New Jersey Mathematics Curriculum Framework
© Copyright 1996 New Jersey Mathematics Coalition

## 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 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

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

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.