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Line Plots Objective To provide experience creating and interpreting line plots with fractional units. www.everydaymathonline.com ePresentations eToolkit Algorithms Practice EM Facts Workshop Game™ Teaching the Lesson Key Concepts and Skills • Identify equivalent fractions. [Number and Numeration Goal 5] • Identify fractions on a number line. [Number and Numeration Goal 6] • Add fractions and mixed numbers with like and unlike denominators. [Operations and Computation Goal 4] • Create a line plot. [Data and Chance Goal 1] • Analyze a data set. [Data and Chance Goal 2] Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Ongoing Learning & Practice Operations with Multidigit Whole Numbers and Decimals Math Journal 2, p. 242B Students practice solving problems involving operations with multidigit whole numbers and decimals. Math Boxes 7 10 Math Journal 2, p. 243 Students practice and maintain skills through Math Box problems. Interactive Teacher’s Lesson Guide Differentiation Options READINESS Identifying Fractions on a Number Line Math Masters, p. 211 Students label fractions on number lines. EXTRA PRACTICE Plotting Rain Gauge Data Math Master, p. 213 Students read measures from pictures of rain gauges, plot the data on a line plot, and then answer questions about data landmarks. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 3. Key Activities Students make a line plot to display and analyze a data set of measurements in fractions of a unit. Students use equivalent fractions and operations of fractions including addition and subtraction to solve problems involving information presented in line plots. [Number and Numeration Goal 6 and Operations and Computation Goal 4] Study Link 7 10 Math Masters, p. 212 Students practice and maintain skills through Study Link activities. Materials Math Journal 2, pp. 242 and 242A Fraction Cards (Math Journal 2, Activity Sheets 5–7) Student Reference Book, pp. 119–121 (optional) calculator Probability Meter Poster (optional) Advance Preparation For Part 1, students need Fraction Cards (Math Journal 2, Activity Sheets 5–7), which were cut out and used in Lesson 5-1. 590 Unit 7 Exponents and Negative Numbers 590_EMCS_T_TLG2_G5_U07_L10_576914.indd 590 3/1/11 11:54 AM Getting Started Mental Math and Reflexes Math Message Have students rename fractions as decimals and as percents. Suggestions: Sort the Fraction Cards to find the cards 3 1 _ 1 _ equivalent to 0, _ 4 , 2 , 4 , and 1. Use your slate to record the number of cards in each group. Be ready to justify your choices. 3 0.3; 30% _ 10 4 _ 0.50; 50% 8 3 _ 0.75; 75% 4 1 0.3 _ ⎯; 33.3 ⎯% 3 2 _ 0.40; 40% 5 1 _ 0.125; 12.5% 8 3 0.375; 37.5% _ 8 2 _ 0.6 ⎯; 66.6 ⎯% 3 4 _ 1.00; 100% 4 Study Link 7 9 Follow-Up Have partners compare answers and resolve differences. 1 Teaching the Lesson WHOLE-CLASS ACTIVITY ▶ Math Message Follow-Up (Math Journal 2, Activity Sheets 5–7) Tell students that the number line will be used to create a line plot displaying the results of the equivalent-fraction card sorting activity. Sorting Fraction Cards Number of Cards Draw a number line on the Class Data Pad, and label the 3 , and 1. Label the line plot Sorting Fraction 1, _ 1, _ benchmarks 0, _ 4 2 4 Cards, the x-axis Fractions, and the y-axis Number of Cards. Place the title, Sorting Fraction Cards, above the number line, allowing enough room for data to be added for a line plot. 0 1 4 1 2 3 4 1 Fractions Ask students to share the results of their card sorting and justify their choices as you record the results on the Class Data Pad. Possible questions to ask include the following: ● 3, _ 2, _ 4, How many cards show a fraction equivalent to 1? eight: _ 2 3 4 5, _ 6, _ 8, _ 10 , _ 16 _ 5 6 8 10 16 ● How do you know these fractions are all equivalent to 1? Sample answers: The front of each of these cards is fully colored. The numerator is equal to the denominator in each fraction. 3, 1 ? seven: _ 1, _ 2, _ How many cards show a fraction equivalent to _ 2 2 4 6 5,_ 6,_ 8 4, _ _ ● 8 10 12 16 ● How many cards show a fraction equivalent to ● How many cards show a fraction equivalent to 0? 4 3? _ 4 4 8 12 3, _ 6, _ 9 three: _ 4 8 12 0, _ 0,_ 0 three: _ 5 10 16 Sorting Fraction Cards Number of Cards ● 1? How do you know these fractions are all equivalent to _ 2 Sample answers: The denominator in each of these fractions is twice the numerator. The number of spaces shaded blue on each of these cards is equal to the number of spaces that are white. 3 1 ? three: _ 1, _ 2, _ How many cards show a fraction equivalent to _ ● X X X X X X X X X X X X X 0 1 4 1 2 X X X 3 4 X X X X X X X X 1 Fractions Lesson 7 10 591-595_EMCS_T_TLG2_G5_U07_L10_576914.indd 591 591 3/21/11 1:20 PM 3, _ 5 , and _ 7 on the line plot. 1, _ Ask volunteers to label the fractions _ 8 8 8 8 Ask students to explain their reasoning for the placement of each fraction as it is recorded. Number of Cards Sorting Fraction Cards X X X 0 X X X X X X X X X X X X X X 1 8 1 4 3 8 1 2 X X X X X X X 5 8 3 4 7 8 X X X X X X X X 1 Have students continue to sort the Fraction Cards to find those that are equivalent to the fractions for eighths that were just added to the number line. Continue discussing students’ findings as you complete the line plot. Ask: ● How many cards did you find for each new fraction we marked 1 is equal to _ 2, on the number line? There are 2 cards for each: _ 8 16 3 is equal to _ 6,_ 5 is equal to _ 10 , and _ 7 is equal to _ 14 . _ 8 16 8 16 8 16 ● How can you use the line plot to find the total number of cards sorted? Count all the Xs in the line plot. Fractions ▶ Using a Line Plot to WHOLE-CLASS ACTIVITY Analyze Data (Student Reference Book, pp.119–121) Refer students to Student Reference Book, pages 119–121 to review data landmarks, if necessary. Number of Cards Sorting Fraction Cards X X X 0 X X X X X X X X X X X X X X 1 8 1 4 3 8 1 2 minimum X X X X X X X 5 8 3 4 7 8 mean median Fractions X X X X X X X X Have students use slates to record information about the data on the Sorting Fraction Cards line plot. Ask students to share their thinking as they analyze the data. As they determine the various landmarks, label them on the line plot on the Class Data Pad. ● Which number on the line plot has the most equivalent fractions? 1 whole, or 1 What do we call the landmark for the value or values that occur with the greatest frequency in a set of data? mode ● What is the maximum value in the set of data? 1 What is the minimum value in the set of data? 0 ● What is the range of data displayed on the line plot? 1 How did you find the range? I subtracted the minimum from the maximum: 1 – 0 = 1. 1 How did you find the median? I started What is the median? _ 1 mode and maximum ● 2 at each end and then worked toward the center until I found the middle number. 592 Unit 7 ● How many cards do we have in our data set? 32 ● If the blue shading on the cards represents blue liquid in a beaker, for how many beakers do we have data? 32 ● What landmark would tell us how much liquid each beaker would have if the liquid were distributed equally among all the beakers? mean ● If you actually had these beakers of liquid, what would you do to distribute the liquid equally among all beakers? Sample answer: I’d pour liquid from the beakers with more liquid into the beakers with less liquid until the liquid levels in all of the beakers were equal. ● How do you find the mean for a set of data? Sample answer: Add all of the data values, and then divide the sum by the number of data values. Exponents and Negative Numbers 591-595_EMCS_T_TLG2_G5_U07_L10_576914.indd 592 3/21/11 1:20 PM Student Page PARTNER ACTIVITY ▶ Finding the Mean Date Time LESSON 7 10 Line Plots and Data in Fractional Units Use the data in the table below to create a line plot. Ask students to work with a partner to find the sum of the data 1 Ask students to share their values using various strategies. 18_ 2 strategy for finding the sum. K I N E S T H E T I C Yuma, Arizona 3 2_ 4 Alamosa, Colorado 1 7_ 8 Las Vegas, Nevada 1 4_ 4 Reno, Nevada 1 7_ 2 Bishop, California 3 5_ 4 Winslow, Arizona 3 7_ 4 Bakersfield, California 3 5_ 4 El Paso, Texas 3 7_ 4 Phoenix, Arizona 1 7_ 8 Winnemucca, Nevada 3 7_ 4 City and State Annual Rainfall (in.) Place an X on the line plot below to represent the annual rainfall in each of the 10 driest cities in the United States. Label the line plot with the mixed numbers for the data. 1. Rainfall in the Driest U.S. Cities Students who need a visual model to find the sum of the fractions may benefit from using the Fraction Cards. Have students match cards that together add to a sum 7 1 _ of 1 whole. For example, _ 8 and 8 add to 1 whole. Using 1 this method, all cards except for one of the _ 2 cards has a “match” to make 1 whole. Annual Rainfall (in.) Source: National Climatic Data Center Adjusting the Activity A U D I T O R Y Rainfall in the Driest U.S. Cities City and State T A C T I L E + Number of Cities X 2 34 3 2 X 4 14 4 X X X X XX 7 71 8 7 18 2 7 34 X X 5 34 6 5 Annual Rainfall (in.) Now analyze the data from the table and line plot to answer the questions about the annual rainfall in the 10 driest U.S. cities. V I S U A L Students then use a calculator to find the mean of the set of data. 1 to a decimal and then use the calculator Students may convert 18_ 2 to divide the sum by 32 to find the mean. Ask students to round the quotient in the calculator display to the nearest hundredth. 18.5 / 32 = 0.578125, or about 0.58 Ask: ● Between which two fractions on your line plot does 0.58 fall? 5 1 and _ Between _ 2 8 ● Which fraction on your line plot is 0.58 closest to? How do 5 because _ 5 = you know? Sample answer: 0.58 is closest to _ 8 8 0.625, and 0.625 - 0.58 = 0.045. This is less than the distance between 0.58 and 0.5, which is 0.08. 2_34 2. What is the minimum amount of annual rainfall? 3. What is the range of annual rainfall? 4. What is the mode of annual rainfall? 5. What is the median annual rainfall? 6. What is the combined annual rainfall for the 10 driest cities? 7. What is the mean annual rainfall for the 10 driest cities? 5 7_34 7_18 in. in. in. in. 63_21 6.35 in. in. Math Journal 2, p. 242 209-247_EMCS_S_MJ2_G5_U07_576434.indd 242 3/18/11 5:26 PM Students then use estimation to determine an approximate placement on the line plot for the mean of the set of data. Adjusting the Activity Students who need a visual model to estimate the placement of the mean on the line plot may benefit from referring to the Probability Meter Poster. After determining the approximate location of the decimal on the decimal side of the probability meter, students look on the opposite side of the meter to determine the fraction that is closest to the one found on the line plot. Students should notice that 0.58 is greater than 0.5 and less than 0.625, and that it is closer to 0.625. Student Page Date Time LESSON 7 10 Line Plots and Data in Fractional Units Rainfall Data Day 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 cont. The table below shows rainfall for 10 days in one of the driest cities. Organize the data in a line plot. Then analyze the data to answer questions about the plotted rainfall. Rainfall (in.) 1 2 3 _ 1 _ 8 2 3 4 0 0 5 6 7 1 _ 5 _ 3 _ 8 8 4 8 9 10 0 0 0 3 _ 5 _ 6 1 _ 2 _ 4 _ 7 _ _ 8. Label the approximate locations on the line plot for 8 , 8 , 8 , 8 , 8 , 8 , and 8 . 9. Plot the rainfall data from the table. Rainfall Data ▶ Displaying and Analyzing PARTNER ACTIVITY Data on a Line Plot Number of Days (Math Journal 2, pp. 242 and 242A) X X X X X X 1 8 0 2 8 X X X X 3 8 4 8 5 8 6 8 7 8 1 Rainfall (in.) Students work with a partner. Have students solve the problems on journal pages 242 and 242A involving plotting data points on a line plot with fractional units and analyzing the data with landmarks. Circulate and assist. 10. Identify the following landmarks for the rainfall data: 0 a. mode: b. range: in. c. median: d. mean (as a decimal, rounded to the nearest hundredth inch): 6 3 _ _ 8 or 4 in. 1 _ 16 in. 0.24 in. Math Journal 2, p. 242A 209-247_EMCS_S_MJ2_G5_U07_576434.indd 242A 3/18/11 5:26 PM Lesson 7 10 591-595_EMCS_T_TLG2_G5_U07_L10_576914.indd 593 593 3/21/11 1:20 PM Student Page Date Time LESSON 7 10 1. 2. 2 º $7,550 is shared equally among 25 people. a. Write an open number sentence to show how much money each person gets. + 7,550 ÷ 25 = m b. c. 3. 2 Ongoing Learning & Practice Operations with Multidigit Whole Numbers and Decimals Use the standard algorithm to complete the multiplication problem shown at right. 0 7 5 4 828 1 0,3 5 0 1 1,1 7 8 4 º 207 50 º 207 ▶ Operations with Multidigit Whole Numbers and Decimals Will a reasonable estimate for the quotient be in the tens, hundreds, or thousands? In the hundreds Each person gets $ 302 INDEPENDENT ACTIVITY (Math Journal 2, p. 242B) . In a store, Denny picks out jeans for $34.98 and a shirt for $16.49. He has $50 with him. a. Does he have enough money to buy the jeans and shirt? b. If not, how much more money does he need? 4. a. no $1.47 Students practice solving problems involving operations with multidigit whole numbers and decimals. The correct digits for the product are given. Write the decimal point in the product without actually multiplying. . 0.98 ∗ 39.39 = 3 8 6 0 2 2 b. Explain how you decided where to place the decimal point. Sample answer: 0.98 is almost 1, so I knew that the product would be close to 39. 5. a. ▶ Math Boxes 7 10 Sample answer: The shading shows 13 groups of 0.05 in 0.65. b. INDEPENDENT ACTIVITY Explain how you could use the shading in the grid to find the quotient 0.65 ÷ 0.05. (Math Journal 2, p. 243) What whole number division problem has the same quotient? Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 7-7. The skills in Problem 6 previews Unit 8. Sample answer: 65 ÷ 5 Math Journal 2, p. 242B 209-247_EMCS_S_MJ2_G5_U07_576434.indd 242B 3/28/11 3:24 PM Math Boxes Problem 3 Ongoing Assessment Recognizing Student Achievement Use Math Boxes, Problem 3 to assess students’ ability to add fractions with unlike denominators and to use benchmarks to compare fractions. Students are making 3 1 _ adequate progress if they can compare the fraction _ 8 to 2 and find the sum of the 1 two fractions. Some students may be able to compare the fraction to _ 4. [Number and Numeration Goal 6 and Operations and Computation Goal 4] Student Page Date Time LESSON Math Boxes 7 10 1. Write the following number in standard notation. 2. Nine billion, one hundred two million, three thousand, sixty-two Fill in the blanks. 8 a. (8 ∗ 7) + (8 ∗ 4) = b. 23 ∗ ( – )= (23 ∗ 3.2) – (23 ∗ 2) 9,102,003,062 ∗ (7 + 4) 3.2 2 c. 6 d. (9 ∗ 9 ∗ (12 + 21) = (6 ∗ 12) + (6 ∗ 21) 11 ) – (9 ∗ 3 )= ∗ (11 – 3) 5 3. 3 Shade _ 8 of the fraction stick. a. b. 219 4. 3 _ 5 1 _ _ a. 4 , 4 , 4 , less Is this more or less than _? more 1 Is this more or less than _ 2? 3 _ c. 8 7 _ 1 +_ 2 = Write the next two numbers in each pattern. 1 4 8 4 6, 12, 24, c. –16, –13, –10, A large jar of drink mix holds 1.75 kg. How much drink mix will 12 jars of the same size hold? 21 kg , –7 56 _ 5 66 67 5. 4 , 48 b. 28 7 _ 14 _ _ d. 5 , 5 , 5 , 9 _ 7 _ 96 –4 , 112 _ , 5 230 6. 36 stamps per package. How many in . . . 27 stamps 30 stamps _ of a package? 8 stamps _ of a package? 21 stamps _ of a package? 24 stamps 3 _ a. 4 of a package? 5 _ b. 6 of a package? 2 c. 9 7 d. 12 2 e. 3 38–40 75 Math Journal 2, p. 243 209-247_EMCS_S_MJ2_G5_U07_576434.indd 243 594 Unit 7 3/28/11 3:24 PM Exponents and Negative Numbers 591-595_EMCS_T_TLG2_G5_U07_L10_576914.indd 594 3/28/11 3:33 PM Study Link Master INDEPENDENT ACTIVITY ▶ Study Link 7 10 Name STUDY LINK 7 10 (Math Masters, p. 212) Date Scouting for Insects Some insects are harmful to farmers’ crops. The insects listed in this table are harmful to alfalfa crops. Farmers scout their crops and randomly collect sample plants. They regularly observe the plants and gather data to determine if the insect population is growing or remaining stable. Home Connection Students practice making line plots with fractional increments using insect data. Time Type of Insect Average Length of Insect (in.) Number of Insects _3 8 4 Alfalfa Weevil Larvae 1 1_4 Fall Armyworm 3 4 _1 8 _1 8 Pea Aphid The table shows the result of one farmer’s insect scouting. 1 _1 Meadow Spittlebug Potato Leafhopper 6 6 Plot and label the lengths of the insects from the scouting sample on the line plot below. Use the line plot to find the following data landmarks: _1 3 Differentiation Options 8 1. Minimum insect length 3. Range of insect lengths 5. Mean insect length _1 in. 1_1 8 4 in. 2. Maximum insect length 4. Median insect length 1_14 _1 8 in. in. in. Insect Lengths SMALL-GROUP ACTIVITY READINESS Number of Insects 5–15 Min ▶ Identifying Fractions on a Number Line (Math Masters, p. 211) 0 X X X X X X X X X X X X X X X X X X X 1 8 1 4 3 8 X 1 2 3 4 1 12 1 14 1 Average Length (in.) Students label fractions on number lines. Math Masters, p. 212 SMALL-GROUP ACTIVITY EXTRA PRACTICE 187-220_EMCS_B_MM_G5_U07_576973.indd 212 2/25/11 9:20 AM 5–15 Min ▶ Plotting Rain Gauge Data (Math Masters, p. 213) Students use rain gauge data to practice creating a line plot with fractional increments and then answer questions about the landmarks of the data set. Teaching Master Name Date LESSON Teaching Master Time Name Identifying Fractions on a Number Line 7 10 LESSON 7 10 Date Time Plotting Rain Gauge Data Label each number line with the correct fractions. in. 3 1. 0 1 3 1 2 3 X 2 0 1 2 Rainfall (in.) 1 3 2. 0 1 5 2 5 3 5 1 4 5 Example: Read the rain gauge pictured above and place an X on the line plot to show how much rain has been collected. 1. 3. 0 4. Draw and label the following points on this number line. The first one is done for you. 2 A: 1_4 1 B: _4 B C 0 5. Read each rain gauge to find out how much rain fell each day. Write the amount of rain on the line below the gauge. 1 1 2 3 4 5 6 7 8 8 8 8 8 8 8 1 4 1 4 4 A 1 24 4 C: _4 8 D: _4 8 4 1 12 5 2 in. 3 Wed in. 3 3 Thu Fri Sat in. in. in. 3 3 3 2 2 2 2 2 2 2 1 1 1 1 1 1 1 2 4 2. Jay Kay Mae 3 4 Tue in. 3 _1 2 343 3 Jay, Kay, and Mae are in a 3-mile walk-a-thon. So far, Jay has walked _4 mi, 5 1 Kay has walked 1_2 mi, and Mae has walked _2 mi. Draw and label points on the number line below to show their approximate locations. 0 Mon in. E: 2_34 D E 2 Sun 3. 3 _1 in. Use the number line to make a line plot recording daily rainfall for 1 week. Label the tick marks on the number line. Place an X to represent each rain amount. 4 in. 1_12 in. 0 _1 in. 4 _1 in. 4 _1 in. 2 in. Daily Rainfall X X X X X X Number of Days 0 1 4 1 2 X 3 4 1 1 14 1 12 1 34 2 Rainfall (in.) Analyze the rain gauge data to answer the questions below. _1 a. What is the mode amount of rain that fell each day? 4 in. _1 2 b. What is the range of the rainfall during the week? in. 1 Math Masters, p. 211 187-220_EMCS_B_MM_G5_U07_576973.indd 211 Math Masters, p. 213 2/25/11 9:20 AM 187-220_EMCS_B_MM_G5_U07_576973.indd 213 3/16/11 2:50 PM Lesson 7 10 591-595_EMCS_T_TLG2_G5_U07_L10_576914.indd 595 595 3/21/11 1:20 PM Name Date LESSON 7 10 Time Identifying Fractions on a Number Line Label each number line with the correct fractions. 1. 0 1 0 1 0 1 2. 3. 4. Draw and label the following points on this number line. The first one is done for you. 2 A: 1_4 1 B: _4 4 C: _4 8 D: _4 E: 2_34 Copyright © Wright Group/McGraw-Hill A 0 5. 1 1 24 2 3 4 3 Jay, Kay, and Mae are in a 3-mile walk-a-thon. So far, Jay has walked _4 mi, 5 1 Kay has walked 1_2 mi, and Mae has walked _2 mi. Draw and label points on the number line below to show their approximate locations. 0 3 211 187-220_EMCS_B_MM_G5_U07_576973.indd 211 2/25/11 9:20 AM Name STUDY LINK 7 10 Date Time Scouting for Insects Some insects are harmful to farmers’ crops. The insects listed in this table are harmful to alfalfa crops. Farmers scout their crops and randomly collect sample plants. They regularly observe the plants and gather data to determine if the insect population is growing or remaining stable. Type of Insect Average Length of Insect (in.) Number of Insects _3 8 4 1 1_4 1 Alfalfa Weevil Larvae Fall Armyworm _1 Meadow Spittlebug 4 _1 8 _1 8 Pea Aphid The table shows the result of one farmer’s insect scouting. Potato Leafhopper 3 6 6 Plot and label the lengths of the insects from the scouting sample on the line plot below. Use the line plot to find the following data landmarks: 1. Minimum insect length 3. Range of insect lengths 5. Mean insect length in. in. 2. Maximum insect length 4. Median insect length in. in. in. Insect Lengths Copyright © Wright Group/McGraw-Hill Number of Insects 0 1 2 3 4 1 1 12 Average Length (in.) 212 187-220_EMCS_B_MM_G5_U07_576973.indd 212 2/25/11 9:20 AM Name Date LESSON 7 10 Time Plotting Rain Gauge Data in. 3 2 0 1 2 Rainfall (in.) 1 3 Example: Read the rain gauge pictured above and place an X on the line plot to show how much rain has been collected. 1. Read each rain gauge to find out how much rain fell each day. Write the amount of rain on the line below the gauge. Sun Mon Tue in. in. in. 3 Copyright © Wright Group/McGraw-Hill 3. in. 3 3 Thu Fri Sat in. in. in. 3 3 3 2 2 2 2 2 2 2 1 1 1 1 1 1 1 in. 2. 3 Wed Use the number line to make a line plot recording daily rainfall for 1 week. Label the tick marks on the number line. Place an X to represent each rain amount. in. in. in. in. in. in. Daily Rainfall Number of Days 0 1 2 Rainfall (in.) Analyze the rain gauge data to answer the questions below. a. What is the mode amount of rain that fell each day? b. What is the range of the rainfall during the week? in. in. 213 187-220_EMCS_B_MM_G5_U07_576973.indd 213 3/16/11 2:50 PM