The Partial-Quotients Division Algorithm, Part 1

Transcription

1 The Partial-Quotients Division Algorithm, Part 1 Objectives To introduce and provide practice with a low-stress division algorithm for 1-digit divisors. 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 and use multiples of 10. [Number and Numeration Goal 3] Add multiples of 10. [Operations and Computation Goal 1] Subtract multidigit numbers. [Operations and Computation Goal 2] Apply extended multiplication facts to long-division situations. [Operations and Computation Goal 3] Solve equal-grouping division number stories. [Operations and Computation Goal 4] Key Activities Students learn and practice a paper-andpencil algorithm for division that permits them to build up the quotient by working with easy numbers. Ongoing Assessment: Informing Instruction See pages 413 and 416. Ongoing Assessment: Recognizing Student Achievement Use journal pages 144 and 145. [Operations and Computation Goal 4] Key Vocabulary dividend divisor partial quotient Materials Math Journal 1, pp. 144 and 145 Study Link 6 2 Math Masters, p. 403; pp. 436 and 438 (optional) slate Reviewing Place Value in Decimals Math Journal 1, p. 146 Students solve problems involving various basic skills with decimals. Math Boxes 6 3 Math Journal 1, p. 147 Students practice and maintain skills through Math Box problems. Study Link 6 3 Math Masters, p. 180 Students practice and maintain skills through Study Link activities. READINESS Playing Beat the Calculator Student Reference Book, p. 233 Math Masters, p. 461 per group: 4 each of number cards 1 10 (from the Everything Math Deck, if available), calculator Students practice extended multiplication facts. ENRICHMENT Determining the Cost of Pens Math Masters, p nickels (optional) Students use clues to solve a division number story. EXTRA PRACTICE Playing Division Dash Student Reference Book, p. 241 Math Masters, p. 471 per partnership: 4 each of number cards 1 9 (from the Everything Math Deck, if available) Students practice dividing 2- or 3-digit dividends by 1-digit divisors. ELL SUPPORT Building Vocabulary chart paper markers Students display and label parts of division number models. Advance Preparation Make 1 or 2 copies of Math Masters, page 403 for each student. Have extra copies available throughout this unit. Teacher s Reference Manual, Grades 4 6 pp Unit 6 Division; Map Reference Frames; Measures of Angles

2 Getting Started Mental Math and Reflexes Pose true or false problems involving equal groups. Suggestions: There are at least 2 [5s] in 11. true There are at least 3 [2s] in 5. false There are at least 4 [3s] in 14. true There are at least 7 [6s] in 45. true There are at least 9 [8s] in 67. false There are at least 8 [9s] in 75. true There are at least 10 [8s] in 92. true There are at least 30 [6s] in 157. false There are at least 40 [9s] in 391. true Math Message How many days are there in 30 weeks? How many weeks are there in 98 days? Study Link 6 2 Follow-Up Partners compare answers. Have volunteers explain different strategies that could be used to solve Problem 3. 1 Teaching the Lesson Math Message Follow-Up PROBLEM SOLVING WHOLE-CLASS Algebraic Thinking In the first problem, the number of weeks is known; students find the number of days by multiplying by 7. In the second problem, the total number of days is given; students divide by 7 to find the number of weeks. Ongoing Assessment: Informing Instruction Watch for students who do not understand where the numeral 7 came from because they did not see a 7 in the problem and did not equate a week with 7 days. Remind students that using another name for an equal quantity is often necessary to solve problems. weeks days per week total days 30 7 t 210 weeks days per week total days w Name Date Time Easy Multiples Teaching Aid Master 17 Introducing the Partial-Quotients Algorithm (Math Masters, pp. 403 and 438) WHOLE-CLASS This lesson formally introduces the partial-quotients algorithm. Begin with an equal-grouping division problem like the following: Amy is 127 days older than Bob. How many weeks older is Amy? Briefly work through the steps mentioned in the previous lessons: Step 1: Decide what you need to find out. How many weeks older is Amy? Write a variable in the weeks column. ELL Math Masters, p. 438 Lesson

3 weeks days per week total days w Step 2: Identify the data you need to solve the problem. The number of days in all and the number of days per week. Write the data in the appropriate spaces in the diagram. (See margin.) Step 3: Decide what to do to find the answer. 127 days must be equally grouped (divided) so that there are 7 days in each week. Write a number model to represent the problem using a variable for the unknown. 127 / 7 = w Step 4: Do the computation. Model the following algorithm while students follow along with paper and pencil. 1. Write the problem in the traditional form:. Point out that the dividend the number that is being divided is 127. The divisor the number that the dividend is being divided by is 7. To support English language learners, label the dividend and the divisor on the board. 2. Draw a vertical line so that the problem looks like the problem below. The vertical line will separate subtractions from partial quotients. Adjusting the Activity Suggest that students first make a list of easy multiples of the divisor. For example, if the divisor is 7, students might make the following list: = = = = = = = 7 Depending on the students and the divisor, the list can be extended or reduced. Students can use their list of easy multiples to take the guesswork out of successive multiples and focus on solving the division problem. Students will realize that they can work more quickly if they begin with a more extensive list of multiples. Math Masters, page 438 provides an optional form for writing multiples. AUDITORY KINESTHETIC TACTILE VISUAL 3. Suggest that one way to proceed is to use a series of at least/not more than multiples of the divisor. A good strategy is to start with easy numbers, such as 100 times the divisor or 10 times the divisor. Are there at least 100 [7s] in 127? No, because = 700, which is more than 127. Are there at least 10 [7s] in 127? Yes, because 10 7 = 70, which is less than 127. Are there at least 20 [7s]? No, because 20 7 = 140, which is more than 127. So there are at least 10 [7s], but not more than 20 [7s]. Try 10. Write 10 7, or 70, under 127. Write 10 at the right. 10 is the first partial quotient. Partial quotients will be used to build up the final quotient (The first partial quotient) 10 * 7 = The next step is to find out how much is left to divide. Subtract 70 from (The first partial quotient) 10 * 7 = Subtract. 57 is left to divide. 414 Unit 6 Division; Map Reference Frames; Measures of Angles

4 5. Now find the number of 7s in 57. Following are two ways to do this: Use a fact family: 8 7 = 56, so there are at least 8 [7s] in 57. Record as follows: (The first partial quotient) 10 7 = Subtract. 57 is left to divide (The second partial quotient) 8 7 = 56 Use at least / not more than multiples with easy numbers. For example, ask: Are there at least 10 [7s] in 57? No, because 10 7 = 70. Are there at least 5 [7s]? Yes, because 5 7 = 35. Next, subtract 35 from 57 and continue by asking: How many 7s are in 22? (The first partial quotient) 10 7 = Subtract. 57 is left to divide (The second partial quotient) 5 7 = Subtract. 22 is left to divide (The third partial quotient) 3 7 = For both ways, the division is complete when the subtraction leaves a number less than the divisor (7 in this example). The final step is to add the partial quotients the numbers of 7s that were subtracted. 18 is the quotient. There is 1 left over. So, 1 is the remainder Have students record the final answer in the traditional position above the dividend. To support English language learners, label the quotient and the remainder. 18 R1 8. Conclude by interpreting the answer: Amy is 18 weeks and 1 day older than Bob. Step 5: Record the answer and write a number model to summarize the number story. Answer: 18 weeks and 1 day older than Bob 127 / 7 18 R1 Lesson

5 Date Time 6 3 Partial-Quotients Division Algorithm 1. There are 6 pencils in each pack. How many packs can be made from 96 pencils? 96 6 = p Answer: 16 packs How many pencils are left over? 0 pencils 96 6 = There are 184 plants to be put into pots. Each pot can hold 8 plants. How many pots are needed? = p Answer: 23 pots How many plants are left over? 0 plants = 23 Sample number models are given. 2. Phil has $79 to purchase books. Books cost $7 each. How many books can Phil buy? 79 7 = b Answer: 11 books How many dollars are left over? 2 dollars R2 4. A waiter distributed 1,325 drinking straws evenly among 9 dispensers. How many straws went in each dispenser? = s Answer: 147 straws How many straws were left over? 2 straws R Lead students through several more problems on the board. Ask: How many [ns] are there in m? Each n should be a 1-digit number; each m should be a 2- or 3-digit number. Some students may be ready for a 2-digit divisor. Suggestions: How many [4s] are there in 92? 23 How many [3s] are there in 87? 29 How many [7s] are there in 301? 43 How many [8s] are there in 925? 115 R5 How many [12s] are there in 588? 49 How many [15s] are there in 556? 37 R1 Have the class pose a division problem, and ask partnerships to try to find the answer. Have volunteers share their work with the class. Look for students who got the same results in different ways. Math Journal 1, p. 144 Ongoing Assessment: Informing Instruction Watch for students using multiples that are not too large and that are easy to work with. Using such multiples may require more steps, but it will make the work go faster. Also, students should not be concerned if they pick a multiple that is too large. If that happens, they will quickly realize that they have a subtraction problem involving a larger number below. Using the Partial-Quotients Algorithm (Math Journal 1, pp. 144 and 145) Date 6 3 Divide Answer: 29 Try This Time Partial-Quotients Division Algorithm cont. Sample number models are given. 7. A factory has 372 boxes of shirts to distribute evenly among 12 stores. Each box holds 15 shirts. How many shirts will each store receive? Number model(s) with unknown: (372 12) 15 = s Answer: 465 shirts How may shirts are left over? 0 shirts Summary number model(s): (372 12) 15 = ,081 7 Answer: 154 R3 Sample answers: 8. There are 250 players in the league. (Write a number greater than 100.) There are 6 players on each team. (Write a number between 3 and 9.) How many teams can be made? = t Answer: 41 teams How many players are left over? 4 players R4 Students use the partial-quotients algorithm to solve number stories and computation problems. Have copies of Math Masters, pages 436 and 438 readily available for students who choose to use them. Encourage students to use the relationship between multiplication and division to check their answers. Ongoing Assessment: Recognizing Student Achievement Journal pages 144 and 145 Problems 1, 2, 5 Use journal pages 144 and 145, Problems 1, 2, and 5 to assess students ability to solve division problems and number stories with 1-digit divisors and 2-digit dividends. Students are making adequate progress if they are able to compute the answers using the partial-quotients division algorithm. Some students may be able to solve Problems 3, 4, and 6 8, which involve 2-digit divisors or 3- or 4-digit dividends. [Operations and Computation Goal 4] Math Journal 1, p Unit 6 Division; Map Reference Frames; Measures of Angles

6 2 Ongoing Learning & Practice Reviewing Place Value in Decimals (Math Journal 1, p. 146) Students solve problems involving ordering of decimals, naming place value in decimals, and reading and writing decimals. Math Boxes 6 3 (Math Journal 1, p. 147) Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 6-1. The skill in Problem 5 previews Unit 7 content. Writing/Reasoning Have students write a response to the following: Explain how the exponent in Problem 4a changes the value of 10. Sample answer: The exponent tells how many times the base 10 is used as a factor. For example, 10 4 = , or 10,000. Date Place Value in Decimals 1. Write these numbers in order from smallest to largest smallest largest 3. Write the smallest number you can make with the following digits: ,467 (whole number) and (decimal) 5. Write each number using digits. a. four and seventy-two hundredths 4.72 b. nine hundred thirty-five thousandths Time Math Journal 1, p A number has 6 in the tenths place, 4 in the ones place, 5 in the hundredths place, and 9 in the tens place. Write the number What is the value of the digit 4 in the numerals below? a tenths b tens c hundredths d thousandths 6. I am a four-digit number less than 10. The digit in the tenths place is the result of dividing 36 by 4. The digit in the hundredths place is the result of dividing 42 by 7. The digit in the ones place is the result of dividing 72 by 8. The digit in the thousandths place is the result of dividing 35 by 5. What number am I? Study Link 6 3 (Math Masters, p. 180) Home Connection Students practice using the partialquotients division algorithm. Study Link Master Name Date Time Date Time STUDY LINK 6 3 Division Sample number models are given. 1. Bernardo divided a bag of 83 marbles 2. The carnival committee has 360 small prizes evenly among five friends and himself. to share equally with 5 carnival booths. How many marbles did each get? How many prizes will each booth get? 83 6 = m = p Answer: 13 marbles Answer: 72 prizes How many marbles are left over? How many prizes are left over? 5 marbles 0 prizes R = Answer: 22 R Answer: 53 R Math Boxes 1. Measure each line segment to the nearest millimeter. a. P S About 9 cm 3 mm b. A B About 6 cm 5 mm 2. Round 5,906,245 to the nearest a. million. 6,000,000 b. ten thousand. 5,910, Multiply. Use a paper-and-pencil method. 3,016 = c. thousand. 5,906,000 d. hundred. 5,906, Practice = = = = 4. Complete. a = 10,000 5 b. 10 = c. 100 = 10 2 d. 10 to the seventh power = 10,000, Circle 1_ 2 of the squares Math Masters, p. 180 Math Journal 1, p Lesson

7 Name Date Time 6 3 A Pen Riddle Mrs. Swenson bought 2 pens for each of her 3 daughters. She gave the clerk a $10 bill. Her change was all in nickels. Her change was less than 50 cents. 1. What was the cost of each pen? Each pen cost the same amount. No sales tax was charged. $1.60 or $ Show or explain how you got your answer. Sample answer: I needed an amount between $9.55 and $9.95 that can be evenly divided by 6. Both $9.60 and $9.90 are evenly divided by 6. $ $1.60; $ $ Differentiation Options READINESS Playing Beat the Calculator (Student Reference Book, p. 233; Math Masters, p. 461) SMALL-GROUP 5 15 Min Math Masters, page 181 To provide experience with extended multiplication facts, students play a version of Beat the Calculator in which the caller attaches a 0 to one or both of the factors shown on the cards. ENRICHMENT Determining the Cost of Pens (Math Masters, p. 181) 5 15 Min To apply students understanding of division, have them use clues to find the cost of several pens. Encourage students to use money to model the problem if necessary. EXTRA PRACTICE PARTNER Playing Division Dash (Student Reference Book, p. 241; Math Masters, p. 471) 5 15 Min To practice dividing 2- or 3-digit dividends by 1-digit divisors, have students play Division Dash. See Lesson 6-4 for additional information. Division Dash Materials number cards 1 9 (4 of each) 1 score sheet Players 1 or 2 Skill Division of 2-digit by 1-digit numbers Object of the game To reach 100 in the fewest divisions possible. Directions 1. Prepare a score sheet like the one shown at the right. 2. Shuffle the cards and place the deck number-side down on the table. 3. Each player follows the instructions below: Turn over 3 cards and lay them down in a row, from left to right. Use the 3 cards to generate a division problem. The 2 cards on the left form a 2-digit number. This is the dividend. The number on the card at the right is the divisor. Divide the 2-digit number by the 1-digit number and record the result. This result is your quotient. Remainders are ignored. Calculate mentally or on paper. Add your quotient to your previous score and record your new score. (If this is your first turn, your previous score was 0.) 4. Players repeat Step 3 until one player s score is 100 or more. The first player to reach at least 100 wins. If there is only one player, the object of the game is to reach 100 in as few turns as possible. Turn 1: Bob draws 6, 4, and 5. He divides 64 by 5. Quotient 12. Remainder is ignored. The score is Turn 2: Bob then draws 8, 2, and 1. He divides 82 by 1. Quotient 82. The score is Turn 3: Bob then draws 5, 7, and 8. He divides 57 by 8. Quotient 7. Remainder is ignored. The score is Bob has reached 100 in 3 turns and the game ends. 64 is the dividend. 5 is the divisor. Quotient Score Games ELL SUPPORT SMALL-GROUP Building Vocabulary 5 15 Min To provide language support for division, have students write division number models on chart paper in the following ways: 42 / 5 8 R2 42_ 5 8 R R2 5 8 R2 42 Have students do the following for each number model: Label and underline the dividend (number being divided) in red. Label and underline the divisor (the number the dividend is being divided by) in blue. Label and circle the quotient in a third color. Label and circle the remainder in a fourth color. Point out that both the quotient and the remainder are part of the answer. Display this chart throughout the division lessons. Student Reference Book, p Unit 6 Division; Map Reference Frames; Measures of Angles

8 Name Date Time STUDY LINK 6 3 Division Bernardo divided a bag of 83 marbles 2. The carnival committee has 360 small prizes evenly among five friends and himself. to share equally with 5 carnival booths. How many marbles did each get? How many prizes will each booth get? Answer: marbles Answer: prizes How many marbles are left over? marbles How many prizes are left over? prizes Answer: Answer: Practice Copyright Wright Group/McGraw-Hill 5. = = = = 180