FRACTIONS Book 1 An Introduction to Fractions for the Adult Learner

Transcription

1 ACADEMIC STUDIES MATH Support Materials and Exercises for FRACTIONS Book An Introduction to Fractions for the Adult Learner SPRING

2 FRACTIONS Fractions are used in our everyday life. We talk about fractions with regards to food (I cut my daughter s sandwich in ), we measure fractions when we are cooking (First, you add cup of flour), even our time has been divided into fractions (I will be back in a of an hour). Since we talk in fractions so much, it is necessary that we understand what fractions are. A fraction is an equal part of a whole. When you cut an apple pie into equal sections, each section stands for an equal part of the whole pie. Each section is a fraction of the whole pie. If you cut the pie, but did not eat any of it you would have of the pie or whole pie. If you ate one section of the pie, you would have eaten of the pie.

3 Exercise : Anything that is divided into equal parts can be referred to in terms of fractions. Look at the following chart. Which of the figures can be divided into equal parts?

4 Parts of a Fraction There are two parts of a fraction. The numerator and the denominator. Let s use the following example to help us understand these two parts: Joe has eaten of the apple pie. numerator - the numerator is the top half of the fraction which lets us know how many equal parts of the whole we are dealing with or how many pieces of the pie Joe has eaten. In our example, the is the numerator. It tells us that Joe has eaten equal parts of the pie. denominator - the denominator is the bottom half of the fraction which tells us how many equal parts there are in the whole. In our example, is the denominator. It tells us that the pie was cut in five equal parts. ± this pie has been cut into five equal parts Joe has eaten three of the five pieces The fraction of the pie that was eaten was.

5 Writing Fractions: We see: We say: three fifths are coloured We write: -number of parts coloured -number of parts in total We see: We say: one half is coloured We write: -number of parts coloured -number of parts in total We see: We say: two thirds are coloured We write: -number of parts coloured -number of parts in total We see: We say: five twelfths are coloured We write: -number of parts coloured -number of parts in total

6 Exercise : Example: Write the correct fraction beside each number word given. five eights ) one half ) two thirds ) four fifths ) three quarters ) six ninths ) seven tenths ) two fourths ) three eights ) fifty-one hundredths ) ten twentieths Exercise : Write the correct number word beside each fraction given. Example: ) ) ) ) ) ) ) seven eights

7 Exercise : Draw a line segment between each fraction and number word that name the same number. five eights one half two thirds four fifths six sixths one quarter one third two fourths five ninths three quarters

8 Look at the following figures. a) b) c) d) e) Exercise : Write a fraction for the coloured part of each figure above. Exercise : What fraction of each of the figures above was not coloured.

9 Exercise : = Try the following. The first one has been done for you. Draw a diagram showing the fraction given. = = = = =

10 Types of Fractions There are three types of fractions: proper, improper, and mixed. proper fraction - in a proper fraction, the denominator is always larger than the numerator, such as,, or because the fraction represents less than the whole thing. Ex: He ate of the chocolate cake. improper fraction - mixed number - in an improper fraction, the numerator is equal to, or larger than, the denominator; for example,,, or because the fraction represents the whole or more than the whole thing. (Improper fractions can always be reduced.) a number that includes both a whole number and a fraction.,, and are all mixed numbers. They represent a whole number plus a fraction. Ex: I will be back in hours.

11 Exercise : For each of the following fractions, write whether they are proper, improper, or mixed. The first one has been done for you. ) = proper ) = ) = ) = ) = ) = 0 ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = 0) =

12 Changing an Improper Fraction into a Mixed Number In math, there are times when we need to change an improper fraction into a mixed fraction such as when we reduce them to lower terms. To do this you simply divide the numerator by the denominator. R = = Three goes into eight, times; therefore, your whole number is. The remainder (also a ) is always placed over the original denominator which in this case is. Let s try another one: R = = Five goes into nineteen, times; therefore, your whole number is. The remainder is placed over the original denominator which is. Here is one more: = = In this case, the denominator goes into the numerator an even amount of times; so, the answer is just a whole number. Notice that you do 0 not put a zero fraction ( ) as this is incorrect.

13 Exercise : Now, you try. Change the following improper fractions into mixed fractions. ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = 0) = 0

14 Changing a Mixed Fraction into an Improper Fraction There are other times, such as when you are multiplying and dividing fractions (which you will learn about later), when you will need to change a mixed fraction into an improper fraction. In order to do this, you multiply the whole number by the denominator, then add the numerator, and then you place your answer over the original denominator. Ex: = x + = over = Let s try one. First, you multiply the whole number by the denominator: x (); then you add on the numerator + (); and then you place your answer over the original denominator which is : Let s do another. = x + = over = If you have only a whole number and you want to make this into a fraction, you simply place the whole number over the denominator one (). Ex: =

15 Exercise : Change the following mixed numbers into improper fractions: ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) 0 = ) = ) = 0) = 0

16 Equivalent Fractions Fractions can appear different because they have different terms (numerator and denominator), yet, they are the same number. We call these fractions equivalent fractions. Equivalent fractions are fractions that are equal in value; such as,, and. All of these are talking about the same amount or the same part of the whole. = = = As you can see, the fractions all represent the same part of the whole. The whole does not change, it is the same size. The difference is that it has been divided into smaller or larger portions. The symbol = stands for equal to; in this case, you can say,, and are equal to each other ( = = ). They are equivalent fractions. We can create equivalent fractions quite easily. You simply multiply the numerator and the denominator by the same number.

17 The most important thing to keep in mind from here on in is that whatever you do to the denominator, you must also do to the numerator. To make an equivalent fraction for, we choose a number to multiply the terms (numerator and denominator) by and then do the multiplication. Let s choose : = = So, is an equivalent fraction for. 0 0 Let s try another one: Find an equivalent fraction for. First, choose a number. I will choose. = = 0 Exercise : Find an equivalent fraction for the following: ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = ) = 0 0

18 ) = ) = 0 ) = ) = Sometimes, you are given the new denominator and you need to find the missing numerator. This equation will look like this =.? The first thing that you must do is to figure out what number you have multiplied the original denominator by to get the new denominator. You do this by dividing the second denominator by the original one: = Now we know that our multiplier will be. The second step is to multiply the original numerator by (remember, whatever you do to the denominator, you must also do to the numerator). = =? = Therefore, =.?

19 Let s try another one. =.?? First, we divide the new denominator by the original one: = Then, we multiply the original numerator by the response to the first step: = =? = Therefore, = The same steps are followed if you are missing the new denominator but you know the numerator. For example:. =? First divide the new numerator by the original one ( =); then, multiply the original denominator by the response to step one (x=).? = = Therefore, =

20 0 Exercise : Make equivalent fractions by finding the missing numerator or denominator ) ) =? 0 ) ) =? =? =? ) ) =? ) ) =? ) ) =? 0 ) ) =? =? =? =? =? ) ) =? ) ) =? =? 0 =? 0 ) ) 0 =? 0 =? 00 ) 0) 0 =? =?

21 You can also find out if two fractions are equivalent by following similar steps. Let us look at the two fractions and 0 to see if these two fractions are equivalent. Step : Step : divide the second denominator (0) by the original denominator () 0 =()* multiply the original numerator () by the product of step ()*. If the numerator of the second fraction equals this number, then the two fractions are equivalent. = = these two fractions are equivalent. 0 How about the following two fractions? Step : = ()* Step : x()*= = Does the numerator of the second fraction equal? No, it is an. Therefore, and are not equivalent. We can use the not equal to symbol ( ) to show that they are not equivalent. Let s try a couple more. Are the following fractions equivalent (equal =) or not equivalent (not equal ),, Step : =()* Step : =()* Step : x()*= Step : x()*= numerators are different numerators are the same therefore, Therefore, =

22 Exercise : Are the following fractions equivalent? Use = or to show their relationship. ), ) ), ) ), ) ), ) ), ) ), ) 0 ), ) 0 0 ), ) 0 ), ) ), 0), 0,, 0,, 0,,, 0,, 0

23 Comparing Fractions with Common Denominators Using the Symbols < and >. We use many symbols in math to show relationships between numbers. Here are two:. <. > < means smaller than > means greater than Let s look at some examples with fractions that use these symbols. Example : < The fraction is smaller than the fraction. It represents less of the whole. <

24 Example : The fraction is greater than the fraction. The first fraction represents more of the whole than the second fraction. Notice! You can read the expression from left to right, or from right to left. It still reads the same. Two sixths is smaller than four sixths, or... Four sixths is greater than two sixths. > < > < Usually though, the expression is read from left to right. Read the following expressions. ) Three fifths is greater than one fifth. ) Two sixths is smaller than four sixths. ) Four sevenths is greater than two sevenths. ) Five ninths is smaller than eight ninths. How do I tell which fraction is the largest?

25 When comparing fractions that have the same denominators, the largest fraction is the one that has the biggest numerator- what we really want to know is, which fraction represents more of the whole? The fraction is larger than, because the numerator is larger than the numerator. Remember...the bottom number (denominator) tells us how many equal parts there are in the whole. In this case, both have five equal parts, so only the numerators are being compared. Read the following expressions. ) is larger than ) is smaller than ) is larger than ) is smaller than ) is larger than ) is smaller than ) is smaller than

26 Exercise : Write the word true or false for each set of fractions. > > < < > > ) ) ) 0 0 > 0 ) ) ) 0 < 0 ) ) ) 0 0 < 0 ) ) ) > ) ) ) 0 > < 0 > 0 > 0 < Exercise : Use the symbol > or < to compare each of the following sets of fractions. ) ) 0 ) ) ) ) 0 ) ) ) ) ) )

27 Exercise : Write a fraction that is smaller than the one given. ) ) ) ) ) ) ) ) ) ) ) ) 0 Exercise : Write a fraction that is larger than the one given. ) ) ) ) ) ) ) ) ) ) ) ) Exercise : Draw a picture that represents more of the whole than the fraction given. ) ) ) ) )

28 AN EASY WAY TO COMPARE FRACTIONS THAT HAVE DIFFERENT DENOMINATORS. Which fraction is larger - or Did you notice? The denominators are different! Not to worry!!! When two fractions have the same numerator but different denominators, the largest fraction is the one with the smallest denominator. The fraction The fraction means that a whole amount has equal parts. means that a whole amount has equal parts. So of course, the parts will be smaller for the fraction. Compare: and The larger fraction is, because " is the smallest denominator, which means that the whole is divided into fewer parts. The only thing you really have to remember in order to use this shortcut is that the numerators must be the same. is greater than is greater than is greater than is greater than

29 Exercise : Put the following fractions in order from least to greatest. Remember to compare the denominators since the numerators are the same! Exercise 0: Put the following fractions in order from least to greatest. / / / / Exercise : Compare these fractions using > or <. ) / / ) / / ) / / ) / / ) / / ) / / ) / / ) / / ) / /

30 0 Exercise : For each fraction given, write one that is smaller and one that is larger. Example: / < / < / ) < / < ) < / --- ) < / < --- ) --- < / < ) < / ) < / < --- ) < / < ) < / < ) < / < --- Exercise Draw a line to the fraction that is larger than the one given. ) / / ) / / / / ) / / ) / / / / ) / / ) / / / /

31 REDUCING FRACTIONS TO THEIR LOWEST TERMS We already know that fractions can look entirely different and still represent the same amount. We called these kinds of fractions equivalent fractions. Here are two examples: and. One half equals two quarters. While we re on the subject of reducing... The word reduce means to make smaller. We reduce our mortgage by making regular payments, we reduce our weight by eating less and exercising, and...we can reduce fractions too! When we reduce a fraction, we create a new one using smaller numbers, but... the value of the fraction remains the same ( we are left with the same amount of the whole as what we started with). Why bother reducing fractions anyway???????? Its easier to understand fraction amounts when they are written using smaller numbers. Besides, would you rather cut a pizza in pieces and eat piece or have to cut it in eight pieces and eat? The second choice is a lot more work!

32 Reducing Fractions Which is easier to understand? I will call you in of an hour. or, I will call you in of an hour. Both of these fractions represent the same amount of time, but the second example makes more sense to us. The fraction is in its reduced form. To create a smaller, but equivalent fraction, just think of a number that both the numerator and denominator can be divided by evenly. Example: 0 Both numbers can be divided by evenly to create the fraction. divided by = divided by = Watch out! If you do not divide the numerator and denominator by the SAME number, you will not end up with an equivalent fraction. Instead, you will be left with a different amount than when you started! ex: divided by = The new fraction is. 0 divided by = 0 In the fraction, is half of 0. In the fraction, is less than half of.

33 Example To create an equivalent but smaller fraction, divide both numbers by. = = Example ' The new fraction is. Both numbers can be divided by evenly. = Example = ' The new fraction is. Both numbers can be divided by evenly. = = The new fraction is. '

34 Greatest Common Factor Sometimes when you divide to reduce a fraction, the new numerator and denominator can still be reduced.. Example: Both numbers can be divided by to create the new fraction This fraction can be reduced again. Both numbers can be divided by. We then end up the fraction. You can save yourself a lot of time and effort by finding the greatest common factor or GCF to start with. This is the largest common number that both numerator and denominator can be divided by evenly. Example: The factors of are: {,,,,,} The factors of are: {,,,,,,,,} Six is the greatest common factor (GCF). Now just divide. divided by = divided by = The fraction has now been reduced to its simplest form.

35 Exercise : Reduce the fractions below to their simplest form. ) ) ) ) ) ) 0 ) ) ) ) ) ) ) ) ) ) ) ) 0 ) 0) Exercise : Think of a number that can be divided evenly by... ) and ) and ) and ) 0 and 0 ) 0 and 0 ) and ) and ) and 0 ) and ) 0 and 00

36 Exercise : For each group of fractions, circle the ones that can be reduced. a. ) b. ) c. ) d. ) e. ) 0 Exercise : True or False. a. If I spend of my day at work, I spend of my day at work. b. If I buy of the books I want, I buy of the books. c. If I find of the cards I lost, I find of the cards. d. If I earn of the money you do, I earn of the money.

37 Exercise : Which of the following fractions have been reduced to their simplest form? ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 0 0 Exercise : Tell which GCF you would use to reduce each fraction. 0 0 ) ) ) ) ) ) ) ) 0 ) ) ) ) ) ) ) )

38 Answer Key Exercise page : see instructor Exercise page : five eights one quarter one half one third two thirds two fourths four fifths five ninths six sixths three quarters Exercise and page : a), b) c), d),, ), Exercise page : see instructor

39 Exercise page : ) proper ) mixed ) improper ) proper ) mixed ) proper ) improper ) improper ) proper ) mixed ) mixed ) improper ) proper ) mixed ) mixed ) proper ) improper ) mixed ) improper 0) proper Exercise page : ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) 0) Exercise page : ) ) ) ) ) ) ) ) ) ) ) ) 0 ) ) ) ) ) ) 0 ) 0) Exercise page : see instructor Exercise page 0: ) = ) 0 = ) = ) 0 = ) = ) = 0

40 ) = ) = 00 ) = ) 0 = ) = ) = 0 ) = ) = 0 ) = ) 0 = 0 ) = ) 0 0 = 00 ) = 0) 0 0 = Exercise page 0: ) ) = ) ) = 0 0 ) = ) = ) = ) = 0 ) = ) ) = ) = 0 ) ) = ) ) = ) ) = ) 0) Exercise page : ) false ) true ) true ) false ) true ) false ) true ) true ) false ) false ) false ) true ) true ) false ) true

41 0 Exercise page : ) < ) < ) > ) > ) > ) < ) < ) < ) > ) > ) < ) < Exercises,, and page : see instructor Exercise Page :,,,, Exercise 0 Page :,,, Exercise Page : < >... < > >... > > <... >

42 Exercise Exercise Page : &&&& Page : see instructor ). &&&& &&&& ). &&&& &&&& ). &&&& Exercise page : ' ' ' 0 ' ' 0 ' 0 0 ' 0 ' 0 ' 00 ' ' ) divide by ) divide by ' ) divide by ) divide by ' ) divide by ) divide by ' ) divide by ) divide by ' ) divide by ) divide by ' ) divide by ) divide by ' ) divide by 0 ) divide by 0 ' ) divide by ) divide by ' ) divide by ) divide by ' ) divide by 0 0) divide by

43 Exercise page : ) ), ) ),, ),, ) ) ) ) ),0,0,0 Exercise page :, ) ) ) ) 0, ) ), 0 0 ) ) 0, ) ) 0,,, Exercise page : ) true ) false ) true ) false Exercise page :,,,,,, 0

44 Exercise page ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )

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