Addition of Whole Numbers and Decimals

Transcription

1 Addition of Whole Numbers and Decimals Objectives To review place-value concepts and the use of the partial-sums and column-addition methods. 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 Write numbers in expanded notation. [Number and Numeration Goal 1] Use paper-and-pencil algorithms for multidigit addition problems. [Operations and Computation Goal 1] Make magnitude estimates for addition. [Operations and Computation Goal 6] Key Activities Students review place-value concepts and write numbers in expanded notation. They review addition of whole numbers and decimals with the partial-sums and columnaddition methods. Ongoing Assessment: Recognizing Student Achievement Use journal page 33. [Operations and Computation Goal 1] Playing Addition Top-It (Decimal Version) Student Reference Book, p. 333 per partnership: 4 each of the number cards 1 (from the Everything Math Deck, if available); 2 counters Students practice place-value concepts, use addition methods, and compare numbers. Math Boxes 2 2 Math Journal 1, p. 34 Students practice and maintain skills through Math Box problems. Study Link 2 2 Math Masters, p. 36 Students practice and maintain skills through Study Link activities. READINESS Building Numbers with Base- Blocks Math Masters, p. 37 base - blocks Students use base - blocks to explore the partial-sums method of addition. ENRICHMENT Using Place Value to Solve Addition Problems Math Masters, p. 38 Students apply place-value and addition concepts to solve problems. ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. 142 Students define and illustrate the term expanded notation. Key Vocabulary place value digit algorithm partialsums method place value expanded notation column-addition method Materials Math Journal 1, pp. 32 and 33 Student Reference Book, pp. 13, 14, 28 30, and 35 Study Link 2 1 Math Masters, p. 415 slate Advance Preparation Plan to spend two days on this lesson. Distribute copies of the computation grid on Math Masters, page 415 for students to use as they do addition problems. Make and display a poster showing expanded notation for a whole number and a decimal. Teacher s Reference Manual, Grades 4 6 pp Lesson

2 Getting Started Mathematical Practices SMP1, SMP2, SMP3, SMP6, SMP7 Content Standards 5.NBT.1, 5.NBT.3a, 5.NBT.3b, 5.NBT.7 Mental Math and Reflexes Read the numbers orally and have students write them in expanded notation on their slates. Remind students that expanded notation expresses a number as the sum of the values of each digit. For example, 906 is equivalent to 9 hundreds + 0 tens + 6 ones, and is equivalent to 7 tenths + 9 hundredths + 6 thousandths. In expanded notation, 906 is written as , and may be written as or as 7 ( 1_ ) + 9 ( 1_ 0 ) + 6 ( 1_ 1,000 ). Encourage students to write the decimal numbers in fraction notation. Sample answers are given ( 1_ ) + 5 ( 1_ 0 ) ( 1_ ) + 2 ( 1_ 0 ) ( 1_ ) + 4 ( 1_ 0 ) + 1 ( 1_ 1,000 ) ( 1_ ) + 6 ( 1_ 0 ) + 7 ( 1_ 1,000 ) ( 1_ ) + 9 ( 1_ 1,000 ) ( 1_ ) + 8 ( 1_ 1,000 ) ( 1_ 0 ) + 4 ( 1_ 1,000 ) 7,904 7, Math Message Use the information on Student Reference Book, pages to solve the Check Your Understanding Problems on the bottom of page 30. Study Link 2 1 Follow-Up Have partners discuss their strategies and identify one thing that they did the same and one thing that they did differently. Have volunteers share their findings. On Day 1 of this lesson, students should complete the Mental Math and Reflexes and the Math Message. They should review and discuss the partial-sums addition method. On Day 2 of this lesson, do the Study Link Follow-Up. Then review and discuss the column-addition method. Finally, have students complete the Part 2 activities. 1 Teaching the Lesson Math Message Follow-Up (Student Reference Book, pp ) WHOLE-CLASS Ask students to use the information they read in the Student Reference Book to think of one true statement they could make about the base-ten number system. Decimals and Percents You use facts about the place-value chart each time you make trades using base- blocks. Suppose that a flat is worth 1. Then a long is worth 1_, or 0.1; and a cube is worth 1_, or For this example: You can trade one long for ten cubes because one 1_ equals ten 1_ 0 s. A flat is worth 1. A long is worth 1_ or 0.1. You can trade ten longs for one flat because ten 1_ s equals one 1. You can trade ten cubes for one long because ten 1_ s equals one 1_ 0. Left to Right in the Place-Value Chart Study the place-value chart below. Look at the numbers that name the places. As you move from left to right along the chart, each number is 1_ as large as the number to its left. one 0 = 1_ of 1,000 one = 1_ of 0 one 1 = 1_ of 1. What is the value of the digit 2 in each of these numbers? a. 20,006.8 b c A cube is worth 1_ 0, or one 1_ = 1_ of 1 one 1_ 0 = 1_ of 1_ one 1_ 1,000 = 1_ of 1_ 0 2. Using the digits 9, 3, and 5, what is a. the smallest decimal that you can write? b. the largest decimal less than 1 that you can write? c. the decimal closest to 0.5 that you can write? Check your answers on page 434. Student Reference Book, p. 30 Adjusting the Activity Refer students to the place-value chart on page 30 of the Student Reference Book. Ask them to look over the headings on the chart and describe any patterns they see. The numbers decrease in size from left to right; the columns on the right side of the chart have a decimal point and the left side does not; the 0s increase by one for each column as you move outward from the center in either direction. 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 Survey the class and use their responses to discuss the following: Each place has a value that is times the value of the place to its right. For example, 1,000 is times as much as 0; 0 is times as much as ; is times as much as 1; 1 is times as much as 0.1; and 0.1 is times as much as Each place has a value that is one-tenth the value of the place to its left. For example, 0 is 1_ of 1,000; is 1_ of 0; 1 is 1_ of ; 0.1 is 1_ of 1; and 0.01 is 1_ of 0.1. Ask students how these relationships guide them in writing the decimals in Problem 2 on Student Reference Book, page 30. Sample answers: Place the largest digits rightmost when forming the smallest decimal; place the largest digits leftmost when forming the largest decimal; place the 5 in the tenths place and the other 86 Unit 2 Estimation and Computation

3 two digits so the larger is to the right to form the decimal that is closest to 0.5. Explain that knowing these relationships also helps with comparing and ordering numbers by their relative sizes. After students have checked their answers to Problem 2, have them write the decimals using number names. Ask students to listen closely as you read the numbers from Problem 1 on Student Reference Book, page 30. Tell them that you will include some mistakes. Read the numbers as 200,068; 0.2; and, For each number, ask students to tell a partner what mistake was made. Then ask volunteers to describe the mistake, read the number correctly, and write it using number names. 200,068 no decimal point; 0.2 decimal point in the wrong position; reverses the tenths and hundredths. Reviewing Algorithms: Partial-Sums Method (Math Journal 1, p. 32; Student Reference Book, pp. 13, 29, and 35; Math Masters, p. 415) WHOLE-CLASS DISCUSSION Most fifth-grade students have mastered an algorithm of their choice for addition. If they are comfortable with that algorithm, there is no reason for them to change it. However, all students are expected to know the partial-sums method for addition. This method helps students develop their understanding of place value and addition. In the partial-sums method, addition is performed from left to right, column by column. The sum for each column is recorded on a separate line. The partial sums are added either at each step or at the end. Ask students to read Student Reference Book, page 29 and then write the numbers 348 and 177 in expanded notation = 348 and = 177 Provide additional examples for students to write in expanded notation if needed. Then refer to page 13 in the Student Reference Book and demonstrate adding using the partial-sums method. Ask students to describe any relationships they see between the expanded notation and the partial-sums method. Sample answer: Both methods use the value of the digits. Ask students to write the numbers 4.56 and 7.9 in expanded notation = 4.56 and = 7.9 Provide additional decimal examples for students to write in expanded notation if needed. Then refer to page 35 in the Student Reference Book and demonstrate adding using the partial-sums method. Ask: What are the similarities and differences between expanded notation and the partial-sums method with whole numbers and with decimals? Sample answers: Both methods for whole numbers and decimals use the value of the digits. With decimals, you have to line up the places correctly, either by affixing 0s to the end of the numbers, or by aligning the digits in the ones place. Have students independently solve the problems on journal page 32 and then check each other s answers. Algorithm Project The focus of this lesson is the partial-sums and column-addition methods for adding whole numbers and decimals. To teach U.S. traditional addition with whole numbers and with decimals, see Algorithm Project 1 on page A1 and Algorithm Project 2 on page A6. Expanded Notation Number Expanded Form Expanded form may also be written with fractions instead of decimals. For example: = ( 1_ ) + 5 ( 1_ 0 ). NOTE Display a poster showing the number in standard form, expanded form, and using number names to provide students with readily accessible examples. Language Arts Link The word algorithm is used to name a step-bystep procedure for solving a mathematical problem. The word is derived from the name of a ninth-century Muslim mathematician, Al-Khowarizimi. Encourage students to research the etymology of other mathematical terms. Lesson

4 Date Time Adding with Partial Sums Write the following numbers in expanded notation : Write an estimate for each problem. Then use the partial-sums method to find the exact answer. Example: Estimate: Estimate: Estimate: Estimate: Estimate: Math Journal 1, p. 32 Date Time Methods for Addition NOTE Remind students always to say aloud, or to themselves, the numbers that they are adding when they use the partial-sums method. For example: , not 5 + 2; , not Solve Problems 1 5 using the partial-sums method. Solve the rest of the problems using any method you choose. Show your work in the space on the right. Compare answers with your partner. If there are differences, work together to find the correct solution , ,312 3, ,416 2, , , , Alana is in charge of the class pets. She spent $ on hamster food, $ on rabbit food, $ 2.01 on turtle food, and $ 7.51 on snake food. How much did she spend on pet food? Estimate: About $90 Solution: $91.06 Reviewing Algorithms: Column-Addition Method (Student Reference Book, pp. 13 and 35; Math Masters, p. 415) SMALL-GROUP The column-addition method is similar to the traditional algorithm most adults know. It can become an alternate method for students who are still struggling with addition. Demonstrate the method using examples like those on pages 13 and 35 of the Student Reference Book. In this method, each column of numbers is added separately, and in any order. If adding results in a single digit in each column, the sum has been found. If the sum in any column is a 2-digit number, it is renamed and part of it is added to the sum in the column on its left. This adjustment serves the same purpose as carrying in the traditional algorithm. Ask students to compare the examples of column addition on pages 13 and 35 of the Student Reference Book. Assign each small group one of the following problems to solve using the column addition method. Encourage students to use concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Have students write about their method and explain the reasoning they used to solve the assigned problem. Volunteers share the group s solution using the board or a transparency , ,481, Adding Whole Numbers and Decimals (Math Journal 1, p. 33; Student Reference Book, pp. 13, 14, and 35; Math Masters, p. 415) Encourage students to estimate and solve the problems independently and then check each other s work. Using the computation grid helps students line up digits and/or decimal points. Encourage students to try the methods described on pages 13, 14, and 35 of the Student Reference Book. Ongoing Assessment: Recognizing Student Achievement Journal Page 33 Problems 1 and 2 Use journal page 33 to assess students ability to solve multidigit addition problems. Students are making adequate progress if they correctly use the partial-sums method to solve Problems 1 and 2. [Operations and Computation Goal 1] Math Journal 1, p Unit 2 Estimation and Computation

5 Sharing Results (Math Journal 1, pp. 32 and 33) Bring the class together to share solutions. Some possible discussion questions include the following: WHOLE-CLASS DISCUSSION What are some of the advantages or disadvantages of different methods for addition? When might a particular method be useful? When might it not be useful? How did students use their estimates on journal page 32? Did they make estimates for any subsequent problems? Ask students to explain the reasoning they used to solve some of the problems on Math Journal 1, page Ongoing Learning & Practice Playing Addition Top-It (Decimal Version) (Student Reference Book, p. 333; Math Masters, p. 493) Addition Top-It (Decimal Version) provides practice adding decimals, comparing numbers, and understanding place-value concepts. Direct students to play a decimal version of Addition Top-It, Student Reference Book, page 333. Use this variation: Each player draws 4 cards and forms 2 numbers that each has a whole-number portion and a decimal portion. Players should consider how to form their numbers to make the largest sum possible. Use counters or pennies to represent the decimal point. Each player finds the sum of the 2 numbers and then writes the sum in expanded form. Each player records his or her sum on Math Masters, page 493 to form a number sentence using >, <, or =. The player with the largest sum takes all of the cards. Math Boxes 2 2 (Math Journal 1, p. 34) INDEPENDENT Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 2-4. The skill in Problem 6 previews Unit 3 content. Writing/Reasoning Have students write a response for the following: Leroy rounded to 60. Rosina said that he was incorrect. Do you agree or disagree with Rosina? Sample answer: It depends on whether Leroy intended to round to the nearest or the nearest whole number; rounded to the nearest is 60; rounded to the nearest whole number is 56. Date Name Date Time STUDY LINK Number Hunt Reminder: A means Do not use a calculator. Use the numbers in the following table to answer the questions below. You may not use a number more than once. 1. Circle two numbers whose sum is Make an X in the boxes containing three numbers whose sum is Make a check mark in the boxes containing two prime numbers whose sum is Make a star in the boxes containing two numbers whose sum is Make a triangle in the boxes containing two numbers whose sum is 5.7. Explain how you found the answer. Solve Problems 6 9 using any method you want. Show your work in the space below. 6. 3, , R2 Sample answers: X X X30 Sample answer: Since the sum has a 7 in the tenths place, look for numbers with tenths that add to 7: ; and Practice Math Boxes 1. Round to the nearest tenth. a b c d e The temperature at midnight was 25 F. The wind chill temperature was 14 F. How much warmer was the actual temperature than the wind chill temperature? 11 F 5. Complete. a. A person 74 in. tall is ft in. b. A person who runs a mile runs 5,280 ft. c. A person who runs 1,760 yd runs 5,280 ft. d. A person who grew 1 2 ft over the summer grew in Study Link Master Math Masters, p Math Journal 1, p. 34 Time 2. Multiply. a b c ,200 d e f , Complete. a b c d. 1,256 1,000 6 e. 6,704 6, Match. a. Straight angle b. Obtuse angle c. Right angle d. Acute angle NOTE Remind students of the benefits of making estimates prior to solving problems. Estimation as an ongoing practice helps students to become flexible with mental computation and to check their answers for reasonableness <90 > Lesson

6 Name Date Time Example: Teaching Master Modeling with Base- Blocks Study Link 2 2 (Math Masters, p. 36) INDEPENDENT Copyright Wright Group/McGraw-Hill 0s s 1s Add 0s Add s 7 0 Add 1s Work with a partner. Choose a problem below. Use the base- blocks to model the problem. Have your partner solve the problem and record the answer using the partial-sums method. Compare your model with your partner's solution. Reverse roles and continue until all problems are solved Add 0s 400 Add s 0 Add 1s Add 0s 800 Add s 120 Add 1s s s 1s Add 0s 900 Add s 60 Add 1s Add 0s 900 Add s 70 Add 1s Home Connection Students practice finding sums. Students can solve problems using any method they choose. 3 Differentiation Options READINESS Building Numbers with Base- Blocks (Math Masters, p. 37) Min Math Masters, p. 37 To explore place value and expanded notation using concrete models, have students use base- blocks to model numbers. After students complete Math Masters, page 37, discuss the relationship between expanded notation and base- block representations. Have students also share how they used the blocks to add the numbers. ENRICHMENT Using Place Value to Solve Addition Problems (Math Masters, p. 38) Min Name Date Time Teaching Master Place-Value Strategies Use your favorite addition algorithm to solve the first problem in each column. Then use the answer to the first problem in each column to help you solve the remaining problems. 1. 3,058 2,182 5,240 a. 3,058 2,282 b. 3,058 2,082 c. 3,058 2,582 d. 3,058 2, ,401 2,659,060 a. 7,401 2,679 5,340,080 b. 7,401 2,669 5,140,070 c. 7,401 2,689 5,640,090 d. 7,401 2,699 5,239,0 3. Explain the strategy you used to solve the problem sets above. I identified which digit in the second number changed. Then I adjusted the sum of the original problem by that amount. To apply students understanding of place value and addition algorithms, have them use the sum of one problem to help them find the solution of other problems. After students complete Problems 1 and 2 on Math Masters, page 38, discuss their strategies. Ask students to explain how these strategies might be useful when solving addition problems. Sample answer: I could find related addends that are easier to work with than those in the original problem and use what I know about place value to help me solve the problem. ELL SUPPORT SMALL-GROUP Building a Math Word Bank (Differentiation Handbook, p. 142) 5 15 Min To provide language support for number notation, have students use the Word Bank Template found on Differentiation Handbook, page 142. Ask students to write the term expanded notation, draw pictures relating to the term, and write other related words. See the Differentiation Handbook for more information. Math Masters, p Unit 2 Estimation and Computation