Middle Value (Median) of a Set of Data Objectives To guide children as they sort numerical data and arrange data in ascending or descending order, and as they find the middle value (median) for a set of numerical data. www.everydaymathonline.com 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 Order and compare numbers. [Number and Numeration Goal ] Collect and organize data. [Data and Chance Goal 1] Find the median of a data set using concrete materials. [Data and Chance Goal 2] Key Activities Children sort their class standing long jump data by arranging the data in ascending order. Children find the median jump length in centimeters for the class. Key Vocabulary median middle value sort (the data) Materials Math Journal 2, p. 13 My Reference Book, pp. 45 and 46 Home Link 6 masking tape ( 20 feet) half-sheets of paper slate number grid (optional) Measuring Objects Math Journal 2, pp. 16 and 1 ruler Children measure pictured objects to the nearest inch and the nearest centimeter. Math Boxes Math Journal 2, p. 18 Children practice and maintain skills through Math Box problems. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 1. [Measurement and Reference Frames Goal 6] Home Link Math Masters, pp. 215 and 216 Children practice and maintain skills through Home Link activities. READINESS Ordering Numbers Math Masters, p. 21 Children sort numbers from the smallest to the largest. ENRICHMENT Finding the Median Math Masters, p. 218 Children find the median of an even set of numbers and explain their solution. ELL SUPPORT Discussing the Word Median Children discuss the word median and find the median for sets of numbers. Advance Preparation For the Math Message, place a stack of half-sheets of paper next to the problem. Before beginning the Sorting the Standing Jump Data activity in Part 1, tape a 20-foot baseline along the floor in a large, open space. Teacher s Reference Manual, Grades 1 3 pp. 125 12 56 Unit Patterns and Rules
Getting Started Mental Math and Reflexes Have children use <, >, or = to compare large numbers. Ask them to explain their answers in terms of place value. Suggestions: 132 and 123 > 234 and 334 < 456 and 45 < 1,054 and 1,154 < 1,243 and 1,233 > 1,522 and 1,622 < 6,989 and 6,99 > 10,001 and 10,010 < 9,8 and 9, > Math Message Turn to Math Journal 2, page 13. Write your longer jump length in centimeters on a half-sheet of paper. Then open My Reference Book to pages 45 and 46. Read range and median with a partner. Home Link 6 Follow-Up Have children name some objects that are the same length as their arm span. Have them share their answers to Problem 4. Example 1 Teaching the Lesson Math Message Follow-Up (My Reference Book, pp. 45 and 46) ELL WHOLE-CLASS Briefly discuss range and median with children, including the examples in My Reference Book, pages 45 and 46. The range is the difference between the minimum and maximum values of a data set. The term median is another widely used name for middle value. Write a list of the children s jump lengths on the board. Be sure that children recorded their centimeter data. Ask children to identify the shortest jump length and the longest jump length from the class data. Then, tape an actual tape measure to the board and mark the longest and shortest jump lengths. (See marg) To support English language learners, remind children that the shortest jump length is called the minimum value and the longest jump length is called the maximum value. Label the shortest and longest jump lengths with the words minimum and maximum. Working together, calculate the difference between the longest and shortest jumps. Draw a comparison diagram on the board. Fill in the known quantities the longest and the shortest jump lengths. The difference is what you want to find, so write a question mark for the difference. Some children may use the number grid to find the difference. Others may subtract by counting up. Links to the Future Longest Jump: 94 Shortest Jump: 34 Quantity Quantity 94 longest difference 39 shortest? Difference 94-39 = 55 The difference is 55. Do not expect children to remember how to find the range of a data set. This is an early exposure. Use the word range but do not expect children to use it at this time. Finding the range of a data set is a Grade 3 Goal. Lesson 5
Date 6 Record of Our Jumps You will make two jumps. For each jump, measure to the nearest centimeter and to the nearest inch. How to Measure Each Jump Place a penny or other marker (or make a dot with chalk) where the Jumper s back heel lands. Measure from the starting line to the marker. 1. Record your two jumps. First jump: centimeters inches Second jump: centimeters inches jump measure 2. Circle the measures for your longer jump. Answers vary. You will complete the next question in Lesson -. 3. A middle value of jumps for our class is centimeters. A Math Journal 2, p. 13 EM3MJ2_G2_U0_161_183.indd 13 39 Date The Lengths of Objects Reminder: means inches; means centimeters Measure each item to the nearest inch. Measure each item to the nearest centimeter. Record your answers in the blank spaces. 1. pencil 1 2. screwdriver 55 60 60 60 Children find their correct positions on the line. 1/29/11 10:59 AM 94 Sorting the Standing Jump Data WHOLE-CLASS Explain to children that they have found the smallest value, the largest value, and the range. Now they are going to find the middle value. Because data values are frequently listed in haphazard order such as the list on the board, a common first step in analyzing numerical data is to sort the data that is, to arrange it in ascending or descending order. This sorting is typically done with paper and pencil or a computer. Children will sort the standing jump data by arranging themselves in a single line from shortest jump length at one end of the line to longest jump length at the other end. Use the following routine: 1. Select a large, open space. Tape a 20-foot baseline on the floor. 2. Have children stand away from the line until they are called. Children should carry their half-sheets of paper (showing their jump lengths in centimeters) with them. 3. Ask the children who made the shortest and the longest jumps to step forward and stand on the line shortest at one end and longest at the other end. 4. One-by-one, ask the remaining children to step forward and find their correct position on the line. They do this by comparing the length of their jump to the lengths of the other children s jumps, as shown on their half-sheets of paper. (You should pause several times during the routine to make sure the children standing on the line are evenly distributed along it. This will allow for spaces for the remaining children to fit themselves into the line.) 5. Several children may have the same jump length. If this is the case, they should be together on the line, but it does not matter which order these children are When all children have found their places on the line, make an oral check that they are sorted by the lengths of their jumps. Begin at the end with the shortest length. Move down the line, in order, asking each child to say his or her jump length. If any child says a length that is smaller than a length already named, that child is not in the correct position. 6 15 Adjusting the Activity ELL 3. pen Have children use the number grid or Class Number Line to figure out the order of their numbers. 5 13 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 Math Journal 2, p. 16 EM3MJ2_G2_U0_161_183.indd 16 1/28/11 4:43 PM 58 Unit Patterns and Rules
Finding the Median Length of the Standing Long Jumps (Math Journal 2, p. 13) WHOLE-CLASS If you have not already done so, you might want to introduce the term median at this time. The routine for finding the median is quite simple. After arranging the data in ascending or descending order, count from either end to the number in the middle. If the data set consists of an even number of data, there are two middle values and the median is the number halfway between those two middle values. At this stage, either value or any number between them is acceptable. To find the median for the standing long jump data, use this routine: 1. Ask the children at each end of the line to take two large steps forward. 2. Identify the two children who are at each end of the line now, and ask them to take two large steps forward. 3. Continue to repeat this process. Each time that a child at each end steps forward, the original line of children is reduced by two. (See marg) 4. If you have an odd number of children in your class, there will be one remaining child on the line. The length of his or her jump is the median length. Each time the routine is repeated, the number of children standing on the line is reduced by two. NOTE Consider having children formulate questions for future data exploration. Data collection activities are usually more meaningful to children if they are connected to a real-life problem or involve situations that children really care. For example: How many children know how to tie their shoes? These questions can lead to a survey, a y, and a display of the collected data, as well as a solution to a problem meaningful to the children in the class. Date 5. If you have an even number of children in your class, there will be two remaining children on the line. The lengths of their jumps are the middle values. The median is officially the number halfway between these two lengths, but either value (or any number between them) is an acceptable answer. 4. bolt The Lengths of Objects continued 2 5 5. dandelion leaf 4 11 Ask children to record the median jump length on journal page 13. Review the usefulness of finding a middle value: Because there are great differences in second graders jumps, it is easier to talk a median length than many different lengths. 6. List the objects in order from shortest to longest. bolt dandelion leaf pen screwdriver pencil. How much longer is the pencil than the bolt? 5 12 8. How much longer is the screwdriver than the pen? 1 2 EM3MJ2_G2_U0_161_183.indd 1 Math Journal 2, p. 1 1/29/11 10:59 AM Lesson 59
Date Math Boxes 1. Draw hands to show 4:15. 12 11 1 10 2 9 8 3 4 6 5 2. Who read the most books (maximum)? Grace Who read the least books (minimum)? Lilly 3. There are 6 rooms. Each room 4. Fill in the missing numbers. has 4 windows. How many windows in all? 24 892 893 windows Draw an array. 5. Here is a picture of a pyramid. What is the shape of one of the faces? triangle EM3MJ2_G2_U0_161_183.indd 18 112 113 Math Journal 2, p. 18 58 Number of Books Books Read Juan Lilly Grace Terell 912 903 913 6. Color 5_ 8 of the rectangle. 8 13 1/28/11 4:43 PM 2 Ongoing Learning & Practice Measuring Objects (Math Journal 2, pp. 16 and 1) Children measure the lengths of various items to the nearest inch and to the nearest centimeter. Then they list the objects in order from shortest to longest and compare lengths. Math Boxes (Math Journal 2, p. 18) Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson -5. The skill in Problem 6 previews Unit 8 content. Writing/Reasoning Have children draw, write, or verbalize their answers to the following: Tell the time 2 hours later than the time shown in Problem 1. How did you figure it out? Sample answer: The time will be 6:15. The hour hand goes around the clock two times, to 5:15 and then to 6:15. Ongoing Assessment: Recognizing Student Achievement Math Boxes Problem 1 Use Math Boxes, Problem 1 to assess children s progress with telling time to the quarter-hour. Children are making adequate progress if they successfully complete the problem. [Measurement and Reference Frames Goal 6] Name Date HOME LINK Find the Middle Value Family Note Home Link Master In this lesson, your child sorted data to find the median. Median is a term used for the middle value. To find the median of a set of data, arrange the data in order from smallest to largest. Count from either end to the number in the middle. The middle value is the median. As your child finds the median in Problems 2 and 3, remind him or her that is the abbreviation for inches and is the abbreviation for centimeters. 46 Please return this Home Link to school tomorrow. Name Date HOME LINK Find the Middle Value continued 2. Jarel: 66 Home Link Master Suki: 0 Peter: 56 Keisha: 3 Cesar: 68 List the data in order from smallest to largest. Draw a circle around the median in your list. 1. 12 3 21 15 20 9 56 smallest 66 68 0 3 largest 3. Jarel: 168 Suki: 18 Peter: 142 Keisha: 185 Cesar: 13 3 smallest 9 12 15 20 21 largest 142 smallest 168 13 18 185 largest Math Masters, p. 215 Math Masters, p. 216 195-226_EMCS_B_G2_MM_U0_56949.indd 215 1/28/11 2:04 PM 195-226_EMCS_B_G2_MM_U0_56949.indd 216 1/26/11 3:49 PM 580 Unit Patterns and Rules
Home Link (Math Masters, pp. 215 and 216) Home Connection Children sort data and find the median. Name Date Ordering Numbers Choose from the number cards pictured below. 8 2 5 92 Teaching Master Match a number card to each of the sets of pictured dots. 3 Differentiation Options READINESS Ordering Numbers (Math Masters, p. 21) 5 15 Min 8 2 92 5 Match a number card to each of the number lines below. Draw a small mark on the number line where you think the number would be. Label your mark with the number. 1. 2. 92 0 100 3. 4. 2 8 0 10 5 To provide experience with comparing and ordering numbers using a visual model, have children match number cards to a quantity and to a location on a number line. When children have completed the master, have them describe how they decided where to put the numbers. 195-226_EMCS_B_G2_MM_U0_56949.indd 21 0 40 Math Masters, p. 21 0 0 1/26/11 3:49 PM ENRICHMENT Finding the Median (Math Masters, p. 218) 5 15 Min To apply their knowledge of finding the median, have children review the data on Math Masters, page 218. Children find the median of an even set of numbers and explain their solution. ELL SUPPORT Discussing the Word Median SMALL-GROUP 5 15 Min To provide language support for data landmarks, have children determine the median for sets of numbers. The median is the middle value of a set of ordered data. Write five numbers in ascending order on the board. Ask children to identify the middle number or median. Circle the number and label it with the synonyms middle number and median. Provide children with lists of five numbers that are out of sequence. Have them put the numbers in order and circle the median. Name Date Teaching Master Find the Median Solve. The track team collected standing long jump data. They are as follows: Our Jumps 93 inches 9 inches 82 inches 96 inches 85 inches 91 inches 89 inches 8 inches Find the median. 90 inches Explain your work. Sample answer: I ordered the numbers from smallest to largest. The two middle numbers are 89 and 91. The median is 90 because it is halfway between 89 and 91. Math Masters, p. 218 195-226_EMCS_B_G2_MM_U0_56949.indd 218 1/26/11 3:50 PM Lesson 581