STANDARD 12 - PROBABILITY AND STATISTICS
K-12 Overview
All students will develop an understanding of statistics and
probability and will use them to describe sets of data, model
situations, and support appropriate inferences and arguments.
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Descriptive Statement
Probability and statistics are the mathematics used to understand
chance and to collect, organize, describe, and analyze numerical data.
From weather reports to sophisticated studies of genetics, from
election results to product preference surveys, probability and
statistical language and concepts are increasingly present in the
media and in everyday conversations. Students need this mathematics
to help them judge the correctness of an argument supported by
seemingly persuasive data.
Meaning and Importance
Probability is the study of random events. It is used in analyzing
games of chance, genetics, weather prediction, and a myriad of other
everyday events. Statistics is the mathematics we use to collect,
organize, and interpret numerical data. It is used to describe and
analyze sets of test scores, election results, and shoppers'
preferences for particular products. Probability and statistics are
closely linked because statistical data are frequently analyzed to see
whether conclusions can be drawn legitimately about a particular
phenomenon and also to make predictions about future events. For
instance, early election results are analyzed to see if they conform
to predictions from pre-election polls and also to predict the final
outcome of the election.
Understanding probability and statistics is essential in the modern
world, where the print and electronic media are full of statistical
information and interpretation. The goal of mathematical instruction
in this area should be to make students sensible, critical users of
probability and statistics, able to apply their processes and
principles to real-world problems. Students should not think that
those people who did not win the lottery yesterday have a greater
chance of winning today! They should not believe an argument merely
because various statistics are offered. Rather, they should be able
to judge whether the statistics are meaningful and are being used
appropriately.
K-12 Development and Emphases
Statistics and probability naturally lend themselves to plenty of
fun, hands-on cooperative learning and group activities. Activities
with spinners, dice, and coin tossing can be used to investigate
chance events. Students should discuss the theoretical probabilities
of different events such as the possible sums of a pair of dice,
andcheck them experimentally. They can choose topics to investigate,
such as how much milk and juice the cafeteria should order each day,
gather statistics on current orders and student preferences, and make
predictions on future use. Connections between these topics and
everyday experiences provide motivation and a sense of relevance to
students.
In the area of probability, young children start out simply
learning to use probability terms correctly. Words like
possibly, probably, and certainly have definite
meanings, referring to the increasing likelihood of an event
happening, and it takes children some time to begin to use them
correctly. Beyond that, though, elementary age children are certainly
able to understand the probability of an event. Starting with
phrases like once in six tosses, children progress to more
sophisticated probability language like chances are one out of six,
and finally to standard fractional, decimal, and percent notation
for the expression of a probability. To motivate and foster that
maturation, students should be regularly engaged in predicting and
determining probabilities.
Experiments leading to discussions about the difference between
experimental and theoretical probability should be done by
older elementary and middle school students. The theoretical
probability is the probability based on a mathematical analysis of
the physical properties and behavior of the objects involved in the
event. For instance, when a fair die is rolled each face is equally
likely to wind up on top, and so the probability of any particular
face showing is one-sixth. Experimental probabilities are
determined by data gathered through experiments. For example,
students may be able to compare the experimental probabilities of
rolling a sum of seven vs. a sum of four with two dice long before
they can explain why the first is twice as likely from a theoretical
point of view.
Older students should understand the difference between simple
and compound events, like rolling one die vs. rolling two dice,
and the difference between independent and dependent events,
like picking five marbles out of a bag of blue and green marbles one
at a time with replacement vs. without replacement. Again, the best
way to approach this content is with open-ended investigations that
allow the students to arrive at their own conclusions through
experimentation and discussion. Eventually, students should feel
comfortable representing real-life events using probability
models.
In statistics, young children can start out as early as
kindergarten with data collection, organization, and
graphing. The focus on those skills, with obviously
increasing sophistication, should last throughout their schooling.
Students must be able to understand the tables, charts, and graphs
used to present data, and they must be able to organize their own data
into formats which make them easier to understand. While young
students can do exhaustive surveys about some interesting question for
all of the members of the class, older students should focus
some time and energy on the questions involved with sampling,
where information is obtained from only some of the members of
a group. Identifying and obtaining data from a well-defined sample of
the population is one of the most challenging tasks of a professional
pollster.
As students progress through the elementary grades, an increased
focus on central tendency and later, on variance and
correlation, are appropriate. Students should be able to use
the average or mean, the median, and the
mode and understand the differences in their uses. Measures of
the variance from the center of a set of data, or dispersion, also
provide useful insights into sets of numbers. These can be introduced
early with the range for the early grades, box-and-whisker
plots showing quartiles of a data distribution for upper
elementary school students, and progress to measures like standard
deviation for older students.
The reason statistics grew as a branch of mathematics, however, was
to provide tools that are helpful inanalysis and inference in
situations of uncertainty, and that focus should permeate
everything students do in this area. Whenever they look at data, they
should be trying to answer a question, support a position, or discover
a pattern. Students at all grade levels should have many
opportunities to look for patterns, draw conclusions, and make
predictions about the outcomes of future experiments, polls, surveys,
and so on. They should examine data to see whether they are consistent
with some hypotheses that a classmate may already have made, and learn
to judge whether the data are reliable or whether the hypothesis might
need revision.
In summary, probability and statistics hold the key for
enabling our students to better understand, process, and interpret the
vast amounts of quantitative data that exist all around them, and to
have a probabilistic sense in situations of uncertainty. To be able
to judge the validity of a data-supported argument presented to them,
to discern the believability of a persuasive advertisement that talks
about the results of a survey of all of the users of a particular
product, or to be knowledgeable consumers of the data-intensive
government and electoral statistics that are ever-present, students
need the skills that they can learn in a well-conceived probability
and statistics curriculum strand.
Note: Although each content standard is discussed
in a separate chapter, it is not the intention that each be
treated separately in the classroom. Indeed, as noted in the
Introduction to this Framework, an effective curriculum is one
that successfully integrates these areas to present students with rich
and meaningful cross-strand experiences.
Standard 12 - Probability and Statistics - Grades K-2
Overview
Students can develop a strong understanding of probability and
statistics from consistent experiences in classroom activities where a
variety of manipulatives and technology are used. The key components
of this understanding in probability for early elementary students, as
identified in the K-12 Overview, are probability terms, the
concept of the probability of an event, and predicting and
determining probabilities. In statistics they key
components for early elementary students are data collection,
organization, and representation.
The understanding of probability and statistics begins with their
introduction and use at the earliest levels of schooling. Children
are natural investigators and explorers - curious about the world
around them, as well as about the opinions and the habits of their
classmates, teachers, neighbors and families. Thus, a fertile setting
already exists in children for the development of statistics and
probability skills and concepts. As with most of the curriculum at
these grade levels, the dominant emphasis should be experiential with
numerous opportunities to use the concepts in situations which are
real to the students. Statistics and probability can and should
provide rich experiences to develop other mathematical content and
relate mathematics to other disciplines.
Kindergarten students can gather data and make simple
graphs to organize their findings. These experiences should
provide opportunities to look for patterns in the data, to answer
questions related to the data, and to generate new questions to
explore. By playing games or conducting experiments related to
chance, children begin to develop an understanding of probability
terms.
First- and second-grade children should continue to collect and
organize data. These activities should provide opportunities for
students to have some beginning discussions on sampling, and to
represent their data in charts, tables, or graphs which
help them draw conclusions, such as most children like
pizza or everyone in the class has between 0 and 4
sisters and brothers, and raise new questions suggested by the
data. As they move through this level, they should be encouraged to
design data collection activities to answer new questions.
They should be encouraged to see how frequently statistical claims
appear in their life by collecting and discussing appropriate items
from advertising, newspapers, and television reports.
Students in these grades should experience probability at a variety
of levels. Numerous children's games are played with random
chance devices such as spinners and dice. Students should have
opportunities to play games using such devices. Games where students
can make decisions based upon their understanding of probability help
to raise their levels of consciousness about the significance of
probability. Gathering data can lead to issues of probability as
well. Students should experience probability terms such as
possibly, probably, and certainly in a variety of
contexts. Statements from newspapers, school bulletins, and their own
experiences should highlight their relation to probability. In
preparation for later work, students need to have experiences which
involve systematic listing and counting of possibilities, such as all
the possible outcomes when three coins are tossed (see Standard 14,
Discrete Mathematics.)
Learning probability and statistics provides an excellent
opportunity for connections with the rest of the mathematics standards
as well as with other disciplines. Probability provides a rich
opportunity for children to begin to gain a sense of fractions.
Geometry is frequently involved through use of student-made spinners
of varying-sized regions and random number generating devices such as
dice cubes or octahedral (eight-sided) shapes. The ability to explain
the results of data collection and attempts at verbal generalizations
are the foundations of algebra. Making predictions in both
probability and statistics provides students opportunities to use
estimation skills. Measurement using non-standard units occurs in the
development of histograms using pictures or objects and in discussions
of how the frequency of occurrence for the various options are
related. Even the two areas of this standard are related through such
things as the use of statistical experiments to determine estimates of
the probabilities of events as a means for solving problems such as
how many blue and red marbles are in a bag.
The topics that should comprise the probability and statistics
focus of the kindergarten through second grade mathematics program
are:
- collecting data
- organizing and representing data with tables, charts and graphs
- beginning analysis of data using concepts such as range and "most"
- drawing conclusions based on data
- using probability terms correctly
- predicting and determining probability of events
Standard 12 - Probability and Statistics - Grades K-2
Indicators and Activities
The cumulative progress indicators for grade 4 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in kindergarten
and grades 1 and 2.
Experiences will be such that all students in grades K-2:
1. Formulate and solve problems that
involve collecting, organizing, and analyzing data.
- Students collect objects such as buttons, books, blocks,
counters, etc. which can be sorted by color, shape, or size. They
classify the objects and color one square of a bar graph for each item
using different colors for each category. Then they compare the
categories and discuss the relationships among them.
- As an assessment following activities such as the
one described above, young students are given a sheet of picture
stickers and a blank sheet of paper. They sort the stickers according
to some classification scheme and then stick them onto the paper to
form a pictograph showing the number in each category.
- At the front of the room is a magnetic board and, for every
child in the class, a magnet with that child's picture. At the
start of each day, the teacher has a different question on the board
and the children place their magnet in the appropriate area. It might
be a bar graph tally for whether they prefer vanilla, chocolate or
strawberry ice cream or a Venn diagram where students place their
magnet in the appropriate area based on whether they have at least one
brother, at least one sister, at least one of both, or neither.
- Students survey their classmates to determine
preferences for things such as food, flavors of ice cream, shoes,
clothing, or toys. They analyze the data collected to develop a
cafeteria menu or to decide how to stock a store.
- Second graders record and graph the times of sunrise and
sunset one day a week over the entire year. They calculate the time
from sunrise to sunset, make a graph of the amount of daylight, and
interpret these weekly results over the year.
- A second grader, upset because she had wanted to
watch a TV show the night before but had to go to bed instead, asks
the teacher if the class can do a survey to find out when most
children her age go to bed.
2. Generate and analyze data obtained
using chance devices such as spinners and dice.
- Students roll a die, spin a spinner, or reach blindly into
a container to select a colored marble, with replacement, a dozen
times. They then color the appropriate square in a bar graph for each
pick. Did some results happen more often or less often than
others? Do you think some results are more likely to happen
than others? They repeat the experiment, this time without
replacement, and compare the results.
- Students spill out the contents of cups containing five
two-colored counters and record the number of red sides and the number
of yellow sides. They perform the experiment twentytimes, examine
their data, and then discuss questions such as Does getting four
red sides happen more often than two red sides? They
explain their reasoning.
- Each student has a 4-section spinner. Working in pairs,
the students spin their spinners simultaneously and together they
record whether they have a match. After doing this several times,
they predict how many times they would have a match in 20 spins. Then
they compare their prediction with what happens when they actually
spin the spinners 20 times. They repeat the activity with a different
number of equal sections marked on their spinners. Students in the
second grade combine the results of all the students in the class, and
compare their predictions with the class total.
3. Make inferences and formulate hypotheses based
on data.
- Students roll a pair of dice 100 times and make a
frequency bar graph of the sums. They compare their results with
those of their classmates. Do your graphs look essentially
alike? Which sum or sums came up the most? Does everyone have
a 'winning' sum? Is it the same for
everyone? Why do some sums come up less than others?
- Children are regularly asked to think about their data.
Is there a pattern in the dice throws, bean growth, weather,
temperature, or other data? What causes the patterns? Are the
patterns in their data the same as those of their classmates?
4. Understand and informally use the
concepts of range, mean, mode, and median.
- When performing experiments, children are regularly asked
to find the largest and smallest outcomes (range) for numerical data
and the outcome that appeared most often (mode). They are asked to
compare the mode they obtained for an experiment with the modes found
by their classmates.
5. Construct, read, and interpret displays of data
such as pictographs, bar graphs, circle graphs, tables, and
lists.
- After collecting and sorting objects, children develop a
pictograph or histogram showing the number of objects in each
category.
- Students design and make tallies and bar graphs to display
data on information such as their birth months.
- Students list all possible outcomes of probability
experiments, such as tossing a penny, nickel, and dime
together.
- Working in cooperative groups, students are given six
sheets of paper each containing an outline of a circle which has been
divided into eight equal sectors. The students color each whole
circle a different color and then cut their circles into individual
sectors so each group has 8 sectors in each of 6 colors. Then they
roll a die eight times keeping a tally of the results using orange for
rolls of 1, blue for rolls of 2, and so on. They use these eight
colored sectors to record their results in a circle graph, which they
put aside. They repeat this twice and get two other circle graphs.
Finally, as a whole class activity, they gather the circle graphs from
all the groups, and rearrange the sectors to make as many solid color
circles as they can. They discuss the results.
- Students regularly read and interpret displays of data;
they also read information from their classmates' graphs and
discuss the differences in their results.
6. Determine the probability of a simple event, assuming
equally likely outcomes.
- Children roll a die ten times and record the number of
times each number comes up. They combine their tallies and discuss the
class results.
- Children predict how often heads and tales come up when a
coin is tossed. They toss a coin ten times and tally the number of
heads and tails. Are there the same number of heads and
tails? They combine their tallies and compare their class results
with their predictions. (See Making Sense of Data, in the
Addenda Series, by Mary Lindquist.)
7. Make predictions that are based on intuitive,
experimental, and theoretical probabilities.
- Second graders are presented with a bag in which they are
told are marbles of two different colors, twice as many of one color
as the other. They are asked to guess the probability for drawing
each color if a single marble is drawn. Is this the same as
flipping a coin? Will one color be picked more often than the
other? The experiment is performed repeatedly and tallies are
recorded. The chosen marble is returned to the bag each time before a
new marble is drawn. The children discuss whether their estimates of
the probabilities made sense in light of the outcome.
- Students are told that a can contains ten beads, some red
ones, some yellow ones, and some blue ones. They are asked to predict
how many beads of each color are in the can. The students attempt to
determine the answer by doing a statistical experiment. One at a
time, each child in the class draws a bead, records the color with a
class tally, and replaces it. At various times in the process, the
teacher asks the children to return to their prediction to determine
if they want to modify it.
- As an informal assessment of the students'
understanding of these concepts, they are presented with a bag in
which they are told there are 10 yellow marbles and 2 blue ones. They
are asked to predict what color marble they will pick out of the bag
if they pick without looking, and about how many students in the class
will pick a blue marble.
8. Use concepts of certainty, fairness, and
chance to discuss the probability of actual events.
- Students work through the Elevens Alive!
lesson that is described in the Introduction to this Framework.
They make number sentences adding up to 11 by dropping 11 chips which
are yellow on one side and red on the other, and writing 11= 4+7 when
four chips land yellow-side-up and seven chips land red-side-up. They
notice that they are writing some number sentences more frequently
than others, and these observations lead into a discussion of
probability.
- Each child plants five seeds of a fast growing plant.
They count the number of seeds which sprout and discuss how many seeds
might sprout if they had each planted ten, or twenty, or a hundred
seeds. They explain their reasoning. (The numbers can be adjusted for
different grade levels.)
- Students predict how many M&Ms of each color are in a
large unopened mystery bag. To help make these predictions,
cooperative groups are given a handful of M&Ms from the bag; they
tally the count of the colors, report their results, and prepare
graphs of their results. Students refine their predictions by looking
at the class totals. The mystery bag is then opened and the colors
counted. Students discuss how their prediction matches theactual
count and how the experiment helped them make their prediction.
- Students examine various types of raisin bran cereal.
They experiment with scoops of cereal and determine the number of
raisins that appear in each scoop. They make inferences about which
brand might be the "raisiniest."
References
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Lindquist, M., et al. Making Sense of Data.
Curriculum and Evaluation Standards for School
Mathematics Addenda Series, Grades K-6. Reston,
VA: National Council of Teachers of Mathematics, 1992.
General References
-
Burton, G., et al. First Grade Book. Curriculum and
Evaluation Standards for School
Mathematics Addenda Series, Grades K-6. Reston, VA:
National Council of Teachers of Mathematics, 1991.
Burton, G., et al. Kindergarten Book. Curriculum and
Evaluation Standards for School
Mathematics Addenda Series, Grades K-6. Reston, VA:
National Council of Teachers of Mathematics, 1991.
On-Line Resources
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http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during Spring
1997. In time, we hope to post additional resources relating to this
standard, such as grade-specific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
Standard 12 - Probability and Statistics - Grades 3-4
Overview
Students can develop a strong understanding of probability and
statistics from consistent experiences in classroom activities where a
variety of manipulatives and technology are used. The key components
of this understanding in probability for elementary school students,
as identified in the K-12 Overview, are: probability terms, the
concept of the probability of an event, predicting and determining
probabilities, and the relationship between theoretical
and experimental probabilities. In statistics the key components
for elementary school students are data collection, organization,
and representation, central tendency, and analysis and
inference.
Based on their earlier experiences with data, third- and
fourth-graders should strengthen their ability to collect,
organize, and represent data. They should build on their informal
discussions of data by developing their ability to analyze data,
formulate hypotheses, and make inferences from the data. As their
numerical skills increase, they should begin to understand and to use
the mean and median, as well as range and mode, as
measures of central tendency. Frequent probability experiments
should help students extend their ability to make predictions
and understand probability as it relates to events around them, and
should provide the intuition they will need in order to determine
probabilities in simple situations.
As in the previous grade levels, probability and statistics
understanding is best developed through frequent opportunities to
perform experiments and gather and analyze data. Such activities are
most valuable when students choose a topic to investigate based on a
real problem or based on an attempt to answer a question of interest
to them. Children should experience new activities, but they should
have the opportunity to revisit problems introduced in grades K-2 when
doing so would allow them to practice or develop new understandings.
Probability and statistics are closely related. Students should
use known data to predict future outcomes and they should grapple with
the concept of uncertainty using probability terms such as
likely, not likely, more likely, and less
likely. Developing an understanding of randomness in probability
is crucial to acquiring a more thorough understanding of
statistics.
Third and fourth grade is a wonderful time for students to see
connections among subjects. Most science programs at this level
involve collection and analysis of data as well as a focus on the
likelihood of events. Social studies programs usually ask children to
begin to develop ideas of the world around them. Discussions might
focus on their school, neighborhood, and community. Such explorations
can be enhanced through analysis and discussion of data such as
population changes over the last century. Third-and fourth-graders
are more attuned to their environment and are more sensitive to media
information than early elementary school children. Discussions about
such things as the claims in TV advertisements or commercials, or
newspaper articles on global warming, help students develop the
ability to use their understandings in real situations.
At all grade levels, probability and statistics provide students
with rich experiences for practicing their skills in content areas
such as number sense, numerical operations, geometry, estimation,
algebra, patterns and functions, and discrete mathematics.
The topics that should comprise the probability and statistics
focus of the mathematics program in grades three and four are:
- collecting, organizing, and representing data
- analyzing data using the concepts of range, mean, median, and mode
- making inferences and formulating hypotheses from their analysis
- determining the probability of a simple event assuming outcomes are equally likely
- making valid predictions based on their understandings of probability
Standard 12 - Probability and Statistics - Grades 3-4
Indicators and Activities
The cumulative progress indicators for grade 4 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 3 and
4.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 3-4 will be such that all students:
1. Formulate and solve problems that involve
collecting, organizing, and analyzing data.
- Students wish to study the differences in temperature
between their hometown and a school they have connected with in Sweden
through the Internet. They exchange highs and lows for each Monday
over the three-month period from January through March. They note
whether the temperatures are given in degrees Celsius or Fahrenheit,
and use a thermometer with both markings to change from one to the
other if necessary. They organize and represent the data and develop
questions about possible differences in lifestyle that are prompted by
the temperature. They then exchange their questions with their sister
school to learn more about their culture.
- While studying about garbage and recycling, children
notice the amount of waste generated in the cafeteria each day. A
variety of questions begin to surface such as: What types
of waste are there? How much of each? Can we measure it? How? How
often should we measure it to get an idea of the average amount
of waste generated each day? How can we help make less
waste? The class considers how it can find answers to these
questions, designs a way to obtain the data, and finds answers to
their questions.
- Students perform experiments such as rolling a toy car
down a ramp and measuring the distance the car rolled beyond the
bottom of the ramp. This experiment is repeated, holding the top of
the ramp at various heights above the ground. Students discuss the
patterns and relationships they see in the data and use their
discoveries to predict the distances obtained for ramps of other
heights.
2. Generate and analyze data obtained using
chance devices such as spinners and dice.
- Each child in the class rolls a die 20 times and records
the outcomes in a frequency table. The class combines the results in a
class frequency table. They discuss which outcome occurred most often
and least often and then whether the class results differ from their
individual results and why that might be.
- Students make their own cubes from cardstock and label the
sides 1, 2, 2, 3, 4, 5. They roll their cubes 20 times each,
recording the results. After combining their results, the class
discusses the experiment and the reasons the results differ from the
results obtained when using a regular die.
- As a question on a class test, students
are told that Sarah rolled a die 20 times and she got twelve 1s,
two 2s, three 3s, and three 6s. They are
asked what they would conclude about Sarah's experiment and what
might have accounted for her
results.
3. Make inferences and formulate
hypotheses based on data.
- Students read A Three Hat
Day by Laura Geringer. They use concrete objects (different
colored beans, hats, or pattern blocks) to show different orders for
wearing three different hats. They investigate how many different
ways there are to wear four different hats.
- After collecting, organizing, and analyzing data on the
favorite sport of the fourth graders in their school, third graders
are asked to interpret the findings. Why do you suppose
soccer was chosen as the favorite sport? How close were other
sports? What if we collected data on the same question from
fourth graders in another county or another state? Do you
think first graders would answer similarly? Why?
- Students read Mr. Archimedes' Bath and
Who Sank the Boat by Pamela Allen and discuss what happens to
the water level in a container as things are added and why.
- The fourth grade class is planning a walking tour of a
local historic district in February. They want to take hot chocolate
but don't know which type of cup to take so that it stays warm as
long as possible after being poured. In the science unit on the
cooling of liquids, the students discussed notions of variables and
constants. They set up an experiment using cups of the same size but
of different materials and measure the temperatures in each at equal
intervals over a 30-minute period. They plot the data and use their
graphs to discuss which cup would be best.
4. Understand and informally use the
concepts of range, mean, mode, and median.
- Before counting the number of raisins contained in each
of 24 individual boxes of raisins, students are asked to estimate the
number of raisins in each box. They count the raisins and compare the
actual numbers to their estimates. Students discover that the boxes
contain different numbers of raisins. They construct a frequency
chart on the blackboard and use the concepts of range, mean, median,
and mode to discuss the situation.
- In a fourth grade assessment, students are asked to
prepare an argument to convince their parents that they need a raise
in their allowance. Students discuss what type of data would be
needed to support their argument, gather the data, and use descriptive
measures as a basis for their argument. In a cooperative effort,
sixth grade students play the part of parents and listen to the
arguments. The sixth graders provide feedback as to whether the
students had enough information to convince them to raise the
allowance and, if not, what more they might use.
5. Construct, read, and interpret displays of data
such as pictographs, bar graphs, circle graphs, tables, and
lists.
- Presented with a display of data from USA TODAY,
students generate questions which can be answered from the display.
Each child writes one question on a 3x5 card and gives it to the
teacher. The cards are shuffled and redistributed to the students.
Each student then answers the question he or she has been given and
checks the answer with the originating student. Disagreements are
presented to the class as a whole for discussion.
- Following a survey of favorite TV shows of students in the
entire third grade, groups of students develop their own pictographs
using symbols of their choosing to represent multiple children.
6. Determine the probability of a simple event assuming
equally likely outcomes.
- Children toss a coin fifty times and record the results as
a sequence of Hs and Ts. They tally the number of heads and tails.
Are there the same number of heads and tails? The children
discuss situations that often lead to misconceptions such as If
three tosses in a row come up heads, what is the chance that
the next toss is a head? Is there a better chance than
there would have been before the other tosses took place? After
what is a lively discussion, the children review their sequence of Hs
and Ts to see what happened on the next toss each time that three
consecutive heads appeared. This analysis should demonstrate that
each result does not depend upon the previous ones.
- Students discuss the probability that a particular number
will come up when a die is thrown, and predict how many times that
number will appear if the die is rolled 50 times. They then toss a die
50 times and compare the results with their predictions.
7. Make predictions that are based on
intuitive, experimental, and theoretical probabilities.
- Fourth-graders are presented with a bag in which there are
marbles of three different colors, the same number of two of the
colors, and twice as many of the third. They are asked what they
would expect to happen if a marble were drawn twelve times and placed
back in the bag after each time. The experiment is performed and the
children discuss whether their estimates of the outcome made sense in
light of the actual outcome.
- During an ecology unit, students discuss the
capture-recapture method of counting wildlife in a local refuge. A
number of animals, say 30 deer, are captured, tagged, and released;
later another group of deer is captured. If five of the twenty-five
recaptured deer are tagged, then you might conclude that about one in
five deer have been tagged, and therefore that the total number of
deer in the refuge is about 5 x 30 or 150. The students perform a
capture-recapture experiment using a large bag of lollipops to
determine the number of lollipops in the bag.
8. Use concepts of certainty, fairness, and
chance to discuss the probability of actual events.
- Students discuss the probability of getting a zero or a
seven on the roll of one die or picking a blue bead from a bag full of
blue beads, and use this as an introduction to a discussion about the
probability of certain events and impossible events.
- Students discuss the relationship between events such as
flipping a coin, a newborn baby being a girl, guessing on a true-false
question, and other events which have an approximately equal chance of
occurring.
References
-
Allen, Pamela. Mr. Archimedes' Bath. New
York: Lothrop, Lee, and Shepard Books, 1980.
Allen, Pamela. Who Sank the Boat. New York: Putnam,
1990.
Lindquist, M., et al. Making Sense of Data.
Curriculum and Evaluation Standards for School
Mathematics Addenda Series, Grades K-6. Reston,
VA: National Council of Teachers of Mathematics, 1992.
General References
-
Burton, G., et al. Third Grade Book. Curriculum and
Evaluation Standards for School
Mathematics Addenda Series, Grades K-6. Reston, VA:
National Council of Teachers of Mathematics, 1992.
Geringer, Laura. A Three Hat Day. New York: Harper and
Row, 1985
On-Line Resources
-
http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during Spring
1997. In time, we hope to post additional resources relating to this
standard, such as grade-specific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
Standard 12 - Probability and Statistics - Grades 5-6
Overview
Students can develop a strong understanding of probability and
statistics from consistent experiences in classroom activities where a
variety of manipulatives and technology are used. The key components
of this understanding in probability for middle school students, as
identified in the K-12 Overview, are: probability terms, the
concept of the probability of an event, predicting and determining
probabilities, the relationship between theoretical and
experimental probabilities, and compound events. In
statistics, the key components are: data collection, organization,
and representation, sampling, central tendency, variance and
correlation, and analysis and inference.
In grades K4, students explored basic ideas of statistics by
gathering data, organizing data, and representing data in charts and
graphs, and then using this information to arrive at answers to
questions and raise further questions. Students in grades 5 and 6 are
keenly interested in movies, fashion, music, and sports. These areas
provide a rich source of real problems for students at this age. The
students should make the decision on how to sample and then
collect and organize data. They should determine how best to
represent the data and begin to develop a more formal
understanding of summary statistics of central tendency such as
the mean, the median, and the mode. They should recognize that for
certain types of data, such as height, the mean is an appropriate
measure, but it is inappropriate for other types of data, such as hair
color. These activities should provide opportunities for students to
analyze data and to make inferences regarding the data
and to communicate their inferences in a convincing manner. They
should further develop their understanding of statistics through the
evaluation of arguments by others, whether they come from classmates,
advertising, political rhetoric, or news sources.
While statistical investigations can be similar to those in earlier
grades, fifth- and sixth-graders should have access to statistical
software on computers or calculators which have statistical
capability. This will allow them to carry out statistical work using
real data without becoming mired in tedious calculations. The
technology will be used to do the manipulation of the data and the
students will focus on developing their skills in interpreting the
data.
Students enter these grades having participated in a wide variety
of activities designed to help them understand the nature of
probability and chance. The emphasis in grades K-4 was primarily on
simple events such as the roll of a die or the flip of one coin. Even
when compound events such as the roll of two dice were considered, the
outcomes were looked upon as a simple event. In grades 5 and 6,
students begin to experiment with compound events such as flips
of several coins and rolls of dice and to predict and evaluate their
theoretical and experimental probabilities. As they develop
their understanding of fractions, ratios, and percents, they should
use them to represent probabilities in place of phrases such as
"three out of four." They begin to model probability
situations and to use these models to predict events which are
meaningful to them.
At all grade levels, probability and statistics provide students
with rich experiences for practicing their skills in content areas
such as number sense, numerical operations, geometry, estimation,
algebra, patterns and functions, and discrete mathematics. Because
most of the activities are hands-on and students are constantly
dealing with numbers in a variety of ways, it assists the development
of number sense as well.
The topics that should comprise the probability and statistics
focus of the mathematics program in grades 5 and 6 are:
- collecting, organizing, and representing data
- analyzing data using range and measures of central tendency
- making inferences and hypotheses from their analysis of data
- evaluating arguments based upon data analysis
- interpolating and/or extrapolating from data using a line of best fit
- representing probability situations in a variety of ways
- modeling probability situations
- predicting events based on real-world data
Standard 12 - Probability and Statistics - Grades 5-6
Indicators and Activities
The cumulative progress indicators for grade 8 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 5 and
6.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 5-6 will be such that all students:
9. Generate, collect, organize, and
analyze data and represent this data in tables, charts, and
graphs.
- Students recognize that this is a time of growth for many
of them. The class measures various lengths associated with a person,
such as height, length of forearm, length of thigh, handspan, length
of foot, and armspan. They enter the data into a spreadsheet and
produce various graphs as well as statistical analyses of their
measures. They update their data every month and discuss the change,
both individually and as a class.
- Students survey another class to determine data of
interest, such as the last movies seen, and then organize the data and
produce reports discussing the interests of the grade level.
- Students work on problems like this one from the
New Jersey State Department of Education's Mathematics
Instructional Guide (p. 7-95):
A fair spinner with 4 congruent regions labeled A, B, C, and D
is spun 20 times by each member of a class of 23 students.
Assume that your class conducted the experiment and obtained
the expected results. Make a bar graph illustrating the
combined class results. Explain why you drew your bar
graph the way you did.
Explain why an individual
student's results might be different from the class
results.
10. Select and use appropriate graphical
representations and measures of central tendency (mean, mode,
and median) for sets of data.
- Students demonstrate understanding of measures of central
tendency by writing a letter to a fictional classmate explaining how
the mean, the median, and the mode each help to describe data. They
then extend their discussion by presenting a picture of an
"average student" in their grade. The picture discusses
height, color of hair, preference in movies, etc. In creating the
picture, the students must choose the appropriate measures of central
tendency based upon the type of data and justify their choice. (For
example, the mean is not appropriate in discussing hair color.) They
will likely want to present pictures of both an "average
boy" and an "average girl" in the class.
- During a social studies unit, students determine a method
to ascertain the value of the homes in their community. They
determine the mean, the median, and the mode for the data and decide
which provides the most accurate picture of the community. They
include in their study homes from different sections of the town.
- Students perform an experiment where one group is
given 10 words in a jumbled order while another group is given them in
a sequence which facilitates remembering them.After giving each group
one minute to study the words, the students are asked to turn their
papers over and write as many of the words as they remember. The
papers are graded by fellow students and the scores reported. After
considering various graphing methods, the students determine that a
box-and-whiskers plot would be the best way to illustrate the
results and compare the two groups.
11. Make inferences and formulate and evaluate
arguments based on data analysis and data displays.
- Students are asked to develop a generalization about their
classmates. They are allowed to make any hypothesis which is
appropriate. For example, some boys might suggest that boys are
stronger than girls or others might say that girls are taller than
boys. They should determine how they would determine the validity of
their hypotheses by designing a data collection activity related to
it.
- The teacher in one fifth-grade class is especially alert
for generalizations that students make about any topic. She writes
them on slips of paper, and keeps them in a box. As an assessment of
the students' ability to develop statistical activities to
validate hypotheses, groups of students pull slips from the box,
develop data collection activities, collect the data, analyze it, and
make reports to the class about the validity of the generalizations
originally made.
- Students are shown a newspaper article which states that
25% of fifth graders have smoked a cigarette. They discuss their
reaction by indicating whether they believe the figure to be correct,
too high, or too low. They then design a survey which they use to
poll their fellow fifth graders in an effort to check the validity of
the claim for the population of their school. They also send a letter
to the newspaper requesting the sources of data for the article and
compare the data in the article with their data.
12. Use lines of best fit to interpolate and predict
from data.
- Given a jar with straight sides and half filled with
water, students drop marbles in five at a time. After each group of
five, they measure the height of the water and record in a table the
number of marbles in the jar and the height of the water. The
students then represent their data in a scatterplot on an x-y plane
and find that the points lie almost exactly in a straight line. They
draw a line through the data and use it to determine answers to
questions like: How high will the water be after 25 marbles have
been added? and How many marbles will it take to have
the water reach the top? Activities like this one form the
foundation for understanding graphs in algebra.
13. Determine the probability of a compound
event.
- Students create a table to show all possible results of
rolling two dice. At the left of the rows are the possible rolls of
the first die and at the top of the columns are the possible rolls of
the second die. They complete the table by putting in each cell the
appropriate sum of the number in the top row and the left column.
Counting the number of times each sum appears in the table, they
determine the probability of getting each possible sum. They then roll
two dice 100 times and compare the sums they get with the sums
predicted from the table.
- Students make a list of all possible outcomes when four
coins are tossed and determine thetheoretical probability of having
exactly two heads and two tails.
14. Model situations involving probability, such as
genetics, using both simulations and theoretical
models.
- Students examine the probability of a family with four
children having two boys and two girls by simulating the situation
using four coins. They first choose which side of the coin will
represent males and which will represent females. They toss the set
of coins 50 times and record their results as the number of boys and
the number of girls in each "family." They compare the
results of their experiment with the prediction based on
probability. They also survey a large sample of students in the school
and record the family composition of all families with four children.
All of these are used to discuss the likelihood of an evenly-matched
family.
- A 25-cent "prize" machine in the grocery store
contains an equal number of each of six plastic containers with Power
Ranger tattoos. Students are asked to determine how many containers
they need to buy to have a good chance of getting all six. They
simulate this situation with a bag containing an equal number of six
different colored marbles. They draw out, record, and replace one
marble at a time until they have drawn marbles of all six colors,
recording the number of times that took. They repeat the simulation
three times. The class results are gathered and discussed. One issue
discussed is whether the model is a good one for the situation or
whether it should be modified in some way to better represent
reality.
- Students read Caps for Sale by Esphyr
Slobodkina. The peddler in the story sells caps and wears his entire
inventory on his head: a checked cap and four each of identical blue,
gray, and brown hats. Students use concrete objects to model some of
the different orders in which the hats can be worn. They come to
realize that there are many ways and try to discover the total number
of different ways. They search for an efficient way to determine the
number of permutations.
- Students work through the Two-Toned Towers
and Pizza Possibilities lessons that are described in the First
Four Standards of the Framework. They make a systematic list
of all the towers built out of four red and blue cubes (or of all the
pizza combinations) and calculate the probability that a tower has
three or four blue cubes.
15. Use models of probability to predict
events based on actual data.
- Students examine weather data for their community from
previous years, and then use their analysis of the data to predict the
weather for the upcoming month. They compare the actual results with
their predictions after the month has passed and then use the
comparison to determine ways to improve their predictions.
- Using data from previous years, students determine the
number of times their favorite professional football team scored a
number of points in each of six ranges of scores (0-5, 6-10, 11-15,
16-20, 21-25, and 26 or more). They determine the fraction or
percentage of games the score was in each of those ranges and make a
spinner whose areas are divided the same way. Each Friday during
football season, they spin their spinners to predict how many points
the team will score and who will win the game. Toward the end of the
season, they discuss the success or failure of their efforts and the
probable causes.
16. Interpret probabilities as ratios and
percents.
- The students are introduced to the game Pass The Pigs
(Milton Bradley) where two small hardrubber pigs are rolled. Each
pig can land on a side where there is a dot showing, a side where the
dot does not show, on its hooves, on its back, leaning forward
balancing itself on its snout, and balancing itself on its left
foreleg, snout, and left ear. The students determine the fairness of
the distribution of points on the sides of the pig by rolling the pig
numerous times, recording the results, and using the ratios of
successes for each, divided by the total number of rolls, to represent
the probability of obtaining each situation.
- Students examine uses of probability expressed as
percentages in such situations as weather forecasting, risks in
medical operations, and reporting the confidence interval of
surveys.
References
-
New Jersey State Department of Education,
Mathematics Instructional Guide: Linking Classroom
Experiences to Current Statewide Assessments.
D. Varygiannes, Coord. Trenton, N.J., 1996.
Slobodkina, Esphyr. Caps for Sale. New York: W.R. Scott,
1947.
General References
-
Stenmark, J. K., et al. Family Math. Berkeley, CA:
Regents, University of California, 1986.
Zawojewski, Judith, et. al. Dealing with Data and Chance.
A component of the Curriculum and
Evaluation Standards for School Mathematics Addenda Series,
Grades 5-8. Reston, VA: National Council of Teachers of
Mathematics, 1991.
On-Line Resources
-
http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during Spring
1997. In time, we hope to post additional resources relating to this
standard, such as grade-specific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
Standard 12 - Probability and Statistics - Grades 7-8
Overview
Students can develop a strong understanding of probability and
statistics from consistent experiences in classroom activities where a
variety of manipulatives and technology are used. The key components
of this understanding in probability for middle school students, as
identified in the K-12 Overview, are: probability terms, the
concept of the probability of an event, predicting and determining
probabilities, the relationship between theoretical and
experimental probabilities, and compound events. In
statistics, the key components are: data collection, organization,
and representation, sampling, central tendency, variance and
correlation, and analysis and inference.
Students should enter the seventh grade with a strong intuitive
understanding of probability and statistics as a result of their
activities in grades K-6, and should have a basic understanding of the
more formal methods which were introduced in grades 5-6. They will
build on this foundation in grades 7 and 8.
Students in grades 7 and 8 present unique challenges. They are
turning to their peer group for leadership and support and, at the
same time, placing a strain on the relationships between themselves
and significant adults in their lives. Some students begin to
experiment with things they associate with being an adult: smoking,
alcohol, drugs, and sex. The quantity of statistics in all of these
areas provides an ideal opportunity to weave together statistical
activities which dovetail with information provided by the the health
and physical education department.
Students at these ages also become more aware of community issues.
Integrating statistics activities with topics in the social studies
curriculum can enhance their work in both areas as well as fit in with
their growing interests. Handson science activities require good
statistical methods and understanding in order to develop accurate and
appropriate conclusions. At the same time, students need to
understand how often statistics and probability statements are
incomplete, misunderstood, or purposely used to mislead. Having
students read books such as How to Lie with Statistics by
Darryl Huff or Innumeracy by John Allen Paulos provides
excellent opportunities to discuss how statistics and probability are
misused.
In statistics, students continue to collect, organize, and
represent data and to use various measures of central
tendency to describe their data. But they should now become more
focused on sampling techniques that justify making
inferences about entire populations. Examples of this appear
frequently in the news media. They also begin to explore variance
and correlation as additional tools in describing sets of
data.
Many of the probability experiments should continue to be related
to games and other fun activities. Students in these grades should
continue to develop their understanding of compound events and
their related probabilities, and should continue to consider and
compare experimental and theoretical probabilities.
Furthermore, the connection between probability and statistics should
help them understand issues such as sampling and
reliability. Students need to develop a sense of the
application of probability to the world around them as well. Everyday
life is rich with "coincidences" which are actually likely
to occur. For example, they should examine the probability that two
people in their class, or any group of 25 or more people, have the
same birthday. The results always stir up considerable interest and
disbelief.
At all grade levels, probability and statistics provide students
with rich experiences for practicing their skills in content areas
such as number sense, numerical operations, geometry, estimation,
algebra, patterns and functions, and discrete mathematics. Because
most of the activities are hands-on and students are constantly
dealing with numbers in a variety of ways, it assists the development
of number sense as well.
The topics that should comprise the probability and statistics
focus of the mathematics program in grades 7 and 8 are:
- collecting, organizing, and representing data
- analyzing data using range and measures of central tendency
- making inferences and hypotheses from their analysis of data
- evaluating arguments based upon data analysis
- interpolating and/or extrapolating from data using a line of best fit
- representing probability situations in a variety of ways
- modeling probability situations
- analyzing probability situations theoretically
- predicting events based on real-world data
Standard 12 - Probability and Statistics - Grades 7-8
Indicators and Activities
The cumulative progress indicators for grade 8 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 7 and
8.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 7-8 will be such that all students:
9. Generate, collect, organize, and analyze data and
represent this data in tables, charts, and graphs.
- Most students in grades 7 and 8 have major physical growth
activity. Students can continue to maintain the statistics related to
their body that they began to collect in the fifth and sixth grades.
They should continually update what the average person in the grade
would look like in terms of this data.
- In the spring, the social studies teacher and the
mathematics teacher plan a unit on the school board elections.
Students are broken into groups to study questions such as What
percent of the registered voters can be expected to vote?
Will the budget pass? and Who will be elected to the
board of education? Students plan their survey, how they will
choose the sample, how best to gather the data, and how best to report
the information to the class.
10. Select and use appropriate graphical
representations and measures of central tendency (mean, mode,
and median) for sets of data.
- Students study the sneakers worn by students in the
school. They form into a human histogram based upon their brand of
sneakers. The data is recorded and a discussion is encouraged about
the distribution of sneakers throughout the school. Students discuss
in their journals which of the measures of central tendency a sporting
goods store would use in determining which brands to stock and in what
proportion. The students gather prices for a variety of brands and
styles and enter the data into a spreadsheet. They respond in their
journal as to whether and why the mean, median, or mode would be most
useful to discuss sneaker prices.
- Presented with a list of OPEC countries and their
estimated crude oil production in a recent year, students determine
how best to report the data. Some present their graphs as box plots,
others use histograms, and others use circle graphs. They use the
three measures of central tendency and discuss what each result would
mean in this situation and which would be best to use in other
situations.
- Students work on problems like this one from the
New Jersey Department of Education's Mathematics Instructional
Guide (p. 7-99):
- A set of test scores in Mrs. Ditkof's class of 20
students is shown below.
62 77 82 88 73 64 82 85 90 75
74 81 85 89 96 69 74 98 91 85
Determine the mean, median, mode, and range for the data.
Suppose each student completes
an extra-credit assignment worth 5 points, which is then added to
his/her score. What is the mean
of the set of scores now if each student received the extra five
points? Explain how you
calculated your answer.
11. Make inferences and formulate and evaluate
arguments based on data analysis and data displays.
- Students are presented with data from The World
Almanac showing the number of cigarettes smoked per year per adult
and the rate of coronary heart disease in 21 countries. They produce
a scatterplot and recognize a relatively high correlation between the
two factors. They write an essay on the possible causes of this
relationship and their interpretation of it.
- Students are asked to predict how many drops of water will
fit on a penny. They write their prediction on a postit note along
with an explanation of their reasoning. The predictions are collected
and displayed on bar graphs or stemandleaf plots. Students perform
the experiment and record their results on another postit note. They
compare their hypotheses with the conclusions. A science lesson on
surface tension can easily be integrated with this lesson.
- Students are studying their community's recycling
efforts in an integrated unit. In getting ready for discussion in
this area, the mathematics teachers ask the students to predict how
many pounds of junk mail comes in to their community in a month. The
students collect all junk mail sent to their house over the course of
a month. They weigh the junk mail weekly and record the results. At
the end of the month, all the students bring in their data. The class
determines the mean, median, and mode for the collected data, decides
which of these measures would be the best to use, and agrees on a
method to use to estimate the amount of junk mail for the entire
community.
12. Use lines of best fit to interpolate and predict
from data.
- Presented with the problem of determining how long it
would take the wave to go around Giants Stadium, students
design an experiment to gather data from various numbers of students.
They produce a scatterplot and use it to determine a line of best fit.
They pick two points on the line and determine the equation for that
line. Last, they estimate the number of people around the stadium and
answer the question.
- Given some of the winning times for the Men's and
Women's Olympic 100 meter freestyle events during the past
century, students plot the data and produce a line of best fit for
each event. They use their equations to estimate the winning times in
those years for which the information was not recorded, and they
predict when the women's winning times will equal the men's
current winning times.
- Students are presented with an article that states that
police have discovered a human radius bone which is 25 centimeters
long. Students perform measurements of the lengths of radius bones of
various-sized people and their heights, produce a scatterplot, fit a
line to the data, and determine their prediction of the height of the
person whose bone was found. They write a letter to the chief of
police, predicting the height of the person, with justifications for
their conclusion.
13. Determine the probability of a compound
event.
- Students watch the longrange weekend weather forecast and
learn that the probability of rain is 40% on Saturday and 50% on
Sunday. They determine that the probability that it will rain on both
days is 20% by multiplying the two percentages together (.40 x .50 =
.20 or 20%), and similarly then find that the probability that it will
not rain on either day is 30%. Following the weekend, they discuss
the success or failure of their prediction methods.
- Two teams are in a playoff for the division title. If the
probability of the Eagles defeating the Falcons in an individual game
is 40%, what is the probability that they will win a three game
playoff? What about a five-game playoff?
14. Model situations involving probability, such as
genetics, using both simulations and theoretical
methods.
- During an integrated unit with their science and health
classes, students discuss the various gender possibilities for
children within a family. For each large family, that is, number of
children, up to 6, they calculate the probability of each possible
gender mix. Three groups of students conduct simulations - one
with coins, one with dice (1, 2, or 3 on a die represent a female) and
one with spinners. They also collect this data for all of the
students in their school. They report their findings and compare the
theoretical possibilities, the simulated probabilities, and the actual
outcomes, and discuss the differences and similarities.
- Students study the chances of winning the New Jersey Pick
3 lottery. They model the problem by using spinners with 10 numbers
and calculate the theoretical probability. They may also use a
computer program to randomly generate a million 3-digit numbers and
see how close to 1 out of 1000 times their favorite number comes
up.
15. Use models of probability to predict events based
on actual data.
- Students are presented with data collected by an ecologist
tallying the number of deer of one species that died at ages from 1 to
8 years. Students use the data to discuss the probability of living
to various given ages and what they would expect the life expectancy
of this species to be.
16. Interpret probabilities as ratios and
percents.
- Students examine uses of probability expressed as
percentages in such things as weather forecasting, risks in medical
operations, and reporting the confidence interval of surveys.
- Students work on problems like this one from the
New Jersey State Department of Education's Mathematics
Instructional Guide (p. 7-103):
A dart board is composed of three concentric circles with radii
2 cm, 10 cm, and 20 cm [as indicated in an accompanying
diagram]. A grand prize is earned if a dart is thrown in the 2
cm circle (bulls-eye). Given that a person is blindfolded and throws
a dart somewhere on the board, find the probability that the
grand prize will be won when the first dart is thrown. Explain
the process you used to solve the problem.
References
-
Huff, D. How to Lie with Statistics. New York: Norton,
1954.
Paulos, J. A. Innumeracy: Mathematical Illiteracy and its
Consequences. New York: Hill and Wang, 1988.
Zawojewski, Judith, et al. Dealing with Data and Chance.
A Component of the Curriculum and
Evaluation Standards for School Mathematics Addenda Series,
Grades 5-8. Reston, VA: National Council of Teachers of
Mathematics, 1991.
On-Line Resources
-
http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during Spring
1997. In time, we hope to post additional resources relating to this
standard, such as grade-specific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
Standard 12 - Probability and Statistics - Grades 9-12
Overview
Students can develop a strong understanding of probability and
statistics from consistent experiences in classroom activities where a
variety of manipulatives and technology are used. The key components
of this understanding in probability for middle school students, as
identified in the K-12 Overview, are: probability terms, the
concept of the probability of an event, predicting and determining
probabilities, expected value, the relationship between
theoretical and experimental probabilities, and compound
events. In statistics, the key components are: data
collection, organization, and representation, sampling, central
tendency, variance and correlation, and analysis and
inference.
The field of statistics is relatively new. Beyond the work of
scientists, Florence Nightingale was the great pioneer in gathering
and analyzing statistical data for public health questions. During
the great cholera epidemic of 1854 in London, England, statistics on
the prevalence of cholera cases in various London neighborhoods were
used to deduce that the cholera originated with a single well. In our
own century statistics touches all of us through such diverse means as
statistical quality control in industry, advertising claims,
pre-election polls, television show ratings, and weather forecasts.
To be successful members of present day society, high-school graduates
need an understanding of statistics and probability which formerly was
rare even among college graduates.
By the time students enter high school, they should have mastered
basic descriptive statistical methods. On the basis of their varied
experience, they should be able to set up a study, gather the data,
and appropriately analyze and report their findings. Throughout
grades 9 to 12, students should have numerous opportunities to
continue to practice these skills in a variety of ways, and also to
extend these skills, in connection with their growth in other
mathematical areas. As students learn new algebraic functions, they
might revisit a problem they had previously modeled linearly and apply
a different model. For example, they may have linearly modeled the
series of winning times of the men's Olympic marathon but now
understand that there would probably be a limiting time and so attempt
to fit a quadratic or logarithmic curve instead. Where appropriate,
the content should be developed through a problem-centered approach.
For example, if students are required to generate a report on two sets
of data which have the same measures of central tendency only
to find later they have very different variance, they should
recognize the need for some way to identify that difference.
John Allen Paulos, in his book, Innumeracy, cites numerous
problems associated with a lack of understanding of probability. If
people are to make appropriate decisions, then they must understand
the relationship of probability to real situations and be able to
weigh the consequences against the odds. As with statistics,
probability needs to be experienced, not memorized. Work done at this
level should provide insight into the use of probability and
probability distributions in a variety of real-world situations. The
normal curve presents interesting opportunities to examine uses and
abuses of mathematics.
Students should have access to appropriate technology for their
work in probability and statistics, not only to simplify calculation
and display charts and graphs, but also to generate appropriate data
for activities and projects. They should make use of data taken from
the Internet and CD-ROMs, and simulate experiments with Calculator
Based Laboratories. Whenever possible, real data gathered from
school, the community, or cooperating businesses should be used.
Probability and statistics offers a rich opportunity to integrate
with other mathematics content and other disciplines. This content
provides the opportunity to generate the numbers and situations which
should be used in other areas such as geometry, algebra, functions,
and discrete mathematics. The goal to have students become effective
members of a democratic society requires them to practice and
participate in decision-making experiences. The ability to make
intelligent decisions rests on an understanding of statistics and
probability, and students should regularly integrate this content with
their experiences in social studies, science, and other
disciplines.
The topics that should comprise the probability and statistics
focus of the mathematics program in grades 9 through 12 are:
- designing, conducting, and interpreting statistical work to solve problems
- analyzing data using range, measures of central tendency, and dispersion
- applying probability dispersions in real situations
- evaluating arguments based upon their knowledge of sampling and data analysis
- interpolating and/or extrapolating from data using curve fitting
- using simulations to estimate probabilities
- determining expected values
- using the law of large numbers
Standard 12 - Probability and Statistics - Grades 9-12
Indicators and Activities
The cumulative progress indicators for grade 12 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 9, 10,
11, and 12.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 9-12 will be such that all students:
17. Estimate probabilities and predict outcomes from
actual data.
- In a standard class test, students are asked to compute
the probability that a given raffle ticket for a senior class raffle
to raise money for the senior trip will win a prize. The class will
be printing 500 tickets that they will sell for $1 each. First prize
is a stereo worth $150. Second prize is a $100 shopping spree in the
local Gap store. Third prize is a $50 gift certificate to The Golden
Goose restaurant. There are ten fourth prizes of a commemorative
T-shirt worth $8 each. Students also compute the expected value of
each ticket.
- Students determine the area of an irregular closed figure
drawn on a large sheet of paper using the Monte Carlo method: Each
person in the group drops a handful of pennies over their shoulder
(without looking) onto the paper containing the figure. They count
the number of coins on the paper (total shots) and the number within
the figure (hits). They thus produce the ratio of hits to total shots
and multiply this fraction by this area of the paper to estimate the
area of the figure.
- Students work through the On the Boardwalk
lesson that is described in the Introduction to the Framework.
In this lesson they explore the probability that a quarter thrown onto
a rectangular grid will land entirely within one of the squares on the
grid, and then discuss how changing the size of the squares will
affect the probability.
- A point P inside a square is selected at random
and is used to form a triangle with vertices A and B of the square.
Students determine the probability that the triangle is acute using a
simulation and a theoretical calculation.
18. Understand sampling and recognize its role in
statistical claims.
- While studying United States history, students read about
the prediction in the 1936 presidential race that Alfred Landon would
defeat incumbent President Franklin Delanor Roosevelt. They raise
questions as to why that prediction was so far off and research how TV
stations can forecast winners of some elections with a very small
percentage of the voting results reported. Students contact local
radio and TV stations and newspapers to discover how they determine
their population sample.
19. Evaluate bias, accuracy, and
reasonableness of data in real-world contexts.
- After reading the chapter on sampling in the book How
to Lie With Statistics by Darryl Huff, students bring in ads,
graphs, charts, and articles from newspapers which all makestatements
or claims allegedly based on data. Students examine the articles for
information about the sample and identify those claims which may have
little or no substantiation. They also discuss how the sample
populations chosen could have influenced the outcomes.
- Students take statements such as "50% of the
students failed the test," and "4 out of 5 dentists
recommend" and discuss what data they would need to know in order
to judge if the conclusions were reasonable. How many students took
the test? How many dentists were queried? How were the students or
dentists selected? What factors can be identified which would bias
the results?
20. Understand and apply measures of dispersion and
correlation.
- Students are presented with data gathered by an
archaeologist at several sites. The data identifies the number of
flintstones found at each site and the number of charred bones. The
archaeologist claimed that the data showed that the flintstones were
used to light the fires that charred the bones. Students produce a
scatterplot, find the correlation between the two sets of figures, and
use their work to support or criticize the claim.
- As an assessment activity using their journals, students
respond to the claim that children with bigger feet spell better.
They discuss whether they believe the claim is true, how statistics
might have led to this claim, and whether it has any importance to a
philosophy of language teaching.
21. Design a statistical experiment to study a
problem, conduct the experiment, and interpret and communicate
the outcomes.
- Based on a discussion among some members of the class, a
question arises as to which are the most popular cars in the
community. The students work in cooperative groups to design an
experiment to gather the data, analyze the data, and design an
appropriate report format for their results.
- Intrigued by the question How long would it take
dominoes set up one inch apart all the way across the room to
fall?, the class designs an experiment to gather data on smaller
sets of dominoes and then extrapolates to estimate the answer.
- Students have just finished a unit in which they discussed
the capturerecapture method for estimating the population of wildlife.
Part of their assessment for the unit is a project where they work in
groups to design and conduct a simulation of the capturerecapture
method. One group uses the method to determine the number of
lollipops in a large bag.
22. Make predictions using curve fitting and numerical
procedures to interpolate and extrapolate from known
data.
23. Use relative frequency and probability, as
appropriate, to represent and solve problems involving
uncertainty.
- After a unit where dependent and independent events were
detailed, students are challenged by a problem containing this
excerpt from The Miami Herald of May 5, 1983.
An airline jet carrying 172 people between Miami and Nassau lost
its engine oil, power, and 12,000 feet of altitude over the
Atlantic Ocean before a safe recovery was made.
When all three engines' low oil pressure warning
lights all lit up at nearly the same time, the
crew's initial reaction was that something was wrong with
the indicator system, not the oil pressure.
They considered the possibility of a malfunction in the
indication system because it's such an unusual
thing to see all three with low pressure indications. The odds are so
great that you won't get three indications like
this. The odds are way out of sight, so the
first thing you would suspect is a problem with the indication
system.
Aviation records show that the probability of an engine failure
in any particular hour is about 0.00004. If the failures of three
engines were independent, what would the probability be of them
failing within one hour? Discuss why the speaker in the article
would refer to such a probability as "way out of sight."
Discuss situations which might make the failures of three engines
not independent events.
- Students keep a record of their trips through the town and
whether or not they have to stop at each of the four traffic lights.
After one month, the data is grouped and studied. Theyuse their data
to determine whether the timing of the lights is independent or
not.
- While discussing the issue of mandatory drug testing in
social studies, students examine the probability of misdiagnosing
people as having AIDS with a test that would identify 99% of those who
are true positives and misdiagnose 3% of those who don't have
AIDS. They examine situations where the prevalence of the disease is
50%, 10%, and 1% using 100,000 people as a base. They discuss the
fact that, at the 1% level, 75% of the people identified as having
AIDS would be false positives, the implications that fact has on
mandatory testing, and potential ways to improve the predictive value
of testing.
24. Use simulations to estimate
probabilities.
- Students derive the theoretical probability of winning the
New Jersey Pick 6 lottery and then write a computer program to
simulate the lottery. The students enter the winning numbers and the
computer generates sets of 6 numbers until it hits the winning
combination. The computer prints out the number of sets generated
including the winning one. Students run the program several times,
attempting to verify experimentally the theoretical probability they
derived.
25. Create and interpret discrete and continuous
probability distributions, and understand their application to
realworld situations.
- Students work on a project where they pick one form of
insurance (life, car, home), and determine the variables which affect
the premiums they would need to pay for this type of insurance and
what it would cost for them to obtain it. Using their research, they
write an essay summarizing how insurance companies use statistics and
probabilities to determine their rates.
- An article in Consumer Reports indicates that 25%
of 5lb bags of sugar from a particular company are underweight. The
class works with the local supermarket to develop and perform a
consumer research project. Each group is given a commodity to study
(e.g., potato chips, sugar). They design a method for randomly
selecting and testing whether the product matches the claimed
specifications or not. They use their data to determine the
probability that a randomly selected bag would be underweight.
- Students repeatedly extracted five marbles from a
bag containing 10 red and 10 blue marbles, and each time record the
number of marbles of each color obtained. They combine the data for
the entire class, tabulating the number of times there were 0, 1, 2,
3, 4, and 5 red marbles, and the percentages for each number. They
compared their percentages to the theoretical percentages for this
binomial distribution, and make the connection to the fifth row of
Pascal's triangle.
26. Describe the normal curve in general terms, and
use its properties to answer questions about sets of data that
are assumed to be normally distributed.
27. Understand and use the law of large numbers (that
experimental results tend to approach theoretical probabilities
after a large number of trials).
- Students are given two dice, each a different color and
roll them repeatedly. For each roll, they record the result for each
individual die as well as the total. After a large number of rolls
they compare their relative frequencies to the expected outcomes.
Then they combine the totals for the entire class and compare the
experimental results with the theoretical predictions.
- Students are presented with a paper containing the
following gambler's formula: When playing roulette,
bet red. If red does not win, double the bet on red. Continue in
this manner. They evaluate whether the formula makes
sense, identify potential problems, and limitations, and discuss the
fallacy that the odds improve for red to appear on the next roll every
time red doesn't win.
References
-
Burrill, Gail, et al. Data Analysis and Statistics Across the
Curriculum. A component of the Curriculum and Evaluation
Standards for School Mathematics Addenda Series, Grades 9-12.
Reston, VA: National Council of Teachers of Mathematics, 1992.
Huff, D. How to Lie with Statistics. New York: Norton,
1954.
Paulos, J. A. Innumeracy: Mathematical Illiteracy and its
Consequences. New York: Hill and Wang, 1988.
The North Carolina School of Science and Mathematics.
Contemporary Precalculus Through
Applications. Providence, RI: Janson Publications,
1991.
On-Line Resources
-
http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/
The Framework will be available at this site during
Spring 1997. In time, we hope to post additional resources
relating to this standard, such as grade-specific activities submitted
by New Jersey teachers, and to provide a forum to discuss the
Mathematics Standards.
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