Box Plots. Objectives To create, read, and interpret box plots; and to find the interquartile range of a data set. Family Letters

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1 Bo Plots Objectives To create, read, and interpret bo plots; and to find the interquartile range 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 Collect and organize data to create bo plots. [Data and Chance Goal 1] Compare and interpret data displayed in bo plots. [Data and Chance Goal 1] Identify landmarks (maimum, minimum, median, lower quartile, upper quartile) of data sets. [Data and Chance Goal 2] Use measures of spread (interquartile range) to answer questions and draw conclusions. [Data and Chance Goal 2] Key Activities Students collect hair length data and use them to create bo plots. They find data landmarks and measures of spread of the data they collect. Ongoing Assessment: Recognizing Student Achievement Use journal page 17B. [Data and Chance Goal 2] Key Vocabulary bo-and-whiskers plot bo plot lower quartile upper quartile interquartile range (IQR) Materials Math Journal 1, pp. 17A and 17B Student Reference Book, pp. 141A 141C Study Link 1 5 tape measure straightedge Class Data Pad Data Landmarks on Line Plots Math Journal 1, p. 17C Students practice creating line plots and finding data landmarks. Math Boes Math Journal 1, p. 17D Students practice and maintain skills through Math Bo problems. Study Link Math Masters, p. 17A Students practice and maintain skills through Study Link activities. READINESS Dividing Data into Quartiles Math Masters, p. 17B per partnership: 1 complete deck of number cards (the Everything Math Deck, if available) Students use the process of finding a median to divide a data set into quartiles. ENRICHMENT Using Technology to Create Bo Plots Student Reference Book, pp. 134 and 135 computer with Internet access Students use online tools to generate bo plots from data sets. ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. 130 Students add the terms bo plot and interquartile range (IQR) to their Math Word Banks. Advance Preparation The data used in this lesson are collected and analyzed by gender. If you teach in a single-gender classroom, consider collecting data from another classroom that has students of both genders. If this is not an option, consider having students collect data from the other gender outside of school and recording it during Part 1 of the lesson. Teacher s Reference Manual, Grades 4 6 pp B Lesson 42A

2 Getting Started Mental Math and Reflees Have students represent real-life situations with positive or negative numbers. For each situation, have them eplain what 0 represents. Suggestions: A temperature of 25 C 25 C; 0 represents 0 C A profit of $25 $25; 0 represents no profit or loss An elevation of 18,000 feet 18,000; 0 represents sea level, or no elevation A debt of $100 dollars $100; 0 represents no debt A $30,000 drop in the median home sales price for a town $30,000; 0 represents no change in the median price A water depth of 70 feet 70 feet; 0 represents the surface of the water Math Message Work with a partner to complete Steps 1 3 on journal page 17A. Be prepared to share your results with the whole class. Study Link 1 5 Follow-Up Twelve-year-old boys tend to be slightly shorter than twelve-year-old girls. The data samples in Study Link 1-5 support this conclusion. Discuss and compare the median and mean for each set of data. NOTE Some students may benefit from doing the Readiness activity before beginning Part 1 of the lesson. See Part 3 for details. Interactive whiteboard-ready epresentations are available at to help you teach the lesson. 1 Teaching the Lesson Math Message Follow-Up (Math Journal 1, p. 17A) WHOLE-CLASS NOTE Depending on your situation, the whole-class data may be rather uniformly distributed around the median (the median may be at or near the center of the segment). Don t press students to draw conclusions like the sample answers if they aren t appropriate. Date Math Message Hair Length Time Work with a partner. Use a centimeter tape measure to measure each other s hair length. Step 1 Place the end of the tape measure at the hairline in the middle of your partner s forehead. Step 2 Measure across the top of your partner s head, along the centerline, and down the back of the head to the end of all the hair. Round your answer to the nearest centimeter. Student Page Start at hairline, center front. Measure to hairline, center back. If hair is longer, continue measuring from back hairline to tips of hair. Step 3 Record your partner s gender and your measure of his or her hair length below. My partner s gender My partner s hair length (cm) Have a volunteer record the gender and hair length data for the class on the Class Data Pad. Then ask each student to calculate and enter the maimum, minimum, median, and range for the whole class (boys and girls together) in the chart on journal page 17A. Check that all students have the correct values. Make a plot to show the median and range of the data. An eample is shown below. 1. Draw a 3 4 foot long horizontal line segment on the board. 2. Plot and label the minimum and maimum hair lengths at the endpoints of the segment. 3. Plot and label the median hair length. Plot it in the correct relative position on the line between the minimum and maimum. 4. Title the plot Class Hair Length Distribution (cm). Class Hair Length Distribution (cm) Whole-Class Hair Length Record the landmarks and measures of spread from your whole-class discussion in the table below. Landmarks and Measures of Spread Minimum Maimum Median Range Lower quartile (Q1) Upper quartile (Q3) Interquartile range (IQR) min median ma Tell students that this is a simple way to show a central landmark and spread (also called variability, or distribution) of the hair length data. Ask students to describe the distribution based on the plot. Sample answer: The median hair length is closer to the minimum than the maimum; most people have shorter hair. Math Journal 1, p. 17A EM3cuG6MJ1_U01_1-44.indd 17A 12/29/10 1:25 PM 42B Unit 1 Collection, Display, and Interpretation of Data

3 Constructing Bo Plots: Hair Length (Math Journal 1, p. 17A) WHOLE-CLASS A bo-and-whiskers plot, commonly called a bo plot, improves the simple minimum/median/maimum plot to show more about how a data set is distributed. The median splits a data set into lower and upper halves. The middle values of those two halves are called the lower and upper quartiles of the data set, respectively. They get their names because, along with the median, they split the data into four quarters. The upper quartile is often referred to as Q3 and the lower quartile is often referred to as Q1. NOTE Quartiles are percentiles. The lower quartile is the 25th percentile and the upper quartile is the 75th percentile of the distribution of a data set. NOTE To find quartiles, students look at values above or below the median. If the median lies at one data value, that one data value is not considered when finding the lower and upper quartiles. For eample, for the data set 2, 3, 5, 6, 6, 6, 8, 12, 13, the median lies at the middle 6. This value is not considered when finding either quartile. The values below the median are 2, 3, 5, and 6, so the lower quartile is 4. The values above the median are 6, 8, 12, and 13, so the upper quartile is 10. Ask students to identify the hair lengths below the median on the Class Data Pad, and then to find the median of those values. Tell them this number is called the lower quartile and have them enter the value in the chart on journal page 17A. Plot and draw a tick mark at this point on the plot on the board. This is the left side of the bo for this bo plot. Class Hair Length Distribution (cm) min lower quartile median ma Repeat the process for the hair lengths above the median to calculate, define, and plot the upper quartile. Complete the bo on the plot by drawing a horizontal segment connecting the tops of the tick marks at the lower and upper quartiles. Do the same for the bottoms of these two tick marks. Erase the part of the original horizontal line that is inside the bo. Ensure that the tick mark at the median touches the top and bottom of the bo. Your finished plot should look something like the eample shown below. Class Hair Length Distribution (cm) min lower quartile median upper quartile ma Lesson 42C

4 NOTE Some data analysis software allows you to make bo plots. It may also allow you to adjust the length of the whiskers to indicate all but the top and bottom 5 or 10 percent of the data. Tell students that this is a bo-and-whiskers plot of the whole-class hair length data. Its name comes from the bo between the lower and upper quartiles and the whiskers, or line segments that etend from the sides of the bo to the maimum and minimum values. It is often simply called a bo plot. Ask students what part of the distribution is in the bo. The middle half Then ask how wide the bo is. Answers vary based on class data. The width is the difference of the upper and lower quartiles. For eample, if the upper quartile is 45 and the lower quartile is 34, then the width of the bo is = 11. Tell them that this value is called the interquartile range of the data set, or IQR. The IQR can be thought of as the distance between the upper and lower quartiles. It is calculated by finding the difference between the upper quartile and the lower quartile, or Q3 Q1. Have students record the interquartile range in the chart on journal page 17A. Again, ask students to describe the distribution based on the plot. Sample answer: The middle half of the hair lengths is closer to the minimum than the maimum length. Ask if it seems fair to use data values contained in the bo to describe the typical hair length of students in this class. Sample answer: Yes, because the middle half of the hair lengths are in the bo, while the other half are either much shorter or much longer than the median. Constructing Bo Plots: Hair Length by Gender (Math Journal 1, p. 17B; Student Reference Book, pp. 141A 141C) PARTNER Date Student Page Time Whole-Class Hair Length by Gender Use the class data divided by gender boys hair lengths compared to girls. With a partner, calculate the landmarks and measures of spread for each gender and complete the table. Landmarks and Measures of Spread Girls Boys Minimum Answers vary. Maimum 141A 141B Tell students that bo plots are often constructed above, beneath, or beside a number line. On this journal page, they will draw two bo plots above the same number line to help them compare the data. Circulate and assist as pairs of students draw bo plots for hair length for each gender. For more information on bo plots, have students refer to Student Reference Book, pages 141A 141C. Median Lower quartile (Q1) Upper quartile (Q3) Range Interquartile range (IQR) Using the information in the table, draw and label two bo plots, one for each gender, above the number line. Remember to label the number line before drawing the bo plots. Then use the completed bo plots to answer the questions below. Hair Lengths by Gender Centimeters 1. Which gender has a wider distribution of hair lengths? Answers vary. 2. What is a typical hair length for girls in your class? Eplain your answer. Answers vary. Ongoing Assessment: Recognizing Student Achievement Journal Page 17B Use journal page 17B to assess students ability to calculate the interquartile range of a data set. Students are making adequate progress if they are able to find the upper and lower quartiles and use these values to calculate the interquartile range. Some students may be able to use the interquartile range to draw conclusions about the two data sets. [Data and Chance Goal 2] 3. Use the data shown in the bo plots to compare the hair lengths of boys and girls in your class. Answers vary. Math Journal 1, p. 17B EM3cuG6MJ1_U01_1-44.indd 17B 12/29/10 1:25 PM 42D Unit 1 Collection, Display, and Interpretation of Data

5 2 Ongoing Learning & Practice Data Landmarks on Line Plots (Math Journal 1, p. 17C) INDEPENDENT The problems on journal page 17C provide practice creating line plots and finding data landmarks. Math Boes (Math Journal 1, p. 17D) INDEPENDENT Mied Practice Math Boes in this lesson are paired with Math Boes in Lessons 1-2 and 1-4. The skills in Problems 4 and 5 preview Unit 2 content. Study Link (Math Masters, p. 17A) INDEPENDENT Date Time Line Plots and Data Landmarks Use the data shown in the tally chart to make a line plot Use the line plot to find the following landmarks. Value Frequency 16 // 17 ////\ 18 ////\/ 19 //// 20 /// 28 // 33 / 35 / 40 / a. Minimum: 16 b. Maimum: 40 c. Range: 24 d. Mode: 18 e. Median: 18 f. Mean: Is the mean or the median a better representation of the center of the data set? Eplain. Sample answer: The median better represents the center of the data. There are several outliers in this data set which makes the mean greater than most of the data in the data set. 3. Describe a situation that might be represented by the data in the line plot. Sample answer: The time in minutes it took for someone to travel to work each morning during the last 5 weeks 4. Using your situation, describe why there might be outliers in the data set. Sample answer: There may have been road construction on certain days, which would have made the commute time longer. Math Journal 1, p. 17C EM3cuG6MJ1_U01_1-44.indd 17C Student Page 1/22/11 1:27 PM Home Connection Students construct bo plots and practice finding data landmarks and measures of spread. Name Date Time STUDY LINK Bo Plots 141A 141B The tables below show the number of automobiles sold each month at two dealerships. Find the data landmarks and measures of spread for each data set. 1. Main Street Autos a. Maimum: 45 b. Minimum: 25 c. Range: 20 d. Median: 34 e. Lower quartile: 29 f. Upper quartile: 37 g. Interquartile range: 8 Study Link Master Main Street Autos Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Park Road Cars Jun Jul Jan Feb Mar Apr May Aug Sep Oct Nov Dec Park Road Cars a. Maimum: 42 b. Minimum: 22 c. Range: 20 d. Median: 35.5 e. Lower quartile: 31.5 f. Upper quartile: 37 g. Interquartile range: 5.5 Date Math Boes 1. Draw line segments having the following lengths. a. 3.7 cm b. 46 mm c. 4.1 cm 3. Subtract. Student Page Time 2. Write a data set that fits the following description. There are 6 numbers in the data set. The maimum is 35. The range is 14. Sample The mode is 21. The median is , 21, 28, 32, 33, 35 answer: a. 3, = 2,588 b = 662 c. 1,797 = 4,609-2, Construct and label a bo plot for each set of sales data. Main Street Park Road Monthly Auto Sales Number of Automobiles Sold 4. Use the bo plots to compare the sales at the two dealerships. Sample answer: The range of sales at both dealerships is the same. The bo on the Park Road Cars plot is narrower. This means that the middle half of their sales is not as spread out as Main Street Autos. Math Masters, p. 17A 4. Divide = 18 Math Journal 1, p. 17D 5. Complete. a = 24,000 b = 48,000 c. 1,600 = d = 5,600 e. 9, = 450, _EMCS_B_G6_MM_U01_ indd 17A 3/6/12 1:43 PM EM3cuG6MJ1_U01_1-44.indd 17D 1/29/11 1:13 PM Lesson 42E

6 Name Date Time Dividing Data into Quartiles Work with a partner. Shuffle a complete deck of number cards and draw at least 10 cards from the deck. Write the numbers below. Answers vary. Teaching Master Put the cards in order from least to greatest. Find the median of your data set. If you have any cards that are equal to the median, take them out of the data set. Place the cards with numbers less than the median in one group and the cards with numbers greater than the median in a second group. Record your work below. Group 1 Median Group 2 3 Differentiation Options READINESS Dividing Data into Quartiles (Math Masters, p.17b) PARTNER Min Net, find the median of the cards in Group 1. This number is called the lower quartile. Remove any cards that are equal to the lower quartile from your data set. Place the cards with numbers less than the lower quartile in one group and those with numbers greater than the lower quartile in a second group. Group 1a Lower Quartile Group 1b Repeat the process for Group 2. The median of the upper half of the data is called the upper quartile. Group 2a Upper Quartile Group 2b Each of the four groups (1a, 1b, 2a, and 2b) is called a quartile. A quartile is about 1_ 4 of the data set. Repeat the process using a different number of cards. What are some things that you notice about the quartiles of a data set? Sample answers: There are two quartiles below and two quartiles above the median of a data set. Sometimes there are data points that do not end up in a quartile. Math Masters, p. 17B To provide eperience with dividing data sets into quartiles, have students use number cards to create a data set. They place the cards in order and use the median to divide the set into two halves. They then find the median of each half, dividing the set into quartiles. Eplain to students that the term quartile can be used in two ways. It can refer to a number (the medians of the upper and lower halves of the data set, called the upper quartile and the lower quartile), or it can refer to a part of the data set (the data contained in each quarter of the set). As partners complete the activity, check to make sure that they are differentiating between the two ways the term is used. EM3MM_G6_U01_ indd 17B 1/21/11 5:49 PM ENRICHMENT SMALL-GROUP Using Technology to Create Bo Plots (Student Reference Book, pp. 134 and 135) Min To etend students understanding of bo plots, have them use an online bo plot maker, such as the one at illuminations.nctm.org, to create bo plots of the data sets on page 134 or 135 of the Student Reference Book. Have students compare and contrast the bo plots that they create. If the technology allows them to do so, suggest that students create and plot several different data sets on the same number line and compare the resulting bo plots. ELL SUPPORT Building a Math Word Bank (Differentiation Handbook, p. 130) INDEPENDENT 5 15 Min To provide language support for constructing and analyzing bo plots, have students use the Word Bank template found on Differentiation Handbook, page 130. Ask them to write the terms bo plot and interquartile range, draw pictures relating to each term, and write other related words. See the Differentiation Handbook for more information. 42F Unit 1 Collection, Display, and Interpretation of Data

7 Name Date Time STUDY LINK Bo Plots 141A 141B The tables below show the number of automobiles sold each month at two dealerships. Find the data landmarks and measures of spread for each data set. Main Street Autos Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Park Road Cars Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Main Street Autos a. Maimum: b. Minimum: c. Range: d. Median: e. Lower quartile: f. Upper quartile: g. Interquartile range: 2. Park Road Cars a. Maimum: b. Minimum: c. Range: d. Median: e. Lower quartile: f. Upper quartile: g. Interquartile range: 3. Construct and label a bo plot for each set of sales data. Monthly Auto Sales Number of Automobiles Sold 4. Use the bo plots to compare the sales at the two dealerships. Copyright Wright Group/McGraw-Hill 17A

8 Name Date Time Dividing Data into Quartiles Work with a partner. Shuffle a complete deck of number cards and draw at least 10 cards from the deck. Write the numbers below. Put the cards in order from least to greatest. Find the median of your data set. If the median falls on one of the numbers of the cards, take that card out of the data set. Place the cards with numbers less than the median in one group and the cards with numbers greater than the median in a second group. Record your work below. Group 1 Median Group 2 Net, find the median of the cards in Group 1. This number is called the lower quartile. Remove any cards that are equal to the lower quartile from your data set. Place the cards with numbers less than the lower quartile in one group and those with numbers greater than the lower quartile in a second group. Group 1a Lower Quartile Group 1b Repeat the process for Group 2. The median of the upper half of the data is called the upper quartile. Group 2a Upper Quartile Group 2b Copyright Wright Group/McGraw-Hill Each of the four groups (1a, 1b, 2a, and 2b) is called a quartile. A quartile is about 1_ 4 of the data set. Repeat the process using a different number of cards. What are some things that you notice about the quartiles of a data set? 17B