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 middle school students, as identified in the K-12 Overview, are: probability terms, the concept of the probability of an event, predicting and determining probabilities, the relationship between theoretical and experimental probabilities, and compound events. In statistics, the key components are: data collection, organization, and representation, sampling, central tendency, variance and correlation, and analysis and inference.

Students should enter the seventh grade with a strong intuitive understanding of probability and statistics as a result of their activities in grades K-6, and should have a basic understanding of the more formal methods which were introduced in grades 5-6. They will build on this foundation in grades 7 and 8.

Students in grades 7 and 8 present unique challenges. They are turning to their peer group for leadership and support and, at the same time, placing a strain on the relationships between themselves and significant adults in their lives. Some students begin to experiment with things they associate with being an adult: smoking, alcohol, drugs, and sex. The quantity of statistics in all of these areas provides an ideal opportunity to weave together statistical activities which dovetail with information provided by the the health and physical education department.

Students at these ages also become more aware of community issues. Integrating statistics activities with topics in the social studies curriculum can enhance their work in both areas as well as fit in with their growing interests. Handson 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 provides excellent opportunities to discuss how statistics and probability are misused.

In statistics, students continue to collect, organize, and represent data and to use various measures of central tendency to describe their data. But they should now become more focused on sampling techniques that justify making inferences about entire populations. Examples of this appear frequently in the news media. They also begin to explore variance and correlation as additional tools in describing sets of data.

Many of the probability experiments should continue to be related to games and other fun activities. Students in these grades should continue to develop their understanding of compound events and their related probabilities, and should continue to consider and compare experimental and theoretical probabilities. Furthermore, the connection between probability and statistics should help them understand issues such as sampling and reliability. 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 are actually likely to occur. For example, they should examine the probability that two people in their class, or any group of 25 or more people, have the same birthday. The results always stir up considerable interest and disbelief.

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. 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 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

making inferences and hypotheses from their analysis of data

evaluating arguments based upon data analysis

interpolating and/or extrapolating from data using a line of best fit

representing probability situations in a variety of ways

modeling probability situations

analyzing probability situations theoretically

predicting events based on real-world data

The cumulative progress indicators for grade 8 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 7 and 8.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 7-8 will be such that all students:

9. 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.

10. Select and use appropriate graphical representations and measures of central tendency (mean, mode, and median) for sets of data.

• 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 discuss in their journals which of the measures of central tendency a sporting goods store would 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 and why the mean, median, or mode would be most useful to discuss sneaker prices.

• 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. They use the three measures of central tendency and discuss what each result would mean in this situation and which would be best to use in other situations.

• Students work on problems like this one from the New Jersey Department of Education's Mathematics Instructional Guide (p. 7-99):

  A set of test scores in Mrs. Ditkof's class of 20 students is shown below.

       62   77   82   88   73   64   82   85   90   75
       74   81   85   89   96   69   74   98   91   85
  

  Determine the mean, median, mode, and range for the data. Suppose each student completes an extra-credit assignment worth 5 points, which is then added to his/her score. What is the mean of the set of scores now if each student received the extra five points? Explain how you calculated your answer.

11. Make inferences and formulate and evaluate arguments based on data analysis and data displays.

• Students are presented with data from The World Almanac showing the number of cigarettes smoked per year per adult and the rate of coronary heart disease in 21 countries. They produce a scatterplot and recognize a relatively high correlation between the two factors. They write an essay on the possible causes of this relationship and their interpretation of it.

• Students are asked to predict how many drops of water will fit on a penny. They write their prediction on a postit note along with an explanation of their reasoning. The predictions are collected and displayed on bar graphs or stemandleaf plots. Students perform the experiment and record their results on another postit 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 junk mail sent to their house over the course of a month. They weigh the junk mail weekly 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 the collected data, decides which of these measures would be the best to use, and agrees on a method to use to estimate the amount of junk mail for the entire community.

12. 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 it to determine a line of best fit. They 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 winning times for the Men's and Women's Olympic 100 meter freestyle events during the past century, students plot the data and produce a line of best fit for each event. They use their equations to estimate the winning times in those years for which the information was not recorded, and they predict when the women's winning times will equal the men's current winning times.

• Students are presented with an article that states that police have discovered a human radius bone which is 25 centimeters long. Students perform measurements of the lengths of radius bones of various-sized people and their heights, produce a scatterplot, fit a line to the data, and determine their prediction of the height of the person whose bone was found. They write a letter to the chief of police, predicting the height of the person, with justifications for their conclusion.

13. Determine the probability of a compound event.

• Students watch the longrange weekend weather forecast and learn that the probability of rain is 40% on Saturday and 50% on Sunday. They determine that the probability that it will rain on both days is 20% by multiplying the two percentages together (.40 x .50 = .20 or 20%), and similarly then find that the probability that it will not rain on either day is 30%. Following the weekend, they discuss the success or failure of their prediction methods.

• Two teams are in a playoff for the division title. If the probability of the Eagles defeating the Falcons in an individual game is 40%, what is the probability that they will win a three game playoff? What about a five-game playoff?

14. Model situations involving probability, such as genetics, using both simulations and theoretical methods.

• During an integrated unit with their science and health classes, students discuss the various gender possibilities for children within a family. For each large family, that is, number of children, up to 6, they calculate the probability of each possible gender mix. Three groups of students conduct simulations - one with coins, one with dice (1, 2, or 3 on a die represent a female) and one with spinners. They also collect this data for all of the students in their school. They report their findings and compare the theoretical possibilities, the simulated probabilities, and the actual outcomes, and discuss the differences and similarities.

• Students study the chances of winning the New Jersey Pick 3 lottery. They model the problem by using spinners with 10 numbers and calculate the theoretical probability. They may also use a computer program to randomly generate a million 3-digit numbers and see how close to 1 out of 1000 times their favorite number comes up.

15. Use models of probability to predict events based on actual data.

• Students are presented with data collected by an ecologist tallying the number of deer of one species that died at ages from 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.

16. Interpret probabilities as ratios and percents.

• Students examine uses of probability expressed as percentages in such things as weather forecasting, risks in medical operations, and reporting the confidence interval of surveys.

• Students work on problems like this one from the New Jersey State Department of Education's Mathematics Instructional Guide (p. 7-103):

  A dart board is composed of three concentric circles with radii 2 cm, 10 cm, and 20 cm [as indicated in an accompanying diagram]. A grand prize is earned if a dart is thrown in the 2 cm circle (bulls-eye). Given that a person is blindfolded and throws a dart somewhere on the board, find the probability that the grand prize will be won when the first dart is thrown. Explain the process you used to solve the problem.

References

Huff, D. How to Lie with Statistics. New York: Norton, 1954.

Paulos, J. A. Innumeracy: Mathematical Illiteracy and its Consequences. New York: Hill and Wang, 1988.

Zawojewski, Judith, et al. Dealing with Data and Chance. A Component of the Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 5-8. 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.