TOPIC 1 The Teaching of Fractions Introduction · TOPIC 1 The Teaching of Fractions Introduction Many children get confused when learning the concept of fractions. As illustrated

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TOPIC 1 The Teaching of Fractions

Introduction Many children get confused when learning the concept of fractions. As illustrated in Figure 1, when paper of different sizes are folded once, both the folded parts represent ½ , although one is bigger than the other.

Figure 1: Different sizes for ½ Therefore, you need to provide opportunities for your pupils to learn fractions meaningfully. One good way to do so is to use concrete manipulative such as pattern blocks, cut-out shapes, fraction strips and fraction discs to involve your pupils actively in your classroom. This topic is aimed at inspiring you to teach fractions innovatively. Learning Outcomes By the end of this topic, you will be able to:

1. list out the pedagogical content knowledge for the topic of fractions; 2. plan teaching and learning activities to recognize, name, write, classify as

well as compare fractions; 3. plan teaching and learning activities to perform addition and subtraction of

fractions; 4. plan teaching and learning activities to perform multiplication of fractions with

a whole number; 5. plan teaching and learning activities to solve daily problems involving

fractions.

1

½

1

½

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2.1 Major Mathematical Skills – Fractions As specified in the Mathematics‟s syllabus in the Integrated Curriculum for Primary Schools, your pupils need to learn the following major mathematical skills related to fractions:

Name, write as well as compare proper and improper fractions.

State fractions as part of a collection of objects.

Find equivalent fractions of proper fractions.

Change improper fractions to mixed numbers and vice versa.

Add and subtract proper fractions and mixed numbers.

Multiply fractions with whole numbers.

Solve daily problems involving fractions and mixed numbers. 2.2 Pedagogical Content Knowledge – Fractions

In your opinion, why is there a need for fractions to exist in our daily lives?

When human beings started to keep count of things, whole numbers were invented. When things do not exist as „wholes‟, whole numbers were no longer sufficient to suit daily needs. Fractions are required when a whole is divided into parts. When

dividing whole numbers such as 38 ÷ , the remainder can be expressed as 3

2. So,

fractions are not just for school mathematics. Instead, they also complement whole numbers as a useful tool in our daily lives. 2.2.1 The Meanings of Fractions

What is a fraction?

Jot down your ideas.

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There are three main interpretations of fractions: (a) fraction as parts of a unit whole; (b) fractions as parts of a collection of objects; and (c) fractions division of whole numbers. Fractions as Parts of a Unit Whole A fraction is usually considered as parts of a unit whole. A unit whole is normally called „one‟ for primary pupils. Figure 2 illustrates one example of this meaning of fractions.

Figure 2: An example of fractions as parts of a unit whole. Four main ideas are associated with this meaning of fractions:

The size and shape of the unit whole may not be the same.

The unit whole is divided into parts of equal size.

The sum of all the equal parts is the unit whole.

The fraction refers to the number of parts under consideration. Pertaining to that, the numerator shows the number of parts under consideration; whereas the denominator shows the number of all the equal parts. Consequently, as the number of equal parts increases, the size of each part decreases. In addition, many pupils may not see the idea that the parts can be collected. As an example, 3/4 is made up of ¼ and ¼ and ¼ as illustrated in Figure 3.

¾ = ¼ ¼ ¼

Figure 3: ¾ = ¼ + ¼ + ¼ Fractions as Parts of a Collection of Objects Fractions can also be considered as parts of a collection of objects as illustrated in Figure 4.

Figure 4: An example of fractions as parts of a collection of objects.

This meaning of fractions will lead to the idea of multiplying a fraction with whole numbers. As an example, ¼ from 8 can also be expressed as (¼ x 8).

¼ from the square is shaded

¼ from 8 eggs are white.

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Fractions as Division of Whole Numbers

A fraction can be expressed as a division of whole numbers in the form of q

p, where

p and q are whole numbers (except zero). The number p is called the numerator and

q is called the denominator. For example: 4÷3=4

3 and 3÷4=

3

4.

2.2.2 Different Ways to Represent a Fraction

Fractions are concisely represented in the symbolic form of q

p as described earlier.

You need to bear in mind that abstract symbols are often difficult for young children to understand. Thus, other forms of representing fractions play a significant role in helping them to develop their conceptual understanding of fractions. Some other forms of representing fractions as illustrated in Figure 5 are (a) fraction disc, (b) fraction strip and (c) fraction number line.

Figure 5(a). A fraction disc representing 8

3.

Figure 5(b). A fraction strip representing 8

3.

Figure 5(c). A fraction number line representing 8

3.

1 0 8

3

8

1

8

2

8

4

8

5

8

6

8

7

8

3

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A common error that children often made is to write 3

2for the fraction

represented by the shaded region here:

Why do you think they do this?

2.2.3 Types of Fractions A proper fraction is a fraction where the numerator is less than the denominator. In other words, its fractional parts are less than a whole. Therefore, its value is less

than 1. Examples of proper fractions are 5

4

3

1; and

11

10. These fractions are read as

“one third”; “four fifths”; and “ten elevenths” respectively. An improper fraction is a fraction where the numerator is equal to or greater than the denominator. In other words, its fractional parts are equal to or beyond a whole. Therefore, its value is equal to or more than 1. Examples of improper fractions are

4

5

2

2; and

12

12. These fractions are read as “two halves”; “five fourths or five quarters”

and “twelve twelfths” respectively. A mixed number is made up of a whole number (except zero) and a fraction part.

An example of mixed number is 22

1 which consists of 2 wholes and one

2

1 or

2

1+2 .

Another example, 7

35 is read as “five and three sevenths”. Mixed numbers and

improper fractions are interchangeable. As an example, 6

13=

6

12 .

Equivalent fractions are fractions with the same value but expressed in different

numerators and denominators. For example, 2

1 =

4

2 =

8

4 =

48

24=

10

5.

What happens to the value of a fraction when its numerator and denominator are multiplied by a same number (except zero)? What happens to the value of a fraction when its numerator and denominator are divided by the same number (except zero)?

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How would you use drawings or concrete manipulatives to show your

pupils that 4

11=

4

5?

How would you use drawings or concrete manipulatives to show your

pupils that 3

2=

9

6?

2.2.4 Addition and Subtraction of Fractions Addition and subtraction of fractions can be demonstrated in many different ways. One easy way for your pupils to understand these operations is to use the fraction strips. Figure 6 shows how it can be done for (a) addition and (b) subtraction.

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

Figure 6 (a). Using fraction strips to perform 8

2+

8

3.

8

5

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8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

Figure 6(b). Using fraction strips to perform 8

5 –

8

2.

The same process can be illustrated using fraction number lines as shown in Figure 7.

Figure 7(a). Using fraction number line to perform 8

2+

8

3.

Figure 7(b). Using fraction number line to perform 8

5 –

8

2.

8

3

8

3

8

5

0 1

8

2

8

3

8

5

0 1

8

2

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2.2.5 Stages of Conceptual Development for Fractions The concept of fractions is first developed as fractional parts of a unit whole among children. This concept of fractions is commonly known as the region model for fractions. In this model, areas of parts are compared. When exploring the ideas of addition and subtraction involving fractions, the concept of fractions as parts of a unit whole is extended to include the measurement model for fractions. In this model, lengths are compared instead of areas. At a later stage, the concept of fractions is extended to be parts of a collection of objects, which is commonly known as the set model for fractions. In this model, quantities of discrete objects are compared. On the whole, the conceptual developmental of fractions among children includes the following basic stages:

Building fraction number sense by eliciting intuitive ideas of fractions existent among children.

Extending understanding of fractions as parts-to-whole relationship based on the region model.

Consolidating the concepts of fractions through understanding the relationships between various types of fractions.

Understanding basic operations (addition and subtraction) involving fractions.

Extending the concepts of fractions using the set model for fractions.

Solving daily problems involving fractions. 2.3 Samples of Teaching and Learning Activities for Fractions As mentioned earlier, active construction of fractional concepts is crucial in ensuring meaningful learning among your pupils. The following are samples of activities to encourage active pupil participation when teaching fractions. Activity 1: Folding Paper Strips Children‟s Learning Outcomes:

To demonstrate the meaning of fractions as parts of a unit whole.

To name proper fractions with denominators not more than 10. Materials:

Paper strips

Colour pencils

Procedure:

1. Children fold paper strips into 2, 4, and 8 equal parts, mark the folded lines and then write down the fractions in each part as shown in Figure 8.

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2

1 2

1

4

1

4

1

4

1

4

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

8

1

Figure 8. Paper strips folded into halves, quarters, and eighths.

2. Ask children to shade one part of the „half strips‟. Guide them to say the fractions out loud: “half is shaded; half is not shaded”. Then, guide them to write the

fractions on the other side of the strips: “2

1 shaded;

2

1 not shaded.”

3. Repeat procedure 2 for the „quarter strips‟ and guide them to say the fractions out loud: “one quarter is shaded and 3 quarters are not shaded”. Then guide them to

write on the other side of the strips: “4

1 shaded;

4

3 not shaded.”

4. Ask children to shade 5 parts of the „eighth strips‟. Guide them to say the fractions out loud: “5 eighths are shaded; 3 eighths are not shaded.” Then, guide

them to write on the other side of the strips: “8

5 shaded;

8

3 not shaded”.

Activity 2: Paper Folding

Children‟s Learning Outcomes:

To name half, fourth, and eighth.

To compare the values of half, fourth, and eighth. Materials: Squared and circular papers Procedure: 1. Guide pupils to fold squared papers as follows:

ONE square fold 1 time fold 2 times fold 3 times

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2. Get pupils to stick each of their folded squared papers on a card and write the corresponding fraction on it.

3. Get pupils to arrange the fractions from the smallest value to the biggest value. 4. Repeat procedure 1, 2 & 3 for circular papers. Activity 3: Fraction Game I Children‟s Learning Outcomes;

To consolidate the concept of fractions as parts of a unit whole. Materials:

A cube with two sides marked ½; two marked 1/3; and two ¼.

A cube with two sides marked “equal to”; two marked “more than”; and two “less than”.

12 red counters; 12 blue counters.

A game board for each group of 2 – 4 players. Procedure:

1. Players take turns to roll the two cubes. If “1/2” and “more than” are thrown, the

player may place a counter on or as the shaded area on both represent more than half the total shape.

2. The winner is the first player to cover three adjacent squares in a row; horizontally, vertically or diagonally.

1 2

1 4

1

8

1

1 2

1

4

1

8

1

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Game Board for Fraction Game I

[Source: The Mathematical Association of Western Australian. (1991). Mathematical games for the classroom. Australian Association of Mathematics Teachers.]

Activity 4: Fraction Game II Children‟s Learning Outcomes:

To develop initial ideas of addition and subtraction of fractions.

To write addition and subtraction fraction sentences.

To exchange equivalent fraction strips. Materials:

1 pair of scissors

1 fraction dice (2

1,

2

1,

4

1,

4

1,

8

1,

8

1,

16

1,

16

1)

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5 fraction strips (6-by-36-cm), cut into strips of ½; ¼; 1/8; 1/16 pieces as shown below:

1

½ ½

¼ ¼ ¼ ¼

1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8

1/16 1/16 1/16 1/16 1/16 1/16 1/16 1/16 1/16 1/16 1/16 1/16 1/16 1/16 1/16 1/16

Procedure: Game 1: Cover Up 1. Pupils play the game in groups of 2 – 4 players. 2. Each player starts with the whole strip. 3. The goal is to be the first to cover the whole strip completely with other pieces of

the fraction kit. No overlapping pieces are allowed. 4. Players take turns to roll the fraction dice. 5. The fraction face up on the dice tells what size piece to place on the whole strip. 6. When the game nears the end and a player needs only a small piece, such as

1/8 or 1/16, rolling ½ or ¼ won‟t do. The player must roll exactly what is needed. Game 2: Uncover 1. Each player starts with the whole strip covered with the two ½ pieces. 2. The goal is to be the first to uncover the strip completely. 3. Players take turns rolling the dice. 4. A player has three options on each turn:

- to remove a piece (only if he or she has a piece the size indicated by the fraction face up on the dice)

- to exchange any of the pieces left for equivalent pieces (A player may not remove a piece and trade on the same turn, but can do only one or the other.)

- to do nothing and to pass the cube to the next player. Extension Get pupils to write addition and subtraction fraction sentences based on any cover-up or uncover situations during the game.

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Activity 5: Creative Cakes Children‟s Learning Outcomes:

To find the fractional values of parts of a whole that are unequal in size. Materials: Cards of Creative Cakes:

Cake 1 Cake 2

Cake 3 Cake 4

Cake 5 Cake 6

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Procedure: 1. Each pair of pupils is given a card of cakes cut into various creative parts.

2. Pose the following problem to pupils:

“Each cake costs RM1. How much will each piece cost?”

3. Pupils cut apart each of the cakes and use the pieces as manipulatives to determine the fractional value and cost of each piece of the cakes.

4. Have pupils explain their strategies for solving the problem and their solutions.

5. Have pupils explain the relationship between and among the pieces of each creative cake.

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Summary 1. The concept of fractions is difficult for many children to understand. Therefore,

concrete experiences with manipulatives and active participation are crucial in facilitating the learning of this concept.

2. Fractions can be interpreted as (a) parts of a unit whole; (b) parts of a collection

objects; and (c) division of whole numbers. 3. Types of fractions that children need to learn include (a) proper fractions; (b)

improper fractions; and (c) mixed numbers. The concept of equivalent fractions is important in determining the relationship between these fractions.

You have come to the end of this topic. Hope you have enjoyed yourself and see you soon in the next topic.

References Cathcart, W. G.; Pothier, Y. M.; Vance, J. H. & Bezuk, N. S. (2003).

Learning mathematics in elementary and middle schools. Columbus, Ohio: Merrill Prentice Hall.

Curcio, F. R. & Bezuk, N. S. (1994). Curriculum and evaluation

standards for school mathematics. Addenda series, Grades 5 – 8. Understanding rational numbers and proportions. Reston, Virginia: National Council of Teachers of Mathematics.

John A. De Walle. (2002). Elementary and Middle school Mathematics-

Teaching Developmentally, Fourth Edition, Addison Wesley Longman, Inc

The Mathematical Association of Western Australian. (1991).

Mathematical games for the classroom. Australian Association of Mathematics Teachers.