Topic Fractions T3

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X INTRODUCTIONThe study of fractions is foundational in mathematics, yet it is among the mostdifficult topics of mathematics for young school-going children. They often getconfused when learning the concept of fractions. Many pupils have difficultyrecognising when two fractions are equal, putting fractions in order by size, andunderstanding that the symbol for a fraction represents a single number. Pupils alsorarely have the opportunity to understand fractions before they are asked toperform operations on them such as addition or subtraction (Cramer, Behr, Post, &Lesh, 1997).

For that reason, it is very important for us to provide opportunities for our pupils tolearn and understand fractions meaningfully. We should use physical materials andother representations to help our children develop their understanding of theconcept 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 discrete models (e.g., counters, sets).

In order to start teaching fractions, it is important for us to have an overview of themathematical skills involved in acquiring the concept of fractions. At the beginning

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33

X Fractions

LEARNING OUTCOMES

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

1. Apply the terms related to fractions correctly for the topic of fraction inKBSR Mathematics Year 3;

2. Adopt teaching and learning activities of reading one whole, one half,one quarter and three quarters;

3. Adopt teaching and learning activities of writing one whole, one half,one quarter and three quarters; and

4. Appraise effective strategies in teaching to recognise43

64 and 6

630



X TOPIC 3 FRACTIONS46

of this topic, we will learn about the pedagogical content knowledge of fractionssuch as the meaning of fractions. In the second part of this topic, we will look at themajor mathematical skills for fractions in Year 3, before learning how to plan and

implement activities to recognise, name, write, classify as well as compare fractions.Since the fractions involved are simple, the important thing is to guide the pupils tohave a proper understanding of fractions.

PEDAGOGICAL CONTENT KNOWLEDGE

In the beginning, when human beings started to keep count of things, they startedto use whole numbers. However, as they realised that things just do not alwaysexist as complete wholes, they have to invent numbers that represent „a wholedivided into parts‰. Since, whole numbers were just not sufficient to suit their dailyneeds, fractions were invented to supplement the gap found in between wholenumbers. We will see that fractions are not only used for the formal purpose in

school mathematics but rather a very important tool in our daily lives tocomplement whole numbers.

3.1.1 Meaning of Fractions

What do you understand by the word „fractions‰? The word „fractions‰ comesfrom the Latin Word for ‰breaking‰. Actually, fractions are about breaking orsplitting things up equally. We use fractions all the time. We might say, „IÊll do half of that job‰, or „Please cut me a quarter of that cake‰. In fact, as you will see,

splitting things into equal parts is so important and useful that a whole system of measurement is based on that idea.

The use of fractions started from the idea that it is very natural to take a length andhalve it, then halve each part, then halve each smaller part, and so on. From this, weshall see, for example, that a quarter is half of a half.

However, a visual introduction to fraction is the preferred approach. We can startthe class using the following activity.

3.1

Can you think of reasons for fractions to exist in our daily lives? Searchthe Internet for information about how man started to use fractions.

ACTIVITY 3.1



TOPIC 3 FRACTIONS W 47

Sample Class Activity 3.1

Ask the pupils to count the number of circles. Next, ask the pupils to count the

number of circles in the following picture.

You start by saying that there are five pieces. You take one out, and four remain.So, you say there are four out of five and write (remember, use simple words).

Now, you take out other parts and ask the pupils similar questions. Initially, thepictorial representation can be very intuitive, but later on this can be a drawbackto their understanding. So, words and symbols should also be stressed.

Now, let us see how fractions are interpreted. There are three ways, as shown inFigure 3.1.

Figure 3.1: Interpretation of fractions

We will see an example for each interpretation so that we will understand better theconcept and apply them in our teaching and learning activities. In fact, it is veryimportant for us to provide opportunities for our pupils to differentiate these threeinterpretations so that they would understand fractions well. However, from thepedagogical point of view, it is best for the pupils to be comfortable with one view

before moving on to other interpretations. We can use physical materials or any




other representation to help our pupils develop the understanding of fractions.Now, let us see the three interpretations in detail.

(a) Fractions as Parts of a Unit WholeA fraction is a share of something. The most common shares are halves, that is,when we split (divide) something into two equal parts, and quarters when wesplit something into four equal parts.

Usually, a fraction is considered as parts of a unit whole. For primary pupils, aunit whole is better known as „one‰. Figure 3.2 illustrates the meaning of fractions as parts of a unit whole or „one‰.

Figure 3.2: Examples of fractions as parts of a unit whole

The following four main ideas are associated with the meaning of fractions:

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

(ii) The unit whole is divided into parts of equal size.

(iii) The sum of all the equal parts is the unit whole.

(iv) The fraction refers to the number of parts under consideration.

The numerator indicates the number of parts under consideration, while thedenominator indicates the number of all the equal parts. Accordingly, as thenumber of equal parts increases, the size of each part decreases.

We may believe that children will automatically understand the concept of

fractions simply as a result of using the various representations ormanipulatives. This is not necessarily the case (Thompson & Lambdin, 1994).Some pupils define a fraction as „a piece of pie to eat‰ because they have onlyseen fractions represented using circle diagrams (Niemi, 1996). Providingmany kinds of representations can help pupils with such problems, as long aswe help pupils connect their understanding of fractions to the differentrepresentations.




For instance, if we ask a pupil to fold paper of different sizes once (see Figure 3.3 below), he/she would find that one is bigger than the other, although both

represent43 .

Figure 3.3: Different sizes for 43

Many pupils may not see the idea that the parts of a unit whole can be collected. For

example,

7

6is actually made up of four „one fifth‰s. Another example is illustrated

in Figure 3.4 where it is shown that three „one quarter‰s make up6

5.

Figure 3.4: Parts of a unit whole can be collected

However, you have to be careful with this. We have to help the pupils understandthat fraction is part of a whole. This is their first introduction to the topic and theymost probably will not realise that this is a number just like 1, 2 and so on. As aresult, there is a very strong likelihood for them to see fractions as two separatewhole numbers and can be manipulated separately. They may say (later on, withmore complex fractions, providing pictures is not easy; anyway, just relying onpictures can deter them from developing abstract thinking) if we can add the




numerator separately, why not also the denominator? So you may get „weird‰answers such as 1/12.

(b) Fractions as Parts of a Collection of Objects Fractions can also be considered as parts of a collection of objects. Thisconcept is clearly illustrated in Figure 3.5.

Figure 3.5: Fractions as parts of a collection of objects

You may have the tendency to reduce the fraction in Figure 3.5 to ó.Reduction is a higher level skill and we must resist the tendency to do this.

On the other hand, to multiply a fraction by a number is acceptable for pupilsat this level. Now, let us try this activity in class.

Sample Class Activity 3.2

Show the class a diagram as shown in Figure 3.6.

Figure 3.6: Example of division

First, ask the pupils to group the eggs into groups of 2. Then, ask them how manygroups there are. Next, say that in these groups which consist of 10 eggs, 4/5 of them are rotten. How can you find the number of rotten eggs? Now, the pupilsknow that four groups are rotten. In a group, there are 2 eggs. Hence, the number of rotten eggs is 4 x 2 = 8.

Try not to hide the symbol and manipulation by writing (4/5) x 10 = 8.

This is because from the pupilÊs point of view, there are two operations taking placeand they might find it difficult. It is important that they do not get the wrong idearight from the beginning.




(c) Fractions as Division of Whole Numbers

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

r,

where p and q are whole numbers. The number p is called the numerator and

q is called the denominator. For example, 767

6y and :9

:

9y .

MAJOR MATHEMATICAL SKILLS FORFRACTIONS IN ELEMENTARYMATHEMATICS

A systematic conceptual development of fractions will be very helpful for ourpupils to learn this topic effectively. We must remember that fractions areintroduced for the very first time in Year 3. It would be advisable to introduce thetopic in a systematic manner. It is important for us to provide opportunities for ourpupils to understand fractions meaningfully. We should use physical materials andother representations to help the children develop their understanding of theconcept of fractions.

The major mathematical skills related to fractions to be mastered by Year 3 pupilsare as follows:

(a) Concept of fractions

(i) Recognise fractions as equal shares of a whole set.

(ii) Recognise one whole, one half, one quarter and three quarters.

(iii) Recognise parts that are not fractions.

(b) Saying and reading fractions

(i) Saying one whole, one half, one quarter and three quarters.

(ii) Reading one whole, one half, one quarter and three quarters.(iii) Saying and reading one whole, one half, one quarter and three quarters

in context.

(c) Writing fractions

(i) Knowing the terms „numerator‰ and ‰denominator‰.

(ii) Writing4

3,6

3,6

5in context.

3.2




(iii) Recognise4

3

6

4 and 3

6

6

(iv) Recognise fractions as equal shares.

TEACHING AND LEARNING ACTIVITIES

Now, let us look at some of the teaching and learning activities involving fractions.

3.3.1 Basic Facts of Fractions

Activity 1: Recognise Fractions

Learning Outcome:

To recognise fractions

Materials:

(a) Task cards

(b) Answer sheets

Procedure:

(a) Divide the class into groups of six pupils and each pupil is given an answersheet.

(b) Pupils write their names on the answer sheet (sample is shown in Figure 3.7).

(c) Cards are shuffled and put face down in a stack at the centre.

(d) Each player begins by drawing a card from the stack.

(e) The player writes all the answers to the questions in the card drawn on theanswer sheet.

3.3

1. Have you ever taught fractions to Year 3 pupils? If yes, share yourexperience and method of teaching with your coursemates. How didyou introduce fractions to your pupils?

2. What kind of teaching aids can you use to teach fractions? Why?Compile and compare your answers with your coursemates.

ACTIVITY 3.2




(f) After a period of time to be determined by the teacher, the pupils in the groupexchange the card with the pupil on their left in clockwise direction.

(g) Pupils repeat steps (e) and (f) until all of them in the group have answered the

questions in all the cards.(h) The winner is the pupil who has the most number of correct answers.

(i) Summarise the lesson on the basic facts of fractions.

Example of an answer sheet is illustrated in Figure 3.7:

Figure 3.7: An example of an answer sheet




Example of a task card is given in Figure 3.8:

Figure 3.8: An example of a task card




3.3.2 Say and Read Fractions

Activity 2: Read Fractions

Learning Outcome:

To read one whole, one half, one quarter and three quarters.

Materials:

(a) 30 different Fraction Cards

(b) Clean writing papers

The following are some examples of the Fraction Cards:

Figure 3.9: An example of fraction cards

Work with your friends in class to prepare five more Task Cards.

There should be four questions in each card. Make sure your cards aresuitable for Activity 1: Recognise Fractions.

ACTIVITY 3.3




Procedure:

(a) Divide the class into groups of three pupils and each group is given a cleanwriting paper.

(b) Pupils write their names on the clean paper given.

(c) Fraction Cards are shuffled and put face down in a stack at the centre.

(d) Player A begins by drawing a card from the stack. He shows the card to Player B.

(e) Player B then reads out the fraction shown in the card. If he reads it correctly,Player C writes two points below Player BÊs name. If he reads it wrongly,Player B is not given any points.

(f) Players repeat steps (d) and (e) until 10 cards are drawn by Player A.

(g) Steps (c) through (f) are repeated until all the players have the opportunity toread 10 Fraction Cards shown to them.

(h) The winner is the group where the pupils have the most number of points.

(i) Summarise the lesson on how to read one whole, one half, one quarter andthree quarters.

3.3.3 Write Fractions

Activity 3: Write Fractions

Learning Outcome:

To write one whole, one half, one quarter and three quarters

Materials:

(a) Task Sheets


(c) Colour pencils




Procedure:

(a) Divide the class into groups of four to six pupils. Each group is given adifferent colour pencil and a clean writing paper.

(b) Set up five stations in the classroom. A Task Sheet is placed at each station.

(c) Instruct pupils to solve the questions in the Task Sheet at each station.

(d) Each group will spend 10 minutes at each station.

(e) At the end of 10 minutes, the groups will have to move on to the next stationin clockwise direction.

(f) At the end of 50 minutes, collect the answer papers.

(g) The group with the highest score is the winner.

(h) Summarise the lesson on how to write one whole, one half, one quarter andthree quarters.

Example of a Task Sheet as illustrated in Figure 3.10.




Figure 3.10: An example of a task sheet




3.3.4 Knowing "Numerator" and "Denominator"

Activity 4: Knowing the terms "Numerator" and "Denominator"

Learning Outcome:To be able to identify the numerator and the denominator of a given fraction.

Materials:

(a) Activity cards


(c) Colour pencils

The following are some examples of the activity cards:

1. How would you introduce the terms "numerator" and"denominator"?

2. What kind of activities would help your pupils to rememberthese two terms correctly? Why?

SELF-CHECK 3.1




Figure 3.11: An example of the activity cards

Procedure:

(a) Divide the class into groups of four pupils. Each group is given a differentcolour pencil and a clean writing paper.

(b) A set of 12 activity cards are shuffled and put face down in a stack at thecentre.

(c) When you signal, pupils will begin solving the questions in the first ActivityCard drawn.

(d) Once they are done with the first card, they may continue with the next

activity card.

(e) At the end of 10 minutes, the groups will stop and hand their answer paper toyou.

(f) The group with the highest score is the winner.

(g) Summarise the lesson on how to identify the numerator and denominator of agiven fraction.




3.3.5 Recognise4

3

6

4 and 3

6

6

Activity 5: Recognising equivalent fractions4

3

6

4

and 36

6

Learning Outcome:

To be able to recognise equivalent fractions4

3

6

4 and 3

6

6 .

Materials:

(a) Exercise Sheets

(b) Colour pencils

Procedure:

(a) Divide the class into groups of two pupils. Each group is given a differentcolour pencil.

(b) Each group is given an Exercise Sheet with five questions.

(c) The group that finishes first with all correct answers is the winner.

(d) Summarise the lesson on how to recognise equivalent fractions.

(e) Figure 3.12 shows an example of the Exercise Sheet.




Figure 3.12: An example of an exercise sheet




3.3.6 Recognise Fractions as Equal Shares

Activity 6: Recognising Fractions as Equal Shares

Learning Outcome:

To be able to recognise fractions as equal shares of a whole set.

Materials:

(a) Fraction Worksheets

(b) A set of four erasers, a set of two pens, a set of four sharpeners and a set of two pencils

(c) Clean writing papers(d) Colour crayons

Procedure:

(a) Divide the class into groups of four pupils. Each group is given a set of fourerasers, a set of two pens, a set of four sharpeners and a set of two pencils.

(b) Instruct the groups to sketch all the four sets of items on the paper provided.

(c) After a period of time to be determined by you, instruct the pupils to colour

6

3of the eraser set, 4

3of the pen set, 6

5of the sharpener set and one whole of

the pencil set.

(d) Each group will spend five minutes to colour.

(e) At the end of five minutes, collect the answer papers and award four pointsfor all correct answers.

(f) Distribute the Fraction Worksheets to each group.

(g) Instruct the groups to complete the Fraction Worksheet by 10 minutes.

(h) At the end of 10 minutes, collect the Fraction Worksheet.(i) The group with the highest score, including the points from the earlier task, is

the winner.

(j) Summarise the lesson on how to recognise fractions as equal shares of thewhole set.

(k) Figure 3.13 illustrates an example of the Fraction Worksheet:




Figure 3.13: An example of a fraction worksheet




Activity 7: A Linear Fraction

Learning Outcome:

To be able to see fractions in a linear fashion

Purpose:

So far pupils see fractions in terms of pictures that involve circles, squares, etc. Toget a more abstract idea of fractions, pupils need to see fractions in other contextsthrough word problems.

(a) Take a length of string. Ask the pupil to cut it in half or quarter.

(b) Make a problem such as: If the distance between Pulau Pinang and Serembanis 400 kilometres, and the distance between Rawang and Seremban is 100kilometres, what is the fraction of the distance between Pulau Pinang andRawang to the whole journey? (Of course you may use simpler language toexpress the problem.)

(c) You may express the problem in terms of time of journey. As a teacher, youneed to lead the pupils by using the problem solving strategy.

x Three interpretations for fractions were discussed, namely, fractions as parts of a unit whole, fractions as parts of a collection of objects and fractions as divisionof whole numbers.

x The focus of this topic is on the teaching and learning of fractions for Year 3KBSR Mathematics.

x Many examples of activities were developed so that pupils can read and writeone whole, one half, one quarter and three quarters.

x Ideas developed in this topic would be used on the same topic on Fractions in

Year 4 of KBSR Mathematics.

x This knowledge is then extended so as to enable pupils to do operations such asaddition and subtraction of fractions.




Denominator

Fraction

Half

Numerator

Quarter

Share

Whole

1. Explain with an example each of the following interpretations for fractions:(a) Fractions as parts of a unit whole;

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

(c) Fractions as division of whole numbers.

2. List out seven major mathematical skills related to fractions to be mastered byYear 3 pupils.

3. Plan a teaching and learning activity that would enable pupils to write one

whole, one half, one quarter and three quarters.

Anne Toh. (2007). Resos pembelajaran masteri: Mathematics Year 3 . Petaling Jaya:Pearson Malaysia Sdn. Bhd.

Clarke, P. et.al (2002). Maths spotlight activity sheets 1 . Oxford, UK: HeinemannEducational Publishers.

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Ng, S. F. (2002). Mathematics in action workbook 2B (Part 1). Singapore: PearsonEducation Asia Pte Ltd.

Niemi, D. (1996). A fraction is not a piece of a pie: Assessing exceptionalperformance and deep understanding in elementary school mathematics.Gifted Child Quarterly, 40 , 70-80.




Thompson, P. W., & Lambdin, D. (1994). Research into practice: Concrete materialsand teaching for mathematical understanding. Arithmetic Teacher , 41 , 556-558.

Yee, S., & Lau, P. H. (2007). A problem solving approach: Mathematics Year 3 .

Subang Jaya, Selangor: Andaman Publication Sdn. Bhd.

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