Addition of Multidigit Numbers

Transcription

1 Addition of Multidigit Numbers Objectives To review the partial-sums algorithm used to solve multidigit addition problems; and to introduce a column-addition method similar to the traditional addition algorithm. 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 Identify places in whole numbers and the values of the digits in those places. [Number and Numeration Goal 1] Apply extended addition facts. [Operations and Computation Goal 1] Use the partial-sums and column-addition algorithms to solve multidigit addition problems; choose an appropriate paper-and-pencil algorithm to solve multidigit addition problems. [Operations and Computation Goal 2] Make ballpark estimates for multidigit addition problems. [Operations and Computation Goal 6] Key Activities Students make ballpark estimates for addition problems. They use the partial-sums and column-addition methods for addition Playing High-Number Toss Student Reference Book, p. 252 Math Masters, p. 487 per partnership: 1 six-sided die Students practice comparing numbers. Math Boxes 2 7 Math Journal 1, p. 44 Students practice and maintain skills through Math Box problems. Study Link 2 7 Math Masters, pp. 57 and 58 Students practice and maintain skills through Study Link activities. READINESS Solving Addition Number Stories Math Masters, pp. 59 and 405 base-10 blocks 3-section paper dinner plates (optional) Students use base-10 blocks to model the partial-sums method for addition. ENRICHMENT Writing Addition Number Stories Students write and solve addition number stories. ELL SUPPORT Building a Math Word Bank Differentiation Handbook, p. 140 Students add the term ballpark estimate to their Math Word Banks. Ongoing Assessment: Recognizing Student Achievement Use journal page 42. [Operations and Computation Goal 2] Key Vocabulary partial-sums method column-addition method ballpark estimate Materials Math Journal 1, pp. 42 and 43 Student Reference Book, pp. 10 and 11 Study Link 2 6 Math Masters, p. 403 or 404 (optional) quarter-sheet of paper base-10 blocks (optional) Advance Preparation Plan to spend a total of two days on this lesson. Place quarter-sheets of paper near the Math Message. If students need computation grids in Part 1, make copies of Math Masters, page 403 or 404. Teacher s Reference Manual, Grades 4 6 pp , Lesson

2 Getting Started Mental Math and Reflexes Pose extended addition-facts problems. Suggestions: = = = 500 2, ,000 = 8, = = 900 3, ,000 = 11,000 70, ,000 = 100, = 1,300 6, ,000 = 11,000 90, ,000 = 170,000 70, ,000 = 120,000 Mathematical Practices SMP1, SMP2, SMP3, SMP5, SMP6, SMP7, SMP8 Content Standards 4.OA.3, 4.NBT.2, 4.MD.2, 4.G.2 Math Message Solve the problems on a quarter-sheet of paper. Show your work Study Link 2 6 Follow-Up Have partners compare answers. Ask students to share the estimated time they spend watching TV each week. Have them compare their estimates to 20 1_ 2 hours, the average viewing time reported by the World Almanac 2004 for children 2 11 years old. 1 Teaching the Lesson Links to the Future In Unit 4 of Fourth Grade Everyday Mathematics students apply these addition algorithms to decimal numbers. Math Message Follow-Up WHOLE-CLASS Have students share their solution strategies. Tell them that in this lesson they will review the partial-sums method and explore the column-addition method. To support English language learners, explain the meaning of the word partial. Some students may have used these algorithms to solve the Math Message problems. ELL Making Ballpark Estimates WHOLE-CLASS Remind students that they should always check their answers to see whether they make sense. This is true for number stories and for computation problems like those in the Math Message. Whether done in advance or as a final check, it is often desirable to make a rough ballpark estimate of the answer. One way to estimate a sum is to change the addends to close-but-easier numbers and then add them. To support English language learners, discuss the mathematical as well as the everyday meanings of the terms ballpark and estimate. Ask students to give ballpark estimates rather than exact answers for sums. (See examples on page 121.) Have them tell how they arrived at their estimates. Encourage students to use terms such as closer to, between, and a little more than to refine their estimates. Note that often more than one estimate is acceptable. ELL 120 Unit 2 Using Numbers and Organizing Data

3 Sample answers: = 120; = 130; = 130; = = 100; = = 210; = = 650; = = 1,250; = 1,300 Algorithm Project The focus of this lesson is the partial-sums and column-addition algorithms for addition. To teach U.S. traditional addition, see Algorithm Project 1 on page A1. Discussing and Practicing the Partial-Sums Method for Addition (Student Reference Book, p. 10; Math Journal 1, p. 42) WHOLE-CLASS The partial-sums method (algorithm) for addition was introduced in Second Grade Everyday Mathematics. Discuss the example of the partial-sums method that appears on page 10 of the Student Reference Book. It involves more steps than some standard algorithms, but it is also more explicit; for this reason, it might be easier to use. Addition is performed from left to right and column by column. The sum of each column is recorded on a separate line. The partial sums can be added following each step or at the end. NOTE Addition by the partial-sums method can be performed from right to left. The advantage of working from left to right is that this is consistent with the approach used in estimating sums. Write several 2-digit and 3-digit addition problems on the board. Have volunteers use and describe the partial-sums method to solve these problems. Remind students that the value of each digit is determined by its place in the numeral. Thus, they should keep in mind what numbers they are adding. For example, in the first problem below, they should think , not ; in the second problem, they should think , not 2 + 1, and , not Addition Using the Partial-Sums Method Add the 10s: Add the 1s: Add the partial sums: NOTE An algorithm is a step-by-step set of instructions for solving a problem. In classroom discussion, simply refer to algorithms as methods. The partial-sums algorithm is an example of what is sometimes called a lowstress algorithm. Such algorithms are not necessarily the most efficient, but they are easy to use, and they reveal important underlying concepts. Adjusting the Activity ELL Have base-10 blocks readily available for students to use while solving the addition problems. AUDITORY KINESTHETIC TACTILE VISUAL Whole Numbers Addition Methods Partial-Sums Method The partial-sums method is used to find sums mentally or with paper and pencil. Here is the partial-sums method for adding 2-digit or 3-digit numbers: 1. Add the 100s. 2. Add the 10s. 3. Add the 1s. 4. Then add the sums you just found (the partial sums). Add using the partial-sums method. Use base-10 blocks to show the partial-sums method. 100s 10s 1s Add the 100s Add the 10s Add the 1s Add the partial sums = 435 Note Larger numbers with 4 or more digits are added the same way. Use base-10 blocks to add Add the 100s: Add the 10s: Add the 1s: Add the partial sums: The total is = = 435 Student Reference Book, p. 10 Lesson

4 Date Time 2 7 Partial-Sums Addition Write a number model for a ballpark estimate. Solve Problems 1 3 using the partialsums method. Solve Problems 4 6 using any method. Compare your answer with your estimate to see if your answer makes sense. Sample estimates: , ,672 3,221 4, Ask students to turn to journal page 42. Have computation grids available (Math Masters, page 403 or 404) for those students who prefer to use them to help keep the digits in the proper columns. Assign Problems 1 7 for students to complete on their own. Have students share their solutions to Problem 7 and indicate thumbs-up if they agree with an answer ,590 1,700 3,200 4, , Try This , Name three 4-digit numbers whose sum is 17,491. Sample answer: 8,000 8,000 1,491 17,491 Math Journal 1, p. 42 Add using the column-addition method. 100s 10s 1s Add the numbers in each column Two digits in the ones place. Trade 15 ones for 1 ten and 5 ones. Move the 1 ten to the tens column Two digits in the tens place. Trade 13 tens for 1 hundred and 3 tens. Move the 1 hundred to the hundreds column = 435 7,854 4,550 12,404 Column-Addition Method The column-addition method can be used to find sums with paper and pencil, but it is not a good method for finding sums mentally. Here is the column-addition method for adding 2-digit or 3-digit numbers. 1. Draw lines to separate the 1s, 10s, and 100s places. 2. Add the numbers in each column. Write each sum in its column. 3. If there are 2 digits in the 1s place, trade 10 ones for 1 ten. 4. If there are 2 digits in the 10s places, trade 10 tens for 1 hundred. Larger numbers with 4 or more digits are added the same way. 7,900 4,600 12,500 Whole Numbers Add , Ongoing Assessment: Recognizing Student Achievement Journal page 42 Problems 4 6 Use journal page 42, Problems 4 6 to assess students ability to solve multidigit addition problems. Students are making adequate progress if they are able to use a paper-and-pencil algorithm to calculate the correct sums. Some students may be able to use more than one method to solve the problems or demonstrate how the relationship between + and - can be used to check their answers. [Operations and Computation Goal 2] Discussing and Practicing the Column-Addition Method (Student Reference Book, p. 11; Math Journal 1, p. 43) WHOLE-CLASS Column addition is another method for adding numbers. It can become a reliable method for students who are still struggling with addition. Column addition is similar to the traditional addition algorithm that most adults know. Discuss the example of the column-addition method that appears on page 11 of the Student Reference Book. In this algorithm, each column of numbers is added separately in any order. If this results in a single digit in each column, the sum has been found. If the sum of any column is a 2-digit number, that column sum is adjusted by trading part of the sum into the column to the left. The concept of trading is equivalent to the idea of carrying in the traditional algorithm. In some cultures, the words used to describe what we call trading translate into making and breaking. For example, when you have ten 1s, you make a 10. When you need more 1s, you break a 10. Write several 2-digit and 3-digit addition problems on the board. Have volunteers use and describe the column-addition method to solve these problems. Before students solve each problem, ask for and record ballpark estimates , Check your answers on page 340. Student Reference Book, p Unit 2 Using Numbers and Organizing Data

5 Ask students to solve Problems 1 3 on journal page 43, using the column-addition method. They can do the remaining problems using any method they choose. Bring small groups of students together to share solutions. Adjusting the Activity Have students use base-10 blocks to model the meaning of trading in the context of addition. 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 ELL Date 2 7 Column Addition Time Write a number model for a ballpark estimate. Solve Problems 1 3 using the columnaddition method. Solve Problems 4 6 using any method. Compare your answer with your estimate to see if your answer makes sense , , ,415 Sample estimates: 3. 2,538 4,179 2,500 4,200 6, ,717 4,614 6,058 10, Ongoing Learning & Practice ,420 4,600 6,100 10,700 Playing High-Number Toss (Student Reference Book, p. 252; Math Masters, p. 487) PARTNER Try This 7. Name four 4-digit numbers whose sum is 15,706. Sample answer: 3,000 4,000 5,000 3,706 15,706 Math Journal 1, p. 43 Students play High-Number Toss to practice comparing numbers. Adjusting the Activity Have students play High-Number Toss in groups of three or more to practice ordering numbers. Have them adjust the scoring accordingly. 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 Boxes 2 7 (Math Journal 1, p. 44) INDEPENDENT Mixed Practice Math Boxes in this lesson are linked with Math Boxes in Lessons 2-5 and 2-9. The skill in Problem 6 previews Unit 3 content. Writing/Reasoning Have students write a response to the following: Shaneel said, I can draw a rhombus, rectangle, square, or kite for Problem 4. Do you agree or disagree Explain your answer. Sample answer: I disagree. A parallelogram has two pairs of parallel sides. A rhombus, rectangle, and square have two pairs of parallel sides, but a kite doesn t have any parallel sides. Date 2 7 Math Boxes 1. A number has 3 in the millions place, 1 in the ones place, 8 in the thousands place, 9 in the ten-thousands place, 0 in the tens place, 6 in the hundred-thousands place, and 5 in the hundreds place. Write the number. 3, 6 9 8, Write,, or to make each number sentence true. a b c d e. 423, , Time 2. Write five names for 100. Sample answers: , Draw a parallelogram. Label the vertices so that side AB is parallel to side CD. Sample answer: D A B C Measure these line segments to the nearest 1 2 centimeter. a. b. About About cm cm 6. Multiply mentally. a b c d e Math Journal 1, p. 44 Lesson

6 Study Link Master Name Date Time STUDY LINK 2 7 Multidigit Addition Make a ballpark estimate. Use the partial-sums method to add. Compare your answer with your estimate to see if your answer makes sense. 10 Study Link 2 7 (Math Masters, pp. 57 and 58) INDEPENDENT Sample estimates: Home Connection Students solve addition problems and show someone at home how to use the methods they used in this lesson. Note that this Study Link consists of two pages students use the partial-sums method on the first page and the column-addition method on the second page , ,788 2,520 4,239 1,508 5,747 3 Differentiation Options , ,800 2,500 4,0001,5005,500 Practice Math Masters, p. 57 READINESS Solving Addition Number Stories (Math Masters, pp. 59 and 405) SMALL-GROUP Min To explore solving addition problems using a concrete model, have students solve parts-and-total number stories using base-10 blocks and Math Masters, page 405. For each number story, students put base-10 blocks in each of the Part sections, then move the Parts into the Total section to solve the problem. Example: The class had 43 blue crayons and 15 red crayons. How many crayons did they have in all Students first put 4 longs and 3 cubes in one of the Part sections and 1 long and 5 cubes in the other Part section. To solve the problem, they move all of the base-10 blocks to the Total section. Study Link Master Name Date Time STUDY LINK 2 7 Multidigit Addition continued Make a ballpark estimate. Use the column-addition method to add. Compare your answer with your estimate to see if your answer makes sense Sample estimates: = = = , ,848 2,356 NOTE Instead of Math Masters, page 405, consider using paper dinner plates divided into three sections. Label each of the two smaller sections Part and the larger section Total = = 1, ,800 = 2,300 Tota l Practice , 21, 26,,,, Rule: , 52,, 104, 130, Rule: Math Masters, p. 58 Pa r t Pa r t EM3MM_G4_U02_ indd 58 2/16/10 3:34 PM 124 Unit 2 Using Numbers and Organizing Data

7 ENRICHMENT Writing Addition Number Stories PARTNER Min To apply students understanding of addition algorithms, have them write and solve addition number stories. Then have them record a number model using a letter for the unknown. Encourage students to write multistep number stories. Stories may look similar to the following: Ian is shelving books in the library. He shelves 25 science fiction books, 18 biographies, and 36 mystery books. How many books did he shelve in all Answer: 79 books; Number model: = b Some students may be interested in writing and solving problems that involve distances, intervals of time, liquid volumes, masses of objects, or money. Stories may look similar to the following: Kendra bought some school supplies. She spent $1.75 on folders, $2.40 on pens, and $3.80 on notebooks. How much did she spend in all Answer: $7.95; Number model: $ $ $3.80 = c Marco wanted to make three different kinds of cookies for the school bake sale. The first recipe called for 2 1_ cups of milk. The 2 second called for 1_ 2 cup of milk. The last called for 1 1_ cups of 2 milk. How much milk did Marco need in all Answer: 4 1_ cups of 2 milk; Number model: 2 1_ 2 + 1_ _ 2 = m Provide opportunities for students to revise and share their writing. Then have partners solve each other s problems. EM3MM_G4_U02_ indd 59 Name Date Time 2 7 Addition Number Stories Use Math Masters, page 405 and base-10 blocks to solve the number stories. Record what you did in the parts-and-total diagrams. Example: The class had 43 blue crayons and 15 red crayons. How many crayons did they have in all 58 crayons 1. Auntie May had 24 fish and 11 hamsters. How many pets did she have altogether 35 pets Lucia had 38 cents and Madison had 29 cents. If they put their money together, how much money would they have 67 cents Teaching Master Math Masters, p Jordan made a flower basket for his mother that had 23 daisies and 8 roses. How many flowers were in the basket 31 flowers Miguel has 54 baseball cards. Janet gave him 47 more baseball cards. How many baseball cards does he have now 101 baseball cards /4/10 9:27 AM ELL SUPPORT Building a Math Word Bank (Differentiation Handbook, p. 140) SMALL-GROUP 5 15 Min To provide language support for estimation, have students use the Word Bank Template found on Differentiation Handbook, page 140. Ask students to write the term ballpark estimate, draw a picture representing the term, and write other related words. See the Differentiation Handbook for more information. Planning Ahead For Part 3 in Lesson 2-8, you will need several baseball caps with adjustable headbands one cap to be used by each small group of students. Ask students to bring baseball caps to school if they can. To be on the safe side, bring in one or more caps. Lesson

8 Name Date Time 2 7 Addition Number Stories Use Math Masters, page 405 and base-10 blocks to solve the number stories. Record what you did in the parts-and-total diagrams. Example: The class had 43 blue crayons and 15 red crayons. How many crayons did they have in all 58 crayons Auntie May had 24 fish and 11 hamsters. How many pets did she have altogether pets 2. Jordan made a flower basket for his mother that had 23 daisies and 8 roses. How many flowers were in the basket flowers Copyright Wright Group/McGraw-Hill 3. Lucia had 38 cents and Madison had 29 cents. If they put their money together, how much money would they have cents 4. Miguel has 54 baseball cards. Janet gave him 47 more baseball cards. How many baseball cards does he have now baseball cards 59