Chapter 15 5-6

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.

5 - 6 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 generating, collecting, and analyzing data through experiments and on-line searches, developing lines of best fit to interpolate and extrapolate information from data, and evaluating arguments based upon data. The key components of probability at this level, as well as in all future levels, are extending the students understanding of probability of simple events to compound events, developing models for probabilistic situations using both simulations and theoretical methods, and extending interpretation of probabilities to ratios, and percents.

In grades K-4, students explored basic ideas of statistics by gathering data, organizing it, and representing it in charts and graphs, and then using this information to arrive at answers to questions and raise further questions. Students must be active participants at all levels of statistical activity. As much as possible, the questions needing answers should come from student interests. Children in grades 5 and 6 begin to focus more on their peers and the images they themselves present. They are keenly interested in movies, fashion, music, and sports. These areas provide a rich source of real problems to children at this age. The children should make the decision on how to collect and to organize data. They should determine how best to represent the data and begin to develop a more formal understanding of summary statistics such as mean, median, and mode. These activities should provide opportunities for students to make conjectures and to communicate them in a convincing manner. They further develop their understanding of statistics through the evaluation of others arguments, whether they come from classmates, advertising, political rhetoric, or news sources.

While statistical investigations can be similar to those in earlier grades, fifth and sixth graders are beginning the growth toward adulthood and their interests and questions tend to make the statistical needs more complicated. It becomes more important to provide students with access to statistical software on computers or calculators which have statistical capability. Students cannot be allowed to become mired in tedious calculations as it will inhibit understanding and push them away from the mathematics. Technology should be used to do the manipulation of the data and the students should be developing the necessary human skills to provide the interpretive capability necessary for proper use of data.

Students enter these grades having participated in a wide variety of activities designed to help them understand the nature of probability and chance. The emphasis in grades K-4 was strictly on simple events such as the role of a die or the flip of one coin. Even when compound events such as the roll of two dice was considered, the outcomes were looked upon as a simple event. In these grades, students begin to experiment with compound events such as flips of coins and rolls of a dice. As they develop their understanding of fractions, ratios, and percents, they use them to represent probabilities in place of phrases such as "three out of four." They begin to model probabilistic situations and to use these models to predict events which are meaningful to them.

Probability and statistics provide a rich opportunity to integrate with other mathematics content and other disciplines. This content provides the opportunity to generate the numbers and situations that are used in the other areas such as numerical operations, geometry, estimation, algebra, and patterns and functions. Because most of the activities are hands-on and students are constantly dealing with numbers in a variety of ways, it assists the development of number sense as well. 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 5 and 6 are:

collecting, organizing, and representing data

analyzing data using range and measures of central tendency

make inferences and hypotheses from their analysis of data

evaluate arguments based upon data analysis

interpolate and/or extrapolate from data using a line of best fit

representing probabilistic situations in a variety of ways

modeling probabilistic situations

predicting events based on real-world data

STANDARD 15: PROBABILITY AND STATISTICS

5-6 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-4 expectations, experiences in grades 5-6 will be such that all students:

J. generate, collect, organize, and analyze data and represent this data in tables charts and graphs.

Students recognize that this is a time of growth for many of them. The class measures various body parts such as height, length of forearm, length of thigh, length of hands, and armspan. They then enter the data into a spreadsheet and produce various graphs as well as a statistical analysis of the class. They update their data every month and discuss the change, both individually and as a class.
Students survey another class to determine data of interest, such as the last movies seen, and then organize the data and produce reports discussing the interests of the grade level.

K. understand and apply measures of central tendency.

Students demonstrate understanding of measures of central tendency by writing a letter to a fictional classmate explaining how the mean, the median, and the mode each help to describe data.
During a social studies unit, students determine a method to ascertain the wealth of their community. They decide to survey some people who live in the community but in different sections of town. They determine the mean, median, and mode for the data set and decide which provides the most accurate picture of the community.

L. select appropriate graphical representations and measures of central tendency for sets of data.

Students present a picture of an "average" student in their grade. The picture discusses height, color of hair, preference in movies, etc. In creating the picture, the students must choose the appropriate measure of central tendency based upon the type of data and justify their choice.
Students perform an experiment where one group is given 10 words in a jumbled order while another group is given them in a sequence which facilitates learning their spelling. After giving each group one minute to study the words, the subjects are asked to turn their papers over and write as many of the words as they remember. The papers are graded by fellow students and the scores reported. After considering various graphing methods, the students determine that a box-and-whiskers plot would be the best way to illustrate the results and compare the two groups.

M. make inferences and formulate and evaluate arguments based on data analysis and data displays.

Prior to activity described above where students make measurements of their body parts, the students are asked to develop a generalization about their classmates. They are allowed to make any hypothesis which is appropriate. For example, some boys might suggest that boys are stronger than girls or others might say that girls are taller than boys. They should determine how they would validate their hypotheses by designing a data collection activity related to it.
The teacher in one fifth-grade class is especially alert for generalizations made by students while working in any subject area. She writes them on slips of paper, and keeps them in a box. As an assessment of the students' ability to develop statistical activities to validate hypotheses, groups of students pull slips from the box, develop data collection activities, collect the data, analyze it, and make reports to the class about the validity of the generalizations originally made.
Students are shown a newspaper article which states that 25% of fifth graders have smoked a cigarette. They discuss their immediate reaction by indicating whether they believe the figure to be correct, too high, or too low. They then design a survey which they use to poll their fellow fifth graders in an effort to check the validity of the claim. They also send a letter to the newspaper requesting the sources of data for the article and review them.

N. use lines of best fit to interpolate and predict from data.

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 answers to questions like: How high will the water be after 25 marbles have been added?" and "How many marbles will it take to have the water reach the top?. Activities like this one, of course, go a long way toward preparing children for algebra.
Students are presented with an article that states that police have discovered a human radius bone which is 25 centimeters long. They are to prepare a letter to the chief of police predicting how tall the person was, based upon the radius bone, with justifications for their conclusions. Students perform measurements of heights and lengths of radius bones of various sized people, produce a scatterplot, fit a line to the data, determine their prediction, and then write the letter.

O. determine the probability of a compound event.

Students conduct experiments with two dice and keep track of the sum of each roll. They also establish a matrix with the possible results for one die along the leftmost column and the possible results for the other die along the top row. They complete the matrix by putting in each cell the appropriate sum of the number in the top row and the left column. Counting the number of times each sum appears in the table, they complete the task by comparing that set of numbers with the actual results of the experiment.
Students roll the pigs from the game Pass the Pigs (Milton Bradley) and determine the number of times they get the results which gather points or where they lose their turn. Students compare their results and determine if the distribution of points makes it a fair game.

P. model probabilistic situations, such as genetics, using both simulations and theoretical models.

Students examine the possibility of a family with four children having two boys and two girls by simulating the situation using four coins. They first choose which side will represent males and which will represent females. They toss the set of coins 25 times and record their results as the number of boys and the number of girls in each "family." The data is then used to inform a discussion about the likelihood of the evenly-matched family.
A grocery store loads one of the 25 cent "prize" machines near the checkout counter with an equal number of each of six plastic containers with Power Ranger tattoos. Students are asked to determine how many containers they need to buy to have a good chance of getting all six. They perform two trials of a simulation using a bag of six different colored marbles which they draw out, record, and replace one at a time until they have drawn all six. The class results are gathered and discussed. One issue discussed is whether the model is a good one for the situation or whether it could be modified in some way to better represent reality.

Q. use probabilistic models to predict events based on real-world data.

Students examine weather data for their community from previous years, and then use their analysis of the data to predict the weather for the upcoming month. They compare the actual results with their predictions after the month has passed and then use the comparison to determine ways to improve their predictions.
Using data from previous years, students determine the number of times their favorite professional football team scored a number of points in each of six ranges of scores (5 or fewer, 6-10,11-15,16-20,21-25, and more than 25). They determine the fraction or percentage of games the score was in each of the ranges and make a spinner whose areas are divided the same way. Each Friday during football season, they spin their spinners to predict how many points the team will score and who will win the game. Toward the end of the season, they discuss the success or failure of their efforts and the probable causes.

R. interpret probabilities as ratios and percents.

The students are introduced to the Milton Bradley game Pass The Pigs where two small hard-rubber pigs are rolled. Each pig can land on a side where there is a dot showing, a side where the dot does not show, on its hooves, on its back, leaning forward balancing itself on its snout, and balancing itself on its left foreleg, snout, and left ear. The students determine the fairness of the distribution of points by rolling the pair of pigs numerous times and using the ratios of successes for each over the total rolls to represent the probability of obtaining each situation.
Students examine uses of probability expressed as percentages in such things as reporting the confidence interval of surveys, weather forecasting, and risks in medical operations.