# Continues Probability Distributions and Theirs Applications by ukc14135

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```									                                                TENNESSEE MATHEMATICS UNIT

COVER SHEET                          >> Go to Content Clarification        >> Go to Performance Assessment              > Go to Curriculum Integrations

UNIT TITLE          Probability: The Chances You Take

AUTHOR              Allison Srubas, Roger Williams University

CONTENT AREA        Mathematics                                                   TOPIC                   Probability

NUMBER OF
TIME FRAME          8-9 days                                                                              7
LESSONS

TARGETED
STANDARDS/          Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases.
BENCHMARKS

Students are assigned a casino game to play and research. As a group, the students must create a brochure for gamblers anonymous. The
SUMMARY OF
brochure must include a description of the game played, the students’ results while playing the game, the probability rules and values in the
PERFORMANCE
game, and a report that cautions gamblers who play the game based on the probabilities that the students determined. On the final day, the
ASSESSMENT
students must present their brochures to the rest of the class.

PROBABILITY: THE CHANCES YOU TAKE                                                                        BY   ALLISON SRUBAS                  Page 1
OVERVIEW OF STUDENT LEARNING ACTIVITIES
>>> LESSON 1
During this lesson, students experience the chances of winning the Connecticut Cash 5 lottery. They also have the opportunity to
ENGAGEMENT:            express their ideas and feelings that are associated with the topic of probability. The lottery part of this lesson is referenced later in
I Won the Lottery       the unit.

>>> LESSON 2
In this lesson, students make predictions about the number of each color M&M in a bag of 50 M&M’s. They work in pairs to actually
count the M&M’s in a bag and record their results. Then they make a graph to represent their findings. There is a class discussion
EXPLORATION:            based on their results and students share their graphs. Also, during the class discussion, the students discuss the probability of
M & M Probability
pulling each color M&M out of the bag with guidance by the teacher.

>>> LESSON 3
During this lesson, students explore their ideas about sample space and probability. In groups, the students move to different
stations throughout the class period. They work with a coin, a spinner with four equally colored parts, one red, one blue, one green,
EXPLORATION:            and one yellow, a die, a jar of marbles with 3 red marbles, 3 blue marbles, 4 green marbles, and 5 yellow marbles, and a deck of
Probability Stations
playing cards. At the end of the lesson, there is a class discussion on the student’s findings.

>>> LESSON 4

EXPLANATION:             This is a direct instruction lesson. The topic is sample space in probability.
What is Sample Space

>>> LESSON 5

EXPLANATION:            The teacher uses direct instruction to explain probability distribution and how it relates to the activity two days ago. Students are
What is Probability      given a homework assignment to figure out three probability rules.
Distribution

>>> LESSON 6

EXTENSION:            In this lesson, the class explores the chances of winning the Connecticut Cash 5 lottery.
the Lottery

>>> LESSON 7

EVALUATION:            In this performance assessment, students make brochures for gamblers anonymous based on ―games of chance‖.
Brochures for Gamblers
Anonymous

PROBABILITY: THE CHANCES YOU TAKE                                                                            BY   ALLISON SRUBAS                   Page 2
CONTENT CLARIFICATION                                   >> Go to Cover          >> Go to                 > Go to Curriculum         >> Go to Lesson…
Targeted Standards and Benchmarks                    Sheet                   Performance              Integrations               1 | 2 | 3| 4 |5 |6| 7|8
Assessment
TENNESSEE

Go To…
>>>K, 1, 2, 3, 4, 5, 6, 7, 8,   Learning expectation 4.7
and 9-12 COURSES
Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases
Go to
State Standards…

Data Analysis and Probability Standard for Grades 9–12
PRINCIPLES AND                   Understand and apply basic concepts of probability.
STANDARDS FOR SCHOOL                 Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in
MATHEMATICS                            simple cases. Use simulations to construct empirical probability distributions.
 Compute and interpret the expected value of random variables in simple cases.
 Understand the concepts of conditional probability and independent events.
 Understand how to compute the probability of a compound event
Go to NCTM…

The Mathematical World
D. Uncertainty

BENCHMARKS FOR SCIENCE                  Even when there are plentiful data, it may not be obvious what mathematical model to use to make predictions from them or
LITERACY                                 there may be insufficient computing power to use some models.

   When people estimate a statistic, they may also be able to say how far off the estimate might be.

   The middle of a data distribution may be misleading-when the data are not distributed symmetrically, or when there are
extreme high or low values, or when the distribution is not reasonably smooth.

   The way data are displayed can make a big difference in how they are interpreted.

Go to Benchmarks…                       Both percentages and actual numbers have to be taken into account in comparing different groups; using either category by

   Considering whether two variables are correlated requires inspecting their distributions, such as in two-way tables or
scatterplots. A believable correlation between two variables doesn't mean that either one causes the other; perhaps some
other variable causes them both or the correlation might be attributable to chance alone. A true correlation means that
differences in one variable imply differences in the other when all other things are equal.

PROBABILITY: THE CHANCES YOU TAKE                                                                             BY   ALLISON SRUBAS                    Page 3
   For a well-chosen sample, the size of the sample is much more important than the size of the population. To avoid intentional
or unintentional bias, samples are usually selected by some random system.

   A physical or mathematical model can be used to estimate the probability of real-world events.

PROBABILITY: THE CHANCES YOU TAKE                                                                      BY   ALLISON SRUBAS                 Page 4
CONTENT CLARIFICATION
Associated K-12 Mathematics Standards and Benchmarks

PRINCIPLES AND STANDARDS FOR SCHOOL MATHEMATICS
Go to NCTM…

Understand and apply basic concepts of
K-2 EXPECTATIONS
probability

Understand and apply basic concepts of        • describe events as likely or unlikely and discuss the degree of likelihood using such words as
probability                                     certain, equally likely, and impossible;
3-5 EXPECTATIONS                                                          • predict the probability of outcomes of simple experiments and test the predictions;
• understand that the measure of the likelihood of an event can be represented by a number
from 0 to 1.

Understand and apply basic concepts of        • understand and use appropriate terminology to describe complementary and mutually
probability                                     exclusive events;
• use proportionality and a basic understanding of probability to make and test conjectures
6-8 EXPECTATIONS                                                            about the results of experiments and simulations;
• compute probabilities for simple compound events, using such methods as organized lists,
tree diagrams, and area models.

Understand and apply basic concepts of        • understand the concepts of sample space and probability distribution and construct sample
probability                                     spaces and distributions in simple cases;
• use simulations to construct empirical probability distributions;
9-12 EXPECTATIONS                                                         • compute and interpret the expected value of random variables in simple cases;
• understand the concepts of conditional probability and independent events;
• understand how to compute the probability of a compound event.

Go to Benchmarks…                                                    BENCHMARKS FOR SCIENCE LITERACY

PROBABILITY: THE CHANCES YOU TAKE                                                                            BY   ALLISON SRUBAS                      Page 5
   Some things are more likely to happen that others. Some events can be predicted well and some cannot. Sometimes people
aren't sure what will happen because they don't know everything that might be having an effect.
K-2 BENCHMARKS
   Often a person can find out about a group of things by studying just a few of them.

   Some predictions can be based on what is known about the past, assuming that conditions are pretty much the same now.

   Statistical predictions (as for rainy days, accidents) are typically better for how many of a group will experience something
than for which members of the group will experience it-and better for how often something will happen than for exactly
when.

   Summary predictions are usually more accurate for large collections of events than for just a few. Even very unlikely events
may occur fairly often in very large populations.
3-5 BENCHMARKS
   Spreading data out on a number line helps to see what the extremes are, where they pile up, and where the gaps are. A
summary of data includes where the middle is and how much spread is around it.

   A small part of something may be special in some way and not give an accurate picture of the whole. How much a portion of
something can help to estimate what the whole is like depends on how the portion is chosen. There is a danger of choosing
only the data that show what is expected by the person doing the choosing.

   Events can be described in terms of being more or less likely, impossible, or certain.

   How probability is estimated depends on what is known about the situation. Estimates can be based on data from similar
conditions in the past or on the assumption that all the possibilities are known.

   Probabilities are ratios and can be expressed as fractions, percentages, or odds.

   The mean, median, and mode tell different things about the middle of a data set.
6-8 BENCHMARKS
   Comparison of data from two groups should involve comparing both their middles and the spreads around them.

   The larger a well-chosen sample is, the more accurately it is likely to represent the whole. But there are many ways of
choosing a sample that can make it unrepresentative of the whole.

PROBABILITY: THE CHANCES YOU TAKE                                                                         BY   ALLISON SRUBAS                 Page 6
   Even when there are plentiful data, it may not be obvious what mathematical model to use to make predictions from them or
there may be insufficient computing power to use some models.

   When people estimate a statistic, they may also be able to say how far off the estimate might be.

   The middle of a data distribution may be misleading-when the data are not distributed symmetrically, or when there are
extreme high or low values, or when the distribution is not reasonably smooth.

   The way data are displayed can make a big difference in how they are interpreted.

   Both percentages and actual numbers have to be taken into account in comparing different groups; using either category by

   Considering whether two variables are correlated requires inspecting their distributions, such as in two-way tables or
scatterplots. A believable correlation between two variables doesn't mean that either one causes the other; perhaps some
other variable causes them both or the correlation might be attributable to chance alone. A true correlation means that
differences in one variable imply differences in the other when all other things are equal.

   For a well-chosen sample, the size of the sample is much more important than the size of the population. To avoid intentional
or unintentional bias, samples are usually selected by some random system.
9-12 BENCHMARKS
   A physical or mathematical model can be used to estimate the probability of real-world events.

PROBABILITY: THE CHANCES YOU TAKE                                                                        BY   ALLISON SRUBAS               Page 7
CONTENT CLARIFICATION
Related Content Knowledge

Understand and apply basic concepts of probability
In high school, students can apply the concepts of probability to predict the likelihood of an event by constructing probability
distributions for simple sample spaces. Students should be able to describe sample spaces such as the set of possible outcomes when
four coins are tossed and the set of possibilities for the sum of the values on the faces that are down when two tetrahedral dice are
rolled. High school students should learn to identify mutually exclusive, joint, and conditional events by drawing on their knowledge
of combinations, permutations, and counting to compute the probabilities associated with such events. They can use their
understandings to address questions such as those in following series of examples. The diagram below shows the results of a two-
question survey administered to 80 randomly selected students at Highcrest High School.
PRINCIPLES AND
STANDARDS FOR SCHOOL
MATHEMATICS

· Of the 2100 students in the school, how many would you expect to play a musical instrument?
· Estimate the probability that an arbitrary student at the school plays on a sports team and plays a musical instrument. How is this
related to estimates of the separate probabilities that a student plays a musical instrument and that he or she plays on a sports team?
· Estimate the probability that a student who plays on a sports team also plays a musical instrument. High school students should
learn to compute expected values. They can use their understanding of probability distributions and expected value to decide if the
following game is fair:
Go to NCTM…               You pay 5 chips to play a game. You roll two tetrahedral dice with faces numbered 1, 2, 3, and 5, and you win the sum of the values on
the faces that are not showing. Teachers can ask students to discuss whether they think the game is fair and perhaps have the
students play the game a number of times to see if there are any trends in the results they obtain. They can then have the students
analyze the game. First, students need to delineate the sample space. The outcomes are indicated in figure 7.25. The numbers on the
first die are indicated in the top row. The numbers on the second die are indicated in the first column. The sums are given in the
interior of the table. Since all outcomes are equally likely, each cell in the table has a probability of 1/16 of occurring.

PROBABILITY: THE CHANCES YOU TAKE                                                                        BY   ALLISON SRUBAS                  Page 8
Fig. 7.25. The sample space for the roll of two tetrahedral dice

Students can determine that the probability of a sum of 2 is 1/16; of a 3, 1/8; of a 4, 3/16; of a 5, 1/8; of a 6, 3/16; of a 7, 1/8; of an 8,
1/8; of a 10, 1/16. The expected value of a player’s "income" in chips from rolling the dice is

If a player pays a five-chip fee to play the game, on average, the player will win 0.5 chips. The game is not statistically fair, since the
player can expect to win. Students can also use the sample space to answer conditional probability questions such as "Given that the
sum is even, what is the probability that the sum is a 6?" Since ten of the sums in the sample space are even and three of those are 6s,
the probability of a 6 given that the sum is even is 3/10. The following situation, adapted from Coxford et al. (1998, p. 469), could
give rise to a very rich classroom discussion of compound events. In a trial in Sweden, a parking officer testified to having noted the
position of the valve stems on the tires on one side of a car. Returning later, the officer noted that the valve stems were still in the
same position. The officer noted the position of the valve stems to the nearest "hour." For example, in figure 7.26 the valve stems are
at 10:00 and at 3:00. The officer issued a ticket for overtime parking. However, the owner of the car claimed he had moved the car
and returned to the same parking place. The judge who presided over the trial made the assumption that the wheels move
independently and the odds of the two valve stems returning to their previous "clock" positions were calculated as 144 to 1. The driver
was declared to be innocent because such odds were considered insufficient—had all four valve stems been found to have returned to
their previous positions, the driver would have been declared guilty (Zeisel 1968).

Fig. 7.26. A diagram of tires with valves at the 10:00 and 3:00 positions
PROBABILITY: THE CHANCES YOU TAKE                                                                             BY   ALLISON SRUBAS                    Page 9
Given the assumption that the wheels move independently, students could be asked to assess the probability that if the car is moved,
two (or four) valve stems would return to the same position. They could do so by a direct probability computation, or they might
design a simulation, either by programming or by using spinners, to estimate this probability. But is it reasonable to assume that two
front and rear wheels or all four wheels move independently? This issue might be resolved empirically. The students might drive a car
around the block to see if its wheels do rotate independently of one another and decide if the judge’s assumption was justified. They
might consider whether it would be more reasonable to assume that all four wheels move as a unit and ask related questions: Under
what circumstances might all four wheels travel the same distance? Would all the wheels travel the same distance if the car was
driven around the block? Would any differences be large enough to show up as differences in "clock" position? In this way, students
can learn about the role of assumptions in modeling, in addition to learning about the computation of probabilities. Students could
also explore the effect of more-precise measurements on the resulting probabilities. They could calculate the probabilities if, say,
instead of recording markings to the nearest hour on the clock face, the markings had been recorded to the nearest half or quarter-
hour. This line of thinking could raise the issue of continuous distributions and the idea of calculating probabilities involving an
interval of values rather than a finite number of values. Some related questions are, How could a practical method of obtaining more-
precise measurements be devised? How could a parking officer realistically measure tire-marking positions to the nearest clock half-
hour? How could measurement errors be minimized? These could begin a discussion of operational definitions and measurement
processes. Students should be able to investigate the following question by using a simulation to obtain an approximate answer:
How likely is it that at most 25 of the 50 people receiving a promotion are women when all the people in the applicant pool from
which the promotions are made are well qualified and 65% of the applicant pool is female? Those students who pursue the study of
probability will be able to find an exact solution by using the binomial distribution. Either way, students are likely to find the result
rather surprising.

In the very earliest grades, learning can begin that will eventually lead to students’ having a good grasp of everyday statistics. Children
at this level can array things they collect by size and weight and then ask questions about them, such as which one is in the middle,
how many are the same, and so forth. From there they can go on to make simple pictographs that show how a familiar variable is
distributed and again ask questions about the distribution. They can begin to find out about sampling in the context, say, of reporting
on the kinds of stones found on the school playground. Children will be keeping track of many different phenomena, some of which
they will come to see have patterns of one kind or another. From time to time they should be asked, working in small groups, to
review their records to see if they can figure out if they can predict some future events. The most important part of such exercises is
K-2 EXPECTATIONS
that the students give reasons for their predictions and for not being able to make predictions. Of course they should follow up to see
if they were right or not.

By the end of the 2nd grade, students should know that
· Some things are more likely to happen that others. Some events can be predicted well and some cannot. Sometimes people aren’t
sure what will happen because they don’t know everything that might be having an effect.
· Often a person can find out about a group of things by studying just a few of them.

The questions about data only explored in the earliest grades can now be made into formal questions. Data distributions should be
made of many familiar features and quantities: heights, weights, number of siblings, or kinds of pets. The important thing to
emphasize at this level is the kind of questions that can be posed and answered by a data distribution: "Where is the middle?" is a
3-5 EXPECTATIONS
useful question; "What is the average?" probably is not. Because there is a persistent misconception, even in adults, that means are
good representations of whole groups, it is especially important to draw students’ attention to the additional questions, "What are the
largest and smallest values?" and "How much do the data spread on both sides of the middle?" Children also should be invited to

PROBABILITY: THE CHANCES YOU TAKE                                                                          BY   ALLISON SRUBAS                 Page 10
suggest some circumstances in their studies that might bias the results— for example, making "random" measurements of student
height just as a basketball team comes along or collecting only the insects that were easy to spot.

By the end of the 5th grade, students should know that
· Some predictions can be based on what is known about the past, assuming that conditions are pretty much the same now.
· Statistical predictions (as for rainy days, accidents) are typically better for how many of a group will experience something than for
which members of the group will experience it— and better for how often something will happen than for exactly when.
· Summary predictions are usually more accurate for large collections of events than for just a few. Even very unlikely events may
occur fairly often in very large populations.
· Spreading data out on a number line helps to see what the extremes are, where they pile up, and where the gaps are. A summary of
data includes where the middle is and how much spread is around it.
· A small part of something may be special in some way and not give an accurate picture of the whole. How much a portion of
something can help to estimate what the whole is like depends on how the portion is chosen. There is a danger of choosing only the
data that show what is expected by the person doing the choosing.
· Events can be described in terms of being more or less likely, impossible, or certain.

Building on previous experience, students can now delve into statistics in greater detail. The work should be directly related to
student investigations and utilize computers. As stated in NCTM Standards: Instruction in statistics should focus on the active
involvement of students in the entire process: formulating key questions; collecting and organizing data; representing the data using
graphs, tables, frequency distributions, and summary statistics; analyzing the data; making conjectures; and communicating
information in a convincing way. Students’ understanding of statistics will also be enhanced by evaluating others’ arguments.
Database computer programs offer a means for students to structure, record, and investigate information; to sort it quickly by various
categories; and to organize it in a variety of ways. Other computer programs can be used to construct plots and graphs to display data.
Scale changes can be made to compare different views of the same information. These technological tools free students to spend more
time exploring the essence of statistics: analyzing data from many viewpoints, drawing inferences, and constructing and evaluating
arguments. Students should make distributions for many data sets, their own and published sets, which have already inspired some
meaningful questions. The idea of a middle to a data set should be well motivated— say, by asking for a simple way to compare two
groups— and various kinds of middle should be considered. The algorithm for the mean can be learned but not without recurrent
questions about what it conveys— and what it does not. In studying data sets, questions like these should be raised: What appears
most often in the data? Are there trends? Why are there outliers? How can we explain the data, and does our explanation allow a
6-8 EXPECTATIONS
prediction of what further data would look like? What difficulties might arise when extending the explanation to similar problems?
What additional data can we collect to try to verify the ideas developed from these data? The distinction between ends and means
should be kept in mind in all of this. The ultimate aim is not to turn all students into competent statisticians but to have them
understand enough statistics to be able to respond intelligently to claims based on statistics; without the kind of intense effort called
for here, that understanding will be elusive. Probability, too, should be continued at this level through the use of tables of actual
frequencies of events, begun in the 3rd through 5th grades. Every time, however, students should be asked to consider whether the
data (necessarily collected in the past) are still applicable. How well, for example, would last year's daily temperatures apply to this
year? After they have had many occasions to count possible outcomes (such as the faces of a die) and discuss their equal probability
(is each face as likely to come up as any other?), students can begin to move to generalizations about theoretical probabilities.
Students' attention should consistently be drawn to the assumptions that all possible outcomes of a situation are accounted for and
are all equally probable. Computers should be used to generate simulated probabilistic data for analysis, but only after students have
worked on problems in which they use their own data.

PROBABILITY: THE CHANCES YOU TAKE                                                                         BY   ALLISON SRUBAS                 Page 11
By the end of the 8th grade, students should know that
· How probability is estimated depends on what is known about the situation. Estimates can be based on data from similar conditions
in the past or on the assumption that all the possibilities are known.
· Probabilities are ratios and can be expressed as fractions, percentages, or odds.
· The mean, median, and mode tell different things about the middle of a data set.

· Comparison of data from two groups should involve comparing both their middles and the spreads around them.
· The larger a well-chosen sample is, the more accurately it is likely to represent the whole. But there are many ways of choosing a
sample that can make it unrepresentative of the whole.

As their mathematical sophistication grows during these grades, students are able to perform and make sense of more subtleties in
collecting, describing, and interpreting data. They should have multiple opportunities to plan and carry out studies of their own
observations and of large databases. Their written reports should include the reasoning that went into decisions about sampling
method and size, about models chosen, about the display used, and about alternative interpretations. They should look for selection
bias, measurement error, and display distortion in news reports as well as in their own studies. Important, too, is frequent discussion
of reports in the news media about scientific studies. Students should identify weaknesses in the studies and offer alternative
interpretations of the results— perhaps writing alternative versions of the news stories or writing letters to the editor about what the
stories may have been missing.

By the end of the 12th grade, students should know that
· Even when there are plentiful data, it may not be obvious what mathematical model to use to make predictions from them or there
may be insufficient computing power to use some models.
· When people estimate a statistic, they may also be able to say how far off the estimate might be.
9-12 EXPECTATIONS
· The middle of a data distribution may be misleading— when the data are not distributed symmetrically, or when there are extreme
high or low values, or when the distribution is not reasonably smooth.
· The way data are displayed can make a big difference in how they are interpreted.
· Both percentages and actual numbers have to be taken into account in comparing different groups; using either category by itself
· Considering whether two variables are correlated requires inspecting their distributions, such as in two-way tables or scatterplots. A
believable correlation between two variables doesn’t mean that either one causes the other; perhaps some other variable causes them
both or the correlation might be attributable to chance alone. A true correlation means that differences in one variable imply
differences in the other when all other things are equal.
· The larger a well-chosen sample of a population is, the better it estimates population summary statistics. For a well-chosen sample,
the size of the sample is much more important than the size of the population. To avoid intentional or unintentional bias, samples are
usually selected by some random system.
· A physical or mathematical model can be used to estimate the probability of real world events.

SCIENCE FOR ALL           We must be alert to possible bias created by how the sample was selected. Common sources of bias in drawing samples include
AMERICANS                 convenience (for example, interviewing only one’s friends or picking up only surface rocks), self-selection (for example, studying only
people who volunteer or who return questionnaires), failure to include those who have dropped out along the way (for example,
testing only students who stay in school or only patients who stick with a course of therapy), and deciding to use only the data that
Go to SFAA…               support our preconceptions. Most of what we learn about the world is obtained from information based on samples of what we are

PROBABILITY: THE CHANCES YOU TAKE                                                                         BY   ALLISON SRUBAS                Page 12
studying— samples of, say, rock formations, light from stars, television viewers, cancer patients, whales, or numbers. Samples are
used because it may be impossible, impractical, or too costly to examine all of something, and because a sample often is sufficient for
most purposes.

Our knowledge of how the world works is limited by at least five kinds of uncertainty:
(1) inadequate knowledge of all the factors that may influence something,
(2) inadequate number of observations of those factors,
(3) lack of precision in the observations,
(4) lack of appropriate models to combine all the information meaningfully, and
(5) inadequate ability to compute from the models. It is possible to predict some events with great accuracy (eclipses), others with fair
accuracy (elections), and some with very little certainty (earthquakes).
Although absolute certainty is often impossible to attain, we can often estimate the likelihood—whether large or small—that some
things will happen and what the likely margin of error of the estimate will be. It is often useful to express likelihood as a numerical
probability. We usually use a probability scale of 0 to 1, where 0 indicates our belief that some particular event is certain not to occur,
1 indicates our belief that it is certain to occur, and anything in between indicates uncertainty. For example, a probability of .9
indicates a belief that there are 9 chances in 10 of an event occurring as predicted; a probability of .001 indicates a belief that there is
only 1 chance in 1,000 of its occurring. Equivalently, probabilities can also be expressed as percentages, ranging from 0 percent (no
chance) to 100 percent (certainty). Uncertainties can also be expressed as odds: A probability of .8 for an event can be expressed as
odds of 8 to 2 (or 4 to 1) in favor of its occurring. One way to estimate the probability of an event is to consider past events. If the
current situation is similar to past situations, then we may expect somewhat similar results. For example, if it rained on 10 percent of
summer days last year, we could expect that it will rain on approximately 10 percent of summer days this year. Thus, a reasonable
ESSAY
estimate for the probability of rain on any given summer day is .1—one chance in ten. Additional information can change our
estimate of the probability. For example, rain may have fallen on 40 percent of the cloudy days last summer; thus, if our given day is
cloudy, we would raise the estimate from .1 to .4 for the probability of rain. The more ways in which the situation we are interested in
is like those for which we have data, the better our estimate is likely to be. Another approach to estimating probabilities is to consider
the possible alternative outcomes to a particular event. For example, if there are 38 equally wide slots on a roulette wheel, we may
expect the ball to fall in each slot about 1/38 of the time. Estimates of such a theoretical probability rest on the assumption that all of
the possible outcomes are accounted for and all are equally likely to happen. But if that is not true— for example, if the slots are not of
equal size or if sometimes the ball flies out of the wheel— the calculated probability will be wrong. Probabilities are most useful in
predicting proportions of results in large numbers of events. A flipped coin has a 50 percent chance of coming up heads, although a
person will usually not get precisely 50 percent heads in an even number of flips. The more times one flips it, the less likely one is to
get a count of precisely 50 percent but the closer the proportion of heads is likely to be to the theoretical 50 percent. Similarly,
insurance companies can usually come within a percentage point or two of predicting the proportion of people aged 20 who will die in
a given year but are likely to be off by thousands of total deaths— and they have no ability whatsoever to predict whether any
particular 20-year-old will die. In other contexts, too, it is important to distinguish between the proportion and the actual count.
When there is a very large number of similar events, even an outcome with a very small probability of occurring can occur fairly often.
For example, a medical test with a probability of 99 percent of being correct may seem highly accurate— but if that test were
performed on a million people, approximately 10,000 individuals would receive false results.

PROBABILITY: THE CHANCES YOU TAKE                                                                           BY   ALLISON SRUBAS                 Page 13
CONTENT CLARIFICATION
Research into Student Learning and Ideas about Teaching

Developing a solid mathematical foundation from prekindergarten through second grade is essential for every child. In these grades,
students are building beliefs about what mathematics is, about what it means to know and do mathematics, and about themselves as
Go to NCTM…               mathematics learners. These beliefs influence their thinking about, performance in, and attitudes toward, mathematics and decisions
related to studying mathematics in later years.

Children develop many mathematical concepts, at least in their intuitive beginnings, even before they reach school age. Infants
spontaneously recognize and discriminate among small numbers of objects, and many preschool children possess a substantial body
of informal mathematical knowledge. Adults can foster children's mathematical development from the youngest ages by providing
environments rich in language and where thinking is encouraged, uniqueness is valued, and exploration is supported.

Children are likely to enter formal school settings with different levels of mathematics understanding, reflecting their opportunity to
have learned mathematics. Some children will need additional support so that they do not start school at a disadvantage. Early
assessments should be used not to sort children but to gain information for teaching and for potential early interventions.

All students deserve high-quality programs that include significant mathematics presented in a manner that respects both the
mathematics and the nature of young children. These programs must build on and extend students' intuitive and informal
mathematical knowledge. They must be grounded in a knowledge of child development and provide environments that encourage
students to be active learners and accept new challenges. They need to develop a strong conceptual framework while encouraging and
developing students' skills and their natural inclination to solve problems.
STANDARDS FOR SCHOOL      At the core of mathematics programs in prekindergarten through grade 2 are the Number and Operations and Geometry Standards.
MATHEMATICS               For example, it is absolutely essential that students develop a solid understanding of the base-ten numeration system in
prekindergarten through grade 2. They must recognize that the word ten may represent a single entity (1 ten) or ten separate units (10
ones) and that these representations are interchangeable. Using concrete materials and calculators in appropriate ways can help
students learn these concepts.

Understandings of patterns, measurement, and data contribute to the understanding of number and geometry and are learned in
conjunction with them. Similarly, the Process Standards of Problem Solving, Reasoning and Proof, Communication, Connections, and
Representation both support and augment the Content Standards. Even at this age, guided work with calculators can enable students
to explore number and patterns, focus on problem-solving processes, and investigate realistic applications. See, for example, the
problem in figure 1.

PROBABILITY: THE CHANCES YOU TAKE                                                                         BY   ALLISON SRUBAS                Page 14
Fig. 1. A calculator activity to help develop understanding of place value

In the elementary grades, it often happens that specific blocks of time are not allotted to instruction in particular subjects. It is
essential for students in the elementary grades to study mathematics for an hour a day under the guidance of teachers who enjoy
mathematics and are prepared to teach it well. This basic requirement takes thoughtful arrangements of scheduling and staffing--
whether by shared teaching responsibilities, the use of mathematics specialists, or other creative administrative means.

Students enter grade 3 with an interest in learning mathematics. Nearly three-quarters of U.S. fourth graders report liking
mathematics, seeing it as practical and important. If mathematics continues to be seen as interesting and understandable, students
will remain engaged. If learning becomes simply a process of mimicking and memorizing, students' interest is likely to diminish.

Interwoven through the Content Standards for grades 3–5 are three crucial mathematical themes--multiplicative thinking,
equivalence, and computational fluency. The focus on multiplicative reasoning develops knowledge that students build on as they
move into the middle grades, where the emphasis is on proportional reasoning. As a part of multiplicative reasoning, students in
grades 3–5 should build their understanding of fractions as a part of a whole and as division.

The concept of equivalence helps students learn different mathematical representations and offers a way to explore algebraic ideas.
Students should develop computational fluency-- efficient and accurate methods for computing that are based on well-understood
properties and number relationships. For example, 298 42 can be thought of as (300 42) – (2 42), or 41 16 can be computed
by multiplying 41 8 to get 328 and then doubling 328 to get 656. When these three themes are emphasized, the expectations for
grades 3–5 reinforce two major objectives of mathematics learning: making sense of mathematical ideas and acquiring the skills and
understandings needed to solve problems.

In grades 3–5, algebraic ideas emerge and are investigated by children. For example, students in these grades are able to make a
general statement about how one variable is related to another variable. If a sandwich costs \$3, you can figure out how many dollars
any number of sandwiches cost by multiplying that number by 3. In this case, students have developed a model of a proportional
relationship: the value of one variable is always 3 times the value of the other, or C = 3 n.

PROBABILITY: THE CHANCES YOU TAKE                                                                             BY   ALLISON SRUBAS             Page 15
Given their central role in shaping the mathematics learning of students in these grades, teachers must recognize the need to develop
mathematical expertise. Some elementary schools identify a "mathematics teacher-leader," who can support other teachers in their

instruction and professional development. Other schools use "mathematics specialists" at the upper elementary grade levels, who
assume primary responsibility for teaching mathematics to larger groups of students.
The middle grades represent a significant turning point in students' lives. During the middle grades, students solidify conceptions
about themselves as learners of mathematics. They arrive at conclusions about their competence in mathematics, their attitudes, their
interest, and their motivation. These conceptions will influence how they approach the study of mathematics in later years, which in
turn will affect their later career and personal opportunities.

If middle school students find both challenge and support in their mathematics classes, they will be drawn to the subject. They will be
able to use their emerging capabilities of finding and imposing structure, conjecturing and verifying, thinking hypothetically,
comprehending cause and effect, and engaging in abstraction and generalization.

As in all the grade bands, students in the middle grades need a balanced mathematics program that encompasses all ten Standards,
including significant amounts of algebra and geometry. Algebra and geometry are crucial to success in the later study of mathematics
and also in many situations that arise outside the mathematics classroom. Students should see that these subjects are interconnected
with each other and with other content areas in the curriculum. For example, students might be asked to explain the number of tiles
that will be needed to make borders around pools of various lengths and widths, as in figure 2. Students might develop various
formulas to express this relationship on the basis of a table or their reasoning about the situation; for example, "You need L + 2 tiles
across the top and the same number across the bottom. And you need W tiles on the left and the right. So all together, the number of
tiles needed is T = 2(L + 2) + 2W."

Fig. 2. The "swimming pool" problem

Students' understanding of these crucial ideas should be developed over all three years in the middle grades and across a broad range
of mathematics content. This approach is a challenging alternative to the practice of offering a select group of middle-grades students
a one-year course that focuses narrowly on algebra or geometry. However, all
middle-grades students will benefit from a rich and integrated treatment of mathematics content. By the end of the eighth grade,
students should have a solid background in algebra and other areas that will prepare them to enter substantive high school courses.

Middle-grades mathematics also needs to prepare students to deal with quantitative situations in their lives outside school. For
example, consumer magazines regularly publish comparisons of characteristics of various consumer products, such as the quality of
peanut butter, the duration of rechargeable batteries, or the cost, size, and gas mileage of automobiles. When using data from such
sources, students need to determine which data are appropriate for their needs, to understand how the data were gathered at the
source, and to consider limitations that could affect interpretation.

PROBABILITY: THE CHANCES YOU TAKE                                                                         BY   ALLISON SRUBAS                 Page 16
Special attention must be given to the preparation and ongoing professional support of middle-grades teachers. They need a deep
understanding of mathematical ideas, pedagogical practices, interdisciplinary teaching approaches, how students learn mathematics,
and adolescent development. States and provinces need to give much more attention to the development of special preparation
programs for teachers of mathematics in the middle grades.
In secondary school, all students should learn an ambitious common foundation of mathematical ideas and applications. This shared
mathematical understanding is as important for students who will enter the workplace as it is for those who will pursue further study
in mathematics and science. All students should study mathematics in each of the four years that they are enrolled in high school.

Because students' interests and aspirations may change during and after high school, their mathematics education should guarantee
experience the interplay of algebra, geometry, statistics, probability, and discrete mathematics. They need to understand the
fundamental mathematical concepts of function and relation, invariance, and transformation. They should be adept at visualizing,
describing, and analyzing situations in mathematical terms. And they need to be able to justify and prove mathematically based ideas.

High school mathematics builds on the skills and understandings developed in the lower grades. For example, students should enter
high school with extensive experience in modeling various patterns and relationships. High school students might explore the
following problem:
A student strained her knee in an intramural volleyball game, and her doctor prescribed an anti-inflammatory drug to reduce the
swelling. She is to take two 220-milligram tablets every 8 hours for 10 days. If her kidneys filtered 60% of this drug from her body
every 8 hours, how much of the drug was in her system after 10 days? How much of the drug would have been in her system if she had
continued to take the drug for a year?

PROBABILITY: THE CHANCES YOU TAKE                                                                       BY   ALLISON SRUBAS                Page 17
Fig. 3. A spreadsheet computation of the "drug dosage" problem

Students might represent the equation informally as NEXT = 0.4(NOW) + 440, start at 440. Entering this relationship in a
spreadsheet (see fig. 3), they could note that an "equilibrium" value of about 733 1/3 milligrams is reached. This investigation might
lead to explorations of finite sequences and series.

High school students can study mathematics that extends beyond the material expected of all students in at least three ways. One is to
include in the curriculum material that extends the foundational material in depth or sophistication. Two other approaches make use
of supplementary courses. In the first, students enroll in additional courses concurrent with those expected of all students. In the
second, students complete a three-year version of the shared material and then take other mathematics courses. In both situations,
students can choose from such courses as computer science, technical mathematics, statistics, and calculus. Each of these approaches
has the essential property that all students learn the same foundation of mathematics but some, if they wish, can study additional
mathematics.

The Standards for high school students are ambitious. The demands made on high school teachers in achieving the Standards will require extended
and sustained professional development and a large degree of administrative support.
Children in K-2 can begin to find out about sampling in the context, say, of reporting on the kinds of stones found on the school
Go To Benchmarks…       playground. In 9-12 students’ written reports should include the reasoning that went into decisions about sampling method and size,

PROBABILITY: THE CHANCES YOU TAKE                                                                             BY   ALLISON SRUBAS                   Page 18
about models chosen, about the display used, and about alternative interpretations. Upper elementary students begin to understand
that there is an increase in regularity of a sample distribution with an increase in the sample size, but they can apply this idea only to
relatively small numbers.

9d Uncertainty

Students' conceptions about uncertainty and students' probabilistic reasoning have been extensively researched, and there are several
literature reviews on the topic (Garfield & Ahlgren, 1988; Hawkins & Kapadia, 1984; Shaughnessy, 1992). The research on
summarizing data, which focuses on students' understanding of different measures of central tendency and dispersion, is less
extensive.

Probability
Research presents somewhat contradictory results on elementary children's understanding of probability. Piagetian research says
lower elementary children have no conception of probability (Piaget & Inhelder, 1975; Shayer & Adey, 1981), but other studies
indicate that even lower elementary-school children have probabilistic intuitions upon which probability instruction can build. Falk
et al. (1980) presented elementary-school students with two sets, each containing blue and yellow elements. Each time, one color
was pointed out as the payoff color. The students had to choose the set from which they would draw at random a "payoff element" to
be rewarded. From the age of six, children began to select the more probable set systematically. The ability to choose correctly
precedes the ability to explain these choices.

Upper elementary students can give correct examples for certain, possible, and impossible events, but cannot calculate the probability
BENCHMARKS FOR SCIENCE    of independent and dependent events even after instruction on the procedure (Fischbein & Gazit, 1984). That is partly because
LITERACY                  students at this age tend to create "part to part" rather than "part to whole" comparisons (e.g., 9 men and 11 women rather than 15%
of men and 10% of women). By the end of 8th grade, students can use ratios to calculate probabilities in independent events, after
adequate instruction (Fischbein & Gazit, 1984).

Upper elementary students begin to understand that there is an increase in regularity of a sample distribution with an increase in the
sample size, but they can apply this idea only to relatively small numbers. It is postulated that to deal with large numbers, children
must first cope with notions of ratio and proportion and that their failure to understand these notions creates "a law of small large
numbers" (Bliss, 1978).

Extensive research points to several misconceptions about probabilistic reasoning that are similar at all age levels and are found even
among experienced researchers (Kahneman, Slovic, & Tversky, 1982; Shaughnessy, 1992). One common misconception is the
idea of representativeness, according to which an event is believed to be probable to the extent that it is "typical." For example, many
people believe that after a run of heads in coin tossing, tails should be more likely to come up. Another common error is estimating
the likelihood of events based on how easily instances of it can be brought to mind.

Summarizing data
The concept of the mean is quite difficult for students of all ages to understand even after several years of formal instruction. Several
difficulties have been documented in the literature: Students of all ages can talk about the algorithm for computing the mean and
relate it to limited contexts, but cannot use it meaningfully in problems (Mokros & Russell, 1992; Pollatsek, Lima, & Well,
1981); upper elementary- and middle-school students believe that the mean of a particular data set is not one precise numerical value
but an approximation that can have one of several values (Mokros & Russell, 1992); some middle-school students cannot use the
mean to compare two different-sized sets of data (Gal et al., 1990); high-school students may believe the mean is the usual or
typical value (Garfield & Ahlgren, 1988); students (or adults) may think that the sum of the data values below the mean is
PROBABILITY: THE CHANCES YOU TAKE                                                                          BY   ALLISON SRUBAS                  Page 19
equivalent to the sum above the mean (rather than that the total of the deviations below the mean is equal to the total above)
(Mokros & Russell, 1992). Research suggests that a good notion of representa-tiveness may be a prerequisite to grasping the
defin-itions for measures of location like mean, median, or mode. Students can acquire notions of representa-tiveness after they start
seeing data sets as entities to be described and summarized rather than as "unconnected" individual values. This occurs typically
around 4th grade (Mokros & Russell, 1992).

Research suggests students should be introduced first to location measures that connect with their emerging concept of the "middle,"
such as the median, and later in the middle-school grades, to the mean. Premature introduction of the algorithm for computing the
mean divorced from a meaningful context may block students from understanding what averages are for (Mokros & Russell, 1992;
Pollatsek et al., 1981).

PROBABILITY: THE CHANCES YOU TAKE                                                                       BY   ALLISON SRUBAS                 Page 20
PERFORMANCE ASSESSMENT

>>> ROLE OR PERSONA…                >>> TARGETED BEHAVIOR…                            >>> AUDIENCE…                       >>> PRODUCT OR ACTION…

You are a…                       …who has been asked to…                  …who has been asked by…                             …who will…

You have been asked by gamblers anonymous to create a brochure for gamblers on the probability used in ―games of chance‖. They are looking
for 4 brochures, one for roulette, one for poker, one for craps, and one for poker dice. They want to know what the probability of winning or
losing is for the games that they have been playing for years. In groups, you will either be playing and researching roulette, poker, craps, or
poker dice. Together you will have to create a brochure for the game you are assigned. First, you will be divided into groups and assigned a
casino game to play and research. You will be given two roles in your groups. There will be one leader to keep the group on task. This person
will also be in charge of setting up the game and cleaning up at the end of the class. There will also be a recorder. This person will take notes
during the game playing. However, everyone should record any individual feeling they have while playing. Then, as a group you will be
required to create a brochure for gamblers anonymous.

PERFORMANCE
· An original title
· A description of the game you are assigned.
· Your results while playing the game (take careful notes of what happens during the game).
· The probability rules and values in the game. You must include the sample space and the probability values of each possible outcome. (Be
sure to explain the meaning of these terms and values. They need you to create this brochure because they do not understand the probability
involved.)
· A report that cautions gamblers who play this game based on the probabilities that you found. Are the gamblers more likely to lose or gain
money?

You will have 2 working days in class to complete this performance assessment. On the third day you will be required to present your
brochure. Your brochure should be creative. You need to make the brochure interesting to read and make sure to be neat!!

PROBABILITY: THE CHANCES YOU TAKE                                                                         BY   ALLISON SRUBAS                 Page 21
PERFORMANCE ASSESSMENT

PERFORMANCE INDICATORS
CRITERIA FOR
SUCCESS
· The game assigned is clearly                                                                               · The game assigned is not
· The game assigned is             · The game assigned is explained
Description of     explained                                                                                                    explained.
somewhat explained                 a little
Game and Rules      · The rules of the game are                                                                                  · Little to none of the rules are
· Most of the rules are given      · Some of the rules are given.
clearly and completely given.                                                                                given

· Your results while playing are   · Most of your results while       · Some of your results while           · Little to none of your results
Game Results
clear and complete.                playing are given.                 playing are given.                     while playing are given.

Probability      · Your sample space is accurate    · Your sample space is mostly      · Your sample space is somewhat        · There is little or no evidence of a
Sample Space      and complete.                      accurate and mostly complete.      accurate and partially complete.       sample space.
and Probability    · Your probability distributions   · Your probability distributions   · Your probability distribution is     · There is little or no evidence of
Distribution     are accurate and complete.         are mostly accurate.               somewhat accurate.                     the probability distribution.

Recommendation
Report
and is compete.                    and is mostly complete.            and is partially complete.             complete.

Creativity                                                                                                                    · Your brochure is not creative.
creative.                          creative.                          creative.

· Your presentation is complete    · Your presentation is mostly      · Your presentation is somewhat        · You have no presentation or it is
Presentation
and interesting.                   complete and interesting.          complete and mostly interesting.       not complete and not interesting.

PROBABILITY: THE CHANCES YOU TAKE                                                                          BY   ALLISON SRUBAS                   Page 22
CURRICULUM INTEGRATIONS
ART

> Go to Art Standards…
> TN Art Standards

McREL NATIONAL STANDARDS
AND BENCHMARKS DATABASE

> Go to McREL…
ENGLISH/
LANGUAGE ARTS
> Go to English/Language Arts
Standards…
> TN English/Language Arts

GEOGRAPHY

> Go to Geography Standards…

HISTORY

> Go to History Standards…

SCIENCE

> Go to National Science Education
Standards…
> Go to Benchmarks…

SOCIAL STUDIES
> Go to Social Studies Standards…
> TN Social Studies Standards
TECHNOLOGY
> Go to Technology Standards…
> TN Technology Standards

PROBABILITY: THE CHANCES YOU TAKE                             BY   ALLISON SRUBAS   Page 23
CURRICULUM WEB LINKS                   >>> Go to Marco Polo…

PROBABILITY: THE CHANCES YOU TAKE                                         BY   ALLISON SRUBAS   Page 24
Lesson Title: I Won the Lottery
FOR THE TEACHER
STAGE OF LEARNING CYCLE:
ENGAGEMENT
>>> Go to FOR THE
STUDENT…                   1
During this lesson, students experience the chances of winning the Connecticut Cash 5 lottery. They also have the opportunity to express
LESSON OVERVIEW
their ideas and feelings that are associated with the topic of probability. The lottery part of this lesson is referenced later in the unit.

INSTRUCTIONAL                 Through experiencing the lottery, become interested in probability.
GOALS                     Activate their prior knowledge.

FORMATIVE                   Check for completion of Right Angle Perspective.
ASSESSMENT                  Monitor class participation.

1. Introduce the performance assessment to the students. Explain that this is how they are assessed for
Opening the Lesson:            this unit.
10 minutes                 2. Explain the lottery task. For the task, students write down 5 numbers and then a Connecticut Cash 5
lottery is played on the VCR. (Note: These are broadcast on local television stations)

LESSON
ORGANIZATION

1. The lesson then begins. The teacher and students pick their 5 numbers (between 1 and 40) for the
Developing the Lesson:          Connecticut Cash 5 lottery. The video of the lottery recorded from the television is then played. After
20 minutes                 the lottery is played, the students compare their numbers with the winning numbers. A class
discussion is held about the students’ results.

PROBABILITY: THE CHANCES YOU TAKE                                                                          BY   ALLISON SRUBAS                 Page 25
1. Students are asked to take 5 minutes to fill out a Right Angle Perspective on Probability.
2. Students pair up and discuss their worksheets for 5-7 minutes.
3. Then the teacher leads a ten-minute discussion about what the students know and how they feel
Closing the Lesson:
25 minutes
4. At the end of class, the worksheets are collected so that the teacher can review them.
5. Students read a short essay about gamblers anonymous and required to summarize the essay in one
paragraph. The paragraph is collected tomorrow and graded as pass/fail.

Connecticut Cash 5 lottery video that was recorded off of the television
Television
VCR
TEACHING
Chalkboard
RESOURCES
Chalk
Thinking about probability from a Right Angle Perspective Worksheets

ENRICHMENT
ACTIVITIES

ACCOMMODATIONS
FOR SPECIAL
LEARNERS

STANDARDIZED
TEST ITEM
GENERATOR

> Go to State Exam
Test Items…

PROBABILITY: THE CHANCES YOU TAKE                                                                 BY   ALLISON SRUBAS               Page 26
Lesson Title: I Won the Lottery
FOR THE STUDENT
>>> Go to FOR THE
TEACHER…                   1

LEARNING GOALS         Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases

ASSESSMENT

You first take part in an activity that deals with the Connecticut Cash 5 lottery. You are asked to choose 5 different numbers between 1
and 40. Then you watch a video of the lottery. You then have time to reflect on the lottery and the numbers that you chose. You also
take part in a class discussion that is based on your reflections and your peers’ reflections. During the second part of this lesson, you are
asked to fill in the Thinking about Probability from a Right Angle Perspective worksheet. After about 5 minutes you take part in a class
LEARNING
discussion. The teacher asks questions of you and the other students. It is your responsibility to take notes. At the end of class you must
ACTIVITY
hand in your worksheet; however, it will not be graded. Then you are given a homework assignment to read an essay about gamblers
anonymous from http://www.gamblersanonymous.org/about.html and write a paragraph to summarize what you have read.
This is collected and graded as pass/fail tomorrow.

ENRICHMENT
ACTIVITIES

Pencil/pen
MATERIALS/
Paper
EQUIPMENT
Thinking about Probability from a Right Angle Perspective Worksheet

PROBABILITY: THE CHANCES YOU TAKE                                                                           BY   ALLISON SRUBAS                 Page 27
PROBABILITY: THE CHANCES YOU TAKE   BY   ALLISON SRUBAS   Page 28
PROBABILITY: THE CHANCES YOU TAKE   BY   ALLISON SRUBAS   Page 29
Lesson Title: M & M Probability
FOR THE TEACHER
STAGE OF LEARNING CYCLE:
EXPLORATION
>>> Go to FOR THE
STUDENT…                   2
In this lesson, students make predictions about the number of each color M&M in a bag of 50 M&M’s. They work in pairs to actually
count the M&M’s in a bag and record their results. Then they make a graph to represent their findings. There is then a class discussion
LESSON OVERVIEW
based on their findings and students share their graphs. Also, during the class discussion, the students discuss the likeliness of randomly
pulling each color out of the bag with teacher guidance.
   Predict and record results.
INSTRUCTIONAL                 Explore their ideas associated with sample spaces and of probability distribution.
GOALS                     Work cooperatively in pairs.
   Participate in the class discussion. ¾_Create a graph to represent their findings.

FORMATIVE
Students turn in their predictions, results, and graphs to be graded as a quiz.
ASSESSMENT

1. First, students participate in a think-pair-share regarding their homework assignment.
2. Homework assignment is collected and graded as pass/fail.
Opening the Lesson:
3. Students are told that they will be making predictions about M&M’s and counting one bag of M&M’s.
15 minutes
4. Students are put into pairs to complete this assignment. The teacher passes out the M&M discovery
guides and brown bags of M&M’s. There should be about 60 M&M’s in each.

LESSON
ORGANIZATION                                            1. Students begin by filling in their predictions. They should write these predictions on the discovery
guide.
2. They then actually count the M&M’s in their bag. This is also recorded on the discovery guide.
3. Next, the students compare their predictions to their actual results. They should write these ideas of
Developing the Lesson:
the discovery guide.
25 minutes
4. The students then create a graph to represent their results. This can be a bar graph, line graph,
M&M graph, or any other graph they choose.
5. There is then a class discussion about their predictions, results, and graphs. Lastly, the class
discusses the probability of pulling each color out of the bag with teacher guidance.

PROBABILITY: THE CHANCES YOU TAKE                                                                         BY   ALLISON SRUBAS                  Page 30
Closing the Lesson:
1. The discovery guides and graphs are collected and graded as a quiz.
5 minutes

M&M Discovery Guides
Chalkboard
TEACHING            Chalk
RESOURCES            Graph Paper
Brown bags with 60 M&M’s each (1 for each pair of students)
Colored Pencils

ENRICHMENT
ACTIVITIES

ACCOMMODATIONS
FOR SPECIAL
LEARNERS

STANDARDIZED
TEST ITEM
GENERATOR

> Go to State Exam
Test Items…

PROBABILITY: THE CHANCES YOU TAKE                                                                      BY   ALLISON SRUBAS   Page 31
Lesson Title: M & M Probability
FOR THE STUDENT
>>> Go to FOR THE
TEACHER…                  2

LEARNING GOALS         Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases.

ASSESSMENT           You are required to complete the M&M Discovery Guide and a graph. They are collected at the end of class and graded as a quiz.

During this class, you are given an M&M discovery guide to complete in pairs. Each student must turn in an individual discovery guide
and graph to be graded as a quiz. On the discovery guide you make predictions about the number of each color M&M in a bag of 50.
LEARNING
Then you actually count the M&M’s in a bag. You need to make comparisons between your predictions and your actual results. There is
ACTIVITY
also a class discussion based on the M&M discovery guide and probability. At the end of class, your discovery guides and graphs are
collected and graded as a quiz.

ENRICHMENT
ACTIVITIES

MATERIALS/
EQUIPMENT

PROBABILITY: THE CHANCES YOU TAKE                                                                      BY   ALLISON SRUBAS                Page 32
PROBABILITY: THE CHANCES YOU TAKE   BY   ALLISON SRUBAS   Page 33
Lesson Title: Probability Stations
FOR THE TEACHER
STAGE OF LEARNING CYCLE:
EXPLORATION
>>> Go to FOR THE
STUDENT…                   3
During this lesson, students have the opportunity to explore their ideas about sample space and probability. In groups, the students
move to different stations throughout the class period. They work with a coin, a spinner with four equally colored parts, one red, one
LESSON OVERVIEW
blue, one green, and one yellow, a die, a jar of marbles with 3 red marbles, 3 blue marbles, 4 green marbles, and 5 yellow marbles, and a
deck of playing cards. At the end of the lesson, there is a class discussion on the student’s findings.

   Activate their prior knowledge of sample spaces and of probability distribution.
INSTRUCTIONAL
   Explore their ideas associated with sample spaces and of probability distribution.
GOALS
   Work cooperatively in groups

FORMATIVE            The teacher assesses the students through the handed-in discovery guides, and the class discussion. The discovery guides are not
ASSESSMENT           graded, but the teacher reviews them.

1. When the students walk into class, there are five stations set up around the edge of the room. Station
one has a coin, station two has a spinner with 4 equally colored parts, stations three has a die, station
four has a jar of marbles with 3 red marbles, 3 blue marbles, 4 green marbles, and 5 yellow marbles,
Opening the Lesson:           and station five has a deck of playing cards. The teacher has already decided the group members for
5 minutes                 each group. As the students enter the room, the teacher tells the students which station they should sit
at. After all of the students are at their appropriate stations the teacher passes out a discovery guide for
each station; therefore, each student is given 5 discovery guides. The teacher explains to the students
that in this lesson they explore their ideas of sample space and probability.

LESSON               Developing the Lesson:
1. The students are given 7-10 minutes at each station to fill in their discovery guides.
ORGANIZATION                 40 minutes

Closing the Lesson:          1. After the activity, there is a short class discussion based on the students’ findings.
10 minutes               2. Discovery guides are collected to be reviewed by the teacher.

PROBABILITY: THE CHANCES YOU TAKE                                                                           BY   ALLISON SRUBAS                 Page 34
Coin
Colored Spinner
Die
TEACHING            Jar with Colored Marbles
RESOURCES            Deck of Playing Cards
Chalk
Discovery guides for each station
Chalkboard

Students are given a homework assignment to write a three-paragraph essay. One paragraph is to be a scenario of coincidence, one is to
ENRICHMENT
be a scenario of probability and the third should describe the difference between coincidences and probabilities. Students should be
ACTIVITIES
creative and the scenarios can contain humor.

ACCOMMODATIONS
FOR SPECIAL
LEARNERS

STANDARDIZED
TEST ITEM
GENERATOR

> Go to State Exam
Test Items…

PROBABILITY: THE CHANCES YOU TAKE                                                                      BY   ALLISON SRUBAS                Page 35
Lesson Title: Probability Stations
FOR THE STUDENT
>>> Go to FOR THE
TEACHER…                   3

LEARNING GOALS         Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases.

You are assessed during the class discussion. This is a very informal assessment. Also, you are required to hand in your discovery guides
ASSESSMENT           from each station. Even though there are right and wrong answers, these discovery guides are not graded. The teacher reviews the
discovery guides.

In the beginning of class, you are put into one of five groups. During the class period you rotate to the different stations. You are
required to fill in a discovery guide for each of the five stations. Near the end of the class period, there is a class discussion based on the
discovery guides and any interesting findings you made during the activity. This discovery guide is collected, but not graded. The teacher
LEARNING           reviews your work. You are also given an enrichment activity. You are asked to write a three-paragraph essay. One paragraph is to be a
ACTIVITY           scenario of coincidence, one is to be a scenario of probability and the third should describe the difference between coincidences and
probabilities. This essay should be fun, creative, and possibly humorous.

ENRICHMENT
ACTIVITIES

MATERIALS/
Pencil/pen
EQUIPMENT

PROBABILITY: THE CHANCES YOU TAKE                                                                           BY   ALLISON SRUBAS                  Page 36
PROBABILITY: THE CHANCES YOU TAKE   BY   ALLISON SRUBAS   Page 37
PROBABILITY: THE CHANCES YOU TAKE   BY   ALLISON SRUBAS   Page 38
PROBABILITY: THE CHANCES YOU TAKE   BY   ALLISON SRUBAS   Page 39
PROBABILITY: THE CHANCES YOU TAKE   BY   ALLISON SRUBAS   Page 40
PROBABILITY: THE CHANCES YOU TAKE   BY   ALLISON SRUBAS   Page 41
Lesson Title: What is Sample Space?
FOR THE TEACHER
STAGE OF LEARNING CYCLE:
EXPLANATION
>>> Go to FOR THE
STUDENT…                   4
LESSON OVERVIEW         This is a direct instruction lesson. The topic is sample space in probability.

INSTRUCTIONAL                 Understand sample space
GOALS                     Apply their understanding of same space to the discovery guides completed the day before.

FORMATIVE
There is no formative assessment for this lesson; however, the students are responsible for taking notes.
ASSESSMENT

1. The discovery guides from the previous day are returned to the students. Tell the students to leave
LESSON                Opening the Lesson:            their discovery guides on their desks because they will be used later in the day.
ORGANIZATION                 8 minutes                  2. The teacher asks the students what they know about probability and she writes their responses on
the board.

PROBABILITY: THE CHANCES YOU TAKE                                                                         BY   ALLISON SRUBAS                 Page 42
1. In the beginning of the class the homework assignment is discussed. Students are given the
opportunity to read their scenarios to the rest of the class. There is then a short discussion based on the
differences between coincidences and probabilities.
2. The teacher then begins the explanation of sample space. The teacher writes two definitions on the
board.
 Sample Space: a set of all the possible outcomes
 Sample spaces are written in { }.
3. The teacher then asks the students questions based on their discovery guides.
 What is the sample space of flipping one coin?
 What is the sample space of rolling one die?
 What is the sample space of the spinner used yesterday?
Developing the Lesson:         What is the sample space of drawing one marble?
35 minutes          4. The teacher writes the students’ correct answers on the board.
 Flipping one coin – {heads, tails} ._Rolling on die – {1, 2, 3, 4, 5, 6}
 Spinner – {red, blue, green, yellow}
 Marbles – {red, blue, green, yellow}
5. The teacher then asks the students about the sample spaces of a deck of cards. She writes the correct
a. Deck of cards – {red, black}
b. Deck of cards – {spades, clubs, hearts, diamonds}
c. Deck of cards – {red 2, red 3, red 4, red 5, red 6, red 7, red 8, red 9, red 10, red jack, red queen, red
king, red ace, black 2, black 3, black 4, black
d. 5, black 6, black 7, black 8, black 9, black 10, black jack, black queen, black king, black ace}
d. Deck of card – {each individual card}

1. Students are put into pairs. They are told to discuss the lesson from yesterday and to compare the
Closing the Lesson:
sample spaces from today to the outcomes that they found yesterday. They should be thinking about
10 minutes
why their answers were correct or incorrect.

PROBABILITY: THE CHANCES YOU TAKE                                                                    BY   ALLISON SRUBAS                 Page 43
Chalkboard
TEACHING
Chalk
RESOURCES
Discovery guides to return

ENRICHMENT
ACTIVITIES

ACCOMMODATIONS
FOR SPECIAL
LEARNERS

STANDARDIZED
TEST ITEM
GENERATOR

> Go to State Exam
Test Items…

PROBABILITY: THE CHANCES YOU TAKE             BY   ALLISON SRUBAS   Page 44
Lesson Title: What is Sample Space?
FOR THE STUDENT
>>> Go to FOR THE
TEACHER…                   4

LEARNING GOALS         Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases.

ASSESSMENT

The teacher first gives a direct lesson on probability and sample space. She asks questions during this time and all of the students are
LEARNING            expected to contribute. Near the end of the class, you reference the discovery guides from yesterday’s activity. You compare some of your
ACTIVITY            answers to the class discussion. For this part of the class you work in pairs. Your discovery guides are collected at the end of class and
are passed out again tomorrow.

ENRICHMENT
ACTIVITIES

Pencil/Pen
MATERIALS/
Paper
EQUIPMENT
Returned discovery guides

PROBABILITY: THE CHANCES YOU TAKE                                                                        BY   ALLISON SRUBAS                 Page 45
Lesson Title: What is Probability Distribution?
FOR THE TEACHER
STAGE OF LEARNING CYCLE:
EXPLANATION
>>> Go to FOR THE
STUDENT…                  5
LESSON OVERVIEW         The teacher uses direct instruction to explain probability distribution and how it relates to the activity two days ago.

INSTRUCTIONAL                 Understand the probability distribution.
GOALS                     Use the concepts of probability distribution to the activity two days ago.

FORMATIVE            The teacher will observe the students during the open discussion. Students are not given a formal grade for this lesson; however, they
ASSESSMENT           are given a class participation grade.

1. The discovery guides from the previous day are returned to the students. Tell the students to leave
Opening the Lesson:            their discovery guides on their desks because they will be used later in the day.
8 minutes                  2. The teacher asks the students what probability distribution means and she writes their responses on
the board.

LESSON
ORGANIZATION                                            1. The teacher then goes through the discovery guides with the class. One student is asked what they
discovered the probability of getting a heads with the flip of one coin. He or she is then asked to explain
if he or she still agrees with his or her answer, why or why not?
Developing the Lesson:          2. Each question should be discussed in this manner. After each question and discussion, the correct
35 minutes                 answer and reasoning is given. Then there is an open discussion between the students and the teacher
is the monitor. Students should continue to raise their hands, but the teacher will not put in a lot of
input. She will just keep the students on the right track. The same method takes place for each of the 5
activities.

PROBABILITY: THE CHANCES YOU TAKE                                                                           BY   ALLISON SRUBAS                Page 46
Closing the Lesson:           1. There is a short discussion at the end of class to summarize the lesson.
5 minutes                 2. Students then reflect on the class discussion in their notebooks.

Chalk
TEACHING
Chalkboard
RESOURCES
Discovery guides to return

The students are asked to use their discovery guides to establish probability rules. They should think about sample space and
probability distribution. They should use their discovery guides to help establish these rules. The students are given some information to
ENRICHMENT
help them determine the probability rules. They are told that there are three rules that they can determine. The first rule deals with how
ACTIVITIES
to determine what the probability of an event is, the second rule deals with events affecting other events, the third rule is based upon the
range that a probability can be, and the last rule deals with successive events.

ACCOMMODATIONS
FOR SPECIAL
LEARNERS

STANDARDIZED
TEST ITEM
GENERATOR

> Go to State Exam
Test Items…

PROBABILITY: THE CHANCES YOU TAKE                                                                          BY   ALLISON SRUBAS                 Page 47
Lesson Title: What is Probability Distribution?
FOR THE STUDENT
>>> Go to FOR THE
TEACHER…                   5
LEARNING GOALS         Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases

ASSESSMENT           You will be given a participation grade during the open discussion.

The teacher first passes out your discovery guides. Then there is a lesson on probability distribution. You are first asked what probability
ACTIVITY           to participate in an open discussion on each activity. After the discussions the teacher explains what the correct answers to the
probability distributions of the activities are and why.

You are to establish probability rules. You should use your discovery guides and your notes on sample space and probability
ENRICHMENT           distribution. There are four rules that you can determine. The first rule deals with how to determine what the probability of an event is,
ACTIVITIES          the second rule deals with events affecting other events, the third rule is based upon the range that a probability can be, and the last rule
deals with successive events.

MATERIALS/           Pencil/Pen
EQUIPMENT            Paper

PROBABILITY: THE CHANCES YOU TAKE                                                                           BY   ALLISON SRUBAS                 Page 48
Lesson Title: Your Chances of Winning the Lottery
FOR THE TEACHER
STAGE OF LEARNING CYCLE:
EXTENSION
>>> Go to FOR THE
STUDENT…                   6
LESSON OVERVIEW         In the lesson today, the class explores the chances of winning the Connecticut Cash 5 lottery.

INSTRUCTIONAL          Explore the chances of winning the Connecticut Cash 5 lottery. They will be able to apply this concept to other probability experiments
GOALS              in the future.

FORMATIVE
The students’ papers from class are handed in and they graded as a pop-quiz.
ASSESSMENT

1. In the beginning of class, the student’s homework assignment is discussed. The students first discuss
their homework in pairs for about five minutes. Then the class comes back together and discusses the
rules that they discovered. The teacher makes sure that the students understand that probability is
based on the number of ways an event can occur divided by the total number of possible outcomes.
This is usually written as a ratio or a fraction. Another rules is that probability is always 0 to 1, where 0
indicates that an event is certain not to occur and 1 indicates that the event is certain to occur. In
addition, the students may have discovered that one event does not affect another. The last rule is the
probability of successive events is the product of each individual event. These rules are important for
the students to know and understand.
Opening the Lesson:           2. The teacher asks the students to think back to the first day of this probability unit. On the first day
10 minutes                they participated in a mock Connecticut Cash 5 lottery. One student is asked to refresh everyone’s
memory.
3. The teacher then goes over the rules of the lottery and writes them on the board:
LESSON                                                    5 different numbers are chosen
ORGANIZATION                                                 The numbers must be between 1- 40
 To win the lottery the person must get all five numbers correct
 The numbers can be in any order
4. The teacher also reminds the students that they need to take notes during this activity. Also, their
work from today is turned in at the end of class.

1. The students are put into 5 groups of 4. They are told to try to figure out what the actual probability
is of winning the Connecticut Cash 5 lottery.
Developing the Lesson:
2. After 25 minutes, the teacher calls the class back together.
30 minutes
3. On the board the class tries to determine the correct answer by putting their work together and by
the teacher’s guidance.

PROBABILITY: THE CHANCES YOU TAKE                                                                          BY   ALLISON SRUBAS                  Page 49
Closing the Lesson:
1. At the end of class, all of the students have the correct answer determined on their papers.
5 minutes

Chalk
TEACHING
Chalkboard
RESOURCES
Lottery rules

ENRICHMENT
ACTIVITIES

ACCOMMODATIONS
FOR SPECIAL
LEARNERS

STANDARDIZED
TEST ITEM
GENERATOR

> Go to State Exam
Test Items…

PROBABILITY: THE CHANCES YOU TAKE                                                                  BY   ALLISON SRUBAS                 Page 50
Lesson Title: Your Chances of Winning the Lottery
FOR THE STUDENT
>>> Go to FOR THE
TEACHER…                    6

LEARNING GOALS          Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases

Your work of determining the probability of winning the Connecticut Cash 5 lottery is collected at the end of class and graded as a pop-
ASSESSMENT
quiz.

First, you go over the probability rules that you established for homework. Then you refer back to the first day of this probability unit
LEARNING                when you participated in a mock Connecticut Cash 5 lottery. One student is asked to refresh everyone’s memory. Then you are given the
ACTIVITY                rules of the lottery. You are put into a group to try to determine the actual probability of winning the lottery. After about 25 minutes, you
work as a class to find the answer. Your work is collected so make sure that you show all of your work.

ENRICHMENT
ACTIVITIES

Pencil/Pen
MATERIALS/
Paper
EQUIPMENT
Past notes

PROBABILITY: THE CHANCES YOU TAKE                                                                          BY   ALLISON SRUBAS                  Page 51
Lesson Title: Brochures for Gamblers Anonymous
FOR THE TEACHER
STAGE OF LEARNING CYCLE:
EVALUATION
>>> Go to FOR THE
STUDENT…              7
LESSON OVERVIEW         The students make brochures for gamblers anonymous.

   Apply sample space and probability distribution to ―games of chance‖.
INSTRUCTIONAL
   Express their ideas in a brochure format
GOALS
   Present the information well.

FORMATIVE
The students are graded based on a rubric that they are given. It counts as a test grade.
ASSESSMENT

The teacher asks the students to reference the performance assessment sheets passed out at the beginning of the lesson. She has extra
sheets for the people who lost theirs. For about ten minutes the performance assessment is discussed and the teacher answers any
LESSON             questions. The teacher puts the students into groups and they spend the rest of the day playing the game they are assigned. If they have
ORGANIZATION          time, they can start researching the games. The students can go to the library or computer lab. On the second day, the students should
continue working on their brochures. This is the last day they have to work on it in class. On the third day, the students present their
brochures to class. Everyone is expected to participate in the presentation.

PROBABILITY: THE CHANCES YOU TAKE                                                                            BY   ALLISON SRUBAS            Page 52
―Games of Chance‖
Paper
Markers
TEACHING
Colored Pencils
RESOURCES
Library Access
Computer Lab Access

ENRICHMENT
ACTIVITIES

ACCOMMODATIONS
FOR SPECIAL
LEARNERS

STANDARDIZED
TEST ITEM
GENERATOR

> Go to State Exam
Test Items…

PROBABILITY: THE CHANCES YOU TAKE      BY   ALLISON SRUBAS   Page 53
Lesson Title: Brochures for Gamblers Anonymous
FOR THE STUDENT
>>> Go to FOR THE
TEACHER…                  7
LEARNING GOALS         Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases

ASSESSMENT           You are assessed according to the rubric that was passed out with the performance assessment assignment. It counts as a test grade.

LEARNING            You work in groups to make brochures for gamblers anonymous based on one of four ―games of chance‖. You are given two working
ACTIVITY            days in class and are required to present on the third day.

You have been asked by gamblers anonymous to create a brochure for gamblers on the probability used in ―games of chance‖. They are
looking for 4 brochures, one for roulette, one for poker, one for craps, and one for poker dice. They want to know what the probability of
winning or losing is for the games that they have been playing for years. In groups, you will either be playing and researching roulette,
poker, craps, or poker dice. Together you will have to create a brochure for the game you are assigned. First, you will be divided into
groups and assigned a casino game to play and research. You will be given two roles in your groups. There will be one leader to keep the
group on task. This person will also be in charge of setting up the game and cleaning up at the end of the class. There will also be a
recorder. This person will take notes during the game playing. However, everyone should record any individual feeling they have while
playing. Then, as a group you will be required to create a brochure for gamblers anonymous.

ACTIVITIES           · An original title
(TASK DESCRIPTION)      · A description of the game you are assigned.
· Your results while playing the game (take careful notes of what happens during the game).
· The probability rules and values in the game. You must include the sample space and the probability values of each possible outcome.
(Be sure to explain the meaning of these terms and values. They need you to create this brochure because they do not understand the
probability involved.)
· A report that cautions gamblers who play this game based on the probabilities that you found. Are the gamblers more likely to lose or
gain money?

You will be given 2 working days in class to complete this performance assessment. On the third day you will be required to present
your brochure. Your brochure should be creative. You need to make the brochure interesting to read and make sure to be neat!!

MATERIALS/           Pencil/Pen
EQUIPMENT            Paper

PROBABILITY: THE CHANCES YOU TAKE                                                                         BY   ALLISON SRUBAS                Page 54
PROBABILITY: THE CHANCES YOU TAKE   BY   ALLISON SRUBAS   Page 55

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