New Jersey Mathematics Curriculum Framework

## 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 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/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.