Family Letters. Assessment Management. Math Boxes 1 4

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1 Objective To calculate and compare the median and mean of a data set. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Calculate and compare the median and mean of a data set. Examine how the median and the mean change as data change. Find the mode of a data set. Determine which landmark best represents a set of data mean, median, or mode. Key Activities Students find and compare the median and the mean of various data sets. Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page 404). Materials Math Journal 1, pp. 14 and 1 Study Link 1 3 Math Masters, p. 404 calculator Math Boxes 1 4 Math Journal 1, p. 16 metric ruler Students practice and maintain skills through Math Box problems. Study Link 1 4 Math Masters, p. 13 Students practice and maintain skills through Study Link activities. READINESS Defining the Mean Math Masters, p. 14 unit cubes (2 per student) Students use manipulatives to represent the mean and write number sentences that model how to find the mean. ENRICHMENT Using Technology to Find the Median and Mean Student Reference Book computer spreadsheet/graphing software Students use computer software to find the median and mean of a data set. EXTRA PRACTICE Solving Custom-Made Math Boxes Math Masters, p. 40 Students complete teacher-generated Math Boxes. Advance Preparation Consider having students use computer software for the optional Enrichment activity in Part 3. Teacher s Reference Manual, Grades 4 6 pp. 168, 169 Lesson

2 Getting Started Mathematical Practices SMP2, SMP3, SMP4, SMP6 Content Standards 6.SP.2, 6.SP.3, 6.SP.c, 6.SP.d Bold SMP = Guiding Questions at everydaymathonline.com Bold = Focus of lesson Mental Math and Reflexes Students solve mental math problems such as the following: Add 2 and 70. Take away 1. Triple the result. 240 Add 20 and 70. Divide by 10. Subtract 0. 0 Add 3 and 6. Multiply by. Double the result. 1,000 Math Message Complete Problem 1 on journal page 14. Study Link 1 3 Follow-Up Briefly review answers. NOTE Some students may benefit from doing the Readiness activity before beginning Part 1 of the lesson. See Part 3 for details. 1 Teaching the Lesson Math Message Follow-Up (Math Journal 1, p. 14) ELL WHOLE-CLASS DISCUSSION Date 1 4 Math Message Interactive whiteboard-ready epresentations are available at to help you teach the lesson. Comparing the A small office supply company has 12 employees. Their yearly salaries appear in the table, as well as the mean, median, and mode salaries. 1. Does the mean, the median, or the mode best represent the typical salary at the company? Explain. Sample answer: The mean and mode salaries are greater than $90,000. Because 9 of the 12 salaries are less than $6,000, the median salary seems the best representation of a typical salary at Fancy Font. Salaries at Fancy Font Office Supplies Job Annual Salary President $27,000 Vice-President: Marketing $18,000 Vice-President: Sales $18,000 Marketing Manager $62,00 Product Manager $9,000 Sales Manager $,00 Promotions Manager $2,00 Salesperson 1 $4,000 Salesperson 2 $43,00 Salesperson 3 $43,000 Administrative Assistant 1 $39,00 Administrative Assistant 2 $36,000 Mean salary $90,12 Median salary $4,000 Mode salary $18,000 Review the salary data in the table. Ask students to explain how the mean salary was calculated. Sample answer: The annual salaries were added together and the sum was divided by the total number of employees. Remind students that mean, median, and mode are all landmarks that are considered measures of the center of a data set. Ask them to share their thoughts about which of these landmarks best represents a typical annual salary at the company. Sample answer: Both the mean and mode are greater than $90,000. Because 9 of the 12 employees earn less than $90,000, the mean and mode do not seem to be good representations of the salaries. Since 7 of the 12 salaries are between $40,000 and $63,000, the median salary of $4,000 best represents the typical annual salary. To support English language learners, discuss the meaning of typical as well as annual salary. Discuss how the mean and the median change as data change. Pose the following situation: Suppose the salary of the company president increased to $30,000. How would this affect the mean salary? The mean salary would increase. How would the increase affect the median salary? The median salary would remain the same. It is less affected by the maximum and minimum salaries than the mean salary is. Explore other situations that would affect the mean and median salaries. Circulate and assist as students complete journal page 14. Bring the class together to share answers. Find the median and mean for each of the following sets of numbers. 2. 6, 9, 10, 1 a. median 9. b. mean , 0.7, 1, 1.2, 0.80 a. median 0.80 b. mean , 6, 92, 90, 88 a. median 90 b. mean 89.8 Math Journal 1, p. 14 EM3cuG6MJ1_U01_1-44.indd 14 1/11/11 :29 PM 34 Unit 1 Collection, Display, and Interpretation of Data

3 Adjusting the Activity Ask students to explain how they could use the mean salary to find the sum of the salaries. A U D I T O R Y K I N E S T H E T I C T A C T I L E V I S U A L Comparing the Median and Mean of a Data Set WHOLE-CLASS DISCUSSION Relate the following situation to students. Imagine that it is the end of the grading period. You have received the following test scores: 90, 88, 100, and 82. You have one more test to take before the end of the grading period. Pose the following questions: If your teacher bases your grade on the median of your test scores, will you be motivated to study for the last test? Sample answer: No. There would not be much incentive to study. No matter how low my last test score is, my median score will not be lower than 88. If your teacher bases your grade on the mean score, will you be motivated to study for the last test? Sample answer: Yes. I will be motivated to study. The better test score I make, the higher my final grade will be. Suppose you score 20 out of 100 on your final test. Will the median or the mean give a better picture of your overall performance? Sample answer: The median (88) seems to give a better overall picture. The mean will be 76, which is 6 points lower than the lowest of my other test scores. Suppose you score 9 out of 100 on your final test. Will the median or the mean give a better picture of your overall performance? Sample answer: Both will be about 90. Most teachers determine final grades by calculating the mean of the scores for each student. Why do you suppose they use the mean? Sample answer: One or more low scores can significantly lower the mean of the scores. Using the mean gives students incentive to get high scores on all tests. Can you think of a situation outside of school in which the mode of the data would be most useful? Sample answers: A small grocery store that can stock only one brand of peanut butter would want to stock only the most requested brand. A political candidate often wants to know the most popular public opinion concerning an issue. Lesson 1 4 3

4 Date The 10 most successful coaches in the history of the National Football League (NFL) are listed in the table at the right, along with the number of games won through the end of the 2002 season. Find the following landmarks for the data set displayed in the table. 1. median 2. maximum 3. minimum 4. mean. mode 6. range Try This Data Landmarks none Denzel s first three test scores in math were 90, 100, and 90. a. What must Denzel score on his fourth test to keep his mean score at 90 or higher? 80 or higher b. What must Denzel score on his fourth test to keep his median test score at 90 or higher? Most Successful NFL Coaches Coach Games Won Don Shula 347 George Halas 324 Tom Landry 270 Curly Lambeau 229 Chuck Noll 209 Dan Reeves 198 Chuck Knox 193 Paul Brown 170 Bud Grant 168 Steve Owen 1 Any score, even 0, will keep the median at 90 or higher. Math Journal 1, p Finding Landmarks (Math Journal 1, p. 1) Ask students to find the median of the following data sets: 10, 14, 12, 11, , 14, 12, 11, 12, PARTNER Draw their attention to the number of values in the second data set. Because the number of values is even, the median is the number halfway between the two middle values, or the mean of the two middle values. Circulate and assist as students complete the journal page. Ongoing Assessment: Recognizing Student Achievement Exit Slip Use an Exit Slip (Math Masters, p. 404) to assess students understanding of median and mean. Students are making adequate progress if they can acknowledge that the median is less affected by outliers than the mean is. Pose the following informal assessment item to students: In 2004, baseball player Derek Jeter earned about $2 million in salary, bonuses, and endorsements. If you were to report the typical annual earnings for baseball players, would it be more accurate to report the mean or median earnings? Why? NOTE To help students put Derek Jeter s salary into context, tell them that the salaries of baseball players in 2004 ranged from $300,000 to $21,000,000. Date Math Boxes 1. Measure the line segment below to the nearest centimeter. a. cm Measure the line segment below to the nearest millimeter. b. 3. Subtract mm Write a data set that fits the following description. There are 7 numbers in the data set. The minimum is 17. The range is 4. The median is 32. The mode is 41. 1, Sample answer: 17, 20, 28, 32, 41, 41, 62 a. 1,000 2 b. 2, c. 7,214 6, Ongoing Learning & Practice Math Boxes 1 4 (Math Journal 1, p. 16) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 1-2 and 1-a. The skills in Problems 4 and are prerequisites for Unit 2. Study Link 1 4 (Math Masters, p. 13) INDEPENDENT Home Connection Students construct stem-and-leaf plots and practice finding the mean and median of data sets Find the quotient Complete. 60 b. 2, c. 1, d. 630, e ,000 a , Math Journal 1, p Unit 1 Collection, Display, and Interpretation of Data

5 3 Differentiation Options READINESS Defining the Mean (Math Masters, p. 14) PARTNER 1 Min To provide experience with modeling and finding the mean, have students use unit cubes to represent data values. They redistribute, or even out, the unit cubes to find the mean value. Students then write a number sentence to model how they redistributed the unit cubes to help determine the mean of the data set. Name Date STUDY LINK Mia s quiz scores are 7, 70, 7, 8, 7, 8, 80, 9, and 80. Nico s quiz scores are, 8, 9, 100, 7, 7, 6, 9, and Find each student s mean score. Mia Nico Make a stem-and-leaf plot for each student s scores. a. Mia s Quiz Scores b. Nico s Quiz Scores Stems Leaves (100s and 10s) (1s) Study Link Master Find each student s median score. Mia Nico Stems Leaves (100s and 10s) (1s) What is the range of scores for each student? Mia Nico. Which landmark, mean or median, is the better indicator of each student s overall performance? Explain. Because Mia s mean and median scores are the same (80), either landmark is a good indicator for her. Nico s median score is the better indicator of his performance. His mean is not as good an indicator because of the range of his scores ENRICHMENT SMALL-GROUP Using Technology to Find the 1 30 Min Assign each group one of the data sets provided in the Data and Probability section of the Student Reference Book. Students enter the data into a spreadsheet or graphing program to sort the values from least to greatest to determine the median. They should then enter a formula to calculate the mean of the same data set. Practice $.82 $30.27 $14.24 $ $4.7 $ $14.49 $ $19.99 $.7 9. $39.2 $18.7 Math Masters, p. 13 EXTRA PRACTICE INDEPENDENT Solving Custom-Made Math Boxes (Math Masters, p. 40) 1 Min To provide extra practice, use Math Masters, page 40 to generate Math Box questions that focus on a particular concept or skill for which students need extra practice. Name Date Teaching Master Defining the Mean The table at the right shows the number of students absent from gym class during the week. 1. Place unit cubes on the line below to show the number of absent students for each day. Day Students Absent Monday 6 Tuesday 2 Wednesday Thursday 4 Friday 8 Monday Tuesday Wednesday Thursday Friday If you redistribute, or even out, the number of absent students so the number is the same for each day, you are finding the mean. The mean is a useful landmark when there are not one or two numbers that are far away from the rest of the data values (outliers). 2. Move the cubes on the line plot so that each day has the same number. After you ve evened out the cubes, how many does each day have? You can use a number sentence to model how you evened out the cubes. You started with cubes. Then you redistributed the cubes so that the total number of cubes (2) was the same for each of the days, or Use the cubes to find the mean of the following number of absent students. Monday: ; Tuesday: 0; Wednesday: 6; Thursday: 2; Friday: 7 Then write a number sentence to model what you did Math Masters, p. 14 Lesson

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