STANDARD 12  PROBABILITY AND STATISTICS
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.

Standard 12  Probability and Statistics  Grades K2
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 K12 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 secondgrade 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 studentmade spinners
of varyingsized regions and random number generating devices such as
dice cubes or octahedral (eightsided) 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 nonstandard 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 K2
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 K2:
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
twocolored 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 4section 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 yellowsideup and seven chips land redsideup. 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

Lindquist, M., et al. Making Sense of Data.
Curriculum and Evaluation Standards for School
Mathematics Addenda Series, Grades K6. 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 K6. 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 K6. Reston, VA:
National Council of Teachers of Mathematics, 1991.
OnLine Resources

http://dimacs.rutgers.edu/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 gradespecific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
