Chapter 15 7-8

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.

7 - 8 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 through the median-median method 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 students understanding of probability of simple events to compound events and independent and dependent events, developing models for probabilistic situations using both simulations and theoretical methods, and extending interpretation of probabilities to ratios and percents.

Having gathered and organized data and developed the skills of analyzing and reporting over grades K through 6, students in these grades will be solidifying the fundamental abilities needed to be successful in an information-driven society. They should have a strong intuitive understanding from all of the informal uses, and a basic understanding of the more formal methods started in grades 5 and 6. It is necessary to build on this foundation so that students understand the connections, but when activities, questions, and concepts from earlier grades are revisited, students must see extensions and growth.

Students in grades 7 and 8 present unique challenges. Students begin to move toward the peer group for leadership and support placing a strain on the relationships between them and significant adults in their lives. Some children start fighting for an adult identity and begin to experiment with things they see as adult: smoking, alcohol, drugs, and sex. The quantity of statistics in all of these areas provides ideal opportunities to weave statistical experiences with the health and physical education departments.

Children at these ages also become more aware of community issues as well as those throughout the nation and the world. Integrating statistics opportunities with the social studies curriculum brings necessary meaning. Hands-on science activities require good statistical methods and understanding in order to develop accurate and appropriate conclusions. At the same time, students need to understand how often statistics and probability statements are incomplete, misunderstood, or purposely used to mislead. Having students read books such as How to Lie with Statistics by Darryl Huff or Innumeracy by John Allen Paulos provide excellent opportunities to discuss the abuse of mathematics.

Many of the probability experiments should continue to be related to games and other fun activities since people of all ages enjoy such activities. However, the types of statistical activities mentioned above also lead to discussions of related probability. Making this connection allows students to begin to understand issues such as sampling, reliability, and applicability to their lives. Students need to develop a sense of the application of probability to the world around them as well. Everyday life is rich with coincidences which can be shown to be more likely than intuitively believed. An examination of the probability that two people in their class have the same birthday usually stirs considerable interest.

Probability and statistics provide a rich opportunity to integrate with other mathematics content as well as 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 environmental 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 7 and 8 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

7 - 8 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-6 expectations, experiences in grades 7-8 will be such that all students:

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

Most students in grades 7 and 8 have major physical growth activity. Students can continue to maintain the statistics related to their body that they began to collect in the fifth and sixth grades. They should continually update what the average person in the grade would look like in terms of this data
In the spring, the social studies teacher and the mathematics teacher plan a unit on the school board elections. Students are broken into groups to study questions such as What percent of the registered voters can be expected to vote?, Will the budget pass?, and Who will be elected to the board of education?. Students plan their survey, how they will choose the sample, how best to gather the data, and how best to report the information to the class.

K. understand and apply measures of central tendency.

Students study the sneakers worn by students in the school. They form into a human histogram based upon their brand of sneakers. The data is recorded and a discussion is encouraged about the distribution of sneakers throughout the school. Students write in their journals describing which of the measures of central tendency a sporting goods store should use in determining which brands to stock and in what proportion. The students gather prices for a variety of brands and styles and enter the data into a spreadsheet. They respond in their journal as to whether the mean, median, or mode would be most useful to them and why.
After performing an experiment and producing a scatterplot of the data, students find the line of best fit using the median-median line fit method. After dividing the data into three groups, they find the median x value and the median y value in each group.

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

Presented with a list of OPEC countries and their estimated crude oil production in a recent year, students determine how best to report the data. Some present their graphs as box plots, others use histograms, and others use circle graphs. Students determine the three measures of central tendency and discuss what each would mean in this situation and which would be best to use for a variety of situations.

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

Students are presented with data from The World Almanac showing, for 21 countries, the number of cigarettes smoked per year per adult and the rate of coronary heart disease. They produce a scatterplot and recognize a relatively high correlation between the two factors. They write an essay on the possible causes of the relationship and their interpretation of it.
Students are asked to predict how many drops of water will fit on the head of a penny. They write their prediction on a post-it note along with an explanation of their reasoning. The predictions are collected and displayed on bar graphs or stem-and-leaf plots. Students perform the experiment and record their results on another post-it note. They compare their hypotheses with the conclusions. A science lesson on surface tension can easily be integrated with this lesson.
Students are studying their community's recycling efforts in an integrated unit. In getting ready for discussion in this area, the mathematics teachers ask the students to predict how many pounds of junk mail comes in to their community in a month. The students collect all the junk mail sent to their house over the course of a month (in one week blocks). They weigh the collection each week and record the results. At the end of the month, all the students bring in their data. The class determines the mean, median, and mode for their own set of data, decides which measure would be best to use, and settle on a method to use to estimate the amount for the entire community.

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

Presented with the problem of determining how long it would take "the wave" to go around Giants Stadium, students design an experiment to gather data from various numbers of students. They produce a scatterplot and use the median-median line fit method to determine a line of best fit. They then pick two points on the line and determine the equation for that line. Last, they estimate the number of people around the stadium and answer the question.
Given some of the times for the Men's and Women's Olympic 100 meter freestyle events over the past century, students plot the data and produce a line of best fit for each event. They use their equations to estimate the times for years which had not been given to them as well as to predict when the women will be swimming the same times as the men.

O. determine the probability of a compound event.

Students observe the long-range weather forecast during the noontime weather report. They see that the probability of rain is 40% on Saturday and 50% on Sunday. They determine the probability of rain for the entire weekend by multiplying the two percentages together. Following the weekend, they discuss the success or failure of their prediction methods.
Students discuss the likelihood of both the Giants and the Jets winning their football games on Sunday. Estimates of the probability of that event are computed using each teams winning percentage to that point in the season. The advantages and disadvantages of using that procedure are discussed and the probability is adjusted if such adjustment is felt necessary

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

During an integrated unit with their science and health classes, students discuss the various possibilities for children within a family. One group begins their study by writing computer programs to randomly select families with four children and collect the data in a matrix where the index represents the number of females (0 through 4). Another group flips four pennies repeatedly and lets heads represent females. Another group roles four dice and lets a 1, 2, or 3 on a die represent a female. The last group chooses to use four spinners having four colored areas and chooses two of the colors for each gender. The groups report their findings using appropriate graphs and charts.
Students study the chances of winning the New Jersey Pick 3 lottery. They attempt to model the problem by using spinners with 10 numbers and by using computer programs to randomly generate 3-digit numbers.

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

Students are presented with data collected by an ecologist which gives the count of the number of one species of deer that died at ages of 1 to 8 years. Students use the data to discuss the probability of living to various given ages and what they would expect the life expectancy of this species to be.

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.