II C. Skill Objectives III.

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1 MATH: Probability and Statistics Grade Level: 7 th grade Math Presented by: Karen Phillips, Yerger Middle School, Hope, Arkansas Length of Unit: Five lessons (Ten days) I. ABSTRACT This unit focuses on probability and statistics used in the 4 th, 5 th, 6 th, and 7 th grades. Probability and statistics are used across the curriculum in math, language arts, science, history, and geography. Student generated projects will show how they can use statistics and probability in every day life. This unit relies heavily upon Mathematics: Applications and Connections Course II. III. OVERVIEW A. Concept Objective: Understand that probability and statistical concepts are used for problem solving and decision-making. (Arkansas State Std. DSP 1, 2, 3, 1, 2, 3, 4, 3.1, 3.2, 3.3.) B. Content from the Core Knowledge Sequence C. Skill Objectives Show the relationship between two variables using a scatter-plot and describe the apparent relationship informally. Find the upper and lower quartiles for a data set. Understand that if p is the probability of an event occurring, 1-p is the probability of the event not occurring. Understand the difference between independent and dependent events. The students will construct line plots. The students will create and use a frequency table. The students will read, interpret, and make predictions from bar graphs, line graphs, and scatter plots. The students will find mean, median, mode and range. The students will construct stem and leaf plots. The students will read a box and whisker plot to find the upper and lower quartiles for a data set. The students will recognize when statistics and graphs are misleading. The students will find the probability of a simple event. The students will find the probability of an event not occurring after finding the probability of an event occurring. The students will explain the difference between independent and dependent events. BACKGROUND KNOWLEDGE A. For Teachers: Mathematics: Applications and Connections Course Westerville, Ohio: Glencoe/McGraw-Hill, ISBN (Teacher s Wraparound Edition) Principles and Standards for School Mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc., ISBN Lombard, B. & Fulton, B. Math Discoveries about Data & Probability, Grades 7-8. Alsip, IL: Ideal School Supply Company, ISBN Probability and Statistics, Grade Core Knowledge Conference 1

2 4. Lawrence, A., From The Giver to The Twenty-One Balloons: Explorations with Probability Mathematics Teaching in the Middle School. Reston, VA: NCTM, Vol. 4, No. 8, May, Fisher, A. Math Mind Stretchers. Grand Rapids, MI: Instructional Fair, Inc., Leonard, J. Math Word Problems. Grand Rapids, MI: Instructional Fair, Inc., Lombard, B. & Fulton, B. Math Discoveries about Data & Probability, Grades 7-8. Alsip, IL: Ideal School Supply Company, Sherman, R.J., Coultas, J. & Swalm, J. Strategies for Success in Mathematics. Austin, TX: Steck-Vaughn Company, 1995, Thompson, F.M. Hands on Math!: ready to use games and activities for grades 4-8. West Nyack, New York: The Center for Applied Research in Education, B. For Students Numbers and Number Sense Grade 4 Probability and Statistics Grade 5,6, and 7 IV. RESOURCES A. Mathematics: Applications and Connections Course Westerville, Ohio: Glencoe/McGraw-Hill, ISBN (Teacher s Wraparound Edition) B. Mathematics: Applications and Connections Course 2: Study Guide and Practice Workbook. Westerville, Ohio: Glencoe/McGraw-Hill, ISBN V. LESSONS Lesson One: Statistics (4 to 5 days) A. Daily Objectives Concept Objective a. Understand that probability and statistical concepts are used for problem solving and decision-making. Lesson Content a. Show the relationship between two variables using a scatter-plot and describe the apparent relationship informally, (p. 174, Core Knowledge Sequence) b. Collect and organize data in graphic form (bar, line graphs), (p. 125, Core Knowledge Sequence, review from Grade 5) 3. Skill Objective(s) a. The student will construct line plots. (Arkansas State Std. DSP 1, 2) b. The student will create and use a frequency table. (Arkansas State Std. DSP 1, 2) c. The student will read, interpret and make predictions from bar graphs, line graphs, and scatter plots. (Arkansas State Std. DSP 3, 3.1, 3.3) B. Materials Textbook, Mathematics: Applications and Connections Course 2 Worksheet, Appendix A Textbook Workbook, Mathematics: Applications and Connections Course 2: Study Guide and Practice Workbook graph paper straight edge Probability and Statistics, Grade Core Knowledge Conference 2

3 pencil overhead projector with graph paper transparency individual dry erase boards dry erase markers and erasers C. Key Vocabulary Scatter-plot: In a scatter plot, two sets of related data are plotted as ordered pairs on the same graph. D. Procedures/Activities Day One Day Two Day Three The student will define vocabulary for their notebook. The teacher will explain what a line plot is and how they are constructed. The teacher will explain how a line plot is used. Use the examples on page from the textbook. The teacher will help the students find the range, the scale, the interest, and any clusters. The teacher will direct the students in creating a line plot on their dry erase boards using a short list of numbers like problem #3 on page 100. The teacher will request students to hold up their boards and quickly walk around observing their work. If necessary, after observation, the teacher will direct the students in another example. Teacher will make a class assignment to be completed by the students on their graph paper. Textbook page 100/5-7. The teacher will monitor work by walking around the room. The teacher will assign homework. Textbook page 100/5-9,12,14,15, Teacher and students will grade homework from previous day, together. Teacher will briefly answer questions or reteach. Teacher will explain what a frequency table is and how to use one. Teacher will review range, scale, and interval and explain how intervals in a frequency table differ from frequency tables in a line plot. Teacher will demonstrate with examples from textbook pages Teacher will introduce culminating activity and explain how to complete the movie survey and frequency table in Part Teacher will lead students in a classwork assignment in the textbook on page 89, problems 1-5. Teacher will observe while walking around the room. Teacher will give a homework assignment from textbook page 90, number 7(give students the intervals: 1-10, 11-20, etc.), 21, Teacher and student will grade homework from the previous day, together. Teacher will briefly answer questions or reteach. Teacher will explain again the differences between intervals on a number line and intervals in a frequency table, explain again that the scale and interval must be compatible. (The number of numbers in the scale must be divisible by the interval you choose.) The teacher will remind students that 3 to 6 lines is usually best in a frequency table although more lines can be used. Teacher leads students in a classwork activity from the textbook page 90, numbers 8-1 Randomly ask students to choose an appropriate interval, and then choose a scale. Probability and Statistics, Grade Core Knowledge Conference 3

4 2 Teacher assigns homework from the textbook page 90, numbers 12, 14, and a teacher made worksheet, Frequency Tables, Appendix B. Day Four E. Evaluation/Assessment B. Materials D. Procedures/Activities Teacher and students will grade homework from the previous day, together. Teacher will briefly answer questions or reteach. Teacher will explain how to read graphs using examples from textbook on page Teacher will use random questioning of students to make sure all students can read the graphs. (Vertical axis, horizontal axis, estimating between the lines, and rounding decimals.) Teacher leads students, working together, in textbook on page 96, numbers 4-7a (have students complete 7a themselves.) Teacher assigns homework from textbook page 96, 4-9 and workbook page 18, Study Guide. Observe student interaction and evaluate informally the process that students use to complete class assignment. Lesson Two: Statistics A. Daily Objectives Concept Objective a. Understand that probability and statistical concepts are used for problem solving and decision-making. Lesson Content a. Given a set of data, find the mean, median, range and mode. 6 th grade page Skill Objective(s) a. The student will find the mean, median, range, and mode when given a set of data. Textbook, Mathematics: Applications and Connections Course 2 Workbook, Mathematics: Applications and Connections Course 2: Study Guide and Practice Workbook Calculators 3. C. Key Vocabulary No new vocabulary E. Evaluation/Assessment Teacher and students will grade homework from the previous day, together. Teacher will briefly answer questions or reteach. The teacher will explain and demonstrate on the overhead projector how to find mean, median, range, and mode. (Note: These terms were introduced in Grade 6). Teacher will lead students through multiple examples from textbook, page 104, numbers 1-7 (especially median and mode). Calculators may be used for speed of process, but teacher will emphasize that work set up must still be shown on paper. Teacher will assign homework from the textbook, page 104, numbers 8-17, 20. Introduce Part 2 of the culminating activity. Probability and Statistics, Grade Core Knowledge Conference 4

5 Collect and record grades from homework. Observe student interaction and evaluate informally the process that students use to solve the problems during classwork. Lesson Three: Statistics A. Daily Objectives Concept Objective(s) a. Understand that probability and statistical concepts are used for problem solving and decision-making. Lesson Content a. Find the upper and lower quartile for a data set, page Skill Objective(s) a. The student will construct and read stem and leaf plots. b. The students will understand the terms outlier, upper and lower extreme, and upper and lower quartile. B. Materials Textbook, Mathematics: Applications and Connections Course 2 Workbook, Mathematics: Applications and Connections Course 2: Study Guide and Practice C. Key Vocabulary Upper quartile- The median of the upper half of a set of numbers. Lower quartile- The median of the lower half of a set of numbers indicated by LQ. D. Procedures/Activities Day One Day Two The student will define vocabulary their notebook. Teacher and students will grade homework from the previous day, together. Teacher will briefly answer questions or reteach. Teacher will lead students through examples in textbook, pages 108 and 109. Teacher will demonstrate how to read a stem and leaf plot using a printed transparency on the overhead projector. Teacher will also explain the key and identify. Teacher and students will work classwork together and grade in class page 109 numbers 4-7. Students will just name the stems on number 5 and number 6. Teacher will show several box-and-whisker plots and explain how to read them using the overhead projector. Teacher will use random questioning to check a classroom assignment in the textbook on page 116, numbers 4-8. E. Evaluation/Assessment Collect and record grades from homework. Observe student interaction and evaluate informally the process that students use to solve the problems during classwork. Lesson Four: Statistics A. Daily Objectives Concept Objective(s) a. Understand that probability and statistical concepts are used for problem solving and decision-making. Probability and Statistics, Grade Core Knowledge Conference 5

6 Lesson Content a. Collect and organize data in graphic form (bar, line, and circle graph) from 5 th grade p125. b. Solve problems requiring interpretation and application of graphically displayed data from 6 th grade p Skill Objective(s) a. The student will read bar and line graphs and recognize when statistics are misleading. B. Materials C. Key Vocabulary No new vocabulary D. Procedures/Activities E. Evaluation/Assessment Textbook, Mathematics: Applications and Connections Course 2 Workbook, Mathematics: Applications and Connections Course 2: Study Guide and Practice Workbook Teacher will explain and show examples, using overhead projector, of misleading statistics. Teacher will explain and show how graphs not beginning at zero can be misleading. Teacher will lead students through examples 1 and 2 on page 119 in the textbook. Teacher will explain parts 3 and 4 on culminating activity. Teacher and students will assess understanding of this skill, in class, using the workbook page 23 Practice, numbers 1,3,and 4 and the textbook page 120, numbers 1,2,and 4. Teacher will assign homework from the textbook on page 120, numbers 1,2,4-9, 11, workbook page 23, Study Guide and workbook page 22, Study Guide. Collect and record grades from homework. Observe student interaction and evaluate informally the process that students use to solve the problems during classwork. Lesson Five: Probability A. Daily Objectives Lesson Content a. Express simple outcomes as fractions from 4 th grade page 10 b. Understand that if p is the probability of an event occurring, 1-p is the probability of the event not occurring. c. Understand the difference between independent and dependent events. Concept Objective(s) a. Understand that probability and statistical concepts are used for problem solving and decision-making. 3. Skill Objective(s) a. The student will find the probability of a simple event. b. The student will explain in writing to another student and the teacher his understanding that if p is the probability of an event occurring, 1-p is the probability of an event not occurring. The student will use examples. c. The student will explain the difference, in writing, between independent and dependent events. Probability and Statistics, Grade Core Knowledge Conference 6

7 B. Materials Textbook, Mathematics: Applications and Connections Course 2 Workbook, Mathematics: Applications and Connections Course 2: Study Guide and Practice Workbook C. Key Vocabulary Independent events Two or more events in which the outcome of one event does not affect the outcome of the other event(s). Dependent event Two or more events in which the outcome of one event does affect the outcome of the other event(s). D. Procedures/Activities Teacher and students will grade homework from the previous day, together. Teacher will briefly answer questions or reteach. 3. Teacher will explain and demonstrate on the overhead projector the probability that an event will happen is somewhere between 0 and Teacher will show that probability of an event can be pictured on a number line. Teacher will demonstrate with examples and lead students through examples from textbook on page 167, numbers Teacher will assign homework from workbook, page 31 Study Guide. Teacher will have students guess the probability of getting both an even number and heads if they were to roll a number cube and toss a coin simultaneously. The teacher will emphasize that with independent events the probability remains the same for the second time regardless of the results of the first. E. Evaluation/Assessment Collect and record grades from homework. Have students write two problems for classmates that involve choosing items from a restaurant menu. One should involve the probability of two independent events and the other of two dependent events. 3. Observe interaction of students and their written work. VI. VII. VIII. CULMINATING ACTIVITY Survey 15 to 20 people. Ask each person too pick his or her favorite movie from the list of top 10 movies in Appendix B. Record the results on a frequency table. Find the mean and median of the ticket sales. Draw a bar graph of the data that could be misleading. Write an essay comparing your survey results to the rating in the table from Appendix B. Include your opinion of which method you think is best for determining the most popular movie. A rubric will be used for assessment of this activity. HANDOUTS/WORKSHEETS Appendix A Teacher made worksheet Frequency Table Appendix B Movie Survey Project BIBLIOGRAPHY A. Fisher, A. Math Mind Stretchers. Grand Rapids, MI: Instructional Fair, Inc., B. Leonard, J. Math Word Problems. Grand Rapids, MI: Instructional Fair, Inc., C. Lombard, B. & Fulton, B. Math Discoveries about Data & Probability, Grades 7-8. Alsip, IL: Ideal School Supply Company, Probability and Statistics, Grade Core Knowledge Conference 7

8 D. Sherman, R.J., Coultas, J. & Swalm, J. Strategies for Success in Mathematics. Austin, TX: Steck-Vaughn Company, 1995, E. Thompson, F.M. Hands on Math!: ready to use games and activities for grades 4-8. West Nyack, New York: The Center for Applied Research in Education, F. Mathematics: Applications and Connections Course Westerville, Ohio: Glencoe/McGraw Hill, X (Student Edition) (Teacher s Wraparound Edition). G. Mathematics: Applications and Connections Course 2: Study Guide and Practice Workbook. Westerville, Ohio: Glencoe/McGraw Hill, H. Principles and Standards for School Mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc., Probability and Statistics, Grade Core Knowledge Conference 8

9 Appendix A- Part 1 Lights, Camera, Action! Page 91, Working on the Chapter Project, Exercise 22 The Top Movies of All Time at the U.S. Box Office Top 10 Movies Tally Frequency In this project you will use statistics to investigate the most popular movies, according to ticket sales. You will also conduct, your own survey about favorites movies. Survey at least 30 people. Ask each person to pick his or her favorite movie from the list of top 10 movies on p. 87. Record the results. Fill in the frequency table above with the data from your survey. 3. Ask each person who participates in your survey to sign a piece of paper titled Survey Participants. At least ten of the people surveyed must be adults. Probability and Statistics, Grade Core Knowledge Conference 9

10 Appendix A- Part 2 Page 105, Working on the Chapter Project, Exercise 19. Use the 10 numbers from the table on page 87 (460.9 million, million, etc.) to find the mean of the ticket sales. See page 10 Find the median of the ticket sales. See page Find the range of the ticket sales. See page Find the mode of the ticket sales. See page You must show all your calculations. You may use a calculator but you must show everything you punch into the calculator on this paper. 6. Record your results here: Mean- Median- Mode- Range- Show work below: Probability and Statistics, Grade Core Knowledge Conference 10

11 Appendix A- Part 3 Ticket Sales (millions $) Top Ten Movies in Ticket Sales Star E.T. Jurassic Forrest The Lion Return of Indep. Empire Home Jaws Wars Park Gump King the Jedi Day Strikes Back Alone Movies Draw a bar graph using the numbers from the table on page 87. You must have a scale on the vertical axis and the intervals must be equally spaced starting at zero. You must draw bars, not just vertical lines. Probability and Statistics, Grade Core Knowledge Conference 11

12 Appendix A- Part 4 Ticket Sales (millions $) Top Ten Movies in Ticket Sales Star E.T. Jurassic Forrest The Lion Return of Indep. Empire Home Jaws Wars Park Gump King the Jedi Day Strikes Back Alone Movies Draw a misleading bar graph using the numbers from the table on page 87. You must have a scale on the vertical axis and the intervals must be equally spaced. You must draw bars, not just vertical lines. See pages Probability and Statistics, Grade Core Knowledge Conference 12

13 Appendix A- Part 5 Write a one-page essay comparing your survey results with the data about ticket sales on page 87. The opening paragraph should explain what you are comparing. It should explain how you arrived at the results in your survey and explain how the results on page 87 were obtained. 3. Next you should have one or more paragraphs comparing the results from page 87 with the results from your survey. Tell how the results are alike and how they are different. Tell which movies were most popular in each. Explain why you think the book and your survey produced different results. 4. Write a conclusion paragraph which gives your opinion of which method for determining the most popular movies is best and why you think so. Probability and Statistics, Grade Core Knowledge Conference 13

14 Appendix B- Part 1 Probability and Statistics, Grade Core Knowledge Conference 14

15 Appendix B- Part 2 Probability and Statistics, Grade Core Knowledge Conference 15

16 Appendix B- Part 3 Probability and Statistics, Grade Core Knowledge Conference 16