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.

In grades K4, students explored basic ideas of statistics by gathering data, organizing data, and representing data in charts and graphs, and then using this information to arrive at answers to questions and raise further questions. Students in grades 5 and 6 are keenly interested in movies, fashion, music, and sports. These areas provide a rich source of real problems for students at this age. The students should make the decision on how to sample and then collect and organize data. They should determine how best to represent the data and begin to develop a more formal understanding of summary statistics of central tendency such as the mean, the median, and the mode. They should recognize that for certain types of data, such as height, the mean is an appropriate measure, but it is inappropriate for other types of data, such as hair color. These activities should provide opportunities for students to analyze data and to make inferences regarding the data and to communicate their inferences in a convincing manner. They should further develop their understanding of statistics through the evaluation of arguments by others, 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 should have access to statistical software on computers or calculators which have statistical capability. This will allow them to carry out statistical work using real data without becoming mired in tedious calculations. The technology will be used to do the manipulation of the data and the students will focus on developing their skills in interpreting the 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 primarily on simple events such as the roll of a die or the flip of one coin. Even when compound events such as the roll of two dice were considered, the outcomes were looked upon as a simple event. In grades 5 and 6, students begin to experiment with compound events such as flips of several coins and rolls of dice and to predict and evaluate their theoretical and experimental probabilities. As they develop their understanding of fractions, ratios, and percents, they should use them to represent probabilities in place of phrases such as "three out of four." They begin to model probability situations and to use these models to predict events which are meaningful to them.

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 5 and 6 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

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 5 and 6.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 5-6 will be such that all students:

9. 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 lengths associated with a person, such as height, length of forearm, length of thigh, handspan, length of foot, and armspan. They enter the data into a spreadsheet and produce various graphs as well as statistical analyses of their measures. 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.

• Students work on problems like this one from the New Jersey State Department of Education's Mathematics Instructional Guide (p. 7-95):

  A fair spinner with 4 congruent regions labeled A, B, C, and D is spun 20 times by each member of a class of 23 students. Assume that your class conducted the experiment and obtained the expected results. Make a bar graph illustrating the combined class results. Explain why you drew your bar graph the way you did. Explain why an individual student's results might be different from the class results.

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

• 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. They then extend their discussion by presenting 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 measures of central tendency based upon the type of data and justify their choice. (For example, the mean is not appropriate in discussing hair color.) They will likely want to present pictures of both an "average boy" and an "average girl" in the class.

• During a social studies unit, students determine a method to ascertain the value of the homes in their community. They determine the mean, the median, and the mode for the data and decide which provides the most accurate picture of the community. They include in their study homes from different sections of the town.

• 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 remembering them.After giving each group one minute to study the words, the students 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.

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

• 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 determine the validity of their hypotheses by designing a data collection activity related to it.

• The teacher in one fifth-grade class is especially alert for generalizations that students make about any topic. 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 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 for the population of their school. They also send a letter to the newspaper requesting the sources of data for the article and compare the data in the article with their data.

12. 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 form the foundation for understanding graphs in algebra.

13. Determine the probability of a compound event.

• Students create a table to show all possible results of rolling two dice. At the left of the rows are the possible rolls of the first die and at the top of the columns are the possible rolls of the second die. They complete the table 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 determine the probability of getting each possible sum. They then roll two dice 100 times and compare the sums they get with the sums predicted from the table.

• Students make a list of all possible outcomes when four coins are tossed and determine thetheoretical probability of having exactly two heads and two tails.

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

• Students examine the probability of a family with four children having two boys and two girls by simulating the situation using four coins. They first choose which side of the coin will represent males and which will represent females. They toss the set of coins 50 times and record their results as the number of boys and the number of girls in each "family." They compare the results of their experiment with the prediction based on probability. They also survey a large sample of students in the school and record the family composition of all families with four children. All of these are used to discuss the likelihood of an evenly-matched family.

• A 25-cent "prize" machine in the grocery store contains 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 simulate this situation with a bag containing an equal number of six different colored marbles. They draw out, record, and replace one marble at a time until they have drawn marbles of all six colors, recording the number of times that took. They repeat the simulation three times. The class results are gathered and discussed. One issue discussed is whether the model is a good one for the situation or whether it should be modified in some way to better represent reality.

• Students read Caps for Sale by Esphyr Slobodkina. The peddler in the story sells caps and wears his entire inventory on his head: a checked cap and four each of identical blue, gray, and brown hats. Students use concrete objects to model some of the different orders in which the hats can be worn. They come to realize that there are many ways and try to discover the total number of different ways. They search for an efficient way to determine the number of permutations.

• Students work through the Two-Toned Towers and Pizza Possibilities lessons that are described in the First Four Standards of the Framework. They make a systematic list of all the towers built out of four red and blue cubes (or of all the pizza combinations) and calculate the probability that a tower has three or four blue cubes.

15. Use models of probability to predict events based on actual 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 (0-5, 6-10, 11-15, 16-20, 21-25, and 26 or more). They determine the fraction or percentage of games the score was in each of those 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.

16. Interpret probabilities as ratios and percents.

• The students are introduced to the game Pass The Pigs (Milton Bradley) where two small hardrubber 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 on the sides of the pig by rolling the pig numerous times, recording the results, and using the ratios of successes for each, divided by the total number of rolls, to represent the probability of obtaining each situation.

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

References

New Jersey State Department of Education, Mathematics Instructional Guide: Linking Classroom Experiences to Current Statewide Assessments. D. Varygiannes, Coord. Trenton, N.J., 1996.

Slobodkina, Esphyr. Caps for Sale. New York: W.R. Scott, 1947.

General References

Stenmark, J. K., et al. Family Math. Berkeley, CA: Regents, University of California, 1986.

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.