Subtraction of Multidigit Numbers

Transcription

1 Subtraction of Multidigit Numbers Objectives To review the trade-first and counting-up methods, and to introduce the partial-differences method of solving multidigit subtraction problems; and to provide practice estimating differences for multidigit subtraction problems. 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 ] Apply extended subtraction facts. [Operations and Computation Goal ] Use the trade-first and partial-differences algorithms to solve multidigit subtraction problems; choose an appropriate paper-and-pencil algorithm to solve multidigit subtraction problems. [Operations and Computation Goal ] Make ballpark estimates for multidigit subtraction problems. [Operations and Computation Goal ] Key Activities Students use the trade-first and partialdifferences methods for subtraction. Playing Subtraction Target Practice Student Reference Book, p. Math Masters, p. 0 calculator each of number cards 0 (from the Everything Math Deck, if available) Students practice place-value and subtraction skills. Math Boxes Math Journal, p. Students practice and maintain skills through Math Box problems. Study Link Math Masters, pp. and Students practice and maintain skills through Study Link activities. READINESS Subtracting by Counting Up Student Reference Book, p. Math Masters, p. Students explore visual models for solving subtraction problems. Solving Number-Tile Problems Math Masters, p. 7 scissors Students find missing digits in addition and subtraction problems. Writing Subtraction Number Stories Students write and solve subtraction number stories. Ongoing Assessment: Recognizing Student Achievement Use journal page. [Operations and Computation Goal ] Key Vocabulary trade-first method partial-differences method Materials Math Journal, pp. and 0 Student Reference Book, pp. and Study Link Math Masters, p. 0 or 0 (optional) slate quarter-sheet of paper base-0 blocks (optional) Advance Preparation Plan to spend two days on this lesson. Place quarter-sheets of paper near the Math Message. If students need computation grids in Part, make copies of Math Masters, page 0 or 0. Teacher s Reference Manual, Grades pp., 0, Unit Using Numbers and Organizing Data

2 Getting Started Mental Math and Reflexes Pose subtraction problems such as the following. Have students estimate the differences = 0; 0-0 = = 0; 70-0 = = 0; 0-0 = = 0; 0-0 = = 0; 0-0 = = 0; = = 0; 0-00 = = 0; = = 00; 00-0 = 0 Math Message Solve the problems on a quarter-sheet of paper. Show your work Study Link Follow-Up Have students compare answers. Ask volunteers to share number models for Problems and. Teaching the Lesson Math Message Follow-Up WHOLE-CLASS Links to the Future In Unit of Fourth Grade Everyday Mathematics students apply these subtraction algorithms to decimal numbers. Have students share their solution strategies. Tell the class that in this lesson they will review the trade-first method and explore the partial-differences method. Some students may have used these algorithms to solve the Math Message problems. Discussing and Practicing the Trade-First Method for Subtraction (Math Journal, p. ; Student Reference Book, p. ) WHOLE-CLASS Students used the trade-first method in Third Grade Everyday Mathematics. Discuss the example on page of the Student Reference Book and point out the following: The problem is written vertically. If each digit of the larger (top) number is greater than or equal to the digit directly below it, no adjustments are required before subtracting. Subtraction is performed column by column. If one (or more) digit of the top number is less than the digit directly below it, then the top number is adjusted by trading before any subtracting is done. For example, we might need to trade one of the tens for 0 ones or one of the hundreds for 0 tens. All adjustments are made from right to left. Algorithm Project The focus of this lesson is the trade-first and partial-differences algorithms for subtraction. To teach U.S. traditional subtraction, see Algorithm Project on page A0. Whole Numbers Subtraction Methods Trade-First Subtraction Method The trade-first method is similar to the method for subtracting that most adults were taught when they were in school. Here is the trade-first method for subtracting -digit or -digit numbers. Look at the digits in the s place. If subtracting these digits gives a negative number, trade ten for 0 ones. Look at the digits in the 0s place. If subtracting these digits gives a negative number, trade hundred for 0 tens. Subtract in each column. Subtract from using the trade-first method. 00s 0s s - Look at the s place. You cannot remove ones from ones. Larger numbers with or more digits are subtracted the same way. Subtract , -,. - 00s 0s s - So trade ten for 0 ones. Now look at the 0s place. You cannot remove tens from tens. Check your answers on page 0. 00s 0s s - So trade hundred for 0 tens. Now subtract in each column. - = In, Johann Widmann wrote the first book that used + and - symbols. Traders and shopkeepers had used both symbols long before this. They used + to show they had too much of something. They used - to show they had too little of something. Student Reference Book, p. Lesson

3 Date Trade-First Subtraction Write a number model for a ballpark estimate. Solve Problems using the tradefirst method. Solve Problems using any method. Use your ballpark estimates to see if your answers make sense. Sample estimates: Time ,, ,7,000,000,000,,7 7 Pose the following subtraction problem: - Ask students to make a ballpark estimate. Sample answers: = 00; 00-0 = 0 Encourage students to use terms such as closer to, between, and a little more than to refine their estimates. Have students examine the columns of digits from right to left. is less than. is less than. is greater than. So the top number () must be adjusted before subtracting Try This ,00, Name two -digit numbers whose difference is 7. Sample answer: 7 Math Journal, p. Trade ten for 0 ones: 00s - 0s 0 / s / 00s 0s s Adjusting the Activity ELL Have base-0 blocks readily available for students to use while solving the subtraction problems. Trade hundred for 0 tens: Now subtract separately in each column: / - 0 0/ / / AUDITORY KINESTHETIC TACTILE VISUAL Partial-Differences Method. Subtract from left to right, one column at a time.. In some cases, the larger number is on the bottom and the smaller number is on the top. When you subtract these numbers, the difference will be a negative number. 7 - =? 7 - Subtract the 00s Subtract the 0s Subtract the s. - Find the total = - 7 = - 7 =? One way: Add. (add ) - 7 (add ) - 0 Same-Change Rules Here are the same-change rules for subtraction problems: Think of 0-0 as first subtracting 0 and then subtracting 0 more = -0 If you add the same number to both numbers in the problem, the answer is the same. If you subtract the same number from both numbers in the problem, the answer is the same. Use these rules to change the second number in the problem to a number that has 0 in the ones place. Make the same change to the first number. Then subtract. Another way: Subtract 7. (subtract 7) 7-7 (subtract 7) - 0 Whole Numbers Ask volunteers to write several -digit and -digit subtraction problems on the board. Have them use and describe the trade-first method to solve the problems. Ask for and record ballpark estimates for each problem before solving. Keep the following points in mind as students write and solve the subtraction problems on the board: Make sure that the numbers students write are properly aligned in columns. Have students draw lines separating the digits into columns to help them keep the digits aligned or provide computation grids. Tell struggling students to write place-value reminders (00s, 0s, and s) above the columns. Although it is convenient to adjust the larger number from right to left, do not expect that all students will do it that way. What is important is that a student trades correctly and that the final adjusted number has each digit greater than or equal to the digit directly below it. Subtract Check your answers on page 0. Student Reference Book, p. Unit Using Numbers and Organizing Data

4 Assign Problems 7 on journal page. Have computation grids available (Math Masters, page 0 or 0) for those students who prefer to use them to help keep digits in the proper columns. Ask students to share their solutions to Problem 7 and indicate thumbs-up if they agree with an answer. Date Time Partial-Differences Subtraction Write a number model for a ballpark estimate. Solve Problems using the partialdifferences method. Solve Problems using any method. Use your ballpark estimates to see if your answers make sense. Sample estimates: Ongoing Assessment: Recognizing Student Achievement Journal page Problems ,,0,77 Use journal page, Problems to assess students ability to solve multidigit subtraction problems. Students are making adequate progress if they are able to use a paper-and-pencil algorithm to calculate the correct differences. 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 ] ,00,00,00.,,0,7 Discussing and Practicing the Partial-Differences Method for Subtraction (Math Journal, p. 0; Student Reference Book, p. ) WHOLE-CLASS Try This Name two -digit numbers whose difference is,0. Sample answer:,77 _,,0 Math Journal, p. 0,000,000,000 The partial-differences method is another way to subtract numbers. This method avoids renaming, which is the major pitfall for many students when using the standard subtraction algorithm. Discuss the example on page of the Student Reference Book and point out the following: In the partial-differences method, subtraction is performed from left to right. For example, when subtracting -digit numbers, the difference of hundreds is calculated first, then the difference of tens, then the difference of ones. Each difference is written on a separate line beneath the problem. After all the differences have been found, the numbers are added to find the total difference. Pose the following subtraction problem: - Ask students to make a ballpark estimate. Sample answers: = 00; 0-0 = 70 Date Math Boxes. A number has in the tens place, in the millions place, in the thousands place, in the ten-thousands place, in the hundred-thousands place, 0 in the ones place, and in the hundreds place. Write the number.,, 0. Which number sentence is true? Fill in the circle next to the best answer. A B C 0 Time. Write five names for,000. Sample answers:,000, , Draw a polygon that has no parallel sides. Sample answer: D 0. Measure these line segments to the nearest centimeter. a. b. About About. cm cm. Multiply mentally a. b. c. d. e. Math Journal, p. Lesson

5 Study Link Master STUDY LINK Multidigit Subtraction Make a ballpark estimate. Use the trade-first subtraction method to subtract. Compare your answer with your estimate to see if your answer makes sense Have students solve the problem using the partial-differences method. - Subtract the 00s: Subtract the 0s: Subtract the s: - + Add the partial differences: 00 + (-0) + Sample estimates: ,, 7,7,7,,00,7, Have volunteers write several -digit and -digit subtraction problems on the board. Ask students to use and describe the partial-differences method to solve these problems. Before students solve each problem, ask for and record ballpark estimates. Assign Problems 7 on journal page 0. Bring small groups of students together to share solutions.,00 00,700,000,000,000 7,000,000,000 Practice Math Masters, p. Adjusting the Activity ELL Use a number line to model - versus -. Record several of these pairs on the board and discuss the patterns students see. AUDITORY KINESTHETIC TACTILE VISUAL Ongoing Learning & Practice Playing Subtraction Target Practice (Student Reference Book, p. ; Math Masters, p. 0) Students play Subtraction Target Practice to practice place-value and subtraction skills. Math Boxes (Math Journal, p. ) INDEPENDENT STUDY LINK Multidigit Subtraction continued Make a ballpark estimate. Use the partial-differences method to subtract. Compare your answer with your estimate to see if your answer makes sense Sample estimates: Study Link Master = = = 00 Mixed Practice Math Boxes in this lesson are linked with Math Boxes in Lessons - and -7. The skill in Problem previews Unit content. Writing/Reasoning Have students write a response to the following: Marisa looked at the multiplication equation she wrote for Problem b and said, I know that 0 is times as many as. I also know that 0 is times as many as. Write similar statements for the multiplication equation in Problem e. Sample answer: I know that is times as many as. I also know that is times as many as.... 0,7 - -,,7,0 -,0, Study Link (Math Masters, pp. and ) INDEPENDENT py g g p = 00,700 -,00 =,00,00 -,00 =,00 Practice ,,,,, Rule:.,,,,, 7, Rule: Math Masters, p. Home Connection Students solve subtraction problems and show someone at home how to use the subtraction methods they practiced in this lesson. Unit Using Numbers and Organizing Data

6 Differentiation Options READINESS Subtracting by Counting Up (Student Reference Book, p. ; Math Masters, p. ) 0 Min To explore visual models for solving subtraction problems using a counting-up method, have students perform counting-up subtraction on number lines. This method was reviewed in Mental Math and Reflexes in Lesson -. Solving Number-Tile Problems (Math Masters, p. 7) 0 Min To apply computation and estimation skills, have students cut out and use number tiles to find missing numbers in addition and subtraction problems. Subtraction by Counting Up Use the counting-up method to solve these problems. Use the number lines if they are helpful. Example: 0? Think: So, Math Masters, p. Teaching Master Writing Subtraction Number Stories 0 Min To apply students understanding of subtraction algorithms, have them write and solve multistep subtraction number stories. Then have them record a number model using a letter for the unknown. Stories may look similar to the following: Ginny bought a 0-foot coil of rope from the store. She used 7 feet of rope to tie her boat to the dock, and another feet to make a rope swing. How many feet of rope are left in on the coil? Answer: feet; Number model: 0 7 = r 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: The delivery truck began the day with,0 lb of dog food. At the first animal shelter they delivered 7 lb of dog food, and at the second animal shelter they delivered lb of dog food. How many ounces are left after the two stops? Answer:, ounces; Number models:,0 7 = and = f Provide opportunities for students to revise and share their writing. Then have partners solve each other s problems. Number-Tile Problems Cut out the 0 number tiles at the bottom of the page. Use them to help you solve the problems.. Use five odd-numbered tiles to. Use five even-numbered tiles (that make the smallest possible includes 0) to make the largest possible difference. difference. Do not use 0 as the first digit.. Use one set of the number tiles 0. Find the missing digits in these addition and subtraction problems. a. b. c. d. 7 7, Teaching Master Math Masters, p. 7 Lesson 7