Addition and Subtraction of Decimals Objective To add, subtract, and round decimals. 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 Apply place-value concepts to round decimals to a given place. Adapt whole-number algorithms to add and subtract decimals. [Operations and Computation Goal 1] Use extended multiplication and division facts to convert between metric units. [Operations and Computation Goal 2] Estimate sums and differences of decimals. [Operations and Computation Goal 5] Key Activities Students add and subtract decimals. They also round decimals and convert metric units. Ongoing Assessment: Recognizing Student Achievement Use the Math Message. Key Vocabulary precise Materials Math Journal 1, pp. 52 and 5 Student Reference Book, pp. 1 Study Link 2 2 Math Masters, p. 1 (optional) Drawing and Interpreting Histograms Math Journal 1, p. 55 Students practice drawing and comparing histograms. Math Boxes 2 Math Journal 1, p. 5 Students practice and maintain skills through Math Box problems. Study Link 2 Math Masters, p. Students practice and maintain skills through Study Link activities. READINESS Modeling Subtraction of Decimals Math Masters, pp. 8 and 11 1 scissors Students model subtraction of decimals using base-1 grids or pictures. ENRICHMENT Exploring Meaningful Zeros Students explore how decimal numbers are meaningful in expressing precision. EXTRA PRACTICE 5-Minute Math 5-Minute Math, pp. 91 and 18 Students practice rounding decimals. Advance Preparation Consider having some students use a place-value chart from Lesson 2 2 when completing the Math Message. Teacher s Reference Manual, Grades 6 pp. 119 126 Lesson 2 11
Getting Started Mental Math and Reflexes Students write numbers from dictation and record numbers that are.1 more than the dictated number. Suggestions:.9 1. 19.8 19.18 1.2 1.82 82.15 82.25 1,19.9 1,2. 6,99.9 6,1 Mathematical Practices SMP2, SMP, SMP, SMP5, SMP6 Content Standards 6.NS., 6.SP. Math Message List the following decimals in a column so the digits are lined up by place value..5; 51; 19.; 9.892;.29; 5.99;,89.1.5 51. 19. 9.892.29 5.99,89.1 Study Link 2 2 Follow-Up Briefly go over the answers. For Problems 15 19, ask students to explain their strategies for identifying points on the number line. 1 Teaching the Lesson Math Message Follow-Up (Student Reference Book, pp. 1 ) WHOLE-CLASS DISCUSSION Algorithm Project To teach U.S. traditional addition and subtraction of decimals, see Algorithm Projects 1 and 2 on pages A1 and A6. Ask a volunteer to present his or her list of numbers and share the strategy used to align the numbers. Encourage students to share alternate strategies. Discuss the advantage of attaching zeros when aligning decimals by place value. Select two decimals from the list of numbers, for example, 19. and 9.892. Ask students to first add them 119.292 and then subtract them. 99.58 Remind students to estimate before adding or subtracting. Discuss their estimation strategies, preferred algorithms, solutions, and checks for reasonableness. Expect that virtually all students will have mastered one algorithm for addition and one for subtraction. If necessary, encourage students to review the algorithms shown on pages 1 of the Student Reference Book. Ongoing Assessment: Recognizing Student Achievement Math Message Use the Math Message to assess students ability to align the digits of whole and decimal numbers by place value. Students are making adequate progress if they correctly align the numbers containing decimal points. Some students might recognize that 51 is equivalent to 51. and align this number accordingly. 11 Unit 2 Operations with Whole Numbers and Decimals
Renaming or Rounding to the Same Number of Decimal Places (Student Reference Book, p. ) WHOLE-CLASS Discuss situations in which numbers to be added or subtracted do not have the same number of decimal places. Example 1: A ribbon is 8 meters long. If you cut off a piece 1.68 meters long, what length of ribbon remains? One approach is to convert meters to centimeters. 8 m = 8 cm and 1.68 m = 168 cm 8 cm - 168 cm = 62 cm, or 6.2 m 6.2 m of ribbon remain. A second approach is to solve 8-1.68 =? The number 1.68 has two decimal places. Put a decimal point and two zeros to the right of 8 so it also has two decimal places. 8 m = 8. m 8. m - 1.68 m = 6.2 m 6.2 m of ribbon remain. The smaller the unit of measure you use, the more precise the measurement will be. Measurements should be rounded so they all have the same precision that of the less precise measurement. Example 2: Aaron has a physical exam each year, just before school begins. Aaron s height in August was 1.8 meters. His height one year later was 1.2 meters. How much had Aaron grown during the year? In August, Aaron s height was measured to the nearest centimeter (hundredth of a meter). One year later, Aaron s height was measured to the nearest millimeter (thousandth of a meter). Because a centimeter is a larger unit than a millimeter, the measure in centimeters is the less precise measurement. Before finding the difference between the two heights, round the more precise measurement (1.2 m) to match the less precise measurement (1.8 m). Round 1.2 m to 1.2 m 1.2 m - 1.8 m =. m Aaron grew about. m, or cm, during the year. For additional practice in renaming or rounding to equalize the number of decimal places, discuss the examples on page of the Student Reference Book. 1. 2.5 19.6 Estimate 5 2.5 19.6 52.1. 1.89.5 Estimate 1.89.5.9 5..61 Estimate 8.95 1.61 8.95 9.66. 9.2 Estimate 2.6 22 9.2 2.6 21.6 9. 8. Estimate.15 8..15.15 Adding and Subtracting Decimals Estimate each sum or difference. Then solve. Math Journal 1, p. 52 Sample estimates given. 2. 5.6 1.8 Estimate 5.6 1.8.8.. 9.8 Estimate. 9.8 1.5 6. 115.9 Estimate 25.9 19.95 1 9 115.9 25.9 19.95 9.1 8. 5.6 Estimate.92.5 5.6.92.61 1. 1.9 Estimate.8 For Problems 12 and 1, round the more precise measurement to match the less precise measurement. 11. In 1999, Maurice Greene set a new world record for the 1-meter dash, running the race in 9.9 seconds. Until then, the record had been held by Donovan Bailey, who ran the race in 9.8 seconds. How much faster was Greene s time? 12. In the 2 Olympic Games, the United States women s -meter medley relay team won this swimming event with a time of minutes, 58. seconds. In 2, the United States women s team took second with a time of minutes, 59.12 seconds. How much faster was the 2 team than the 2 team? 1. In the 2 Olympic Games, Kamila Skolimowska of Poland won the women s hammer throw with a distance of 1.2 meters. In 2, Olga Kuzenkova of Russia won with a distance of 5.2 meters. a. Which woman threw the greater distance? b. By how much? 1. Find the perimeter of the rectangle in meters or centimeters. 85 cm Perimeter 2.6 m.8 meters 6.9 m, or 69 cm 1.5 1.9.8 1.52 Adding and Subtracting Decimals continued.5 seconds.8 seconds Olga Kuzenkova Math Journal 1, p. 5 Lesson 2 115
2 Drawing and Interpreting Histograms The students in Ms. Baxter s class kept track of the amount of time they spent reading during a month. The table below shows the number of hours, rounded to the nearest.1 hour, each student in her class spent reading. Use the data in the table to create two different histograms with bar widths as indicated below. Number of Students Spent Reading (hours) 1.6 2. 1.5 11..8 11... 1. 12.1 1.6 12. 1. 1.5 2.6 2...6 1..5 1. 1.. 1.1 Graph A 8 6 5 2 1 2 6 8 1 12 1 16 Spent Reading (hours) (Bar width = 2 hours) Use the histograms to answer the following questions. Number of Students Graph B 8 6 5 2 1 8 12 16 Spent Reading (hours) (Bar width = hours) 1. How are Graphs A and B alike? Sample answer: Both graphs are histograms. They both display the reading data for Ms. Baxter s class. 2. How are Graphs A and B different? Sample answer: The intervals on the horizontal axes are different. This makes the distribution of the data look different.. Suppose Ms. Baxter wanted to give the school principal information about her students reading habits. Which graph do you think she should use? Explain your answer. Sample answer: Ms. Baxter should use Graph A because it gives more detailed information about the class. The bar widths on Graph A are smaller and show the gap in the middle of the data. Graph B does not show this information. Math Journal 1, p. 55 Adding and Subtracting Decimals (Math Journal 1, pp. 52 and 5; Math Masters, p. 1) Adjusting the Activity PROBLEM SOLVING Assign the problems on journal pages 52 and 5. Circulate and assist as needed. Reserve some time for students to discuss their solutions. Problems 12 1 include numbers that do not have the same units. Remind students that for addition or subtraction of measurements all the numbers must have the same unit. Have students use a computation grid (Math Masters, p. 1) when adding and subtracting decimals. Use money to model those problems dealing with whole numbers, tenths, and hundredths. 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 2 Ongoing Learning & Practice NOTE You can use the fix function of some calculators to round numbers to ten-millionths. See the Student Reference Book, pages 282 28. Drawing and Interpreting Histograms (Math Journal 1, p. 55) PROBLEM SOLVING Students draw histograms of a data set. They compare two histograms that display the same data with different bar widths. 2 Math Boxes Math Boxes 2 (Math Journal 1, p. 5) 1. Write each number in expanded form. a. 6.52 ( 1) + ( 1) + (6 1) + (5.1) + (2.1) b..9 ( 1) + (.1) + (9.1). Make a double-stem plot for the data below. Inauguration Age of U.S. Presidents Since 1861 52 56 6 5 9 5 55 55 5 2 51 56 55 51 5 51 6 62 55 56 61 52 69 62 6 5 Use your double-stem plot to find the following landmarks. 2. Convert the numbers given in numberand-word notation to standard notation. a. The least distance that Pluto is from Earth is about 2. billion miles. 2,,,miles b. Earth is about 15 million kilometers from the Sun. 15,, kilometers Stems Leaves (1s) (1s) 2 6 6 9 5 1 1 1 2 2 5 5 5 5 5 6 6 6 6 1 2 2 6 9 a. range 2 b. median 5 c. mode(s) 5 and 55 15 16 Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 2-1. The skills in Problems and 5 preview Unit content. Study Link 2 (Math Masters, p. ) Home Connection Students practice estimating and finding sums and differences of decimals.. Complete the What s My Rule? table. Rule: Multiply by 1. in out 8 8. 9.2 92 2.6 26 5. Solve. Alma is 1_ the age of her father. Alma s father is 6 years old. a. How old is Alma? 15 years old b. Alma s brother Luke is 1_ 5 his father s age. How old is Luke? 12 years old 25 2 Math Journal 1, p. 5 116 Unit 2 Operations with Whole Numbers and Decimals
Differentiation Options READINESS Modeling Subtraction of Decimals (Math Masters, pp. 8, 11 1) PARTNER Students use modeling to gain experience with subtraction of decimal numbers. Provide each pair with one copy of Math Masters, page 8, at least two copies each of Math Masters, pages 11 1, and scissors. Students should think of the flat as the ONE, the longs as tenths, the units as hundredths, and the fractional parts of the units as thousandths. They can then use the cut-apart grids when completing journal pages in Part 1 or the Study Link in Part 2 of the lesson. ENRICHMENT SMALL-GROUP Exploring Meaningful Zeros Many tools, such as Allen wrenches, are sized as 2. mm,. mm, and. mm. Most students would drop the zeros and report these sizes as 2 mm, mm, and mm. However, a measurement of. mm is not necessarily the same as mm. The zero(s) at the end of decimal numbers such as. and. are meaningful in expressing precision. To extend their knowledge of place-value concepts, have students identify and compare the range of numbers that can be rounded to. Numbers greater than 2.5 and less than.5; range: 1 to.. Numbers greater than 2.95 and less than.5; range:.1 to.. Numbers greater than 2.995 and less than.5; range:.1 Name STUDY LINK Solve. Sports Records 1. The fastest winning time for the New York Marathon (Tesfay Jifar of Ethiopia, 21) is 2 hours,.2 minutes. The second fastest time is 2 hours, 8.1 minutes (Juma Ikangaa of Tanzania, 1989). How much faster was Jifar s time than Ikangaa s?.29 minutes 2. In the 198 Olympic Games, Erik Lemming of Sweden won the javelin throw with a distance of 5.825 meters. He won again in 1912 with a distance of 6.6 meters. How much longer was his 1912 throw than his 198 throw? 5.815 meters. Driver Buddy Baker (Oldsmobile, 198) holds the record for the fastest winning speed in the Daytona 5 at 1.62 miles per hour. Bill Elliott (Ford, 198) has the second fastest speed at 16.26 miles per hour. How much faster is Baker s speed than Elliott s? 1.9 miles per hour. The highest scoring World Cup Soccer Final was in 195. Teams played 26 games and scored 1 goals for an average of 5.8 goals per game. In 195, teams played 22 games and scored 88 goals for an average of goals per game. What is the difference between the 195 and the 195 average goals per game? 1.8 goals Sample estimates given. 1.6 169.1.62 5. 6.9 12. Estimate 6..12.5 Estimate 6.9 12..12.5 Practice Solve mentally. $.1. $.6 $.29 $.6 $2..85 8.. 1.5 1. 5 1.5 9. 9.225 8.5 5.5 25 $6.5 Study Link Master 1. $.69 $8.1 $6.25 $25 Math Masters, p. Name Teaching Master Modeling Subtraction of Decimals You can model subtraction of decimals using base-1 grids or pictures. For example, to solve 1.2.65, first represent 1.2, adjust by trading, and then subtract. 1 25 EXTRA PRACTICE 5-Minute Math SMALL-GROUP To offer more practice with rounding decimals, see 5-Minute Math, pages 91 and 18. trade trade Subtract.65.592 left Use base-1 grids or pictures to find each difference. Show your work. 1..6 2.9 2. 2..61. 1..8.62 1.29 1.61 Math Masters, p. 8 2 15 6 9 1 2 9 6 2 2 1 9 9 1 6 1 1 2 9 1 6 9 1 8 1 6 1 Lesson 2 11