NS538 Remainders and NS539 Dividing with Remainders


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1 :1 PAGE NS538 Remainders and NS539 Dividing with Remainders GOALS Students will divide with remainders using pictures, number lines and skip counting. Draw: 6 3 = = 2 Remainder = 2 Remainder 2 PRIOR KNOWLEDGE REQUIRED Relationships between division and multiplication, addition, skip counting, number lines VOCABULARY remainder quotient R divisor 9 3 = = 3 Remainder 1 Ask your students if they know what the word remainder means. Instead of responding with a definition, encourage them to only say the answers for the following problems. This will allow those students who don t immediately see it a chance to detect the pattern. 7 2 = 3 Remainder 11 3 = 3 Remainder 12 5 = 2 Remainder 14 5 = 2 Remainder Challenge volunteers to find the remainder by drawing a picture on the board. This way, students who do not yet see the pattern can see more and more examples of the rule being applied. SAMPLE PROBLEMS: What does remainder mean? Why are some dots left over? Why aren t they included in the circles? What rule is being followed in the illustrations? [The same number of dots is placed in each circle, the remaining dots are left uncircled]. If there are fewer uncircled dots than circles then we can t put 95
2 :1 PAGE one more in each circle and still have the same number in each circle, so we have to leave them uncircled. If there are no dots left over, what does the remainder equal? [Zero.] Introduce your students to the word quotient : Remind your students that when subtracting two numbers, the answer is called the difference. ASK: When you add two numbers, what is the answer called? In = 11, what is 11 called? (The sum). When you multiply two numbers, what is the answer called? In 2 5 = 10, what is 10 called? (The product). When you divide two numbers, does anyone know what the answer is called? There is a special word for it. If noone suggests it, tell them that when you write 10 2 = 5, the 5 is called the quotient. Have your students determine the quotient and the remainder for the following statements. a) 17 3 = Remainder b) 23 4 = Remainder c) 11 3 = Remainder Write 2 friends want to share 7 apples. What are the sets? [Friends.] What are the objects being divided? [Apples.] How many circles need to be drawn to model this problem? How many dots need to be drawn? Draw 2 circles and 7 dots. To divide 7 apples between 2 friends, place 1 dot (apple) in each circle. Can another dot be placed in each circle? Are there more than 2 dots left over? So is there enough to put one more in each circle? Repeat this line of instruction until the diagram looks like this: How many apples will each friend receive? Explain. [There are 3 dots in each circle.] How many apples will be left over? Explain. [Placing 1 more dot in either of the circles will make the compared amount of dots in both circles unequal.] Repeat this exercise with 5 friends want to share 18 apples. Emphasize that the process of division and placing apples (dots) into sets (circles) continues as long as there are at least 5 apples left to share. Count the number of apples remaining after each round of division to ensure that at least 5 apples remain. 96 Copyright 2007, JUMP Math TEACHER S GUIDE
3 :1 PAGE Have your students illustrate each of the following division statements with a picture, and then determine the quotients and remainders. Number in each circle a) 11 5 = Remainder Number left over b) 18 4 = Remainder c) 20 3 = Remainder d) 22 5 = Remainder e) 11 2 = Remainder f) 8 5 = Remainder g) 19 4 = Remainder Then have your students explain what the following three models illustrate Have them explain how these models are the same and how are they different? Have your students complete several division exercises using number lines, and then have them draw number lines for several division statements. Can skip counting show that 14 3 = 4 Remainder 2? Why does the count stop at 12? [Continuing the count will lead to numbers greater than 14.] How can the remainder be determined? [Subtract 12 from = 2.] Have your students complete several division exercises by skip counting. Instruct them to now write R as the abbreviation in equations for remainder. EXAMPLE: 17 5 = 3 R 2. Assign the following exercise to students who have difficulties learning when to stop counting, when skip counting to solve a division statement. Using a number line from 0 to 25, ask your student to skip count out loud by five and to stop counting before reaching 17. Have them point to the respective number on the number line as they count it. This should enable your student to see that their finger will next point to 20 if they don t stop counting at 15, passing the target number of 17. You may need to put your finger on 17 to stop some students from counting further. Repeat this exercise with target numbers less than 25. After completing this exercise, most students will know when to stop counting before they reach a given 97
4 :1 PAGE target number, even if they are counting by numbers other than 5. With a few students, you will have to repeat the exercise with counting by 2s, 3s, etc. Extensions 1. Which number is greater, the divisor (the number by which another is to be divided) or the remainder? Will this always be true? Have your students examine their illustrations to help explain. Emphasize that the divisor is equal to the number of circles (sets), and the remainder is equal to the number of dots left over. We stop putting dots in circles only when the number left over is smaller than the number of circles; otherwise, we would continue putting the dots in the circles. See the journal section below. Which of the following division statements is correctly illustrated? Can one more dot be placed into each circle or not? Correct the two wrong statements = 4 Remainder = 3 Remainder = 4 Remainder 3 Without illustration, identify the incorrect division statements and correct them. a) 16 5 = 2 Remainder 6 b) 11 2 = 4 Remainder 3 c) 19 6 = 3 Remainder 1 2. Explain how a diagram can illustrate a division statement with a remainder and a multiplication statement with addition = 4 Remainder = 14 Ask students to write a division statement with a remainder and a multiplication statement with addition for each of the following illustrations. 3. Compare mathematical division to normal sharing. Often if we share 5 things (say, marbles) among 2 people as equally as possible, we give 3 to one person and 2 to the other person. But in mathematics, if we divide 5 objects between 2 sets, 2 objects are placed in each set and the leftover object is designated as a remainder. Teach them that we can still use division to solve this type of problem; we just have to be careful in how we interpret the remainder. Have students compare the answers to the reallife problem and to the mathematical problem: a) 2 people share 5 marbles (groups of 2 and 3; 5 2 = 2 R 1) b) 2 people share 7 marbles (groups of 3 and 4; 7 2 = 3 R 1) c) 2 people share 9 marbles (groups of 4 and 5; 9 2 = 4 R 1) 98 Copyright 2007, JUMP Math TEACHER S GUIDE
5 :1 PAGE ASK: If 19 2 = 9 R 1, how many marbles would each person get if 2 people shared 19 marbles? Emphasize that we can use the mathematical definition of sharing as equally as possible even when the answer isn t exactly what we re looking for. We just have to know exactly how to adapt it to what we need. 4. Find the mystery number. I am between 22 and 38. I am a multiple of 5. When I am divided by 7 the remainder is Have your students demonstrate two different ways of dividing a) 7 counters so that the remainder equals 1. b) 17 counters so that the remainder equals As a guided class activity or assignment for very motivated students, have your students investigate the following division statements = 5 Remainder 2, and 17 5 = Remainder? 22 3 = 7 Remainder 1, and 22 7 = Remainder? 29 4 = 7 Remainder 1, and 29 7 = Remainder? 23 4 = 5 Remainder 3, and 23 5 = Remainder? = 2 Remainder 5, and 27 2 = Remainder? What seems to be true in the first four statements but not the fifth? 27 2 = 13 Remainder 1, and = Remainder? = 3 Remainder 4, and 40 3 = Remainder? Challenge your students to determine the conditions for switching the quotient with the divisor and having the remainder stay the same for both statements. Have students create and chart more of these problems to help them find a pattern. You could start a class chart where students write new problems that they have discovered belongs to one or the other category. As you get more belonging to one of the categories challenge them to find more examples that belong to the other category. Be sure that everyone has a chance to contribute. ANSWER: If the remainder is smaller than the quotient, the quotient can be switched with the divisor and the remainder will stay the same for both statements. Note that 23 4 = 5 Remainder 3 is equivalent to = 23. But this is equivalent to = 23, which is equivalent to 23 5 = 4 Remainder 3. Note, however, that while = 55 is equivalent to 55 8 = 6 Remainder 7, it is not true that = 55 is equivalent to 55 6 = 8 Remainder 7, because in fact 55 6 = 9 Remainder 1. Journal The remainder is always smaller than the divisor because 99
6 :1 PAGE NS540 Long Division 2Digit by 1Digit GOALS Students will use long division to divide twodigit numbers by a onedigit number. PRIOR KNOWLEDGE REQUIRED Tens and ones blocks Division as sharing Write Ask your students if they recognize this symbol. If they know what the symbol means, have them solve the problems. If they don t recognize the symbol, have them guess its meaning from the other students answers to the problems. If none of your students recognize the symbol, solve the problems for them. Write more problems to increase the chances of students being able to predict the answers R1 Explain that 2 6 is another way of expressing 6 2 = 3, and that 2 7 is another way of expressing 7 2 = 3 Remainder 1. Ask your students to express the following statements using the new notation learned above. a) 14 3 = 4 Remainder 2 b) 26 7 = 3 Remainder 5 c) = 204 Remainder 3 To ensure that they understand the long division symbol, ask them to solve and illustrate the following problems. Bonus Then demonstrate division using base ten materials Have students solve the following problems using base ten materials Copyright 2007, JUMP Math TEACHER S GUIDE
7 :1 PAGE Then challenge them to solve 3 72, again using base ten materials, but allow students to trade tens blocks for ones blocks as long as the value of the dividend (72) remains the same. ASK: Can 7 tens blocks be equally placed into 3 circles? Can 6 of the 7 tens blocks be equally placed into 3 circles? What should be done with the leftover tens block? How many ones blocks can it be traded for? How many ones blocks will we then have altogether? Can 12 ones blocks be equally placed into 3 circles? Now, what is the total value of blocks in each circle? [24.] What is 72 divided by 3? Where do we write the answer? Explain that the answer is always written above the dividend, with the tens digit above the tens digit and the ones digit above the ones digit Have your students solve several problems using base ten materials The following problems will have remainders Tell your students that they are going to learn to solve division problems without using base ten materials. The solutions to the following problems have been started using base ten materials. Can they determine how the solution is written? As in the previous set, illustrate the following problems. Have your students determine how to write the solution for the last problem Challenge students to illustrate several more problems and to write the solution. When all students are writing the solution correctly, ask them how they determined where each number was written. Which number is written above the dividend? [The number of tens equally placed into each circle.] Which number is written below the dividend? [The number of tens placed altogether.] Then, using the illustrations from the two previous sets of problems, ask your students what the circled numbers below express. 101
8 :1 PAGE ASK: What does the number express in relation to its illustration? [The number of tens not equally placed into circles.] Why does the subtraction make sense? [The total number of tens minus the number of tens equally placed into circles results in the number of tens blocks left over.] Teach your students to write algorithms without using base ten materials. Remind them that the number above the dividend s tens digit is the number of tens placed in each circle. For example, if there are 4 circles and 9 tens, as in 4 94,the number 2 is written above the dividend to express that 2 tens are equally placed in each of the 4 circles. Explain that the number of tens placed altogether can be calculated by multiplying the number of tens in each circle (2) by the number of circles (4); the number of tens placed altogether is 2 4 = 8. Ask your students to explain if the following algorithms have been started correctly or not. Encourage them to illustrate the problems with base ten materials, if it helps Explain that the remaining number of tens blocks should always be less than the number of circles, otherwise more tens blocks need to be placed in each circle. The largest number of tens blocks possible should be equally placed in each circle. Display the multiplication facts for 2 times 1 through 5 (i.e. 2 1 = 2, 2 2 = 4, etc.), so that students can refer to it for the following set of problems. Then write 2 75 ASK: How many circles should be used? If 1 tens block is placed in each circle, how many tens blocks will be placed altogether? [2 1 = 2] What if 2 tens blocks are placed in each circle? [2 2 = 4] What if 3 tens blocks are placed in each circle? [2 3 = 6] And finally, what if 4 tens blocks are placed in each circle? How many tens blocks need to be placed? [Seven.] Can 4 tens blocks be placed in each circle? [No, that will require 8 tens blocks.] Then explain that the greatest multiple of 2 not exceeding the number of tens is required. Have them perform these steps for the following problems. 3 3 tens in each circle = 6 tens place 1 1 tens block left over 102 Copyright 2007, JUMP Math TEACHER S GUIDE
9 :1 PAGE Then display the multiplication facts for 3 times 1 through to 3 times 5 and repeat the exercise. Demonstrate the steps for the first problem. 2 2 tens in each circle = 6 tens place 1 1 tens block left over Emphasize that the number above the dividend s tens digit is the greatest multiple of 3 not exceeding the number of tens. Then, using the illustrations already drawn to express leftover tens blocks (see the second page of this section), explain the next step in the algorithm. ASK: Now what do the circled numbers express? The circled number expresses the amount represented by the base ten materials not placed in the circles. Using base ten materials, challenge students to start the process of long division for 85 3 and to record the process (the algorithm) up to the point discussed so far. Then ask students to trade the remaining tens blocks for ones blocks, and to circle the step in the algorithm that expresses the total value of ones blocks. Ensure that students understand the algorithm up to the step where the ones blocks are totalled with the remaining (if any) tens blocks ones to be placed Illustrate all of the placed tens and ones blocks and the finished algorithm, and then ask your students to explain the remaining steps in the algorithm. Perform this for the examples already started (63 3, 64 2, 92 4, etc.). For example, Remainder So, 94 4 = 23 Remainder
10 :1 PAGE Ask your students to explain how the circled numbers are derived. How is the 3 derived? The 12? [Dividing the 14 ones blocks into 4 circles results in 3 blocks in each circle, for a total of 12.] results in a remainder of 2. Then challenge students to write the entire algorithm. Ask them why the second subtraction makes sense. [The total number of ones blocks subtracted by the number of ones blocks placed into circles equals the number of ones blocks left over.] Using base ten materials, have students complete several problems and write the entire algorithms ones in each circle ones to be placed ones placed 0 Remainder (no ones left over) Some students will need all previous steps done so that they can focus on this one. If you prefer, you may use an example for a problem that has leftover ones. Have students finish the examples they have already started and then complete several more problems from the beginning and use base ten materials only to verify their answers With practice, students will learn to estimate the largest multiples that can be used to write the algorithms. When they are comfortable with moving forward in the lesson, introduce larger divisors Note that at this point in the lesson, the dividend s tens digit is always greater than the divisor. Extension Teach students to check their answers with multiplication. For example: R = Copyright 2007, JUMP Math TEACHER S GUIDE
11 :1 PAGE NS541 Long Division 3 and 4Digit by 1 Digit GOALS Students will use the standard algorithm for long division to divide 3 and 4digit numbers by 1digit numbers. Teach your students to use long division to divide threedigit numbers by onedigit numbers. Using base ten materials, explain why the standard algorithm for long division works. EXAMPLE: Divide 726 into 3 equal groups. STEP 1. Make a model of 726 units. PRIOR KNOWLEDGE REQUIRED The standard algorithm for dividing 2digit numbers by 1digit numbers Remainders Division as finding the number in each group Tens and ones blocks 7 hundreds blocks 2 tens 6 ones blocks blocks STEP 2. Divide the hundreds blocks into 3 equal groups. VOCABULARY standard algorithm remainder quotient Keep track of the number of units in each of the 3 groups, and the number remaining, by slightly modifying the long division algorithm hundred blocks, or 200 units, have been divided into each group units (200 3) have been divided units still need to be divided NOTE: Step 2 is equivalent to the following steps in the standard long division algorithm
12 :1 PAGE Students should practise Steps 1 and 2 from both the modified and the standard algorithms on the following problems Students should show their work using actual base ten materials or a model drawn on paper. STEP 3. Divide the remaining hundreds block and the 2 remaining tens blocks among the 3 groups equally. There are 120 units, so 40 units can be added to each group from Step 2. Group 1 Group 2 Group 3 Keep track of this as follows: new units have been divided into each group (40 6) new units have been divided 6 6 units still need to be divided NOTE: Step 3 is equivalent to the following steps in the standard long division algorithm Students should carry out Step 3 using both the modified and standard algorithms on the problems they started above. Then give students new problems and have them do all the steps up to this point. Students should show their work using either base ten materials or a model drawn on paper. 106 Copyright 2007, JUMP Math TEACHER S GUIDE
13 :1 PAGE STEP 4. Divide the 6 remaining blocks among the 3 groups equally. Group 1 Group 2 Group 3 There are now 242 units in each group; hence = new units have been divided into each group (2 3) new units have been divided 0 There are no units left to divide NOTE: Step 4 is equivalent to the following steps in the standard long division algorithm Students should be encouraged to check their answer by multiplying Students should finish the problems they started. Then give students new problems to solve using all the steps of the standard algorithm. Give problems where the number of hundreds in the dividend is greater than the divisor. (EXAMPLES: 842 2, 952 4) Students should show their work (using either base ten materials or a model drawn on paper) and check their answers using multiplication. When students are comfortable dividing 3digit numbers by 1digit numbers, introduce the case where the divisor is greater than the dividend s hundreds digit. Begin by dividing a 2digit number by a 1digit number where the divisor is greater than the dividend s tens digit. (i.e. there are fewer tens blocks available than the number of circles):
14 :1 PAGE ASK: How many tens blocks are in 27? Into how many circles do they need to be divided? Are there enough tens blocks to place one in each circle? How is this different from the problems you just did? Illustrate that there are no tens by writing a zero above the dividend s tens digit. Then ASK: What is 5 0? Write: Number of ones blocks (traded from tens blocks) to be placed Have a volunteer finish this problem, and then ask if the zero needs to be written at all. Explain that the algorithm can be started on the assumption that the tens blocks have already been traded for ones blocks. 5 Number of ones blocks in each circle 5 27 Number of ones blocks (traded from tens blocks) to be placed 25 Number of ones blocks placed 2 Number of ones blocks left over Emphasize that the answer is written above the dividend s ones digit because it is the answer s ones digit. Have students complete several similar problems Then move to 3digit by 1digit long division where the divisor is more than the dividend s hundreds digit. (EXAMPLES: 324 5; 214 4; 133 2) Again, follow the standard algorithm (writing 0 where required) and then introduce the shortcut (omit the 0). When students are comfortable with all cases of 3digit by 1digit long division, progress to 4digit by 1digit long division. Start by using base ten materials and going through the steps of the recording process as before. Some students may need to practise 1 step at a time, as with 3digit by 1digit long division. ASK: How long is each side if an octagon has a perimeter of a) 952 cm b) 568 cm c) 8104 d) 3344 Extension 1. By multiplying the divisors by ten and checking if the products are greater or less than their respective dividends, students can determine if the answers to the following problems will have one or two digits For example, multiplying the divisor in 72 3 by ten results in a product less than 72 (3 10 = 30), meaning the quotient for 72 3 is greater than 10 and has two digits. On the other hand, multiplying the divisor in 38 4 by ten results in a product greater than 38 (4 10 = 40), meaning the quotient for 38 4 is less than 10 and has one digit. 108 Copyright 2007, JUMP Math TEACHER S GUIDE
15 :1 PAGE Then have students decide how many digits (1, 2, 3 or 4) each of these quotients will have: BONUS: (Answer: = is less than the dividend and = is more than the dividend, so the quotient is at least but is less than , and so has 7 digits.) NS542 Topics in Division (Advanced) GOALS Students will use division to solve word problems. PRIOR KNOWLEDGE REQUIRED Long division of 2, 3, and 4digit numbers by 1digit numbers Remainders Word problems VOCABULARY divisor quotient remainder divisible by Explain that division can be used to solve word problems. SAY: Let s start with a word problem that doesn t require long division. A canoe can hold three kids. Four kids are going canoeing together. How many canoes are needed? (2) ASK: What happens when long division is used to solve this question? Does long division give us the same answer? Remainder Long division suggests that three kids will fit in one canoe and one kid will be left behind. But one kid can t be left behind because four kids are going canoeing together. So even though the quotient is one canoe, two canoes are actually needed. How many canoes are needed if 8 kids are going canoeing together? 11 kids? 12 kids? 14 kids? Students should do these questions by using long division and by using common sense. BONUS: How many canoes are need for 26 kids? 31 kids? 85 kids? How does long division help when we have larger numbers? If Anna reads two pages from her book every day, and she has 13 pages left to read, how many days will it take Anna to finish her book? If Rita reads three pages every day, how long will it take her to read a) 15 pages? b) 17 pages? c) 92 pages? d) 84 pages? e) 67 pages? If Anna reads eight pages every day, how long will it take her to read a) 952 pages? b) 295 pages? c) 874 pages? d) pages? e) pages? 109
16 :1 PAGE 99 (Adapted from the Atlantic Curriculum Grade 4) Students should understand that a remainder is always interpreted within the context of its respective word problem. Students should understand when a remainder child receives 2 pieces, and the remaining piece is further divided into thirds. So each child receives 2 and a third licorice pieces. enough money to buy five notebooks, the 25 is ignored. children because none of the children can be left behind. be transported in 3 buses, 30 students will ride in two buses, and 31 students will ride in the other. To guide students through question 2 on the worksheet, have them solve the following problems by long division: 72 3, 88 3, 943 3, , Then ASK: Which of these numbers are divisible by 3: 72, 88, 943, 1011, 8846? How can you tell? (Look at the remainder if the remainder when dividing by 3 is 0, then the number is divisible by 3.) To guide students through question 3 on the worksheet, have them find the pattern in the remainders when dividing by 4. Use a Tchart: Answer when Number divided by R R R R R R R R 0 ASK: What is the pattern in the remainders? (1, 2, 3, 0 then repeat) What is the next number that will have remainder 3 when divided by 4? (11) And the next number after that? (15) How can we continue to find all the numbers that have remainder 3 when divided by 4? (Skip count by 4 starting at 3 to get: 3, 7, 11, 15, and so on). Have students circle all the numbers on a hundreds chart that have remainder 3 when divided by 4. Repeat with numbers that have remainder 2 when divided by 3, but have them use a coloured pencil to circle the numbers on the same hundreds chart as before. ASK: What is the first number that has a remainder of 3 when divided by 4 and a remainder of 2 when divided by 3? 110 Copyright 2007, JUMP Math TEACHER S GUIDE
17 :1 PAGE 99 To guide students through question 4 on the worksheet, begin with dividing only 1 type of snack evenly into packets. Start with 12 raisins. ASK: How many ways can we divide 12 raisins evenly into more than one packet (6 each into 2 packets; 4 each into 3; 3 each into 4; or 1 each into 12). If we want to divide 18 pretzels evenly into more than one packet, how many ways can we do so? (9 each into 2; 6 each into 3; 3 each into 6; 2 each into 9; or 1 each into 18) How can we divide 18 pretzels and 12 raisins evenly into more than one packet? (9 pretzels and 6 raisins into 2 packets; or 6 pretzels and 4 raisins into 3 packets) What is the greatest number of packages we can make if we want to divide 18 pretzels and 12 raisins evenly? (3) What is the greatest number of packages we can make if we want to divide evenly into packages a) 32 raisins and 24 pretzels b) 55 raisins and 77 pretzels c) 54 raisins and 36 pretzels d) 28 raisins, 32 pretzels, and 44 dried cranberries e) 52 peanuts, 40 almonds, and 28 cashews ACTIVITY Distribute a set of base ten blocks and three containers large enough to hold the blocks to each student. Ask students to make a model of 74. Point out that each of the tens blocks used to make the model is made up of ten ones blocks. Then ask them to divide the 74 ones blocks into the three containers as evenly as possible. Ensure that they understand leftover blocks are to be left out of the containers. Their understanding of as evenly as possible could mean that they add the leftover blocks to some of the containers. Play this game with different numbers, in teams, and score points for correct answers. Track the steps your students use to divide the blocks by writing the respective steps from the algorithm. Ask your students to explain how the steps they used match the steps from the algorithm. 111
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