Frequency Distributions

Transcription

1 Frequency Distributions Objectives To guide children as they make frequency tables, and as they find the median, mean, and mode of data sets. 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 whole numbers. [Number and Numeration Goal 6] Collect and organize data to create a frequency table. [Data and Chance Goal 1] Find the median and mode of a data set. [Data and Chance Goal 2] Key Activities Children record their waist-to-floor measurements on the Class Data Pad. They make a frequency table and use it to find the median, mean, and mode of the data set. Informing Instruction See page 855. Key Vocabulary frequency table mode Materials Math Journal 2, pp. 251 and 261 Student eference ook, p. 81 (optional) Home Link 10 8 Class Data Pad calculator slate (optional) counters (optional) Making a ar Graph of Measurements Math Journal 2, p. 262 graphing software (optional) Children make a bar graph of data from a frequency table. Informing Instruction See page 856. Math oxes Math Journal 2, p. 263 Children practice and maintain skills through Math ox problems. ecognizing Student Achievement Use Math oxes, Problem 3. [Number and Numeration Goal 2] Home Link Math Masters, p. 351 Children practice and maintain skills through Home Link activities. EADINESS Organizing Data per group: calculator, numbered index cards, two 8 1_ 2 " by 11" sheets of paper Children create a physical display of data. ENICHMENT Comparing Waist-to-Floor Measurements per partnership: 2 copies of Math Masters, pp. 350 and 352 Math Message data calculator Children make two frequency tables and bar graphs. They compare the landmarks for two sets of data. EXTA PACTICE Minute Math + Minute Math +, pp. 3 7 and Children practice ordering numbers. Advance Preparation For Part 1, divide a page of the Class Data Pad into three columns. Write the heading Waist-to-Floor Measurement (in.) above the first column. Write the heading Frequency above the second and third columns. (See Math Journal 2, page 261.) For the optional eadiness activity in Part 3, write a 2-digit number on an index card for each child. There should be some duplicates. Label two 8 1_ " 2 by 11" sheets of paper with smallest and largest. 852 Unit 10 Measurement and Data

2 Getting Started Mathematical Practices SMP1, SMP2, SMP4, SMP5, SMP6 Content Standards 3.OA.8, 3.MD.3 Mental Math and eflexes Children find the ordinal number for the middle value in a set of data. They may make a list on slates or use counters to help them solve each problem. Have them share solution strategies. Which measurement would be the middle in a list of 5 measurements? The third measurement Which measurement would be the middle in a list of 15 measurements? The eighth measurement Which measurements would be the middle in a list of 16 measurements? The eighth and ninth measurements Math Message Look up your last waist-to-floor measurement on journal page 251. Write it on the Class Data Pad. Do not write your name. Home Link 10 8 Follow-Up Have partners read to each other the basic facts that are suggested by the Fact Triangles in the Home Link problems. 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION Count the number of entries on the Class Data Pad to check that all children have entered their measurements. Making a Frequency Table of Waist-to-Floor Measurements (Math Journal 2, p. 261) SMALL-GOUP ELL NOTE If you plan on having children do the optional Part 3 Enrichment activity, have all the boys write a next to their waist-to-floor measurements and have all the girls write a G next to theirs. Divide the class into groups of three or four. In the second column of the Class Data Pad, draw a frequency table, a chart on which data are tallied to find the frequency of given events or values. To support English language learners, explain that the word frequency is used in different contexts to mean different things. A frequency table can be used to determine how often (or the number of times) an event or value occurs. Children complete the frequency table on page 261 in their journals for the measurements listed on the Class Data Pad. Guide them as follows: 1. Fill in the first column of the table. The first entry should be the smallest measurement on the Class Data Pad, followed by all other possible measurements in ascending order (to the nearest inch), up to the largest measurement. 2. Make a tally mark in the second column next to the appropriate measurement for each time it is listed on the Class Data Pad. 3. After all measurements have been tallied, write a number in the third column that represents each set of tallies. 4. To check that no measurements have been omitted, add the numbers in the third column and compare the sum to the number of measurements listed on the Class Data Pad. Waist-to-Floor Measurement (in.) Frequency Tallies Number // 2 0 ////\ 5 ////\ /// 8 ////\ // 7 //// 4 Lesson 853

3 Date Frequency Table Time 1. Fill in the table of waist-to-floor measurements for the class. This kind of table is called a frequency table. Waist-to-Floor Measurement (inches) Student Page Tallies Answers vary. Frequency Number Finding the Median and Mean of the Data Set (Math Journal 2, p. 261) WHOLE-CLASS Total = 2. What is the median (middle value) of the measurements? in. 3. What is the mean (average) of the measurements? in. 4. The mode is the measurement, or measurements, that occur most often. What is the mode of the waist-to-floor measurements for the class? in. Math Journal 2, p. 261 Each member of each group finds the median and the mean of the set of data. Children should use a calculator to find the mean. Group members compare answers, resolve discrepancies, and record the group s answers on the journal page. ring the class together to share results and strategies. Did anyone find the median by listing the data on the Class Data Pad from smallest to largest? How could you find the median using only the frequency table? One way is to cross off one tally from the high end and then cross off one from the low end, repeating until only one or two tallies remain. Another way is to find the ordinal number for the middle measurement. For example, if there are 25 measurements, the middle measurement is the 13th measurement. If there are 26 measurements, there are two measurements in the middle the 13th and 14th measurements. Starting at the top of the table, add the numbers in the third column until the sum is equal to or greater than the ordinal number for the middle measurement. The measurement in that row is the median. Which is more efficient finding the median from the unordered data on the Class Data Pad or from the frequency table? Using the frequency table; if you use the data on the Class Data Pad, you must first order the data from smallest to largest. Did anyone use the memory keys on the calculator to find the mean? If no one has done so, describe how to use the memory keys to find the sum of all the measurements. For example, if there are three measurements, each of 26 inches, press 3 X 26 M+. This adds the product, 78, to the number in the calculator s memory. Ask children to enter the total for each measurement into the calculator s memory. When all data have been entered, press the memory recall key to display the number in memory. This is the sum of all measurements. Then divide the sum by the total number of tallies. The result is the mean height for the class. emind children to clear the calculator s memory. Ask: Which is easier, finding the mean on the calculator with or without the help of the memory keys? If you don t use the memory keys, the number displayed after entering and adding is the total to that point, not the value just added. This increases the likelihood of losing track of data and having to start over, or losing the total if an error is made pressing the keys. Also, with a large set of data containing many repeated values, using the memory keys reduces the number of keystrokes needed. Compare the median and mean of the data set. Often, the median and the mean of a set of data are the same or almost the same. 854 Unit 10 Measurement and Data

4 eviewing the Mode of the Set of Data (Math Journal 2, p. 261; Student eference ook, p. 81) WHOLE-CLASS Ask children to explain the mode of a set of data. The mode is the value(s) that occurs most often. Have children find and record the mode of the set of waist-to-floor measurements in their journals. Informing Instruction Watch for children who have difficulty finding the mode of the data set. Have them review page 81 in the Student eference ook. Ask: When might it be useful to know the mode of a set of data? Sample answer: To determine which brands of a product are most popular in a store When children have finished, bring the class together to discuss where the mode falls in relation to the data set. Adjusting the Activity If your school has more than one third-grade class, you might pool the data of all of the classes in a table. A U D I T O K I N E S T H E T I C T A C T I L E V I S U A L 2 Ongoing Learning & Practice Making a ar Graph of Measurements (Math Journal 2, p. 262) INDEPENDENT Date ar Graph Student Page Time Make a bar graph of the data in the frequency table on journal page 261. Make sure children copy the measurements onto the graph (journal page 262) correctly. Have them record the scale in increments of 2 and complete the graph. Help children verify that the number of data entries on their bar graphs matches the number of data in the frequency table. Adjusting the Activity If you pooled data for more than one third-grade class, make a bar graph. Class Waist-to-Floor Measurements Waist-to-Floor Measurements (in.) A U D I T O K I N E S T H E T I C T A C T I L E V I S U A L NOTE If available, have children create bar graphs using graphing software. Number of Children Math Journal 2, p. 262 Lesson 855

5 Date 1. Use the partial-products algorithm to solve ,652 Math oxes 3. Jerry has 16 fish in a tank. Color 3_ 8 of the fish red, 1_ 4 of the fish blue, and the rest yellow. What fraction of the fish are yellow? 6_ 16, or 3_ , Weight in pounds of newborn babies: 11, 8, 8, 7, 6 The mean (average) weight is 8 pounds. The median weight is 8 pounds. Student Page Time Math Journal 2, p pint = 16 fluid ounces 3 pints = 48 fluid ounces 1 half-gallon = 2 quarts 3 half-gallons = 6 quarts 1 liter = 1,000 milliliters 4. Fill in the missing factors = = 5, ,000 = 24, = 54, On the first day of spring, the lengths of the day and night are equal. If the sun rises at 6:51 A.M. on that day, at what time would you expect it to set? 6 : 51 P.M. 37 Informing Instruction Watch for children who have difficulty comparing the total number of entries on their graphs to the total number of data on the table. Have them perform a one-to-one correspondence check to find out if measurements are missing. Ask: How does a bar graph make it easier to find the mode of a set of data? ou only need to look for the highest bar or bars. ou and the children pose how many more and how many less questions about the bar graph data. Math oxes (Math Journal 2, p. 263) INDEPENDENT Mixed Practice Math oxes in this lesson are paired with Math oxes in Lesson The skill in Problem 6 previews Unit 11 content. Writing and easoning Have children write an answer to the following: Explain how you found the mean weight of newborn babies in Problem 5. Sample answer: I used my calculator and found the sum of the babies weights and divided by the total number of babies, 40 5 = 8. ecognizing Student Achievement Math oxes Problem 3 Home Link Master Name Date Time Use Math oxes, Problem 3 to assess children s ability to solve problems involving fractional parts of sets. Children are making adequate progress if they are able to color the fish as directed. Some children may be able to write the fraction that represents the number of yellow fish. [Number and Numeration Goal 2] HOME LINK A Frequency Table Family Today we learned how to organize data in a frequency table. For today s Home Link, help Note your child count the number of electrical outlets in at least 8 different rooms. It would be best if the rooms were all in the same kind of building for example, rooms in a house or apartment; rooms in the local library; or rooms in a school Please return this Home Link to school tomorrow. Home Link (Math Masters, p. 351) INDEPENDENT ELL 1. Make a frequency table for the number of electrical outlets in at least 8 different rooms. Answers vary. Number of Electrical Outlets Frequency oom Tallies Number Home Connection Children count the number of electrical outlets in at least eight different rooms. They record the data in a frequency table and find the median, mean, and mode for the data. To support English language learners, show them an electrical outlet. 2. What is the median (middle) number of outlets? 3. What is the mean (average) number of outlets? (ou may use a calculator to calculate the answer. Drop any digits to the right of the tenths place.) 4. What is the mode of the data in the table? (eminder: The mode is the number that occurs most often in a set of data.) Math Masters, p Unit 10 Measurement and Data

6 3 Differentiation Options EADINESS Organizing Data SMALL-GOUP 5 15 Min To explore organizing data, have children create a physical display of data. Distribute the numbered index cards to children. (See Advance Preparation.) Have volunteers suggest what the numbers might represent. Sample answer: Pieces of candy each child has; number of marbles each child has, and so on. Ask children to stand in line holding their cards facing you. Display the smallest sign on the left end of the line (as you face the children), and the largest sign on the right end of the line. Explain to children that they are going to put themselves in order from smallest to largest based on their number cards following these rules: Children can only move based on comparisons with the child next to them. (No one should be moving about the room comparing numbers randomly.) Children will stack up if they have the same number. They will be forming a human bar graph. When they have finished putting themselves in order, discuss which numbers are the most frequent, least frequent, and not represented at all. Ask children to figure out the median and ask them how they would calculate the mean. Name Date Time Comparing Waist-to-Floor Measurements 1. Use the floor-to-waist data from the Math Message. Divide the data into two groups, one for boys data and one for girls data. 2. Make two frequency tables (one for each set of data) on the back of this page. 3. Make a graph for each data set on copies of Math Masters, page Find and record the landmarks (median, mean, and mode) for each data set. Use your calculator to help you. Answers vary. Median: Girls Mean: Girls Mode: Girls Teaching Master oys oys oys 5. Compare the two graphs and the landmarks. What do you know from these results? Sample answers: The longest girls measurements are longer than the longest boys measurements. More boys than girls are in the inch range. oys have a larger range of measures than girls. Math Masters, p. 350 ENICHMENT Comparing Waist-to-Floor Measurements (Math Masters, pp. 350 and 352) PATNE Min Name Date Time Teaching Master ar Graph To apply their understanding of frequency tables and bar graphs, have children use the coded data from the Math Message (G for girls data and for boys data) and make two frequency tables and two bar graphs: one each for boys and one each for girls. They determine landmarks (median, mean, and mode) and compare the two sets of data. Directions for the activity are on Math Masters, page 350. Children record their work on Math Masters, page 352. EXTA PACTICE Minute Math+ SMALL-GOUP 5 15 Min To offer children more experience with ordering whole numbers see the following pages in Minute Math+: asic outines: pp. 3 7 Counting: pp Title: Math Masters, p. 352 Lesson 857