Illustrated Fractions

Transcription

1 Illustrated Fractions Jetser Carasco Copyright 008 by Jetser Carasco All materials on this book are protected by copyright and cannot be reproduced without permission. 1

2 Table o contents Lesson #0: Preliminaries Lesson #1: Introduction to ractions Lesson # : Equivalent ractions Lesson # : Simpliying ractions Lesson # : Proper, improper ractions and mixed numbers----0 Lesson # 5: Renaming mixed numbers/improper ractions Lesson # 6: Comparing ractions Lesson # 7: Adding ractions Lesson # 8: Subtracting ractions Lesson # 9: Multiplying ractions Lesson # 10: Dividing ractions Answers or practice questions

3 Lesson # 0: Preliminaries Greatest common actor The greatest common actor (GCF) is the largest actor o two numbers. An understanding o actor is important in order to understand the meaning o GCF. What are actors? When two or more numbers are multiplied in a multiplication problem, each number is a actor in the multiplication. Take a look at the ollowing multiplication problem: 8. is a actor. 8 is also a actor. You can ind all actors o a number by inding all numbers that divide the number. Find all actors o 6: Start with 1. Since 1 divide 6, 1 is a actor. divides 6, so is actor. divides 6, so is a actor. I you continue with this pattern, you will ind that 1,,,, 6, 9, 1, 18, and 6 are all actors o 6.

4 An easier way to handle the same problem is to do the ollowing: 1 6 = 6 18 = 6 1 = 6 9 = = 6 9 = 6 Note: When the actors start to repeat, you have ound them all. In our example above, the actors started to repeat at 9 = 6 because you already had 9 = 6. Thereore, we have ound them all. Now that you have understood how to get the actors o a number, it is going to be straightorward to get the greatest common actor. Whenever you are talking about greatest common actor, you are reerring to, numbers, or more. Here, we will concern ourselves with just. The GCF o two numbers is the largest actor o the two numbers.

5 Examples #1 Find GCF o 16 and written as GCF (16,). Using the method described above, The actors or 16 are 1,,, 8, and 16. The actors or are 1,,,, 6, 8, 1, and. The largest actor both numbers have in common is 8, so GCF (16,) = 8. Notice that is also a common actor. However, is not the greatest. Examples # Find GCF (7,1) The actors or 7 are 1 and 7. The actors or 1 are 1,,,, 6, and 1 The largest number both actors have in common is 1, so GCF (7,1) = 1 Note: I number a divides b, the greatest common actor is a. 5

6 Examples # Find GCF (,6) Since divides 6, GCF (,6) = Resources: A calculator that will list all actors or any number entered. A calculator that will compute the greatest common actor or you with the click o a button. A un greatest common actor game that will help you test your knowledge o the GCF. 6

7 Least common multiple An understanding o multiples is important in order to understand the meaning o least common multiple. The multiples o a number are the answers that you get when you multiply that number by the whole numbers. Remember that the whole numbers are all numbers rom 0 to ininity. Whole number = {0, 1,,,, 5, 6, 7, 8,...} For example, to get the multiples o, multiply by 0, 1,,,, 5,.... I put the dots to show that the set o whole numbers continues orever. Ater you perorm these multiplications you get: {0,, 8, 1, 16, 0,.....} The multiples o 9 are all the numbers that you get when you multiply 9 by 0, 1,,,, 5, 6,..... Ater you perorm these multiplications, you get: {0, 9, 18, 7, 6, 5, 5,..} The least common multiple (LCM) o two numbers is the smallest number that is a multiple or both numbers. 7

8 Example #1: Find LCM o 6 and 9 First list all the multiples o 6 You get {0, 6, 1, 18,, 0, 6,, 8, 5, 60,...} Next, list all the multiples o 9 You get {0, 9, 18, 7, 6, 5, 5, 6, 7, 81, 90...} Careully examine the two lists and you will see that the smallest number that is a multiple o both 6 and 9 is 18. You can also write LCM (6, 9) = 6 O course, 6 is also a common multiple o 6 and 9.However, it is not the smallest. Example #: Find LCM o,and Multiples o are {0,,, 6, 8, 10,...} Multiples o are {0,, 6, 9, 1, 15...} Again, careully examine the two lists and you will see that the smallest number that is a multiple o both and is 6. You can also write LCM (, ) = 6 Note: When two numbers are prime, the least common multiple is ound by simply multiplying the numbers. However, what is a prime number? 8

9 A prime number is a number that can be divided evenly only by 1 and itsel. This means that the number has exactly two actors. For example, 1 is a prime number because 1 can only be divided evenly by 1 and 1. Any other number we try to divide 1 by will yield a decimal number. I the number has more than two actors, we say that the number is composite. For instance, 1 is a composite number because 1 can be divided by numbers other than 1 and 1, such as,, and. Note that in example #, since and are prime numbers, we could have just multiplied the numbers to get the least common multiple. Example #: Find LCM o 5 and 7 Since 5 and 7 are prime numbers, we can just multiply 5 and 7 to get the least common multiple 5 7 = 5, so the least common multiple is 5. Note: I number a divides b, the least common multiple is b. Example #: Find LCM o and Since divide, LCM (, ) = You can always list all multiples o. 9

10 Multiples o are {,, 6, 8, 10, 1.} Then list all multiples o. Multiples o are {, 8, 1, 16, 0,.} However, listing all the multiples or both numbers can take time. I you remember the act above, it takes a lot less time. Resources: A calculator that will compute the least common multiple or you with the click o a button. A un least common multiple game that will help you test your knowledge o the LCM. A calculator that will tell you i a number is prime 10

11 Divisibility Rules Divisibility rules o whole numbers are very useul because they help us quickly determine i a number can be divided by,,, 5, 6, 7, 8 9, and 10 without doing long division. Divisibility means that you are able to divide a number evenly. For instance, 8 can be divided evenly by because 8/ =. However, cannot divide 8 evenly. To illustrate the concept, let's say you have a cake and your cake has 8 slices. You can share that cake between you and more people evenly. Each person will get slices. However, i you are trying to share those 8 slices between you and more people, there is no way you can do this evenly. One person will end up with less cake. In general, a whole number x divides another whole number y i and only i you can ind a whole number n such that x n = y For instance, 1 can be divided by because = 1 When the numbers are large, use the ollowing divisibility rules: Rule #1: divisibility by A number is divisible by i its last digit is 0,,, 6, or 8. For instance, is divisible by because the last digit is. 11

12 Rule # : divisibility by : A number is divisible by i the sum o its digits is divisible by. For instance, 11 is divisible by because = 9 and 9 is divisible by. Rule # : divisibility by A number is divisible by i the number represented by its last two digits is divisible by. For instance, 890 is divisible by because 0 is divisible by. Rule #: divisibility by 5 A number is divisible by 5 i its last digit is 0 or 5. For instance, is divisible by 5 because the last digit is 5. Rule # 5: divisibility by 6 A number is divisible by 6 i it is divisible by and. Be careul! It is not one or the other. The number must be divisible by both and beore you can conclude that it is divisible by 6. Rule # 6: divisibility by 7 To check divisibility rules or 7, study careully the ollowing two examples: Is 8 divisible by 7? Remove the last digit, which is 8. The number becomes. 1

13 Then, double 8 to get 16 and subtract 16 rom. 16 = 18 and 18 is not divisible by 7. Thereore, 8 is not divisible by 7. Is 7961 divisible by 7? Remove the last digit, which is 1. The number becomes 796. Then, double 1 to get and subtract rom = 79. Since it is still too big, repeat the process. Remove the last digit, which is. The number becomes 79. Then, double to get 8 and subtract 8 rom = 71. Since it is still too big, repeat the process. Remove the last digit, which is 1. The number becomes 7. Then, double 1 to get and subtract rom 7. 7 = 5 and 5 is divisible by 7. Thereore, 7961 is divisible by 7. Rule #7: divisibility by 8 A number is divisible by 8 i the number represented by its last three digits is divisible by 8. For instance, 5870 is divisible by 8 because 0 is divisible by 8. Rule #8: divisibility by 9 A number is divisible by 9 i the sum o its digits is divisible by 9. For instance, 11 is divisible by 9 because the sum o its digits is divisible by 9. 1

14 Rule # 9: divisibility by 10 A number is divisible by 10 i its last digits is 0. For instance, 580 is divisible by 10 because the last digit is 0. Resources: Use this calculator to determine i a number is divisible by any other number. A game that will help you test how well you know the divisibility rules. We have now covered all preliminaries. Although the primary goal o this book is to help you visualize ractions, it is also important to have in hand a quick and eective way to solve raction problems. This is what the GCF, LCM, and divisibility rules will help you do. We now introduce the concept o ractions next. 1

15 Lesson # 1: Introduction to ractions A raction is part o a whole number. Look at the ollowing igures. The shaded portion o each igure is part o the whole igure. Figure 1 Figure You can also say that the shaded portion is a raction o the whole igure. Figure 1 has parts and only 1 part is shaded. We say that one-ourth o the igure is shaded and We call the number on top, which is 1, the numerator. We call the number at the bottom, which is, the denominator. We call the line that separates the two numbers raction bar. 15

16 Similarly, igure has shaded parts out o 8 parts. Once again the number on top, which is is called the numerator and 8 is the denominator. A real lie example o ractions is when you buy a pizza and eat only some o that pizza. Since you did not eat the whole pizza, you only ate a raction o the pizza. I your pizza has 10 slices and you eat slices, you can express this as 10 Another good example is a chocolate bar. A chocolate bar is usually made up o pieces a chocolate shaped like rectangles. I your chocolate bar has 1 pieces and you eat only 1 piece, you can express this as

17 Practice: Introduction to ractions Write the ractions or each o the ollowing igure: 1) ) ) ) 5) 17

18 Lesson # : Equivalent ractions Look at the ollowing illustration: Rectangle 1 Rectangle 1 = Rectangle 1 has parts and only 1 part is shaded, so we can write 1 Rectangle has parts and only parts are shaded, so we can write However, since rectangle 1 is equal to rectangle and the portion that is shaded is the same or both rectangles, we can conclude that 1 = We call 1 and 1 equivalent ractions. Notice that = =

19 Equivalent ractions are ractions that are equal, but have dierent numerators and denominators. Now, what will happen, i in each rectangle, we cut the shaded and the non-shaded area in hal? We do this by drawing two lines in each rectangle. The two lines are shown in black Rectangle 1 Rectangle = 8 Rectangle 1 has now parts and only parts are shaded, so we can write Rectangle has now 8 parts and only parts are shaded, so we can write 8 Rectangle 1 is still equal to rectangle and the portion that is shaded is still the same or both rectangles. We can conclude that = 8 19

20 We call and 8 equivalent ractions. Notice that = = You could also break rectangle in 6 parts. Just notice that now parts out o 6 parts are shaded. Rectangle 1 Rectangle 1 = 6 Once again, we call 1 and 1 equivalent ractions. Notice that = = Here is what we got so ar: 1 = = 0.5, = 8 = 0.5, and 1 = = Since all these ractions are equal to 0.5, they are equal to one another. Putting it all together, we conclude that 1 = = = 6 8 0

21 Notice that to get rom 1 to,, and we can just multiply both 6 8 the numerator and the denominator o 1 by,, and. It is thus not hard to ind equivalent ractions or a given raction. Just multiply the numerator and the denominator o the given raction by any but the same natural number greater than 1 such as,,, 5, 6,. Example #1 Find two equivalent ractions or the ollowing ractions: and 5 a) B B = 6 and B5 B5 10 = 15 So, two equivalent ractions or are 10 and 6 15 b) 5 B 1 = 5B 15 B and 5B 16 = 0 So, two equivalent ractions or 5 1 are 16 and

22 Practice: Equivalent ractions Write the equivalent ractions or each pair o igure 1) ) Write two equivalent ractions or each number ) 5 = ) = 6 10 Note: It is important to know how to get equivalent ractions when adding and subtracting ractions with unlike denominators, which will be covered in lesson #8

23 Lesson # : Simpliying ractions Simpliying a raction simply means that you will look or an equivalent raction that has the smallest numbers you can get or the numerator and the denominator. Note: This lesson will use the GCF as a shortcut. Make sure you understand how to ind the GCF. Simpliying raction is then the reverse o inding equivalent ractions. I outlined some steps below beore using the GCF: Step #1 Just divide the numerator and the denominator by until the numerator and the denominator are no longer divisible by or cannot be divided by anymore. I the numerator and the denominator cannot be divided by at the same time, go to Step #. Step # Then, divide the numerator and the denominator by until the numerator and the denominator are no longer divisible by or cannot be divided by anymore. I the numerator and the denominator cannot be divided by at the same time, go to Step #.

24 Step # Divide the numerator and the denominator by until the numerator and the denominator are no longer divisible by or cannot be divided by anymore. I the numerator and the denominator cannot be divided by at the same time, go to Step #... And so orth As you may have noticed, you just keep dividing by the next higher number. I the next higher number cannot divide the numerator and the denominator evenly, you try the next higher number and so orth Example #1 Simpliy 18 6 Step #1: Divide 18 and 6 by 18 18D = 9 = 6 6D 18 Since cannot divide 9 and 18 at the same time, go to step #. Step #: Divide 9 and 18 by.

25 Step # 9 9D = = 18 18D 6 Since can divide and 6 at the same time, repeat step #. D = 1 = and you are done since 1 and cannot be divided 6 6D anymore evenly by any number. Example # 1) Simpliy 8 16 Step #1: Divide 8 and 16 by Since cannot divide and 6 at the same time, go to step #. Step #: Divide and 6 by 6 = D 6D 1 = 1 Since cannot divide 1 and 1 at the same time, go to step #. Step #: Divide 1 and 1 by Since cannot divide 1 and 1 at the same time, go to step #. Step #: Divide 1 and 1 by 5 Since 5 cannot divide 1 and 1 at the same time, go to step #5. Step #5: Divide 1 and 1 by 6 5

26 Since 6 cannot divide 1 and 1 at the same time, go to step #6. Step #6: Divide 1 and 1 by 7 1 1D 7 = = and you are done since and cannot be divided 1 1D 7 anymore evenly by any number. The method outlined above works very well. However, it can quickly become time consuming, especially i you are working with big number. This is where the greatest common actor comes into place. In example #1, simpliying 18 1 gave 6 We get 1 by dividing 18 and 6 by 18 and 18 is the greatest common actor or 18 and 6. We already explained in lesson #0 (greatest common actor) how to ind the greatest common actor. We will briely review it here. Find greatest common actor or 18 and 6 written as GCF (18, 6) First, list all actors o 18: 1,,, 6, 9, and 18 Second, list all actors o 6: 1,,,, 6, 9, 1, 18, and 6 GCF (18, 6) = 18 because 18 is the greatest actor 18 and 6 have in Common. 6

27 Note that and 9 are also common actors, but they are not the greatest. Notice that 18 divides 6, so as we said beore, so we can immediately conclude that 18 is the greatest common Factor. In example #, simpliying 8 gave 16 We get by dividing 8 and 16 by and is the greatest common actor or 8 and 16 Find greatest common actor or 8 and 16 written as GCF (8, 16) First, list all actors o 8: 1,,,, 6, 7, 1, 1, 1, 8,, and 8 Second, list all actors o 16: 1,,, 6, 7, 1, 18, 1,, 6 and 16 GCF (8, 16) = because is the greatest actor 8 and 16 have in common. Examples # Simpliy 15 5 Find GCF (15, 5) Factors o 15 are 1,, 5, and 15 Factors o 5 are 1, 5, and 5 GCF (15, 5) = 5 because 5 is the greatest actor 15 and 5 have in common. Just divide 15 and 5 by GCF (15, 5). We already know the GCF. It is 5. We just explained this above. 7

28 So, 15D5 = 5D5 5 Examples # Simpliy 1 First, ind GCF (1, ) Factors o 1 are 1,,,, 6, and 1 Factors o are 1,,, 6, 7, 1, and GCF (1, ) = 6 Just divide both 1 and by 6 We get: 1D6 = D6 7 Resources: A calculator that will simpliy any raction. Simpliying ractions is useul when we try to make an answer look simpler or make the numerator and the denominator look small. Sometimes, simpliying ractions beore adding, subtracting, dividing, and/or multiplying can make calculation less complicated and conusing. 8

29 Practice: simpliying ractions Exercises: Simpliy the ollowing ractions 1) 6 ) 6 1 ) 9 ) 8 5) 1 6) ) 7 1 8) 16 9

30 Lesson # : Proper, improper ractions and mixed numbers A proper raction is a raction that has a smaller numerator than a denominator. The ollowing are all proper ractions: An improper raction is a raction that has a bigger numerator than a denominator. The ollowing are all improper ractions: A mixed number is a number that is a combination o a whole number and a proper raction. The ollowing are mixed numbers

31 Notice that 1 1 means and 1 or + Thereore, 1 1 is per se a simpliied ormat or + Nonetheless, know this once and or all that there is an addition sign between the whole number and the proper raction. Mixed numbers can be converted to improper ractions and vice versa. More on this is coming in the next lesson. When adding and subtracting ractions, it is not necessary to convert mixed numbers to improper ractions beore adding or subtracting. Depending on the problem, one or the other could be used. However, when multiplying and dividing ractions, it is imperative to convert all mixed number(s) to improper ractions beore multiplying or dividing. The only exception is when you are using rectangles or circles to perorm the multiplication. We show this in lesson 9. 1

32 Lesson # 5: Renaming mixed numbers and improper ractions Sometimes, you may need to rename or write an improper raction as a mixed number. Here is a simple scenario: You have a cake to share between people and your cake has 10 slices. Question: How many slices can each person get? To answer this question, you will need to convert 10 into a mixed number. You can argue that each person can get complete slices since B= 9 Then, there is 1 slice to share between people or 1 Combining and 1 1, we get the mixed number You can get the same answer when doing long division. In this case, the quotient is the whole number part and the remainder over the divisor is the proper raction. This is illustrated the next.

33 Long division gives: Quotient = and remainder divisor = 1 1 so, we get the same answer In general, use the ollowing ormula to convert improper ractions to mixed numbers. Quotient remainder divisor

34 Examples Convert and into mixed numbers. 5 a) 19 Long division gives: So 19 reminder = quotient divisor 1 = 9 b) 18 5 Long division gives:

35 So, 18 5 = 5 Renaming mixed numbers to improper ractions is a lot easier. Just multiply the whole number part by the denominator. Then, add the result to the numerator. Keep the same denominator. Examples Convert 1 and 9 into improper ractions 5 a) B6+ = 18+ = 1 = b) 9 1 9B+1 = 18+1 = 19 = c) B5+ = 0+ = =

36 Practice: proper, improper ractions, and mixed numbers Exercise A: Find the mixed number or each o the ollowing improper raction. 1) 19 5 ) 6 ) 15 ) 5) ) 5 Exercise B: Rename these mixed numbers as improper ractions 1) 1 ) 5 ) 1 5 ) 5) ) 5 6

37 Lesson # 6: Comparing ractions We compare ractions using greater than ( > ) and less than sign ( < ). For example, < 6 6 or > 6 6 How do we know < 6 6? Take a look at the ollowing illustrations with chocolate bars. 6 = Eating bars out o 6 bars (Represented as 6 ) is less than eating bars out o 6 bars (Represented as 6 ) 7

38 So, we say that < 6 6 Notice that the denominator is the same or both ractions. We draw rom the above the ollowing conclusion: When two ractions have the same denominator, the smaller raction is the one with the smaller numerator. Example #1 Compare and is bigger than because is bigger than. Thus, > You can also write < which means that is smaller than And again the reason is smaller than is because it has a smaller numerator 8

39 Example # Compare the ollowing ractions and put them in order rom least to greatest < and <, so the right order is 1 7 < 7 < 7 Now, what do we do when the ractions have dierent denominators? For example, given and, how do we know which one is greater? The trick is to look or equivalent ractions or both have the same denominators. and that To get the same denominator, ind LCM (,). At this point, it is expected that you already know how to get the least common multiple. I you are still struggling, ollow the steps outlined in lesson #0 and use the calculator listed as a resource to check your answer. Anyway, LCM (,) = 1. Since LCM (,) = 1, 1 is the common denominator. Notice that now, you should multiply the denominator o that is by and the denominator o that is by to get the same denominator. 9

40 Just remember that you have to multiply your numerator by the same number you multiplied the denominator. You need to do B B and B B And you will get equivalent ractions with the same denominators B 8 = B 1 and B 9 = B 1 Since 8 9 < then < 1 1 Here is an illustration or the situation above 0

41 Equivalent raction or Equivalent raction or 8 1 < 9 1 Once again, since 8 9 is smaller than then is smaller than 1 1 A good understanding o how to get equivalent is important when adding or subtracting ractions with unlike denominators. Make sure you understand this lesson very well. We will practice with a ew more examples to make sure you understand. 1

42 Example #1 Compare 1 1 and Since LCM (,) = 6, equivalent ractions are 1B = B 6 and 1B = B 6 > 1 1 so > 6 6 There is a quicker way to compare 1 1 and 1 1 and have the same numerator and it is 1. 1 means to break 1 into equal pieces 1 means to break 1 into equal pieces Let s say 1 is a dollar, which share will you choose? I am pretty sure you will choose to break a dollar into equal pieces because your share will be bigger than breaking a dollar into pieces. In general, when the ractions have the same numerator, the smaller raction is the one with the bigger denominator Example # Compare and 5 Since LCM (5,) = 15, equivalent ractions are

43 B 9 = and 5B 15 B5 10 = B < so < Example # Compare 5 1 and 17 Since LCM (17, ) = 51, equivalent ractions are 5B 15 = and 17B 51 1B17 17 = B < 5 so 1 < Useul tips to keep in mind when comparing ractions. Tip #1: When two ractions have the same denominators, the smaller raction is the one with the smaller numerator. Tip #: When the ractions have the same numerators, the smaller raction is the one with the bigger denominator Resources: A calculator that will compare two ractions and tell you which one is the bigger. A comparing ractions game to see how well you can compare ractions.

44 Practice: Comparing ractions Exercises: Compare the ollowing ractions. Tell whether the irst raction is smaller (<) or greater (>) than the second raction. Replace the question mark with your answer 1)? )? 1 ) 7? 6 ) 5? 6 5) 5? 6) 6 7? 7 8

45 Lesson # 8: Adding ractions Our goal in this lesson is to illustrate and then show a quick way to get the same answer. Take a close look at the ollowing two igures. Figure 1 Figure 5

46 In igure 1, the circle is broken down into parts So each part is hal or 1 Notice that two-halves is equal to one You could get the same answer i you do = = = 1 Similarly, three-thirds is equal to one and adding the three parts in igure gives: = = = 1 Let us explore this concept urther with modeling. Study the models given below careully! We will call these parts o a circle pattern blocks. Notice that we have not modeled 1 1, 1, and so orth The goal is not to model every raction, but to give you a oundation you can build upon. 6

47 Here is the model: Orange = 1 Blue = 1 = hal Red = 1 = one- ourth Brown = 1 = one-third Green = 1 6 = one-sixth 7

48 Now, let s play with these pattern blocks by doing addition. Quick important acts: Putting two red blocks next to each other is equal to one-ourth + one-ourth = two-ourths Putting two brown blocks next to each other is equal to one-third + one-third = two-thirds Putting 5 green blocks next to each other is equal to: one-sixth + one-sixth + one-sixth + one-sixth + one-sixth = ive-sixths Putting 6 green blocks next to each other is equal to six-sixths = one Examples #1 : = = 1 Notice that putting green blocks next to one another is equal 1 big blue block or hal. 8

49 You can get the same answer i you do = = D = 1 = D Examples # : As shown in example #1, to get the answer or , put green blocks next to each other = = Notice that putting green blocks next to each other is equal 1 big brown block and big brown blocks is equal two-thirds. 9

50 You could get the same answer i you do = = = D 6D = From these examples, we draw the ollowing rules: We can only add blocks with the same size. This will result in ractions having the same denominator. Fractions can be added only i the denominators are the same When adding ractions that have the same denominators, just add the numerators. We don t add denominators. Other examples: 1) = ) 1 + =

51 ) = 7 ) = ) = Resources: A calculator that will add two ractions and display the answer. See how well you can add ractions with this game. 51

52 Practice: adding ractions with the same denominator 1 ) 1 + = ) 5 + = ) + 6 = ) 1 + = 5) = 6) + = 7) = 8) 8 + = ) = 10) = 5

53 Adding ractions with unlike denominators Now, how do we add ractions with dierent denominators? The ollowing rules already mentioned above are so important, I shall emphasize them again. Rule #1: Fractions can be added only i the denominators are the same Rule #: When adding ractions that have the same denominators, just add the numerators. Let add 1 1 and First, let us add with pattern blocks = = 6 6 5

54 Notice that the blue block is traded or swapped with green blocks and the brown block is traded or swapped with green blocks. You could get the answer i you use the least common multiple. This is done next! According to rule #1, the denominators should be the same beore adding, so we need to look or equivalent ractions that have the same denominators. The process o looking or equivalent ractions was already demonstrated in the lessons about comparing Fractions. We shall repeat them here or the sake o making this crystal clear. Since LCM (,) = 6, equivalent ractions are 1B = B 6 and 1B = B 6 According to rule #, you can just add the numerators since both ractions have the same denominators So, = + + = 5 =

55 Here is another illustration or adding 1 1 and Example #1 + 5 Since LCM (5, ) = 15, equivalent ractions are B 9 = and 5B 15 B5 10 = B5 15 So, + 9 = = 19 = Example #

56 Notice that 17 and are prime, so the LCM is 17 = 51 Since LCM (17, ) = 51, equivalent ractions are 5B 15 = and 17B 51 1B17 17 = B17 51 So, = = = Example # 5 + We saw in the lesson about LCM that i number a divides b, the least common multiple is b Since divides, LCM (,) =. Thus the common denominator is Just multiply by to get the same denominator. There is no need to multiply by any number since it is already. Equivalent ractions or is B 6 = B So, 5 + = = = 56

57 Practice: Adding ractions with dierent denominators 1 ) + = ) = ) + = ) = 5) = 6) + = ) = 8) + = ) = 10) = 57

58 Renaming mixed numbers and improper ractions: Revisited with patterns blocks Converting an improper raction into a mixed number is the process o trading as many parts as you can into wholes. In order words, you will try to make as many 1s as you can with the parts and were already converted into mixed numbers using long division. Let s do it now with pattern blocks. The trick is to look at the denominator o the raction so you know which pattern block to use. For example, the denominator o 10 is. Thereore, you will try to make as many 1s as possible with the brown block or one-third Then, since the numerator is 10, you need to lay down 10 brown blocks. 58

59 Then, group them into piles o 1s 59

60 Notice that three brown blocks make 1. Since there are groups o brown blocks, it is equal to. The letover is 1 Your mixed number is 1 We will now do the same thing or 19 The denominator o 19 is. Thereore, you will try to make as many 1s as possible with the blue block or hal Then, since the numerator is 19, you need to lay down 19 blue blocks. This is shown on the next page. 60

61 Then, group them into piles o 1s 61

62 =1 =1 =1 =1 =1 =1 =1 =1 =1 Two blue blocks make 1. Since there are 9 groups o blue blocks, it is equal to 9. The letover is 1 Your mixed number is 9 1 6

63 Converting a mixed number into an improper raction is the reverse process. Instead o making 1s, you are trying to make as many parts as possible with the 1s. Convert 1 6 into a mixed number. Since you mixed number has one-sixth, you should break everything down into sixths. 1 1 = A model or this mixed number is shown below. Notice that 1 is equal to six-sixths. 6

64 All you have to do now is to count how many sixths there are. Since there are thirteen-sixths, the improper raction is 1 6 Let s do the same thing or Since you mixed number has two-thirds, it is a good indication that you should break everything down into thirds = A model or this mixed number is shown on the next page. Notice that 1 is equal to three-thirds 6

65 All you have to do now is to count how many thirds there are. Since there are ourteen-thirds, the improper raction is 1 65

66 Adding mixed numbers We can also add mixed numbers with pattern blocks. Add First model the problem. The three orange circles will add up to because each is equal to 1 66

67 Add now the blue block to the red block. Then trade in 1 blue block or two red blocks + = + + = So, = You could also solve the problem this way. First, add the whole numbers. Then, add the ractions The whole numbers are and 1, so + 1 =. The ractions are 1 1 and Since the least common multiple o and is, the common denominator is = 1B B 1 + = + 1 Again, =. = 67

68 Next, we outlined a general approach we can take when adding mixed numbers Recall mixed numbers have the ollowing ormat: Whole number numerator, denominator where numerator is a proper raction denominator For example, 5 is a mixed number. When adding mixed numbers, add all whole numbers portions. Then, add all ractional parts. For example,

69 Add we said beore in the lesson about mixed numbers, 1 1 = + and 5 5 = + So, = Add the whole number portions + = 5 Add the ractional parts: = 6 = The answer is or 5 + Notice though that 6 =. Thereore, you can simpliy the answer i you do 5 + = 8. In act, you should always simpliy an answer whenever possible. Let us make a small change in the problem above. The problem is now

70 Once again, + = 5 but = The answer is 5 7 Notice though that the ractional part 7 is not a proper raction. Furthermore, it cannot be simpliied to give a whole number. What you can do is to convert 7 into a mixed number and add again We know that 7 = 1 7, so 5 1 = 5 + Add the whole number portions: 5 + = 8 Add the ractional parts. There is no ractional part or 5. The only ractional part we have is 1 The answer is 8 1 you could also express the answer as an improper raction to get 8B+1 17 = 70

71 Example # = Add the whole number portions: + 5 = 9 Add the ractional parts We already added by looking or equivalent ractions with the same denominators. Equivalent ractions or 1 1 and are respectively and and + 5 = So, the answer is Example #

72 9 + 6 = Add the whole number portions: = 15 Add the ractional parts + 5 This one two was already added beore in example #1 (adding ractions with unlike denominators). + B = B5 + 9 = = 19 = 5 5B B So, we get Since 19 is improper, convert it to a mixed number and add it to = 1 so, = 15+ = 15+1 = = 16+ = Thus, =

73 Practice: Adding mixed numbers 1 ) 1 + = ) = ) + = ) = 5) = 6) + 1 = ) = 8) + = ) = 10) = 5 9 7

74 Subtracting ractions The rules or subtracting ractions are no dierent than the one or adding ractions, with the exception that instead o adding the numerators, you now subtract them. Rule #1: Fractions can be subtracted only i the denominators are the same Rule #: When subtracting ractions that have the same denominators, just subtract the numerators. We will illustrate with examples: Example 1 = = = 1 Illustration or example #1 First, model 7

75 Then, cross out one hal The answer is two-halves or 1. Example #: 7@ = 5 = Example #: 1 = = 1 = Illustration or example #: First, model hal 75

76 Then, remove one-third. (One-third is shown with black lines) The portion that is let is a portion that you can probably recognize easily. We show it below. It is 1 6. Example #: = Subtract whole number portions: = 1 Subtract ractional parts: 5 = 10 = 10@1 9 = 76

77 We get 1 9 Since 9 is an improper ractions, you could convert it into a mixed number and add it to 1 9 = 1, so 1 9 = = Example #5: Subtract whole number portions: 6 1 = 5 Subtract ractional parts: 1 = = 1 = We get Resources: A calculator that will subtract two ractions and display the answer A game to practice subtraction o ractions 77

78 Practice: subtracting ractions 1 = ) = ) 1 = ) 1 = 6 5 = = 5 7 7) 8 = 8) = ) = = 5 78

79 Multiplying ractions Why do we just multiply the numerators and the denominators? We will challenge the concept with illustrations and a real lie example. Our approach is to illustrate and then show a quick way to get the same answer Get ready to sweat rom this point on! Look at the ollowing illustration: What we are trying to do is 1 B 1 and it means o First, we draw by shading the igure with the orange color and it is the igure on the let. Second, we break the whole igure in hal by shading it with blue lines. 79

80 The answer to the multiplication is where the igure is shaded twice with the orange color and the blue lines. We show this below with red dots. This shaded portion represents 1 o the igure. Thus, B = 8 8 But, you can get the answer i you do 1 B 1B = = B 8 Illustration # 1 1 B Again, 1 B 1 1 means o or o. 80

81 Let s model o. First, you can model Then, break the same igure into equal parts. Finally, shade 1 o the igure. This is shown with red lines The answer to the multiplication is where the igure is shaded twice with the orange color and the red lines. We show this above with blue dots. 81

82 That area is 1 out o 1 or so 1 B 1 = 1 You can get the same answer i you do: 1 B 1 1B1 = B 1 = 1 Illustration # B 5 Again, B means o or o. Let s model o First, you can to model Then, break the same igure into 5 equal parts. 8

83 Then shade 5 o the igure. This is shown with blue lines The answer to the multiplication is where the igure is shaded twice with the orange color and the blue lines. We show this above with red dots That area is 6 out o 15 or 6, so B 6 = You could get the same answer i you do: B B = 6 = 5 B5 15 8

84 Let us talk about pizzas now! Say that you bought a medium pizza and your pizza has 8 slices. I someone eats hal o your pizza, or slices, you are let with slices. The letover can be expressed as 8 1 or I you decide that you are only going to eat 1 slice out o the slices remaining, you are eating 1 o the letover. Remember that the letover is 1 1 1, so you are eating o This means 1 1 B You can also argue that you only ate 1 slice out o 8 slices or 1 8 since the pizza has 8 slices. Thus, we can see that eating 1 1 o 1 is the same as eating 8 Another way to get 1 8 is to perorm the ollowing multiplication: 8

85 1 1 B 1B1 = B 1 = because 1 times 1 is 1 and times is 8 8 This is an interesting result that conirm what we have been doing so ar, but all you need to remember is the ollowing: When you multiply ractions, multiply the numerators together and then multiply the denominators together as already shown in the three illustrations above. Multiplying ractions is thus a very straightorward concept. Suppose you are multiplying the ollowing Fractions: a c B b d Just multiply the numerators Just multiply the denominators So, a b c B abc = d bbd Example #1: 1 B 6 85

86 1 B 6 1B = = B6 1 Example # 5 B 5 B = B5 B 10 = 1 Example # 5 B B 5 7 = B5 6B7 0 = Resources: A calculator that will multiply two ractions and display the answer A game to see how well you can multiply ractions 86

87 Multiplying mixed numbers Again, we will warm up the topic with good illustrations. This time, we will play around with the pattern blocks used to add ractions. Our approach is to illustrate and then show a quick way to get the same answer Beore we start, let us make an important point crystal clear. B 6 means to accumulate 6 twice or B 7 means to accumulate 7 our times or Illustration #1 B1 1 B1 1 means to accumulate or add 1 1 twice or

88 orange blocks equal to and red blocks equal to = 5 = You could also get the same answer i you do B1 1 = 1B+1 B = B = 10 10D = 5 = D 88

89 Illustration # 1 1 B1 1 1 B1 1 means to accumulate 1 hal time. Simply lay down a representation o 1 1 and take hal 1 1 = Take hal. This is shown with the red line 89

90 We get Since we can trade or swap hal or three-sixths, = + = = You could also get the same answer i you do 1 1 B1 1 = 1B+1 B 1 = B = 6 = D 6D = 90

91 Illustration # 1 1 B1 1 1 B1 1 means to accumulate 1 twice and then hal time. 1 1 = I we accumulate 1 1 twice or double the amount, we get: 91

92 I we accumulate 1 1 hal or take hal, we get: Let us put it all together Trade in hal or three-sixths 9

93 Trade in two-sixths or one-third 9

94 Trade in three-thirds or 1 whole So, 1 1 B1 1 = You could also get the answer i you do 1 1 B1 B+1 = 1B+1 B 5 = B 0 = 0D = 10 = 1 = 5 6 6D 9

95 In practice, to multiply mixed numbers, convert the mixed numbers into improper ractions beore perorming multiplication ) 1 B 5 1 B 5 = B5+ 5 B+1 B 10+ = 16+1 B 1 = 17 B 1B17 = 1 = 5 5 5B 0 ) B1 B1 B+ 1B+ = B 1+ = + B 1 = 5 B 70 = 5 95

96 Practice: Multiplying ractions 1 ) B = ) 1 B = 5 ) B = ) 6 1 B = 5) B 6 = 6) 6 B = ) B7 8 = 8) B = 9) 6 8 B 5 = 10) 8 B 9 = 96

97 Dividing ractions Just like multiplying ractions, we can illustrate the process o dividing ractions with patterns blocks. Let s start with an easy example. Our approach is to illustrate and then show a quick way to get the same answer Illustration #1: 1 1 D D 6 1 = 1 6 The problem above means how many times 1 1 will it into 6 Get a green blocks and try putting them together as show below. You should notice that it is going to equal the blue block. = 97

98 So, 1 green block or one-sixth will it into the blue block or hal times. 1 1 D 6 1 = 1 = 6 You could get the same answer i you do 1 1 D 6 1 = 1 1 = 6 6 B 1B6 = 6 = = 1 B1 Notice that to get the same answer we change the operation into multiplication and we change 1 6 to 6 1 Illustration #: 1 1 D

99 The problem above means how many times 1 1 will it into 6 6 First, model 1 6 Just count how many sixths there are. I see = 19 So, 1 green or block or will it into times. 1 1 D =

100 You can get the same answer i you do the ollowing: 1 1 D B6+1 1 = D 19 = 1 D 19 = 6 B 11 = = Again, notice that 1 6 was changed into and the division sign was 6 1 changed to a multiplication. This is what we always do. We explain why later! Illustration #: D 1 D 1 = 1 The problem above means how many times 1 will it into Get a red blocks and try putting them together as shown below. You should notice that it is going to equal the orange block or

101 We need then 8 red blocks to make orange blocks. = So, 1 red block or one-ourth will it into orange blocks 8 times. D 1 = =

102 You could get the same answer i you do D 1 = g = 1 1 = B g = B 1B1 8 = = 8 1 Notice again that to get the same answer we change the operation into multiplication and we change 1 to 1 Notice also that we change into since = 1 1 Illustration #: D D = The problem above means how many times will it into 1 First, model 1. This is shown on the next page 10

103 = 1 Then model. This is shown below: = It is not hard to see that it into once. So, we have part o the answer. However, it is complicated to see how many times will it into 1 10

104 Our strategy will be to put 1 or the brown block inside one o the two circles. Then, you can see that is made up o 6 brown blocks Since is bigger than 1 1, the amount o times will it into is a raction o How much o do I need to take to get 1? You need to take one-sixth o or 1 6 o. Thus, can it into one-third 1 6 time. 10

105 Putting everything together, can it into 1 once and one-sixths time or time 1 1 D = 1 = 1 6 You could get the same answer i you convert 1 into an improper raction and then do the same thing you did in illustration #1 and # 1 B+1 D = D 7 = D = B 1 7 = 1 = 1 6 There is an easier way to handle the same problem. 6 means to break 6 down into equal groups. These two equal groups are and 8 means to break 8 down into equal groups. These our equal groups are,,, and 105

106 By the same token 1 1 D means to break into two equal groups or take hal o everything. The process is shown below: First, model 1 Then, draw the blue lines below that will break everything in hal. Pull out hal 106

107 The two orange halves is equal to 1 and the resulting brown piece has the same size as the green piece. = = 1 6 We get 1 and 1 1 or We can use the latter technique to solve D Illustration #5: First model D 107

108 First, model Then, draw the blue lines below that will break everything in hal. Break down and make equal groups 108

109 The equal groups are shown below: Take 1 group. Trade hal or three-sixths = = = 6 109

110 Once again, you can get the same answer i you do the ollowing: B+ D = D 8 = 1 D 1 = 8 1 B 8 = 8D = 1 1D = Dividing ractions is also a very straightorward concept. Suppose you are dividing the ollowing ractions: a c D b d Invert c d d a by writing it as and times it by. c b Thus, a b D c d = a b d B c Now, why do we invert c d? It is a good question. To make this clear, we will use common denominator. This way we will view division o ractions as an extension o wholenumber division. For example, suppose you are perorming this division: 9 1 D 110

111 This answer can be obtained by asking yoursel the question: How many 1 9 are in? Since 9 1 = 9B 1, there are nine 9 in You could have also asked yoursel this question: How many 1s are in 9 or what is 9 1? Since there are nine 1s in 9, there are also nine 1 9 in. Similarly, i we perorm the ollowing division: 16 D, you can just ask yoursel how many s are there in 16 or what is 16? Since 16 =, there are our s in 16, so there are our 16 in Notice that the answer can be ound simply by dividing the numerators in the correct order(this means to divide the numerator on the let by numerator on the right) 111

112 In the case o 7 7? 5 D 5, we can ask ourselves how many are there in It means the same to ask ourselves how many 5s are there in 7 or what is 7 5? Since 7 5 does not give a whole number, it is perectly ok to express the answer as a raction and say that 7 is the answer. 5 In general, when two ractions have the same denominator a b c D a = b c I the ractions have dierent denominator, rewrite the ractions so they have the same denominator beore using the ormula above. For example, D 5 D 5 = B5 B5 B D 5B = D 10 =

113 In general, a b D c d = ad bd D bc bd ad = bc Notice also that a b D c d = ad bc and a b B d c ad = bc Note: When quantities are equal to the same thing, they are equal to each other a b D c d so a b and a b D c d = a b B d c d B c ad are equal to the same thing bc Thus, when dividing ractions, the procedure is to invert the divisor and multiply by the dividend. There is something important that you should be aware o also. Here it is: a c D = b d a b c d The let side a b D c d means the same thing as a b c d 11

114 It is just a matter o notation or a dierent way o writing the same thing. Example #1: D 5 D 5 = 5 B 10 = 1 It means the same i you had 5 = 5 5 B 10 = 1 Example #: 1 1 D D 1 = 6 B 6 = = 6 1 To divide mixed numbers, convert the mixed numbers into improper ractions beore perorming division. 11

115 Example #: 1 D 5 1 D 5 = B5+ 5 B+1 B 10+ = 16+1 D 1 = 17 D 1 = B 5 = Example #: B1 D 1 = B+ 1B+ D 1+ = + D 1 = 5 D 1 = B 8 =

116 Practice: dividing ractions 1 ) 1 D = ) 1 D = 6 ) 5 1 D 8 = ) 1 D = 5) 5 D 6 = 6) 1 D 5 10 = 7) 1 D 5 = 8) D = 9) 1 D 1 7 = 10) D = 116

117 Answers: Introduction to ractions 1 a 6 a 1 6 a a 16 Answers: Equivalent ractions 1 a 1 and a and a and a 8 16 and 0 0 Answers: Simpliying ractions 1 a 1 a 1 a 1 a 1 a 1 5 a 1 a 1 7 a Answers: Renaming mixed numbers and improper ractions 1 a 5 a 5 6 Exercise A: a 5 a 7 1 a 1 5 a Exercise B: 1 a 11 a 1 a 5 a 1 a 1 5 a

118 Answers: Comparing ractions 1 a < a > 1 a 7 < 6 a 5 < a 5 < a < 8 Answers: Adding ractions Practice: adding ractions with the same denominator 1 a = a 7 a 9 a 5 a 5 5 a 1 = 1 6 = 5 7 a a1 8 a 9 9 a9 10 =

119 Practice: Adding ractions with dierent denominators 1 a 17 a 9 a a 7 6 a 9 5 a a 0 5 = 8 1 a a 5 a Practice: Adding mixed numbers 1 a 6 a 8 = 7 5 a 11 = 9 6 a 7 5 a ) 5 8 a a a a

120 Answers: Subtracting ractions 1 a 1 a a 1 a 1 6 a 11 5 a a 1 1 a 8 a 1 9 a Answers: Multiplying ractions 1 a 6 a a 1 a = a a a 68 a 5 8 a 7 9 a Answers: Dividing ractions 1 a a 1 a 0 a 1 a 50 = = a 18 6 = a 68 a 5 8 a 7 9 a