Chapter 15 K-2

New Jersey Mathematics Curriculum Framework - Preliminary Version (January 1995)
© Copyright 1995 New Jersey Mathematics Coalition

STANDARD 15: PROBABILITY AND STATISTICS

All students will develop their understanding of probability and statistics through experiences which enable them to systematically collect, organize, and describe sets of data, to use probability to model situations involving random events, and to make inferences and arguments based on analysis of data and mathematical probabilities.

K-2 Overview

Probability and statistics hold the key for enabling our students to understand, process, and interpret the vast amounts of quantitative data that exist all around them. To be able to judge the truth 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.

The key components of statistics, applicable here as well as at every other grade level, are collection of data pertaining to problems of interest to children, organizing the data to enable them to make better sense of the information, and presenting the data in graphs. The key components of probability which begin here and extend throughout other grade levels are using probabilistic terms correctly, understanding what is meant by the probability of an event, and using this understanding to predict and determine probabilities.

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 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 study the findings to determine if there are any patterns, 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 make inferences 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 examining numerous societal uses such as advertising, newspapers, and television reports.

Children at this level should experience probability at a variety of levels. Numerous children games are played with random chance devices such a spinners and dice. Students should have numerous 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 probabilistic terms such as probably, possibly, and certainly in a variety of contexts. Statements from newspapers, school bulletins, and their own experiences should highlight their relation to probability.

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 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 octagonal 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 discussing 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 of each blue and red marble is 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 spread and "most"

making inferences justified by their analysis

using probabilistic terms correctly

predicting and determining probability of events

STANDARD 15: PROBABILITY AND STATISTICS

K-2 Expectations and Activities

The expectations for these grade levels appear below in boldface type. Each expectation is followed by activities which illustrate how the expectation can be addressed in the classroom.

Experiences will be such that all students in grades K-2:

A. collect, organize, and analyze data.

Students collect objects such a 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.
At the front of the room is a magnetic board and a magnet with the child's picture for every child in the class. At the start of each day, the teacher has a different question and the children place their magnet in the appropriate area. It might be a bar graph tally for whether they like vanilla, chocolate or strawberry ice cream or a Venn diagram where students identify whether they have at least one brother, at least one sister, at least one of both, or neither.
Second graders can record and graph the sunrise and sunset times one day a week over the entire year. They can even subtract the sunrise time from the sunset time and graph the length of the day and interpret the results.

B. generate and analyze data obtained using chance devices such as spinners and dice.

Students throw a die, spin a spinner, or reach blindly into a container to select a colored marble a dozen times. They then color the appropriate square in a bar graph for each pick. Did some results occur more often or less often than others? Do you think some results are more likely?
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 process ten times, examine their data, and then discuss questions such as Would one result occur more often than others? and explain their reasoning.
Pairs of students each have two spinners. Each student spins his or her spinner simultaneously and together they record whether they have a match. They then predict how many times they would have a match in 20 spins and try to verify their prediction by actually spinning 20 times each.

C. make inferences and formulate hypotheses based on data.

Students throw a pair of dice 100 times and make a bar graph of the sums. They then compare their results with those of their classmates. Do your graphs look alike? Which sum came up the most? Does everyone have the same 'winning' sum? Why did some sums come up less than others?
Children are regularly asked to think about their data. Was there a pattern in their dice throw, bean growth, weather, temperature, or other data? What causes the patterns? Are the patterns in their data the same as those in their classmates' data?

D. understand and informally use the concepts of range, mean, mode, and median.

When performing experiments, children are regularly asked to find the outcome that appeared most often. The teacher provides the necessary language for that concept, the mode, and defines it. Children are asked to compare what they found as the mode for an experiment with the modes found by their classmates.

E. construct, read, and interpret displays of data such as pictographs, bar, and circle graphs.

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.
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 them into individual sectors. Then they roll a die eight times keeping a tally of the results. Finally, they take the appropriate number of different colored sectors to make a circle graph of their results.
Students regularly read information from their classmates' graphs and discuss the differences in their results.

F. formulate and solve problems that involve collecting and analyzing data.

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.
A second grader, upset because she had wanted to watch a TV show the night before but instead had to go to bed, asks the teacher if the class can do a survey to find out when most children her age go to bed.

G. determine the probability of a simple event assuming equally likely outcomes.

Children toss a coin ten times and 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 there were five heads in a row, what is the chance that the next flip is a head? Is there a better chance than there would have been before the other flips took place? After what is a lively discussion, the children flip their coins to demonstrate that each event has no dependence upon the previous ones.
he students predict the chance that a particular number will come up when a die is thrown and then perform an experiment to check their predictions. They respond to their prediction based upon their sample and to a whole class sample.

H. 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 then are asked what they would expect to be the probability for each color if a single marble is drawn. The experiment is performed and the children discuss whether their estimates of the probabilities made sense in light of the outcome.
Students are told that a can contains some red beads, some yellow beads, and some blue beads and are asked to predict how many of each color bead is 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 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.

I. develop intuition about the probability of various events in the real world.

Each child plants five seeds of a fast growing plant. They then count the number of seeds which sprout and discuss how many seeds would sprout if they were each given ten.
Students predict how many m&m's of each color is in a mystery bag. To help refine the guesses, cooperative groups are given different bags which they open, tally the count of the colors, report their results, and prepare a graph of their results. Students refine their guesses about the usual presence of certain colors by looking at their results, all the graphs, and the class totals. The mystery bag is then opened and the colors counted. Students discuss how their prediction matches the actual count and how the experiment helped.
Students examine various types of raisin bran cereal. They should experiment with scoops of cereal and the number of raisins that appears in each scoop.