Chapter 15 3-4

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.

3 - 4 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 making inferences and formulating hypotheses based on data and using the concepts of range, mean, median and mode in analysis of data. The key components of probability, applicable here as well as at every other grade level, are identifying the probability of a simple event given equally likely outcomes, making predictions based upon intuitive, experimental, and theoretical probabilities, and developing an intuitive sense of probabilities of real-world events.

In K-2, students had frequent opportunities to collect and organize data. The activities provided a forum for discussion which developed the foundation for the analysis of data. The children experienced similar opportunities for probability and used both areas simultaneously many times. They should enter this level with the ability to effectively collect, organize, and represent data. They should have a some understanding of concepts such as spread of results, those things which occurred most frequently, and the middle of the data which can be further developed into the more formal concepts of range, mean, median, and mode. The discussions conducted on collected data should provide a background for formulating hypotheses and making inferences. The frequent probability experiments should provide the foundation to extend their ability to use it to make predictions and understand probability as it relates to events around them.

As in the previous grade levels, probability and statistics understanding is best developed through frequent opportunities to perform experiments and gather data. Statistics and probability 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 students. Children should see new activities, but they should have the opportunity to revisit problems from 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 uncertainty of probability using terms such as likely, not likely, more likely, and less likely. Developing an understanding of randomness and probability is crucial to acquiring a more thorough understanding of statistics and information.

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 alert to their environment and are more sensitive to media information than children in the K-2 level. Discussions about such things as the truth of claims in TV ads or newspaper articles on global warming allow students to 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, and patterns and functions. The methods used at this level support all four process standards (problem solving, communication, reasoning, connections) as well as the four environment standards (equity, mathematics as a dynamic activity, technology, assessment).

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 15: PROBABILITY AND STATISTICS

3 - 4 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.

Building upon K-2 expectations, experiences in grades 3-4 will be such that all students:

A. collect, organize, and analyze data.

While studying nutrition and the importance of a good breakfast, students determine who had a good breakfast that morning. They recognize the need to define what they mean by "good breakfast" and then design a method of data collection. They then collect the data, organize it through tables and charts, and analyze the results.
Students wish to study the differences in temperature between their hometown and the 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 organize and represent the data and develop questions about differences in lifestyle prompted by the temperature. They then exchange their questions with their sister school to learn more about their culture.

B. 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 then role 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 using a regular die.

C. make inferences and formulate hypotheses based on data.

After collecting, organizing, and analyzing data on the favorite sport (soccer) 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? state? Do you think first graders would answer similarly? Why?
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 when the water would overflow. Activities like this one, of course, go a long way toward preparing children for algebra.
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 cooling of liquids, the students discussed notions of variables and constants. They set up an experiment using cups of the same size but different material and measuring the temperatures in each at equal intervals over a 30-minute period. They then plot the data and use their graphs to discuss which cup would be best.

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

Before examining the number of raisins contained in each of 24 individual boxes of raisins, students are asked to open the boxes and make predictions as to the number of raisins in each box. They then count the raisins and compare the actual numbers to their predictions. Students discover that boxes contain different numbers of raisins. They construct a frequency chart on the blackboard and further analysis leads to an investigation of the concepts of range, mean, median, and mode.
In a fourth grade assessment, students are told they are to prepare an argument to convince their parents 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 need.

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

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.

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

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? Answers and solutions to these questions are formulated by the group.
Students perform experiments such as rolling toy cars down a ramp of variable height. The distance rolled from the bottom of the ramp for each height is collected. Students discuss the patterns and relationships they see in the data and use their discoveries to determine what height would give the greatest distance.

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.
Students discuss the probability that a particular number will come up when a die is thrown. While most often when this problem is presented, the number requested will be possible, a number such as seven or ten should be used to generate discussions about the probabilities and impossible events.

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 happen if twelve marbles were drawn. 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. They perform a capture-recapture experiment using a large bag of lollipops to determine the number of lollipops in the bag.

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

Students discuss the probability of getting a seven on the roll of one die or picking a blue bead from a bag full of blue beads.
Students discuss the relationship between events such as flipping a coin, a new baby being a girl, guessing on a true-false question, and other events which have a 50-50 split.
Students are intrigued by the notion of winning money. They design a model for the New Jersey Pick-3 Lottery by starting with slips of paper containing the digits 0-9 across a long table. Under each slip, they place another set of the digits 0-9 and then under each slip in that row they place another set of digits 0-9. They use this tree diagram to identify that there are 1000 different possibilities and that each is equally likely.