Topic 2 Fractions

5/24/2018 Topic 2 Fractions

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INTRODUCTIONWhat do you understand by fractions? Are you able to explain the concept offractions clearly to your students?

As mathematic teachers, we need to address the above questions seriously. Thestudy of fractions is an essential part of mathematics. But many teachers view theteaching of fractions as a challenging task. It is difficult to explain to students that1

4is equal to

3.

12It is also not easy to convince students that

3

14is greater than

5

28since 3 over 14 seems smaller than 5 over 28 respectively.

LEARNING OUTCOMES

By the end of this topic, you should be able to:

1. Explain fractions as parts of a whole by representing them withdiagrams;

2. Find the equivalent fractions of a given fraction;3. Compare fractions and arrange fractions in order, according to

values;

4. Simplify fractions;5. Recognise mixed numbers, proper and improper fractions; and6. Perform basic operations (+, , and ) involving mixed numbers.proper and improper fractions; and7. Solve problems involving proper and improper fractions.

TTooppiicc

22 Fractions


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TOPIC 2 FRACTIONS 63

In this topic, we will share with you some of the strategies employed to teach theconcepts and skills related to the learning of fractions.

CONCEPTS OF FRACTIONS

In this section, we will focus on the following areas:

(a) The concept of fractions as parts of a whole (using representations);(b) The concepts of equivalent fractions, comparing and simplifying fractions;

and

(c) The concept of mixed numbers.2.1.1 Fractions as Parts of a Whole

A fractionis a number that represents parts of a whole.Students initial learning experience with parts-whole relationships of fractionsusually involves investigating parts of shapes or regions in relation to the wholeshape. However, students need to understand that parts and whole offractional thinking can also include sets or collections of discrete objects.

The explanation of the meaning of fractions as parts of a whole involves thefollowing steps:

(a) Identify the whole of an object. This can be demonstrated by using variousshapes such as squares, rectangles, circles and so on. Alternatively, thewhole can also be represented by a set of objects;

(b) Divide the object into equal parts;(c) Use appropriate colour to shade one or more of the parts;(d) Determine the fraction by comparing the proportion of the shaded

part/parts to the whole;

(e) For example, in Figure 2.1, the circle is defined as a whole object. The circleis divided into four equal parts. In Figure 2.2, the set or collection of fourcircles is defined as the whole. And, the four circles are equal parts of theset;

2.1


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TOPIC 2 FRACTIONS64

Figure 2.1: The circle is divided into 4 different parts

Figure 2.2: A collection of 4 circles(f) 1 out of the 4 equal parts of a circle is shaded in Figure 2.1 while 1 circle

out of 4 circles is shaded in Figure 2.2. Thus, the shaded part in both

representations can be described as1

4of a whole; and

(g) The fraction 14

is read as one quarter or one-fourth. It is also common to

read the fraction as one over four.


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The examples shown in Table 2.1 should help you to understand fractions better.

Table 2.1: Fractions

Diagrammatic Representation DividedIntoNumberof PartsShaded

SymbolicRepresentationShaded Region) Read As

2equalparts

1part 12 One-half

4equalparts 2parts 24

Two-fourths or

twoquarters

4equalparts

3parts 34 Three-

fourths orthree

quarters

5 equalparts

3parts 35 Three-fifths

6equalparts

2parts 26 Twosixths


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TOPIC 2 FRACTIONS66

2.1.2 Writing Fractions

A fraction is written in the form of ,

a

b where a and b are whole numbersseparated by a horizontal line called the vinculum but which is commonlycalled the fraction bar. The number above the fraction bar, that is a, iscalled the numerator. The number below the fraction bar, that is b, is calledthe denominator. The numerator represents a number of equal parts and thedenominator indicates how many of those parts make up a whole.

If ais greater than b, then the fraction ab

is called a proper fraction. On the otherhand, if

bis greater than a, then the fractiona

b is called an improper fraction.SELF-CHECK 2.1

What do you think is the result when a and b are the same?


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Teaching Sample Activity 2.1: Representing fractions.Objectives: Students are able to divide a piece of paper into equal parts.Students are able to represent a given fraction.

Students are able to read fractions represented in diagrams.

1. Instruct the students to work in groups.2. Each group is given several pieces of paper with various shapes as shown

in Figure 2.3.

Figure 2.3:Various shapes


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TOPIC 2 FRACTIONS68

1. Instruct the students to cut the paper to obtain the various shapes.2.

Tell the students to fold the paper into two equal parts. Encourage themto use all the 10 shapes with different ways of folding.

3. Repeat the process by folding the shapes into 3, 4, 5 or 6 equal parts. Itmay not be possible for some shapes.

4. Complete Table 2.2 by drawing their outcomes in the correspondingspaces.

5. Guide the students to read the shaded parts.

Upon completion of the activity, students should be able to state the correctfractions represented by the shaded parts in any figure which is divided intoequal parts.

For Example:

Note:Students should be given opportunity to draw, identify and justifyrepresentations that are exemplars and non-exemplars of a particular fraction.

For example, Figure 2.4 shows two non-exemplars for1

.3

Figure 2.4:Example of non-exemplars


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Table 2.2:Shading ActivityDivide the following shapes into equal parts and shade the parts to represent thefractions given.

1

2

1

3

14

1

5

1

6


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2.1.3 Equivalent Fractions

Equivalent fractions are fractions that have the same value. The followingactivity may help students understand the concept of equivalent fractions.

Teaching Sample Activity 2.2:Understanding the concept of equivalentfractions.

1. Get the students to fold a piece of rectangular paper into two equal partsand then colour 1 part as shown in Figure 2.5.

Figure 2.5:Coloured rectangular paper with two equal parts2. Tell the students to write down the fraction represented by the coloured

part. The answer given by the student should be1

.2

3. Then ask the students to fold the paper along the dotted line in the centre

as shown in Figure 2.6.

Figure 2.6:Coloured rectangular paper with four equal parts4. The paper now has four equal parts with two coloured parts.

5. Tell the students to write down the fraction represented by the coloured

parts. Now the correct answer should be2

.4


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6. Emphasise to the students that in this activity, both fractions,1

2and

2

4

refer to the same coloured parts, hence 12

and 24

have the same value andthey are known as equivalent fractions. We can write

1 2.

2 4

7. Guide them to arrive at the conclusion that1 2

3 6 by using a similar

technique.

8. Ask students to explain why

1 2 3 4 5 6 7 8

.... 1.1 2 3 4 5 6 7 8

To find the equivalent fractions of a fraction, just multiply or divide both thenumerator and the denominator by the same number.

Example:1 1 3 3 1 1 7 7

4 4 3 12 4 4 7 28

2 2 5 10 16 16 8 2

3 3 5 15 24 24 8 3

3 3 2 6 18 18 6 3

4 4 2 8 24 24 6 4

Therefore,1 3

,4 12

and

7

28 are equivalent fractions.

SELF-CHECK 2.2

Can you think of other activities that can help to explain whatequivalent fractions are?


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Similarly2 10

,3 15

and16

24 are equivalent fractions.

Likewise, 3 6,4 8

and 1824

are equivalent fractions.

The procedure of multiplying or dividing the numerator and denominator toget equivalent fractions should not be taught mechanically without anunderstanding of the concept behind the procedure. Therefore, it is importantthat students are able to make the connection between the pictorialrepresentation as seen in Teaching Sample Activity 2.2 and the symbolicrepresentation of its calculation.

For example, for the equivalent fraction of: 1 1 2 2 ,2 2 2 4

students should be

guided to explain the meaning behind the calculation, as shown in Table 2.3.Being able to make the connection between the pictorial and symbolicrepresentations of the calculation and explain the mathematical process involved,will mean that they are communicating mathematical ideas meaningfully and notmerely learning by rote.

Table 2.3:Explanation of the Calculation

Pictorial RepresentationSymbolic

Representation Explanation

1

2

One coloured part out of two equalparts is shaded.

1 1 2 2

2 2 2 4

The two parts of the whole arepartitioned into two equal parts,

giving 2 2 smaller parts making thewhole. This means the coloured partis partitioned into two smaller equal

parts, giving 1 2 smaller colouredparts. The region of the two smallerparts and the original one colouredpart are still the same. Hence, thevalues are the same.


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Once the students understand the meaning of equivalent fractions, it will beuseful to guide them to build a table of equivalent fractions like the one shown inTable 2.4.

Table 2.4:Table of Equivalent Fractions11

22

33

44

55

66

77

12

24

36

48

510

612

714

13

26

39

412

515

618

721

14 28 312 416 520 624 728 23

46

69

812

1015

1218

1421

There is a simple method to determine if any two given fractions are equivalentfractions. What you have to do is just to multiply the numerator of one fraction

by the denominator of the other fraction, and vice-versa (cross-multiply), thencompare the two products to see whether they are of the same value.

SELF-CHECK 2.3

1. How do you determine if two given fractions are equivalentfractions?

2. Can you draw pictorial representations of the equivalent fractionsin Table 2.4?


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For example, to check whethera

b and

c

dare equivalent fractions, multiply aby

d, and then multiply b by c. If both values are the same, then

a

b and

c

d areequivalent fractions.

The above explanation can be illustrated in the mathematical form as follows:

Ifa

b= c

d

Then,a d = c bFor example, the given fractions 3

4and 6 ,

8

the cross-multiplications are: 3 8 = 24 and 4 6 = 24.

Since both products are the same,3

4and 6

8are equivalent fractions.

Another example:

The fractions 25

and 13 ,30

the cross-multiplications are: 2 30 = 60 and 5 13 = 65.Since both products are different,

2

5and 13

30are notequivalent fractions.


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2.1.4 Comparing the Values of Two Fractions

Let us now compare the values of two fractions.

(a) If two fractions to be compared have the same denominator, then thefraction with the bigger numerator is greater in value than the otherfraction.

(b) If two fractions to be compared have the same numerator, then the fractionwith the smaller denominator is greater than the other fraction.

(c) If two fractions to be compared have different numerators anddenominators, then we have to change the fractions into equivalent

fractions with a common denominator before the comparison can be made.

(d) For example, 57

is greater than3

7since 5 is greater than 3 and

5

8is greater

than5

9since 8 is smaller than 9.

(e) It is good to illustrate the differences in value using pictorialrepresentations. For example, the shaded parts in Figure 2.7 for

5

8is greater

than the shaded parts for 5 .9

Figure 2.7:Shaded figure


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Teaching Sample Activity 2.3: Compare the values of two fractions withdifferent numerators and denominators.

1. Write down the fractions as shown in Figure 2.8 on two flash cards. Thenask the class to guess which of the two fractions is greater in value.

Figure 2.8:Flash cards2. Guide your students to draw two identical rectangles on two separate

transparencies and shade the regions as shown in Figure 2.9 andFigure 2.10.

Figure 2.9: Rectangular transparency

Figure 2.10: Rectangular transparency3. Ask the students again: Are you able to judge which of the two shaded

regions is larger in size?


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4. Guide the students to place 1 transparency on top of the other as shownin Figure 2.11.

Figure 2.11: Rectangular transparency5. Guide students to discover through observation of equivalent fractions

that

3

5= 9

15and 2

3= 10

15

6. By now, students should be able to tell that2

3is greater than 3 .

5

7. Once the students understand the concept, you need to guide themto make the connection between the pictorial representations, the

mathematical procedures and the reasoning involved in getting theanswer.

3 3 3 9

5 5 3 15

and

2 2 5 10

3 3 5 15

Since10

15is greater than

9,

15 therefore

2

3 is greater than

3.

5


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Number lines can also be used to compare values of two fractions. Figure 2.12

illustrates how2

3is greater than 3.

5

Figure 2.12:Number lines to show which value is greater

2.1.5 Arranging Fractions in Order

Fractions with different numerators and denominators can be arranged in anincreasing or decreasing order. First, change them to equivalent fractions with acommon denominator. This common denominator is actually the lowest commonmultiple (LCM). After that, you just need to compare the values by looking at thenumerators.

SELF-CHECK 2.4

Describe 2 ways to compare the fractions3

7and

2.

5Which fraction has

a greater value?


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Teaching Sample Activity 2.1: Arranging fractions in ascending or descendingorder.

1. Divide your class into groups of six.2. Each group is given a set of six cards written with different fractions.3. Each student in the group is given a card.

4. Appoint a group leader for each group.5. Lead the students in discussing how they can determine which student is

holding the card with the biggest or the smallest fraction.

6. You can suggest they start off by everyone choosing a partner in thegroup. Then the two of them would compare the fractions on their cards.

7. Next they can exchange partners and repeat the process.8. You will find that after a while, the students will be able to apply the

skills that they have learned earlier to compare fractions by using LCM.

9. Finally, with the help of the group leader, the students in each groupshould arrange themselves in a row, based on the values of the fractions

written on their cards in an increasing or decreasing order.

10. The group that completes the task in the shortest time will be the winnerand shall be awarded a prize.

2

3

4

5

5

7

3

4 56

7

8


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2.1.6 Simplifying Fractions

We now know that

1 2 3 4 5 6

, , , , ,2 4 6 8 10 12

7

14 are equivalent fractions because all

these fractions have the same value, that is1.

2In this case, we can say that

1

2 is

the fraction in the lowest term. Note that any equivalent fraction can besimplified to its lowest term by dividing both the numerator and denominator by

their highest common factor (HCF).

For example,

3 3 3 1and

6 6 3 2

7 7 7 1

14 14 7 2

8 8 4 2

28 28 4 7

In simplifying fractions, it is common practice not to show the division explicitly.Instead, the division is performed as cancellation as follows:

8

28

2

7

Another example:

24

54

4

9

HCF of 3 and 6 is 3.



2

7

4

9


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Make sure your students understand that the concept underlying the cancellationshort cut is the division of the numerator and denominator with a commonfactor.

THE BASICS OF WHOLE NUMBERS

A proper fraction is a fraction where the numerator is smaller than thedenominator. This means the entire value of a proper fraction is less than one.

Below are examples of proper fractions:

2 4 9 2 7 11, , , , ,

5 7 11 3 9 15

2.2.1 Addition and Subtraction of Proper Fractions

Before teaching students about addition and subtraction of fractions, we have tobe clear about the steps involved in carrying out such operations. The steps canbe summarised as follows:

(a) Case 1: The Denominators of Both the Fractions are the SameExample 2.1:Calculate the value of 1 3 .

5 5

(i) Explain to students that in this problem, the basic unit is 1 ;5

(ii) 15

means one unit of1

,5

and3

5means three units of

1;

5

(iii) So the addition in this case is one unit1

5 adding to three units of1

,5

and the result is four units of1

;5

(iv) Mathematically, it can be written as 1 3 1 3 4 ;5 5 5 5

2.2


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(v) You can add fractions easily if the bottom number (the denominator)is the same. We can use representations with diagrams to illustrate

the concept. Figure 2.13 illustrates the addition of1

4 and

1

4 while

Figure 2.14 shows the addition of5

8 and

1.

8

Figure 2.13: The addition of 14

and1

4

Figure 2.14: The addition of 58

and1

8

The subtraction of a proper fraction from another proper fraction with thesame denominator works on the same principle.


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Example 2.2:Calculate the value of 5 2 .7 7

5 2 5 2 37 7 7 7

Similarly, it is important to use diagrammatic representations to illustratethe concept of subtraction of fractions. For example, subtraction can mean

the difference between the values of5

7and

3

7as shown in Figure 2.15.

Figure 2.15: Differences between the values of 57

and3

7

Subtraction can also mean take away. Try to illustrate this concept usingdiagrammatic representations for the example below.

Example 2.3 : 11 7 11 712 12 12

4

12

13


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(b) Case 2: The Denominators of the Fractions are Different.Table 2.5 clearly details the steps that are used to explain the method to addtwo fractions with different denominators. The example used in this case is

1 2 .3 5

Table 2.5:Steps to Add 2 FractionsInstructional Procedure Mathematical Steps

1. Find the lowest common multiple(LCM) of the two denominators.

The LCM of 3 and 5 is 15.

2. Change each of the fractions to itsequivalent fraction with the LCM

as its denominator.

1 1 5 5and

3 3 5 15

2 2 3 6

5 5 3 15

3. Add the fractions. 1 2 5 63 5 15 15

5 6

15

11

15


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Figure 2.16 can illustrate the concept clearly.

3 1 3 2

8 4 8 83 2

8

5

8

Figure 2.16:Adding two fractionsYou can also apply the same principle when subtracting two fractions withdifferent denominators.Example 2.4:Find the value of 2 4 .

3 9

Since the denominators are different, we need to first make them the same,before being able to subtract them.

2 2 3 6

3 3 3 9

2 4 6 4

3 9 9 9

6 4

9

2

9


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In principle, the common denominator that we want to find is the LCM ofthe two denominators. But sometimes, it is easier to obtain a commondenominator by just multiplying the two common denominators.Example 2.5: Calculate 7 3 .

8 4

Here, the two denominators are 8 and 4.7 7 4 2 8

8 8 4 3 2

3 3 8 2 4

4 4 8 3 2

7 3 2 8 2 4

8 4 3 2 3 2

2 8 2 4

3 2

4

3 2

1

8

2.2.2 Multiplication of Proper Fractions

Multiplication involving proper fractions can be done in various forms. Theseinclude:

(a) Multiplication of a fraction with a whole number(b) Multiplication of two fractions


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To help you understand the different principles employed in carrying out thesecalculations, we have categorised the various forms in Table 2.6:

Table 2.6:Principles Employed in Carrying Out MultiplicationsCategory Steps ExplanationMultiplication of a fractionwith a whole number.

Example:3

2 .8

3

4

3 2 32

8 1 8

2 3

1 8

6

8

3

4

Write 2 as a fraction. Multiply the numerators

and the denominators.

Simplify the resultingfraction by dividing bythe highest commonfactor (HCF).

1

4

3 2 32

8 1 8

1 3

1 4

3

4

Alternative Simplify the fraction

from the start (divide byHCF). Then multiply thenumerators followed bythe denominators.

Multiplication of two

fractions.

Example:3 2

.4 9

13 2 3 2

4 9 4 9

6

36

6

1

6

Multiply the numeratorwith numerator, anddenominator withdenominator.

Then simplify theresulting fraction.

1 1

2 3

3 2 3 2

4 9 4 9

1 1

2 3

1

6

Alternative Simplify first. Then multiply.


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To enhance the understanding of mathematical concepts, you should guidestudents to make the connection between the symbolic representations as shownin the calculation procedures with the visual representations using concrete

materials or diagrams. Table 2.7 illustrates the connection between therepresentations of the two types of multiplication of proper fractions.

Table 2.7:Connection between Symbolic and Diagrammatic Representations of FractionMultiplication

Symbolic Representation Diagrammatic RepresentationMultiplication of afraction by a wholenumber.

Example:

3 2 32

8 1 8

2 3

1 8

6

8

3

4

Two sets of3

8 which equals

3

4

Multiplication of twofractions.

Example:

3 2 3 2

4 9 4 9

636

1

6

The intersection of3

4 and

2

9 shows the product of

6

36which

is equivalent to1

6

3

4

2

9

6

36

34

38 38

6

8


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2.2.3 Division Involving Proper Fractions

Let us begin this section by looking at the division of a proper fraction with a

whole number. For example:

3 32

4 8

How can the above answer be illustrated using concrete objects?(a) Imagine this to be a piece of rectangular cake.

(b) Cut the cake into four equal pieces. This is how it looks.

(c) If you are given 34

portion of the cake, how much will you get?(d) The coloured region shows the portion of the cake that you will get.

(e) Now, if the portion that you get is divided between two people (you andyour friend), how much will each of you get?


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(f) Figure 2.17 shows how the division is made, and how much each of youwill get.

Figure 2.17: The division of the cake(g) From the illustration, it is now quite obvious that each of you will get 3

8of

the whole cake. In other words,3

24

is equal to3

.8

(h) In terms of algorithmic calculation, to divide a fraction by a whole number,we can multiply the fraction by the reciprocal of the whole number asshown below:

3 3 12

4 4 2

3 14 2

3

8


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Note that in the above example, the reciprocal of 2 is1.

2

In general, the reciprocal of a whole number, for example y is 1 ,y

and when a

whole number is multiplied by its reciprocal, the product is always equivalent to1. Whereas, when a fraction is divided by another fraction, we can convert theoperation of division to multiplication by its reciprocal.

Example: or

4 8 4 15

9 15 9 860

5

726

5

6

1 44 8

9 15

3 9

15

5

82

1 5

3 2

5

6

However, how can you convince students that division of fractions can beconverted to multiplication by its reciprocal? Memorising the procedural steps incalculation without knowing the conceptual basis will only make students learn

by rote and weaken their understanding and thinking of mathematics.

You can guide students to understand why inverting fractions works fordivision, by examining the patterns of fraction division using appropriate modelsor representations. Fraction division problems can be viewed as measurementor

quotitivedivision problems. For example, for1

12

you are asking how many1

2 are there in 1.


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How Do We Illustrate the Fraction-division Rule?Step 1:Begin with division of 1 by fractions.

Diagrammatic Representation Symbolic Representation1

12

means how many1

2are there in 1

1 21 2

2 1

1 2 21 1

2 1 1

which is equivalent to

11

3 means how many

1

3are there in 1

1 31 3

3 1

1 3 31 1

3 1 1


11

4 means how many

1

4are there in 1

1 41 4

4 1

1 4 41 14 1 1


1

41

Four1

4are in 1

1

4

1

4

1

4

1

4

1

31

Three1

3are in 1

One1

3

One1

3

One1

3

1

21

One1

2

Two1

2are in 1

One1

2


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21

3 means how many

2

3are there in 1

2 1 31 1

3 2 2

2 3 31 1

3 2 2


31

4 means how many

3

4are there in 1

3 1 41 1

4 3 3

3 4 41 1

4 3 3


34

1One and one third of

3

4are in 1,

I.e.,4

3of

3

4are in 1.

One3

4

3One third of a

4

2

31

One2

3and half a

2

3are in 1

I.e., one and a half2

3are in 1

One2

3

2Half a3


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Step 2:Extend to division of fractions with fractions.Diagrammatic Representation Symbolic Representation

1 1

2 4 means how many

1

4are there in

1

2

1 1 22

2 4 1

1 1 1 4 2

2 4 2 1 1


2 1

3 2 means how many

1

2are there in

2

3

2 1 1 41

3 2 3 3

2 1 2 2 4

3 2 3 1 3


2

3

1 One and one third of

12

are in 2 ,3

I.e.,4

3of

1

2are in

2.

3

1

2

One1

2

1One third of

2

1

21

Two1

4

are in1

2

1

4

One1

4

One1

4


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Step 3:Examine patterns of fraction multiplication to generalise fraction-divisionrules using inductive reasoning.

1 21 12 1

1 31 1

3 1

1 41 1

4 1

2 31 1

3 2

3 4

1 14 3

1 1 1 4

2 4 2 1

2 1 2 2

3 2 3 1

Hence,a c a d

b d b c

2.2.4 Problem Solving Involving Proper Fractions

One major objective of involving students in problem solving activities is to helpthem see the application of abstract mathematic problems in real-life situations.In other words, problem solving activities bridge the gap between mathematicswhich appears to be abstract and the real world. As such, the problems designedshould be relevant to students real life experiences. The following examplesillustrate this.

Example 2.6: Adding Proper FractionsHanifs mother made a cake for him. Hanif ate 3

10 of the cake, and his friend

Wee Kiat ate1

2of the cake. What is the fraction of the cake both of them have

eaten?


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Understand the problem: Hanif ate 310

of the cake and Wee Kiat ate1

2of the

cake. How much have they eaten?

Pictorial Representation:

3

10

1

2

Devise a strategy: Use addition of fractions.Carry out the strategy: 4

5

3 1 3 5 8

10 2 10 10 10

5

4

5

Note that1 1 5 5

2 2 5 10

Check your answer: We may use a suitable diagram to check youranswer. In this particular example, we may use arectangle (representing the cake) which has beendivided into 10 equal parts. First shade three parts

out of the 10 parts (light grey region represents3

10),

then shade five parts (dark grey region represents1 5

or2 10

).

Now, the total amount of cake eaten is represented

by the total shaded region, which is8

10or

4.

5


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Example 2.7: Subtraction of Proper FractionsMary brought 1 litre of water to school. She drank

1

2 litre of water during recess.

Later, she drank another 38

litre of water. How much water is left?

Understand the problem: Mary brought 1 litre of water. She drank 12

litre,

followed by another3

8 litre. How much water is

left?

Devise a strategy: Use subtraction of fractions.Carry out the strategy: 1 3 8 4 31

2 8 8 8 8

8 4 3

8

1

8

Note that8 1 4

1 , .8 2 8

Check your answer:1

2 38

1

81


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TOPIC 2 FRACTIONS98

MIXED NUMBERS AND IMPROPERFRACTIONS

We will now discuss mixed numbers and improper fractions.

2.3.1 Concept of Mixed Numbers and ImproperFractions

A mixed numberconsists of a whole number and a proper fraction.For example,

3 4 1 51 , 3 , 2 , 8

4 7 6 7

are mixed numbers.

An improper fractionis a fraction where the numerator is larger than or equal tothe denominator.

Below are some examples of improper fractions:

6 9 13 3 7 14 6 15, , , , , , ,

5 7 11 2 4 11 6 15

Figure 2.18 illustrates the diagrammatic representation of a mixed number by

combining a whole number representation and a proper fraction representation.Notice that the representation for an improper fraction is the same as therepresentation for the equivalent mixed number.

Figure 2.18:Diagrammatic representation of a mixed number

2.3


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A mixed number can be converted into the form of an improper fraction.

Note that the mixed number3

14

and the improper fraction7

,4

though of

different forms, are equal in value.

To convert the mixed number3

14

to improper fraction, you may use the steps as

shown in Table 2.8.

Table 2.8:Steps to convert 314

to improper fraction

Steps Example1. Multiply the whole number portion (1) by the denominator (4) 1 4 = 4

2. Then add the value obtained from step 1 to the numerator (3).The resulting value is used as the numerator of the improperfraction. The denominator remains the same.

4 3 7

3 4 1 31

4 4

4 3

4

7

4

3. Thus the improper fraction obtained is7

.4

At this point, you should be cautioned that the use of diagrams to illustrate thefractions which have values more than 1 (like in the case of improper fractions)may confuse some of the students.

Referring to the diagrammatic representation which was used to illustrate the

concept of3

14

or 7 ,4

some students may interpret the diagram as a

representation of the fraction7

.8

These students may look at the diagram as

having 8 parts and that there are 7 shaded parts as shown in Figure 2.19.Therefore it appears to them that this is a case of 7 out of 8.


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TOPIC 2 FRACTIONS100

Figure 2.19:Seven shaded parts out of eight parts

Thus, it is important to help your students see that the denominator in animproper fraction refers to the number of equal parts in a one whole, in this case4 parts and not 8.

To avoid the problem mentioned above, the illustration of concepts should becarried out as a developmental process. Students need to have a goodunderstanding of the meaning of the whole as compared to parts. To make thispoint clear, let us look at the following method used (Table 2.9) to explain the

meaning of7

4using representations.

SELF-CHECK 2.5

If you encounter students with this problem of misinterpretation, howwould you help to rectify it?


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2.3.2 Addition and Subtraction Involving MixedNumbers and Fractions

There are two basic methods to this.

Example 2.8:Calculate 1 22 1 .4 3

Method 1:Separate whole numbers from the fractions, as shown in Table 2.10.Table 2.10:Separate Whole Numbers from Fractions

1 22 1

4 3

1 22 1

4 3

Separate the whole number from the respective fractions.

1 22 1

4 3

Rearrange/regroup into whole numbers and fractions.

3 83

12 12

Add the whole numbers; and Change each of the fractions into its equivalent fraction so

that both fractions have a common denominator.

1 1 3 3 2 2 4 8;

4 4 3 12 3 3 4 12

3 8312

Add the fractions.

113

12

Combine the whole number and the proper fraction to form amixed number as the final answer.

So, a complete solution to the above question can be presented as follows:

1 2 1 22 1 2 1

4 3 4 3

3 83 12 12

113

12

113

12


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Method 2:Change mixed numbers into improper fractions as in Table 2.11..Table 2.11:Change Mixed Numbers into Improper Fractions

Steps Explanation

1 22 1

4 3

9 5

4 3

Change each of the mixed number into an improper fraction.

27 20

12 12

Change each of the fractions into its equivalent fraction so thatboth the fractions have a common denominator.

9 9 3 27 5 5 4 20;

4 4 3 12 3 3 4 12

27 20

12

47

12

Add the fractions.

36 11

12 12

113

12

Convert the improper fraction to a mixed number.

A written solution to the problem above can be in the following form:

1 2 9 52 1

4 3 4 3

27 20

12 12

47

12

113

12


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To convert an improper fraction into a mixed number, you can divide thenumerator by the denominator as follows:

312 47

36

11

Therefore,47 11

3 .12 12

2.3.3 Multiplication and Division Involving Mixed

Numbers

Multiplication and division questions involving mixed numbers and fractionsmay be asked in various forms. Some of these forms include:

(a) Multiplication of a mixed number with a fraction or vice-versa;(b) Multiplication of two mixed numbers;(c) Division of a mixed number by a fraction; and(d) Division of a mixed number by another mixed number.


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Table 2.12 categorises the various forms and their calculations.

Table 2.12:Various Forms of Multiplication and Division Involving Mixed NumbersCategory Steps Explanation

Multiplication of amixed numberwith a fractionExample:

1 32 .

2 4

1 3 5 32

2 4 2 4

5 3

2 4

15

8

71

8

Convert the mixed number toimproper fraction;

Multiply numerator withnumerator, and denominator withdenominator; and

Convert improper fraction tomixed number.

Multiplication oftwo mixednumbersExample:

1 23 1

3 5

1 2 10 73 1

3 5 3 5

10 7

3 5

70

15

104

2

15 3

24

3

Convert both mixed numbers toimproper fractions;

Multiply. Then simplify. You canalso choose to simplify first beforethe multiplication of thenumerators and the denominators;and

Convert the improper fraction to amixed number.

Division of a mixednumber by afractionExample:

7 265

9 27

2

7 26 52 265

9 27 9 27

52

1 9

327

26 1

2 3

1 1

61

6

Convert the mixed number to animproper fraction;

Multiply by the reciprocal of thefraction;

Simplify the numerators and thedenominators;

Perform multiplication; and Write the answer in the simplest

form.


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Division of a mixednumber by anothermixed number.Example :

2 13 2

5 25

1

2 1 17 513 2

5 25 5 25

17

1 5

25

5

51 3

1 5

1 35

32

13

Convert the mixed numbers toimproper fractions;

Change division intomultiplication by the reciprocal ofthe second fraction;

Simplify the numerators and thedenominators;

Perform multiplication; and Convert the improper fraction into

a mixed number.

2.3.4 Problem Solving Involving Mixed Numbers andImproper Fractions

By doing appropriate problem solving activities, students will be able tounderstand the differences between proper fractions, improper fractions andmixed numbers better. Here are some examples.

Example 2.9Alis mother makes five cakes. She wants the cakes to be shared equally betweenAli and his two friends. How much of the cake does each person get?

Understand the problem: Total number of cakes = 5Number of people to share the cakes equally = 3


Devise a strategy: Use the division method.Five cakes shared equally by three people can bewritten as 5 3

Ali

Friend 1

Friend 2


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Carry out the strategy: 55 33

3 2

3 3

21

3

213

Check your answer: If the answer is correct, then the answer for2 2 2

1 1 13 3 3

should be 5.Example 2.10Mee Fah brought 2 and

1

3kg of flour to her school for making cookies during her

home science practical lesson. She used up a total amount of 13

4kg of flour. How

much of the flour was left?

Understand the problem: Initially, Mee Fah has 123

kg of flour

She then used3

14

kg of flour.


Devise a strategy: Using the subtraction method 1 32 1 .3 4

Difference

12

3

31

4


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Carry out the strategy:1 3 7 7

2 13 4 3 4

7 4 7 3

3 4 4 3

28 21

12 12

7

12

(Change to an improper fraction)

(Change to equivalent fractions

with a common denominator)

Check your answer: If the answer is correct, then7 3

112 4

should equal

the initial value of1

2 .3

Example 2.11Jaafar is

31

5m tall. His younger sister Fatin is

7

10of his height, whereas his elder

brother Hilmis height is5

18

that of Fatins. Calculate Hilmis height.

Understand the problem: Height of Jaafar = 31 m5

Height of Fatin =7

10of Jaafars height

Height of Hilmi =5

18

of Fatins height.


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Devise a strategy: Use multiplication.Find Fatins height, then find Hilmis height.

Carry out the strategy: Fatins height:

5

7 31

10 5

7

10

8

4

5

7 4

5 5

28m

25

5 28Hilmi s height 1

8 25

13

8

2

28

7

25

13 7

2 25

91

50

411 m

50

Fatins height 7

10 of Jaafars

height

Jaafars height3

1 m5

Hilmis height5

18

of Fatins height

10

7

51

8


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Check your answer: The ratio of Hilmi s heightFatin s height

should be

51

8

13

41 28 91 28150 25 50 25

91

2 50

25

1

28 4

13

8

51

8

Similarly, you can show that

the ratio ofFatin s height

Jaafar s height

is

7.

10

Example 2.12In a mathematics test, Brian is given

31

4hours to complete all the 20 questions in

the test. However, he intends to spend

1

4 hour to check his answers. Calculatehow much time he should spend on each question. Give your answer in minutes.

Understand the problem: Total time = 314

hours

Time for checking =1

4 hour

Use the remaining time to answer 20 questions and

find the duration for each question.

The answer has to be in minutes.


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Devise a strategy: Using mixed operations: subtraction, followed bydivision, then multiplication.

Time left to answer 20 questions =3 1

14 4

hours.

Time for each question divide by 20.

Answer has to be in minutes multiply by 60.

Carry out the strategy: Time left to answer 20 questions.3 1

14 4

7 1

4 4

6

4

3hours

2

31

4

Time to answer 20 questions1

4hour


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Time for each question

2

320

2

3 1

2 20

3hours

40

3

40

603

minutes

9= minutes

2

1=4 minutes2

Check your answer: Total time = 3 1 320 140 4 4

hours.

A fraction is a number that represents part of a whole. A fraction is written in the form of ,a

bwhere ais called the numerator and b

is called the denominator. Equivalent fractionsare fractions that have the same value. A proper fraction is a fraction where the numerator is smaller than the

denominator.

A mixed numberconsists of a whole number and a proper fraction. Representations are essential in understanding fractions including

performing basic operations with fractions and solving word problems.

Making connections between various representations enhances meaningfullearning.


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Factors

Highest Common Factor (HCF)

Lowest Common Factor (LCM)

Number patterns

Number operations

Prime factors

Prime numbers

Whole numbers

Rounding

Sequences

Topic 2 Fractions

Documents

Transcript of Topic 2 Fractions