Topic 2 fractions

$Page 1: Topic 2 fractions$
INTRODUCTION

Hello, and welcome to the topic on fractions. The basis of mathematics is the study of fractions, yet it is among the most difficult topics for school-going children. They often get confused when learning the concept of fractions as many of them have difficulty recognising when two fractions are equal, putting fractions in order by size, and understanding that the symbol for a fraction represents a single number. Pupils also rarely have the opportunity to understand fractions before they are asked to perform operations on them such as addition or subtraction (Cramer, Behr, Post, & Lesh, 1997). For that reason, we should provide opportunities for children to learn and understand fractions meaningfully. We could use physical materials and other representations to help children develop their understanding of the concept of fractions. The three commonly used representations are area models (e.g., fraction circles, paper folding, geo-boards), linear models (e.g., fraction strips, Cuisenaire rods, number lines), and

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

1. Use vocabulary related to fractions correctly as required by the Year 5 and Year 6 KBSR Mathematics Syllabus;

2. List the major mathematical skills and basic pedagogical contentknowledge related to fractions;

3. Use the vocabulary related to addition, subtraction, multiplication and division of fractions correctly;

4. List the major mathematical skills and basic pedagogical contentknowledge related to addition, subtraction, multiplication and division of fractions; and

5. Plan basic teaching and learning activities for addition, subtraction,multiplication and division of fractions.

TTooppiicc

22

Fractions

TOPIC 2 FRACTIONS 21

discrete models (e.g., counters, sets). We introduced these representations to our pupils in Year 3 and Year 4. It would be useful to show them again these representations to reaffirm their understanding about fractions. In order to start teaching fractions in Year 5 and Year 6, it is important for us to have an overview of the mathematical skills pupils need in order to understand the concept of improper fractions and mixed numbers. It is also important to acquire the mathematical skills involved in adding, subtracting, multiplying and dividing fractions. At the beginning of this topic, we will learn about the pedagogical content knowledge of fractions such as the meanings of proper fractions, improper fractions and mixed numbers. In the second part of this topic, we will look at the major mathematical skills for fractions in Year 5 and Year 6. Before we finish this topic we will learn how to plan and implement basic teaching and learning activities for addition, subtraction, multiplication and division of fractions.

PEDAGOGICAL CONTENT KNOWLEDGE

Do you know how fractions came to be used? When human beings started to count things, they used whole numbers. However, as they realised that things do not always exist as complete wholes, they invented numbers that represented “a whole divided into equal parts”. In fact, fractions were invented to supplement the gap found in between whole numbers. We have discussed the meanings of fractions comprehensively in the Year 3. We have seen that there are three interpretations of fractions:

(a) Fractions as parts of a whole unit;

(b) Fractions as parts of a collection of objects; and

(c) Fractions as division of whole numbers. In fact, it is important for us to provide opportunities for our children to differentiate these three interpretations in order to understand fractions better. In the following section, we will look at the pedagogical content knowledge of fractions such as the types of fractions; namely, proper fractions, improper fractions and mixed numbers.

2.1

Can you think of five reasons why fractions exist in our life? List downthe reasons before comparing them with the person next to you.

ACTIVITY 2.1

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22

2.1.1 Types of Fractions

You can introduce the meaning of fraction to teach them the types of fractions. A fraction is a rational number which can be expressed as a division of numbers in

the form of q

p, where p and q are integers and q ≠ 0. The number p is called the

numerator and q is called the denominator. For example, 545

4 and 87

8

7 .

Let us look at the different types of fractions in the next section. (a) Proper Fractions A proper fraction is a fraction where its numerator is less than the

denominator.

For example : ,....245

123,

24

13,

15

7,

7

5,

4

3,

2

1,

4

1

4

1

2

1

4

3

(b) Improper Fractions An improper fraction is a fraction where its numerator is equal to or

greater than the denominator.

For example : ,....245

523,

24

33,

15

15,

7

9,

4

5,

4

4


4

4

4

5

(c) Mixed Numbers A mixed number consists of an integer (except 0) and a proper fraction.

For example: ,....245

133122,

24

1322,

15

25,

7

23,

4

31

2

11

4

31

Pupils should have ample opportunity to identify and represent the different types of fractions as well as to name and write them down in symbols and words.

2.1.2 Equivalent Fractions

Similar to whole numbers, fractions too have various terms and names. For

example, 8

4,

6

3,

4

2,

2

1 and

10

5all represent the same amount. They are called

equivalent fractions. In other words, fractions with identical values are called equivalent fractions.

TOPIC 2 FRACTIONS

24

Note that to find an equivalent fraction, we multiply or divide both the numerator and the denominator by the same number. For example: (i) Multiplying both numerator and denominator by the same number.

6

3

32

31

2

1

Therefore, 2

1 and

6

3 are equivalent fractions.

(ii) Dividing both numerator and denominator by the same number.

3

1

515

55

15

5

Therefore, 15

5 and

3


Use models to verify the generalisation:

2

1

4

2

6

3

12

6

Equivalent Fractions

Since, 10

5

8

4

6

3

4

2

2

1

Therefore,8

4,

6

3,

4

2,

2

1 and

10



2.1.3 Simplifying Fractions

Now we move on to simplifying fractions. Remind your pupils that the ability to change a fraction to its equivalent fraction is an important skill that is required to understand the characteristics of fractions and to master other skills concerning basic operations of fractions. We should provide various activities for our pupils to master this skill. These activities should involve all the three stages of learning: concrete, spatial concrete and abstract.

A fraction with its numerator and denominator without any common factors

(except 1) is said to be in its simplest form. For example: 15

7,

7

5,

4

3,

3

2,

4

1

and25

9. Conversely, ,

15

5,

10

4,

6

2,

4

2and

28

7 are not in their simplest form

because their numerators and denominators have common factors. The process of changing a fraction to its simplest form is called simplifying a fraction. Simplifying should be thought of as a process of renaming and not cancellation.

In the example below, 8

4and

4

2 are renamed or simplified to

2

1 .

2

1

24

22

4

2

28

24

8

4

2

1 is the simplified form of

4

2 and

8

4.

As a teacher you need to tell your pupils that before they can master the skill of simplifying fractions, they must first understand the concept of proper fractions, improper fractions, mixed numbers and equivalent fractions.

SELF-CHECK 2.1

1. Describe briefly with examples the three types of fractions.

2. Explain the two ways of finding equivalent fractions for a givenfraction.

3. What is meant by simplifying a fraction?

TOPIC 2 FRACTIONS

26

MAJOR MATHEMATICAL SKILLS FOR FRACTIONS

A systematic conceptual development of fractions will be very helpful for our pupils to learn this topic effectively. It would be advisable for teachers to introduce the topic in a less stressful manner. It is important for us to provide opportunities for our pupils to understand improper fractions and mixed numbers meaningfully. We should use physical materials and other representations to help our children develop their understanding of these concepts. We should also provide opportunities for our children to acquire mathematical skills involved in adding, subtracting, multiplying and dividing fractions.

The major mathematical skills to be mastered by pupils studying the topic of fractions in Year 5 and Year 6 are as follows:

(a) Name and write improper fractions with denominators up to 10.

(b) Compare the value of the two improper fractions.

(c) Name and write mixed numbers with denominators up to 10.

(d) Convert improper fractions to mixed numbers and vice versa.

(e) Add two mixed numbers with the same denominators of up to 10.

(f) Add two mixed numbers with different denominators of up to 10.

(g) Solve problems involving addition of mixed numbers.

(h) Subtract two mixed numbers with the same denominators of up to 10.

(i) Subtract two mixed numbers with different denominators of up to 10.

(j) Solve problems involving subtraction of mixed numbers.

(k) Multiply any proper fraction with a whole number up to 1,000.

(l) Add three mixed numbers with the same denominators of up to 10.

(m) Add three mixed numbers with different denominators of up to 10.

(n) Subtract three mixed numbers with the same denominators of up to 10.

(o) Subtract three mixed numbers with different denominators of up to 10.

(p) Solve problems involving addition and subtraction of fractions.

(q) Multiply any mixed numbers with a whole number up to 1,000.

(r) Divide fractions with a whole number and a fraction.

(s) Solve problems involving multiplication and division of fractions.

2.2


TEACHING AND LEARNING ACTIVITIES

Now let us look at several activities that could help pupils not only to understand improper fractions and mixed numbers, but also to acquire the mathematical skills involved in adding, subtracting, multiplying and dividing fractions.

2.3.1 Improper Fractions

2.3 ACTIVITY 2.3

ACTIVITY 2.2

Learning Outcomes:

To write the improper fractions shown by the shaded parts.

To write the improper fractions in words.

To compare the value of the two improper fractions. Materials:

Task Cards

Answer Sheets Procedure:

1. Divide the class into groups of six pupils and give each pupil anAnswer Sheet.

2. Ask pupils to write their name on the Answer Sheet.

3. Six Task Cards are shuffled and put face down in a stack at the centre.

4. Each player begins by drawing a card from the stack.

5. The player writes all the answers to the questions in the card drawn onthe Answer Sheet.

6. After a period of time (to be determined by the teacher), each pupil inthe group exchanges the card with the pupil on their left in clockwisedirection.

7. Pupils are asked to repeat steps (5 and 6) until all the pupils in thegroup have answered questions in all the cards.

8. The winner is the pupil that has the most number of correct answers.

9. Teacher summarises the lesson by recalling the basic facts of improperfractions.

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28

Example of an Answer Sheet:

Name :________________________ Class :______________________

Card A Card B Card C

1.________________ 1.________________ 1.________________

2.________________ 2.________________ 2.________________

3.________________ 3.________________ 3.________________

Card D Card E Card F

1.________________ 1.________________ 1.________________

2.________________ 2.________________ 2.________________

3.________________ 3.________________ 3.________________

Example of a Task Card:

Card A

1. Write the improper fractions of the shaded parts.

=

2. Write in words.

4

5=

3. Circle the larger improper fraction.

4

7

4

9

1. Work with a friend in class to prepare five more Task Cards.

2. There should be three questions in each card.

3. Make sure your cards are based on the learning outcomes of Activity2.2.

ACTIVITY 2.3


2.3.2 Mixed Numbers

Learning Outcomes:

To write the mixed numbers shown by the shaded parts

To convert improper fractions to mixed numbers

To convert mixed numbers to improper fractions Materials:

30 different Flash Cards

Clean writing paper Procedure:

1. Divide the class into groups of three pupils and give each group aclean writing sheet.

2. Instruct the pupils to write their names on the clean paper.

3. Flash Cards are shuffled and put face down in a stack at the centre.

4. Player A begins by drawing a card from the stack. He shows thecard to Player B.

5. Player B then reads out the answers within the stipulated time(decided by the teacher).

6. Player C writes the points obtained by Player B below his name.Each correct answer is awarded one point (a maximum of 3 pointsfor each Flash Card).

7. Players repeat steps (4 and 5) until 10 cards have been drawn byPlayer A.

8. Players now change roles. Player B draws the cards, Player C readsout answers and Player A keeps the score.

9. Repeat steps (3 through 6) until all the players have had theopportunity to read the 10 Flash Cards shown to them.

10. The winner in the group is the student that has the most number ofpoints.

11. Teacher summarises the lesson on the basic facts of mixednumbers.

ACTIVITY 2.4

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30

Example of a Flash Card:

Flash Card 1 1. Write the mixed number shown by the shaded parts. 2. Convert this improper fraction to a mixed number.

4

15 =

3. Convert this mixed number to an improper fraction.

7

33 =

ACTIVITY 2.5

1. Work with three friends in class to prepare another 29 FlashCards.

2. There should be three questions in each Flash Card.

3. Make sure your cards are based on the learning outcomes ofActivity 2.4.


2.3.3 Addition of Fractions

Learning Outcomes:

To add two mixed numbers

To add three mixed numbers

To solve problems involving addition of mixed numbers. Materials:

Task Sheets

Clean writing papers

Colour pencils Procedure:

1. Divide the class into groups of four to six pupils. Provide eachgroup with a different colour pencil and a clean writing sheet.

2. The teacher sets up five stations in the classroom. A Task Sheet isplaced at each station.

3. Instruct the pupils to work together to solve the questions in theTask Sheet at each station.

4. Each group will spend 10 minutes at each station.

5. At the end of 10 minutes, the groups will have to move on to thenext station in the clockwise direction.

6. At the end of 50 minutes, the teacher collects the answer papers.

7. The group with the highest score (highest number of correctanswers) is the winner.

8. The teacher summarises the lesson on how to add mixed numberswith the same denominators and different denominators.

ACTIVITY 2.6

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32

Example of a Task Sheet:

STATION 1

1. Add the following two mixed numbers. Express your answers in the simplest form.

(a) 4

33

4

32

(b) 3

24

5

31

2. Add the following three mixed numbers. Express your answers in the

simplest form.

(a) 5

12

5

22

5

31

(b) 4

33

3

21

2

12

3. Encik Ahmad sold 7

33 kg of prawns to Mr. Chong and

5

22 kg of

prawns to Mr. Samuel. Find the total mass of prawns sold by Encik Ahmad.

The total mass of prawns sold is kg.

ACTIVITY 2.7

Work with two of your friends to prepare another four Task Sheets for the other stations. There should be three questions in each sheet. Makesure your sheets are based on the learning outcomes of Activity 2.6.


2.3.4 Subtraction of Fractions

Learning Outcomes:

To subtract two mixed numbers

To subtract three mixed numbers

To solve problems involving subtraction of mixed numbers Materials:

Activity Cards

Clean writing papers


1. Divide the class into groups of four pupils. Provide each group with a different colour pencil and a clean writing sheet

2. A set of 12 Activity Cards are shuffled and put face down in astack at the centre.

3. When the teacher signals, pupils will begin solving the questions in the first Activity Card drawn.

4. Once they are done with the first Card, they may continue with the next Activity Card.

5. At the end of 10 minutes, the groups will stop and hand theiranswer paper to the teacher.

6. The group with the highest score is the winner.

7. The teacher summarises the lesson on how to subtract mixed numbers with the same denominators and different denominators.

ACTIVITY 2.8

TOPIC 2 FRACTIONS

34

Example of an Activity Card:

1. Subtract the following two mixed numbers. Express your answers in the simplest form.

(a) 4

32

4

14

(b) 3

22

5

34

2. Find the difference of the following mixed numbers. Express your

answers in the simplest form.

(a) 7

11

7

22

7

44

(b) 4

32

3

21

2

15

3. A container holds 8

36 litres of water. Abu Bakar pours

5

22 litres of

water from the container into a jug while his brother Arshad pours

3

21 litres of water from the container into a bottle. How much water,

in fractions, is left in the container?

The amount of water left is litres.

ACTIVITY 2.9

Prepare another 11 Activity Cards for the group. There should be threequestions in each card. Make sure your cards are based on the learning outcomes of Activity2.8.


2.3.5 Multiplication of Fractions

Learning Outcomes:

To multiply proper fractions with whole numbers

To multiply mixed numbers with whole numbers

To solve problems involving multiplication of mixed numbers Materials:

Exercise Sheets


1. Divide the class into groups of two pupils. Give each group adifferent colour pencil.

2. Give each group an Exercise Sheet with five questions.

3. The group that finishes fastest with all correct answers is thewinner.

4. The teacher summarises the lesson on how to multiply fractionswith whole numbers.

ACTIVITY 2.10

TOPIC 2 FRACTIONS

36

Example of an Exercise Sheet:

1. Solve the following multiplication

(a) 324

1

(b) 2005

3

2. Solve the following multiplication

(a) 287

44

(b) 4004

15

3. There are 440 apples in a box. 4

3 of the apples are green apples.

The remaining apples are red. How many red apples are there in the box?

There are red apples in the box.

4. Muthu drinks 4

31 litres of water a day. How much water in litres,

will he drink in two weeks? Muthu drinks litres of water in two weeks.

5. Shalwani spends 4

31 hours watching television in a day. How much time

does she spend watching television in three weeks?

Shalwani spends hours watching television in three weeks.


2.3.6 Division of Fractions

ACTIVITY 2.11

Learning Outcomes:

To divide fractions with whole numbers

To divide fractions with fractions

To solve problems involving division of fractions Materials:

Division Worksheets

Clean writing paper


1. Divide the class into 10 groups. Give each group a DivisionWorksheet, clean writing paper and a colour pencil.

2. Instruct the groups to answer all the questions in the DivisonWorksheet.

3. The groups write their answers on the clean writing paper.

4. After a period of time (to be determined by the teacher), theteacher instructs the groups to exchange the Division Worksheets.

5. Repeat steps 2 to 4.

6. Once all the 10 Division Worksheets have been answered, teachercollects the papers and corrects the answers.

7. The group with the highest score is the winner.

8. The teacher summarises the lesson on how to divide fractions withfractions and with whole numbers.

TOPIC 2 FRACTIONS

38

Example of a Division Worksheet:

WORKSHEET 1

1. Solve the following division of fractions.

(a) 28

1

4

1

(b) 25

9

5

3

2. Solve the following division of fractions.

(a) 334

32

(b) 10

3

5

31

3. A company wants to donate RM 4

32 million equally to eight charities.

How much money will each charity receive? Each charity receives RM

million.

4. The total length of 7 similar ropes is 2

110 m. Find the length of one

rope.

The length of one rope is m.

ACTIVITY 2.12

Prepare another nine Division Worksheets for the group. There shouldbe four questions in each worksheet.

Make sure your worksheets are based on the learning outcomes ofActivity 2.11.


The three commonly used representations for fractions are area models (e.g., fraction circles, paper folding, geo-boards), linear models (e.g., fraction strips, Cuisenaire rods, number lines), and discrete models (e.g., counters, sets).

The three interpretations for fractions are (i) fractions as parts of a unit whole, (ii) fractions as parts of a collection of objects, and (iii) fractions as division of whole numbers.

It is important to provide opportunities for our children to differentiate these three interpretations so that they can understand fractions better.

A fraction is a rational number which can be expressed as a division of

numbers in the form of , where p and q are integers and q ≠ 0. The number p is called the numerator and q is called the denominator.

Pupils in Year 5 and Year 6 should be able to identify proper fractions, improper fractions and mixed numbers. They should be able to simplify the given fractions into its simplest form.

A proper fraction is a fraction where its numerator is less than the denominator.

An improper fraction is a fraction where its numerator is equal to or greater than the denominator.

A mixed number consists of an integer (except 0) and a proper fraction.

Fractions with identical values are called equivalent fractions.

The process of changing a fraction to its simplest form is called simplifying a fraction.

Pupils should be able to acquire the mathematical skills involved in adding, subtracting, multiplying and dividing fractions.

Pupils should also be able to solve daily life problems involving basic operations on fractions.

p

q

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40

Story problems are set in real-life situations. Children are able to determine the reasonableness of their answers when story problems are based on familiar contexts.

Addition

Denominator

Division

Fraction

Half

Multiplication

Numerator

Quarter

Share

Subtraction

Whole

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year 5. Kuala Lumpur: Penerbitan Fargoes. Nur Alia bt. Abd. Rahman & Nandhini (2008). Siri intensif: Mathematics KBSR

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Educational Publishers. Sunny Yee & Lau P.H. (2007). A problem solving approach : Mathematics year

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