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Archived Information

Mathematics Curriculum Framework
Achieving Mathematical Power - January 1996

Strand 4: Statistics and Probability

All students should analyze, develop and act on informed opinions about current economic, environmental, political and social issues affecting Massachusetts, the United States and the world.
-- Massachusetts Common Core of Learning

Statistics and probability confront us every day. The ability to understand variability and uncertainty is necessary every time one reads the results of a Gallup poll or a report on the latest medical research findings. What does it mean to test positive for HIV? What are the chances that a nuclear power plant will have an accident? Is angioplasty as effective as coronary artery bypass surgery? If we are to interpret the arguments made by those on each side of an issue and make our own informed decisions on questions like these, then we must rely on our understanding of statistical inference, misuses of statistics, and probability.

Level by Level

Sorting and classifying by attributes are likely to be the first formal introduction to statistics that a student encounters. At a very young age, students begin to describe, analyze, evaluate, and make decisions about the attributes of familiar items such as blocks, shapes, or clothing. Students in early grades should explore probability and data analysis by working with one variable and learning to make, read, and interpret simple graphs. Students learn that a sample can be representative of a population. The more opportunities students have to do hands-on activities with probability and data, the better base they have from which they develop a deeper understanding.

In the middle grades, students use scatter plots to explore the relationship between two variables, construct circle graphs to represent proportional amounts and histograms for data that are grouped into intervals, and construct box and whisker plots to represent data spreads. They also explore differences between theoretical and experimental probability. They extend their data analysis and probability repertoire to include sampling bias and randomness.

Students in high school extend their knowledge to fitting nonlinear graphs to data. They study in depth sampling methods and the role of sampling in making predictions and judging the validity of statistical claims. The topics explored should be integrated with other mathematics courses.

Role of Critical Thinking

The development of critical thinking in statistics should be emphasized. Students at all levels should formulate appropriate questions; gather and explore data; organize and describe data, using graphs, charts, and tables; interpret results; and develop a critical attitude toward the use of statistics. They should investigate real-life problems that require them to employ sampling techniques. These investigations help them realize the relative applicability of statistics to solving problems.

Emphasis on Interpretation

Technology has changed dramatically the way we deal with statistical data. Statisticians spend relatively more time interpreting what their data suggests, using exploratory techniques, and relatively less time applying standard inferential techniques. What computers have done for statisticians, inexpensive statistical calculators and classroom software have done for students. More emphasis should be placed on interpretation of summary statistics--both student-generated and from daily life.

Adult Investigations

Adult learners should become keenly aware that decisions made on the basis of statistics affect them daily. Involve them in discussions about average scores required for tests, percentage of attendance at work or school, weekly price of transportation, cost of child or health care, or the number of immigrants allowed to become legal residents. Adult mathematics investigations should evolve around such statistical representations. They should understand how statistical information may be misused.

Statistics and Probability
Learning Standards Grades PreK-4

Statistics and Probability

PreK-4 Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

  • collect, organize, and describe data.
  • construct, read, and interpret displays of data.
  • formulate and solve problems that involve collecting and analyzing data.
  • explore and describe the concepts of chance.

Examples of Student Learning

  • Give each pair of students 25 assorted pattern blocks. Have them sort the blocks and design a representation of the sorting. Discuss how they sorted and constructed their representations.
  • Students make a floor graph that displays data gathered from classmates. The teacher marks with masking tape on the floor two outlines of bars as in a bar graph.
  • Students each trace one foot on paper and cut it out. Next, they place their cutouts on the bar that identifies whether each is wearing sneakers or not wearing sneakers. Discuss what the graph shows. Extend the activity by gathering the data by using tally marks, and graphing the findings on inch-square graph paper.
  • Students check newspapers and find the time of sunset for Monday through Friday. The data are recorded in a table and a line graph is constructed. Students interpret their graphs and predict the times of sunset for the following week. They record them on the graph in a contrasting color. They check their predictions in the newspaper and discuss their conclusions. Is there a pattern? If so, what is it?
  • Groups of four children choose four different types of candy they will use in a survey. Each group makes a chart to show the possible selections. Each group surveys another class. One student explains the survey, another explains the selection chart, one counts the votes, and another tallies the votes. The groups graph their data.
  • Place two apples in a bag--one green apple, one red apple. Have the students predict what will happen when choosing an object always? never? sometimes? Next, have the students predict which color apple will come out most often. Investigate. Discuss.

How it Looks in the Classroom

A class of second grade students has been working with an integrated mathematics and science program, which introduces the concepts of attributes, estimation, and probability. Each is working with a partner to play a game. Students select a number from 1 to 12. Each pair takes two foam dice and 12 frog playing pieces with a game board. Directions are given to throw the dice, add the two numbers, and move the frog whose number matches the total, one space on the board. The game continues until a frog reaches the other side of the "pond."

As play begins, Briana and Greta observe that frog 1 will never win. The teacher asks why. They respond that since players must throw two dice, the lowest number possible is 2, and therefore frog 1 will never get a chance to move forward.

The teacher keeps track of the winning numbers on the board with stick-on tabs, eventually forming a graph showing frogs 6, 7, and 8 winning more often than the others. Again, she asks the students why they think this is so. Remembering what Briana and Greta said about frog 1, Justin suggests that the only way a player can move frog 3 is by rolling a 1 with a 2. The teacher challenges the class to determine all the combinations that will give a total of 7. The students discover 1 + 6, 5 + 2, and 3 + 4. Since there were more possible combinations for 7, they concluded it would probably be rolled more often. When the students next play the game, most choose 6, 7, or 8 as the winning frog!

Statistics and Probability
Learning Standards Grades 5-8

Statistics

Grades 5-8 Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

  • collect, organize, and describe data systematically.
  • construct, read, and interpret tables, charts, and graphs.
  • make inferences and convincing arguments that are based on data analysis.
  • evaluate arguments that are based on data analysis.
  • develop and explain why statistical methods are powerful aids for decision making.

Examples of Student Learning

After reading an article, students interpret the data and evaluate the conclusions expressed by the reporter. Students write letters to the editor, questioning or approving the reporter's conclusions. Students find other articles that use statistics and their interpretations and discuss them with someone at home.

Students conduct research to learn more about the Consumer Price Index (CPI)--how it is expressed, what it tells us, and how it is determined. Through whole class discussion, they analyze its appropriateness. Probing questions are asked in an attempt to prompt students to realize that the CPI does not take into account the decisions that people make based upon price changes. Working in groups, students create their own measurement of the cost of living for a middle grades student. Will they accommodate price changes? How? What costs will they consider? Will they estimate or be exact? Why? Groups will prepare and deliver oral presentations of their ideas.

Probability

Grades 5-8 Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

  • model situations by devising and carrying out experiments or simulations to determine probabilities.
  • construct a sample space to determine probabilities.
  • describe the power of using a probability model by comparing experimental results with mathematical expectations.
  • make predictions that are based on experimental or theoretical probabilities and determine their reasonableness.
  • develop and explain an appreciation for the pervasive use of probability in the real world.

Examples of Student Learning

  • At the beginning of every mathematics class, each student picks a four-digit number and records it on a chart. At the end of class, the teacher selects a four-digit number by randomly drawing numbered tiles from a bag containing ten tiles numbered from 0-9. After each draw, the tile is returned to the bag before the next draw.
  • A winning number has any three digits. Keep track of the number of times there are winners. Periodically discuss informally with the class what they are discovering about chance and why they believe it is so. How would they change the classroom lottery to make more winners? Fewer winners? What happens to the odds if each tile is not returned to the bag?

How It Looks in the Classroom

Eighth graders at an urban school in the Commonwealth devised their own survey questionnaire. Questions ranged from topics of general interest (height, foot size, number of siblings, favorite shoe brand, and type of music) to those relevant to their urban experience (curfew, suspension from school, rules at home). They filled out the questionnaire during one class period, while taking turns at height and foot measuring stations. To display categorical data, a hypothetical frequency table was made on the board.

Favorite MusicNumber of students
Reggae9
Rock2
Rap8
Other6

One student suggested displaying the results in a bar graph, and the class discussed ways to construct, label, and title such a graph. Several times during this discussion, Shandra insisted a pie chart would be better, so the discussion turned to the subject of pie charts.

Teacher: How do we construct a pie chart?

Student: Draw a circle. [The teacher draws a circle on the board.]

Teacher: How do we divide up the pie?

Student: Reggae gets the largest piece. [The teacher shades in a piece roughly equal to three-quarters of the pie.]

Teacher: Is that okay?

Student: No!

Teacher: Then how do I decide what portion of the circle to shade for reggae?

Student: Guess.

Teacher: I did guess.

Student: You can guess better than that.

Teacher: Maybe I can't. Why don't each of you draw the section as you guess it should be, and we will check them after we figure this out more precisely. [Students draw what they think will be reggae's portion of the circle graph.] Now that we have guessed, tell me a way we could divide the pie that uses mathematics.

Student: Well, a circle is 360.

Teacher: A circle is 360 what?

Student: 360 degrees.

Teacher: The entire way around the circle is 360 degrees. That's a good starting point.

Student: Where is that protractor you gave us? [He refers to a giant cardboard circle marked in degrees, used during discussion of angles in geometry. We locate this protractor and place it in front of the room.]

Teacher: Now what?

Student: There are 25 students.

Teacher: What do we do with 25?

Student: Divide; divide 360 by 25. [Teacher writes the problem on the board.]

Student: But 25 won't go into 360 evenly.

Teacher: That's okay, we know how to use decimals. [There are lots of puzzled looks.] What if we had $360 and 25 students; if we divide 360 by 25, what do we have?

Student: How much money each student gets.

Teacher: Right. Now if we have 360 degrees and 25 students, what do we get? [Some make number guesses.] You don't have to tell me a numerical value. Just tell me what the number will represent.

Student: You'll get the size of a piece of pie for each student.

Teacher: Good. The size of the piece, or angle for each student. Now, how would you show that 9 students prefer reggae.

Student: Use 9 pieces.

Teacher: And rap?

Student: Use 8 pieces.

The class ends with the promise to continue the discussion of pie charts and to organize the work for analyzing their survey data. Students later create personal versions of the circular protractor with a ribbon through the middle that helps them mark out angles when constructing pie charts for their survey data.

Statistics and Probability Learning Standards
Grades 9-10

Statistics

Grades 9-10 Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

  • construct, draw inferences, and reason with charts, tables, and graphs that summarize data from real-world situations.
  • use sampling to recognize and describe its role in statistical claims.

Examples of Student Learning

  • Students bring to class examples of statistics in the media. They analyze and critique the use of the statistics, then write alternative uses of the statistics, contrasting the impression given by the advertisement with what the text actually says. For example, a recent advertisement included the statement that more than 98% of one maker's trucks sold in the last 10 years are still on the road. Students should understand that a quick reading might give the impression that 98% of the trucks sold 10 years ago are still on the road, or that all of that maker's trucks last at least 10 years. With careful reading students determine that all of the trucks still on the road might have been sold in the last two or three years. They understand that the figure quoted may be accurate or inaccurate, but that even if it is reasonably accurate it is potentially misleading as used.
  • Students gerrymander the state to reconfigure political party distributions so that one party gets an overwhelming majority of seats. They then work out a fair redistricting. Extend the activity by asking students to research the Massachusetts basis for the term gerrymander.

Probability

Grades 9-10 Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

  • use simulations to estimate probabilities.
  • determine the likelihood of outcomes, using theoretical probabilities.

Examples of Student Learning

  • Students drop a penny in a grid of 50 mm squares. They win if it does not cross a line, and lose if it does. They compute experimental probability and determine theoretical probability. Study can be expanded to other types of grids.
  • Students use probability to determine fairness of a group birthday game. Two guests enter a room of people and ask everyone to write their birthdays on a piece of paper. If two people have the same birthday, the first guest wins. If no two people have the same birthday, the other guest is the winner. How many people should be in the room for this to be a fair game? Students first guess the number of people that should be in the room, then try it with their class. Finally, they explore it further with the aid of a computer program.

Statistics and Probability
Learning Standards Grades 11-12

Statistics

Grades 11-12 Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

  • use curve fitting to predict from data.
  • apply measures of central tendency, variability, and correlation.
  • design a statistical experiment to study a problem, conduct the experiment, and interpret and communicate the outcomes.
  • analyze the effects of data transformations on measures of central tendency and variability.
  • transform data to aid in data interpretation and prediction.
  • test hypotheses by using appropriate statistics.

Examples of Student Learning

  • Students each have two register tapes from grocery stores. They plot the number of items and total cost on the tapes as ordered pairs on a rectangular coordinate system. They draw a straight line through the middle of the points on the graph by using a clear plastic ruler, a piece of uncooked spaghetti, or some other device that allows them to see the points on both sides of the line. They adjust the device until the line goes through the center of the scatter of points and there are approximately the same number of points on each side of the line. Students then determine an equation for the line they drew by finding two points on the line whose coordinates can be read fairly accurately. In small groups, they compare lines and discuss reasons for the differences. To extend the activity, students use a statistical technique, such as the least squares regression or the median fit line, to find the line that best fits the data, rather than fitting a line to the data by eye. Students use graphing calculators or computer software to fit the line to the data.
  • Students can try fitting nonlinear functions such as quadratic, exponential, or logarithmic curves to sets of data, which connects statistics with algebra.

Probability

Grades 11-12 Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

  • use experimental or theoretical probability, as appropriate, to represent and solve problems involving uncertainty.
  • apply the concept of a random variable.
  • create and interpret discrete probability distributions.
  • describe, in general terms, the normal curve and use its properties to answer questions about sets of data that are assumed to be normally distributed.

Examples of Student Learning

With the aid of a computer, students explore the rush at their local post office during winter holiday season, using experimental and theoretical probability generated by random numbers in simulated situations. Modeling is used when students interpret random numbers and simulate queues.

Statistics and Probability
Learning Standards Adult Basic Education

Statistics and Probability

ABE Learning Standards

Students engage in problem solving, communicating, reasoning, and connecting to:

  • collect, organize and describe data.
  • construct, read and interpret tables, charts and graphs.
  • make inferences and convincing arguments that are based on data analysis.
  • evaluate arguments that are based on data analysis.
  • develop and explain an appreciation for statistical methods as a powerful means for decision making.

Example of Student Learning

  • Students speculate on the percentages of men and women, ethnic groups, or age groups that appear in magazine advertisements. After collecting data and presenting their conclusions, students write about marketing/advertising bias.

How It Looks in the Classroom

At a Massachusetts adult learning center, students conducted a survey. It asked respondents to choose from a list those three issues that concerned them the most. Issues were grouped into the categories of home, school, and community. The survey was distributed to approximately thirty-five teachers at a staff development workshop. Completed surveys were collected, tallied, and graphed by students.

One month later, at a community mathematics forum, students gave attending parents of schoolchildren the same questionnaire. These results were graphed alongside the teachers' responses. Later, teachers and parents were given the opportunity to discuss and analyze the findings.



Last Updated: January 1, 1996
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