Division Landscape What is division? Division is the operation inverse to multiplication; the finding of a quantity, the quotient, that when multiplied by a given quantity, the divisor, gives another given quantity, the dividend; the process of ascertaining how many times one number or quantity is contained in another. There are four terms which describe the four numbers in a division problem. The dividend is the number that is being divided. The divisor is the number that the dividend will be divided by. The quotient is the number of times the divisor will go into the dividend. The remainder is a number that is less than the divisor and is too small to be divided by the divisor to form a whole number. Ways to write division: 3 2)6 = 3 = 3 6 2 = 3 How is division related to other mathematical operations? Division is the opposite or reciprocal operation of multiplication. In other words, division undoes multiplication. 4 X 3 = 12 and 12 3 = 4 as well as 12 4 = 3 Division can also be thought of as multiple subtraction. 12 3 = 4 is also 12 3 = 9, 9 3 = 6, 6 3 = 3 and 3 3 = 0 In other words, you can subtract 3 from 12 a total of 4 times. How is division useful in everyday life? Division is very useful when trying to estimate rate and distance in real life. For example, if you are on a trip and you are driving from San Francisco to Los Angles and you know the distance is 403 miles, you would be able to surmise that if you drove 50 miles per hour, your trip would take about 8 hours. You would know this because 400 50 = 8. Division is also useful as figuring percentages. Let s say you are shopping and you find an item that is on sale for 50 percent off. The item usually sells for 50 dollars. You could find the sale price of the item by dividing 50 dollars by 2.
Types of division problems: In division there are two types of problems: Partitive and Measurement. This section will talk about both types of problems as well as some models, strategies and manipulatives that students might employ to solve these division problems. There will also be some discussion on how rules of operations apply to division problems in general. Partitive Division Problems: In partitive division problems, these are the things you know. You know the total amount of what you are working with and the number of groups you want to make. The thing that you don t know is the size of your groups. This can also be called the fair sharing concept. Here is an example of a partitive division story problem. Chuck has 12 pieces of pizza to share. He wants to share his pizza equally among his four friends. How many pieces of pizza will each of Chuck s four friends get? Strategies: Direct modeling: Students may use a manipulative such as 12 counter disks or blocks. They would then gather 12 of their chosen manipulative and distribute them into 4 different piles (one for you, one for you ). They would then count the number of disks or blocks in their piles when they had distributed all of their pizza slices to determine the result. Counting strategies: A student may use a skip counting strategy by counting how many fours are contained in their 12 manipulatives. Derived facts: A student may know another set of multiplication facts like their 5s better than their 4s and use that to figure out the answer. I know that there are 3 5s in 15, maybe that can help me. Models: 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4
Measurement Division Problems: Here is what you know in measurement problems. You know the total amount you are working with and you know the size of the groups. What you don t know is the number of groups to make. Here is an example of a measurement division problem: Sarah had 12 apples to hand out to her class. Each group of students in the class got 3 apples. How many groups were there in the class? Strategies: Direct modeling: Students may use a manipulative such as 12 counter disks or blocks. They would then gather 12 of their chosen manipulative and distribute them into different piles meant to represent each group. They would count the number of piles to determine the number of groups. Counting strategies: A student may use a skip counting strategy by counting how many set of 3s are contained in their set of 12 manipulatives. Derived facts: a student may know another set of multiplication facts like their 5s, better than their 3s and use that to figure out the answer. I know that there are 3 5s in 15, maybe that can help me. Models: 0 1 2 3 4 5 6 7 8 9 10 11 12 3 4 12 Properties of Division: Some to the properties of mathematics only work one direction when it comes to division. For example, the identity works when 12 is divided by 1 because the answer is 12, but when you divide 1 by 12 you get 1/12. The commutative property does not work under division. 12 divided by 4 is 3 but 4 divided by 12 is 4/12 or 1/3. The closure property does not work under division for integers. If you divide the integer 1 by the integer 3 you do not get an integer as a quotient; you get the rational number 1/3.
DIVISION ALGORITHMS 1. 15 3 1 2 3 4 5 Count out as many counter blocks as the number in the dividend. Arrange the blocks into groups the size of the divisor. Count the number of groups to get the quotient 2. A. Dividing a 2 digit number by a 1 digit number without grouping: 44 2 Start with 4 ten blocks and 4 one blocks, this represents the dividend. Create 2 groups, this represents the divisor. Distribute both the tens and the ones equally to the groups, the number inside each group represents the quotient. 22 22 B. Dividing a 2 digit number by a 1 digit number with regrouping. 44 3 14 14 14 Start with 4 tens and 2 ones blocks to represent the dividend. Create 3 groups to represent the divisor. Share 3 tens equally with the 3 groups. Unpack the remaining ten into ones blocks. Distribute the ones equally to the groups. The number inside each group represents the quotient
3. Dividing using the standard algorithm. 5897 32 184 r. 9 32)5897 32 269 256 137 128 9 1. Find out how many 32s are in 58 2. Write that number above the dividend 3. Multiply result by divisor 4. Write that product under the dividend and then subtract the product from the dividend. 5. Is subtraction less than the divisor? If no, then you have made a mistake and have chosen a number that is too small. If yes, bring down the next digit in the dividend. 6. Repeat steps 1 5 until you reach 0 or after all digits have been brought down and the number is smaller than the divisor. If you have a number left over, that is your remainder! 4. Dividing using the alternative algorithm, Scaffolding 5897 32 32 5897-3200 100 2697 + - 1600 50 1097 + = 184 r. 9 960 30 137 + 128 4 9 1. Estimate how many 32s are in 5897 (ex. 100) 2. Make note of the number you use; this is called the partial quotient 3. Multiply 32 by 100. Subtract product from dividend. 4. Repeat steps 1 3 until you reach 0 or a number smaller than the divisor. 5. Add the partial quotients to find your answer. If you have anything left over this is the remainder.
Division Facts Dividing by 0 Numbers cannot be divided by 0 because it is impossible to make 0 groups of a number. Dividing by 1 Any number divided by 1 equals that number. If you divide by 1 you have one group and everything is in that group. Dividing by 2 and 3 The division facts for twos and threes are listed in the following table: TWO 0 2 = 0 2 2 = 1 4 2 = 2 6 2 = 3 8 2 = 4 10 2 = 5 12 2 = 6 14 2 = 7 16 2 = 8 18 2 = 9 THREE 0 3 = 0 3 3 = 1 6 3 = 2 9 3 = 3 12 3 = 4 15 3 = 5 18 3 = 6 21 3 = 7 24 3 = 8 27 3 = 9 The division facts for four, five and six are listed in the following table: FOUR 0 4 = 0 4 4 = 1 8 4 = 2 12 4 = 3 16 4 = 4 20 4 = 5 24 4 = 6 28 4 = 7 32 4 = 8 36 4 = 9 FIVE 0 5 = 0 5 5 = 1 10 5 = 2 15 5 = 3 20 5 = 4 25 5 = 5 30 5 = 6 35 5 = 7 40 5 = 8 45 5 = 9 SIX 0 6 = 0 6 6 = 1 12 6 = 2 18 6 = 3 24 6 = 4 30 6 = 5 36 6 =6 42 6 = 7 48 6 = 8 54 6 = 9
The division facts for seven, eight and nine are listed in the following table: SEVEN 0 7 = 0 7 7 = 1 14 7 = 2 21 7 = 3 28 7 = 4 35 7 = 5 42 7 = 6 49 7 = 7 56 7 = 8 63 7 = 9 EIGHT 0 8 = 0 8 8 = 1 16 8 = 2 24 8 = 3 32 8 = 4 40 8 = 5 48 8 = 6 56 8 = 7 64 8 = 8 72 8 = 9 NINE 0 9 = 0 9 9 = 1 18 9 = 2 27 9 = 3 36 9 = 4 45 9 = 5 54 9 = 6 63 9 = 7 72 9 = 8 81 9 = 9 The division facts for ten, eleven and twelve are listed in the following table: TEN 0 10 = 0 10 10 = 1 20 10 = 2 30 10 = 3 40 10 = 4 50 10 = 5 60 10 = 6 70 10 = 7 80 10 = 8 90 10 = 9 ELEVEN 0 11 = 0 11 11 = 1 22 11 = 2 33 11 = 3 44 11 = 4 55 11 = 5 66 11 = 6 77 11 = 7 88 11 = 8 99 11 = 9 TWELVE 0 12 = 0 12 12 = 1 24 12 = 2 36 12 = 3 48 12 = 4 60 12 = 5 72 12 = 6 84 12 = 7 96 12 = 8 108 12 = 9 Common Mistakes in Division Sometimes when doing division, students will ignore place value and think of all digits as ones. This mistake can sometimes lead to correct answers, but most often not. Examples: 213 142 Notice that nothing is written under the divisor. 3) 639 4) 518 This is a red flag that something is wrong. Correct this time! Incorrect this time!
Sometimes when doing division students will record their answer right to left just as they learned to do in multiplication, addition and subtraction. Examples: 52 68 3) 75 6) 516 60 480 15 36 15 36 0 0 Sometimes while doing division students will forget to use 0 as a place holder in a quotient. Sometimes students will remember to use a 0, but put it in the wrong place. Examples: No 0 s as place holder 32 r. 3 78 r. 2 9) 2721 6) 4250 27 42 21 50 18 48 3 2 0 s used incorrectly as place holders. 830 530 r. 4 6) 4818 7) 3525 4800 3500 18 25 18 21 0 4
Quick Division Tricks! Dividing by 3 Add up the digits: If the sum is divisible by 3 then the number is as well Examples: 1. 111111: 1+1+1+1+1+1 = 6 so 111111 is divisible by 3 because 6 is divisible by 3. 2. The sum of the digits of 87687687 is 57. The sum of the digits of 57 is 12. 12 is divisible by 3 so both 56 and 87687687 are divisible by 3. Dividing by 4 Look at the last two digits. If the number formed by the last two digits is divisible by 4, the original number is as well. 1. 124 is divisible by 4 because the last two digits are 24 and 24 is divisible by 4. 2. 17328296458312 is divisible by 4 because the last 2 digits are 12 and 12 is divisible by 4. Dividing by 5 If the last digit is a 0 or a 5 then the number is divisible by 5 Dividing by 6 Check 3 and 2. If the number is divisible by both 3 and 2 than it is divisible by 6 as well. Dividing by 7 To find out if a number is divisible by seven, take the last digit, double it, and subtract it from the rest of the number. If the result is divisible by 7 then the original number is divisible by 7. Example: If you had 203, you would take the last digit 3, double it to get 6 and you would then subtract 6 from 20 (the other digits in the number). The result is 14 which is divisible by 7 so 203 is also divisible by 7. If you don t know the new number s divisibility, you can apply the rule again. Dividing by 8 Check the last 3 digits. If the last 3 digits are divisible by 8 then the whole number is divisible by 8. Example: 332645888 is divisible by 8 because 888 is divisible by 8.
Dividing by 9 Add the digits together. If the sum of the digits is divisible by nine, then the original number is as well. Dividing by 10 If the number ends in 0, then it is divisible by 10. Dividing by 11 To check if a number is divisible by 11 take the last digit and subtract it from the remaining digits. If the result is divisible by 11 then the number is divisible by 11. If you aren t sure the first time you can repeat the steps until you get a number within the common 11 multiple range. Example: 4059 The last digit is 9 and if you subtract that from 405 you get 396. The last digit of 396 is 6 and if you subtract that from 39 you get 33. 33 is divisible by 11 so 4059 is divisible by 11. Dividing by 12 Check for division by 3 and 4.