanish Additional Practice:
Investigation 3
Use these resources, in Spanish, to assess students’ understanding of the
mathematics taught in this investigation, to create additional worksheets,          Spanish Partner Quiz
or to provide additional practice problems.                                             Spanish Multiple Choice
These questions can also be found on the ExamView CD-ROM.                               Items
Spanish Question Bank
Spanish Assessment
Spanish ExamView CD–ROM
Questions

How Likely Is It?, Investigation 4, Problem 4.1                                                                  Completed
Genetic Traits
Mathematical Goals                                                        National Standards         State Standards
• Use class and survey data to find the experimental                      NAEP                         6NJ 4.4.B.2,
probabilities for certain genetic traits.                                  D4b, D4d, D4e, D4f,       6NJ 4.4.B.5
D4g, D4j
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,                       CAT6
Student Activity CD-ROM, www.PHSchool.com                      LV16.15
Materials:    Student notebooks, Overhead projector                       CTBS
Pacing:       45 minutes                                                     LV16.53
ITBS
LV12.PS
S10
Int2.DSP, TV
LV16.15

1. LAUNCH (10 minutes)                                                                    Targeted Resources

Discuss each of the characteristics described in the Student Edition with               Transparency 4.1 Generic
your class. Collect your class data for these four traits and record the data           Traits
in a table like the one in the Student Edition. Students will use this data to
complete Question B.
Let students work in groups of two or three.

2. EXPLORE (20 minutes)                                                                   Targeted Resources

Parts (3) and (4) of Questions B are proportional scaling questions that
some students at this level may find difficult. Since they have not formally
been taught the strategies needed to solve these problems, note student
reasoning for these questions. Help students make connections to their
work with equivalent fractions in Bits and Pieces I, their work with percents
in Bits and Pieces III, and to any other novel solution strategies that
students in your class may offer.

3. SUMMARIZE (15 minutes)                                                                 Targeted Resources

As a class, discuss Question B, part (1). Ask students about one of the
traits that had a relatively high probability of occurring within your
3
classroom (for example, say attached earlobe had a probability of        .)
4
• Do you feel confident that a student selected at random from this
school will have an attached earlobe?
You want students to realize that their classroom data serves as an
estimate of the distribution of the trait in the larger school population. You
• What could we do to increase our confidence that a student
selected at random from this school will have an attached earlobe?
Students may see that a larger sample would be more predictive and may
Discuss this last point in the context of Question C, part (2) about the
national data.
• Is our class representative of the nation as a whole?

4. ASSIGNMENT GUIDE                                                                      Targeted Resources
Core 1, 2                                                                              4ACE Exercise 2 Adapted
Version
Other Connections 13–17
exercises, see the CMP Special Needs Handbook.

How Likely Is It?, Investigation 4, Problem 4.2                                                                 Completed
Tracing Traits
Mathematical Goals                                                       National Standards         State Standards
NAEP                         6NJ 4.4.B.4,
• Appreciate the power of probability for making predictions.
D4b, D4d, D4e, D4f,       6NJ 4.4.B.2
• Develop strategies (for example, using a chart or table) for              D4g, D4j
finding theoretical probabilities involving genetics.                  CAT6
LV16.15
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,                      CTBS
Student Activity CD-ROM, www.PHSchool.com                      LV16.53
Materials:   Student notebooks, Overhead projector                       ITBS
Pacing:      45 minutes                                                     LV12.PS
S10
Int2.DSP, TV
LV16.15

1. LAUNCH (10 minutes)                                                                   Targeted Resources

Discuss what a geneticist is and does. The Student Edition presents a                  Transparency 4.2 Tracing
summary of the genetics information students need to know. Have                        Traits
students read the genetics information in the Student Edition, or study the
information yourself and present it to them. They may already be familiar
with the term gene. An allele is a certain form of a gene and is the more
correct term to use when discussing the different forms of a gene, such as
dominant and recessive. You may want to recommend further reading for
students who are especially interested in this topic.
As you talk about the example of Bonnie and Evan, emphasize that the
four possibilities are equally likely, and therefore we can conclude that the
2   1
probability of the baby having attached earlobes is  or , and the
4   2
1
probability of having nonattached earlobes is also .
2
Students may wonder how someone could figure out what his or her
parents’ alleles were. One possibility: suppose that Dasan has attached
earlobes and both of his parents do not. Dasan’s earlobe alleles must be
ee, which means that both of his parents must have at least one e allele.
Since both parents have nonattached earlobes, each must also have an
E, giving them both Ee.
Have students work in pairs or groups of three.

2. EXPLORE (20 minutes)                                                           Targeted Resources

Have students analyze each family situation in the problem. Encourage
groups to make a chart as shown in the example about Bonnie and Evan
to help them list the possibilities.

3. SUMMARIZE (15 minutes)                                                         Targeted Resources

Review the groups’ charts. Make sure students understand how to
analyze the possibilities. Any combination containing the dominant allele E
1
means the child will have nonattached earlobes. There is only a
4
probability that the child will inherit the ee combination and have attached
earlobes. In the general population, having attached earlobes is more rare
than having nonattached earlobes.
After Question C, you may want to ask:
• What if Eileen’s parents had nonattached earlobes? Can you find
the probability that Eileen has nonattached earlobes?
Students may be interested in knowing that these charts or tables are
called Punnett Squares in biology. It is an analysis tool for simple genetic
inheritance problems and is a common topic in middle–school biology.

4. ASSIGNMENT GUIDE                                                               Targeted Resources
Core 3–7, 18                                                                      Answers to ACE and
Mathematical Reflections
Other Connections 19; Extensions 27; unassigned choices from previous
problems
Special Needs Handbook.
Connecting to Prior Units 19: Shapes and Designs

How Likely Is It?, Investigation 4, Problem 4.3                                                          Completed
Roller Derby
Mathematical Goals                                                     National Standards         State Standards
NAEP                         6NJ 4.4.D.1,
• Appreciate the power of probability in determining strategies
D4b, D4d, D4e, D4f,       6NJ 4.4.B.5
for winning a game.
D4g, D4j
• Develop strategies (i.e. using a chart or table) for finding         CAT6
both experimental and theoretical probabilities of winning a            LV16.15
game.                                                                CTBS
LV16.53
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM,                    ITBS
Student Activity CD-ROM, www.PHSchool.com                    LV12.PS
Materials:   Student notebooks, Overhead projector, Two number         S10
cubes, Game markers                                          Int2.DSP, TV
Pacing:      45 minutes                                                   LV16.15

1. LAUNCH (10 minutes)                                                                 Targeted Resources

Discuss the directions for the game. Illustrate the game using                       Transparency 4.3 Roller
Transparency 4.3. Explain that if the column corresponding to a roll is              Derby
empty, no marker is removed.
• What is your best guess at a strategy for placing your markers so
that you will be able to remove them before your opponents remove
their markers?
Try to remain neutral when students make their suggestions. As they play
they will become more aware of which strategies work.
Remember the goal of the game is to remove all markers.
Refer students to the Roller Derby rules. Divide the class into teams of
two, and then pair teams to play against one another. Tell students to
discuss strategies with their teammates, not their opponents.

2. EXPLORE (20 minutes)                                                                Targeted Resources

While students play the game, observe the strategies they use and the                Labsheet 4.3
knowledge they display about sums of the numbers on the cubes. Take
note of the strategies they use.

3. SUMMARIZE (15 minutes)                                                              Targeted Resources

After students have played the game a few times and have had time to
• What was the first strategy your team used for placing markers in
the 12 columns?
• Did you change your strategy the second time you played? Why or
why not? If you were going to play the game again, what strategy
would you use now? Why?
Question C asks students to list all the possible outcomes when they roll
two number cubes. The list should allow students to determine all the
possible sums and to see all the ways each sum can occur.
• What strategies did you use for making your list?
One way to see all 36 sums is to use a table with possible outcomes of
one number cube down the side and possible outcomes of the other
number cube along the top. Once the class has made the table, you may
• When you roll two number cubes, how many different number pairs
are possible? Are these pairs equally likely?
• How many different sums are possible? Are these sums equally
likely?
• When you roll two number cubes, what is the probability that the
sum will be 12? That the sum will be 1? That the sum will be 0?
• How is the table in this problem like the chart in Problem 4.2?

4. ASSIGNMENT GUIDE                                                              Targeted Resources
Core 8–12                                                                       Answers to ACE and
Mathematical Reflections
Other Connections 2–26; unassigned choices from previous problems
Special Needs Handbook.
Connecting to Prior Units 20–23: Prime Time

How Likely Is It?, Investigation 4                                                                      Completed
Completing the Investigation
National Standards
NAEP
Technology: ExamView CD-ROM, TeacherEXPRESS CD-ROM, Student Activity CD-ROM,                  D4b, D4d, D4e, D4f,
www.PHSchool.com                                                                  D4g, D4j
Pacing:     40 minutes                                                                      CAT6
LV16.15
CTBS
LV16.53
ITBS
LV12.PS
S10
Int2.DSP, TV
LV16.15

1. ASSESSMENT (40 minutes)                                                       Targeted Resources

Investigation 4
Use these resources to assess students’ understanding of the
mathematics taught in this investigation, to create additional worksheets,      Skill Practice: Experimental
or to provide additional practice problems.                                     and Theoretical Probability
Multiple Choice Items
These questions can also be found on the ExamView CD-ROM.
Question Bank
ExamView CD–ROM
Questions
ExamView CD–ROM

2. SPANISH ASSESSMENT                                                            Targeted Resources
Assessment and Practice                                                      Spanish Additional Practice:
Investigation 4
Use these resources, in Spanish, to assess students’ understanding of the
mathematics taught in this investigation, to create additional worksheets,   Spanish Skill Practice:
or to provide additional practice problems.                                  Experimental and Theoretical
Probability
These questions can also be found on the ExamView CD-ROM.
Spanish Multiple Choice
Items
Spanish Question Bank
Spanish Assessment