Fractions

Scope of this document

The following is a background literature for teachers. It summarises the things to be known to a teacher to teach this topic more effectively . This literature is meant to be a ready reference for the teacher to develop the concepts, inculcate necessary skills, and impart knowledge in fractions from Class 6 to Class 10.

It is a well known fact that teaching and learning fractions is a complicated process in primary and middle school. Although much of fractions is covered in the middle school, if the foundation is not holistic and conceptual, then topics in high school mathematics become very tough to grasp. Hence this documents is meant to understand the research that has been done towards simplifying and conceptually understanding topics of fractions.

It is also very common for the school system to treat themes in a separate manner. Fractions are taught as stand alone chapters. In this resource book an attempt to connect it to other middle school topics such as Ratio Proportion, Percentage and high school topics such as rational, irrational numbers and inverse proportions are made. These other topics are not discussed in detail themselves, but used to show how to link these other topics with the already understood concepts of fractions.

Also commonly fractions are always approached by teaching it through one model or interpretation namely the part-whole model where the whole is divided into equal parts and the fraction represents one or more of the parts. The limitations of this method, especially in explaining mixed fractions, multiplication and division of fractions be fractions has led to educators using other interpretations such as equal share and measure. These approaches to fraction teaching are discussed.

Also a brief understanding of the common errors that children make when it comes to fractions are addressed to enable teachers to understand the child's levels of conceptual understanding to address the misconceptions.

Syllabus

Class 6

Class 7

Fractions: Revision of what a fraction is, Fraction as a part of whole, Representation of fractions (pictorially and on number line), fraction as a division, proper, improper & mixed fractions, equivalent fractions, comparison of fractions, addition and subtraction of fractions Review of the idea of a decimal fraction, place value in the context of decimal fraction, inter conversion of fractions and decimal fractions comparison of two decimal fractions, addition and subtraction of decimal fractions upto 100th place. Word problems involving addition and subtraction of decimals (two operations together on money,mass, length, temperature and time)

Fractions and rational numbers: Multiplication of fractions ,Fraction as an operator ,Reciprocal of a fraction Division of fractions ,Word problems involving mixed fractions Introduction to rational numbers (with representation on number line) Operations on rational numbers (all operations) Representation of rational number as a decimal. Word problems on rational numbers (all operations) Multiplication and division of decimal fractions Conversion of units (lengths & mass) Word problems (including all operations) Percentage- An introduction w.r.t life situation. Understanding percentage as a fraction with denominator 100 Converting fractions and decimals into percentage and vice-versa. Application to profit & loss (single transaction only) Application to simple interest (time period in complete years)

Concept Map

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Theme Plan

	THEME PLAN FOR THE TOPIC FRACTIONS
CLASS	SUBTOPIC	CONCEPT DEVELOPMENT	KNOWLEDGE	SKILL	ACTIVITY
6	Introduction to Fractions	A fraction is a part of a whole, when the whole is divided into equal parts. Understand what the numerator represents and what the denominator represents in a fraction	Terms - Numerator and Denominator.	To be able to Identify/specify fraction quantities from any whole unit that has been divided. Locate a fraction on a number line.	ACTIVITY1
6	Proper and Improper Fractions	The difference between Proper and Improper. Know that a fraction can be represented as an Improper or mixed but have the same value.	Terms – proper, improper or mixed fractions	Differentiate between proper and improper fraction. Method to convert fractions from improper to mixed representation	ACTIVITY2
6	Comparing Fractions	Why do we need the concept of LCM for comparing fractions	Terms to learn – Like and Unlike Fractions	Recognize/identify like /unlike fractions. Method/Algorithm to enable comparing fractions	ACTIVITY3
6	Equivalent Fractions	Why are fractions equivalent and not equal	Know the term Equivalent Fraction	Method/Algorithm to enable comparing fractions	ACTIVITY4
6	Addition of Fractions	Why do we need LCM to add fractions. Understand Commutative law w.r.t. Fraction addition	Fraction addition Algorithm	Applying the Algorithm and adding fractions. Solving simple word problems	ACTIVITY5
6	Subtraction of Fractions	Why we need LCM to subtract fractions.	Fraction subtraction Algorithm	Applying the Algorithm and adding fractions. Solving simple word problems	ACTIVITY6
6	Linking Fractions with Decimal Number Representation	The denominator of a fraction is always 10 and powers of 10 when representing decimal numbers as fractions	Difference between integers and decimals. Algorithm to convert decimal to fraction and vice versa	Represent decimal numbers on the number line. How to convert simple decimal numbers into fractions and vice versa	ACTIVITY7
6	(Linking to Fraction Topic) Ratio & Proportion	What does it mean to represent a ratio in the form of a fraction. The relationship between the numerator and denominator – proportion	Terms Ratio and Proportion and link them to the fraction representation	Transition from Additive Thinking to Multiplicative Thinking	ACTIVITY8
7	Multiplication of Fractions	Visualise the quantities when a fraction is multiplied 1) whole number 2) fraction. Where is multiplication of fractions used	“of” Operator means multiplication. Know the fraction multiplication algorithm	Apply the algorithm to multiply fraction by fraction	ACTIVITY9
7	Division of Fractions	Visualise the quantities when a fraction is divided 1) whole number 2) fraction .Where Division of fractions would be used 3) why is the fraction reversed and multiplied	Fraction division algorithm	Apply the algorithm to divide fraction by fraction	ACTIVITY10
7	Linking Fractions with Percentage	The denominator of a fraction is always 100.	Convert from fraction to percentage and vice versa	Convert percentage	ACTIVITY11
8	(Linking to Fraction Topic) Inverse Proportion	The relationship between the numerator and denominator – for both direct and inverse proportion	Reciprocal of a fraction	Determine if the ratio is directly proportional or inversely proportional in word problems	ACTIVITY12
8	(Linking to Fraction Topic) Rational & Irrational Numbers	The number line is fully populated with natural numbers, integers and irrational and rational numbers	Learn to recognize irrational and rational numbers. Learn about some naturally important irrational numbers. Square roots of prime numbers are irrational numbers	How to calculate the square roots of a number. The position of an irrational number is definite but cannot be determined accurately	ACTIVITY13

Curricular Objectives

1. Conceptualise and understand algorithms for basic operations (addition, subtraction, multiplication and division) on fractions.
2. Apply the understanding of fractions as simple mathematics models.
3. Understand the different mathematical terms associated with fractions.
4. To be able to see multiple interpretations of fractions such as in measurement, ratio and proportion, quotient, representation of decimal numbers, percentages, understanding rational and irrational numbers.

Different Models used for Learning Fractions

Part-Whole

The most commonly used model is the part whole model where where the whole is divided into equal parts and the fraction represents one or more of the parts.

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Half (½) : The whole is divided into two equal parts.

One part is coloured, this part represents the fraction ½.

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One-Fourth (1/4) : The whole is divided into four equal parts.

One part is coloured, this part represents the fraction ¼.

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One (2/2 or 1) : The whole is divided into two equal parts.

Two part are coloured, this part represents the fraction 2/2

which is equal to the whole or 1.

File:Fractions-Resource Material Subject Teacher Forum September 2011 html 2faaf16a.gifTwo Fifth (2/5) : The whole is divided into five equal parts.

Two part are coloured, this part represents the fraction 2/5.

File:Fractions-Resource Material Subject Teacher Forum September 2011 html 9e5c77.gifThree Seventh (3/7) : The whole is divided into seven equal parts.

Three part are coloured, this part represents the fraction 3/7.

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Seven tenth (7/10) : The whole is divided into ten equal parts.

Seven part are coloured, this part represents the fraction 7/10 .

Terms Numerator and Denominator and their meaning

File:Fractions-Resource Material Subject Teacher Forum September 2011 html 3bf1fc6d.gifThree Eight (3/8) The whole is divided into eight equal parts.

Three part are coloured, this part represents the fraction 3/8 .

3/8 is also written as numerator/denominator. Here the number above the line- numerator tells us HOW MANY PARTS are involved. It 'enumerates' or counts the coloured parts.

The number BELOW the line tells – denominator tells us WHAT KIND OF PARTS the whole is divided into. It 'denominates' or names the parts.

Equal Share

In the equal share interpretation the fraction m/n denotes one share when m identical things are shared equally among n. The relationships between fractions are arrived at by logical reasoning (Streefland, 1993). For example 5/6 is the share of one child when 5 rotis (disk-shaped handmade bread) are shared equally among 6 children. The sharing itself can be done in more than one way and each of them gives us a relation between fractions. If we first distribute 3 rotis by dividing each into two equal pieces and giving each child one piece each child gets 1⁄2 roti. Then the remaining 2 rotis can be distributed by dividing each into three equal pieces giving each child a piece. This gives us the relations

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The relations 3/6 = 1⁄2 and 2/6 = 1/3 also follow from the process of distribution. Another way of distributing the rotis would be to divide the first roti into 6 equal pieces give one piece each to the 6 children and continue this process with each of the remaining 4 rotis. Each child gets a share of rotis from each of the 5 rotis giving us the relation

File:Fractions-Resource Material Subject Teacher Forum September 2011 html m39388388.gif

It is important to note here that the fraction symbols on both sides of the equation have been arrived at simply by a repeated application of the share interpretation and not by appealing to prior notions one might have of these fraction symbols. In the share interpretation of fractions, unit fractions and improper fractions are not accorded a special place.

Also converting an improper fraction to a mixed fraction becomes automatic. 6/5 is the share that one child gets when 6 rotis are shared equally among 5 children and one does this by first distributing one roti to each child and then sharing the remaining 1 roti equally among 5 children giving us the relation

File:Fractions-Resource Material Subject Teacher Forum September 2011 html m799c1107.gif

Share interpretation does not provide a direct method to answer the question ‘how much is the given unknown quantity’. To say that the given unknown quantity is 3⁄4 of the whole, one has figure out that four copies of the given quantity put together would make three wholes and hence is equal to one share when these three wholes are shared equally among 4. Share '''''interpretation is also the quotient interpretation of fractions in the sense that 3⁄4 = 3 ÷ 4 and this is important for developing students’ ability to solve problems involving multiplicative and linear functional relations.

Introducing Fractions Using Share and Measure Interpretations

One of the major difficulties a child faces with fractions is making sense of the symbol m/n. In order to facilitate students’ understanding of fractions, we need to use certain models. Typically we use the area model in both the measure and share interpretation and use a circle or a rectangle that can be partitioned into smaller pieces of equal size. Circular objects like roti that children eat every day have a more or less fixed size. Also since we divide the circle along the radius to make pieces, there is no scope for confusing a part with the whole. Therefore it is possible to avoid explicit mention of the whole when we use a circular model. Also, there is no need to address the issue that no matter how we divide the whole into n equal parts the parts will be equal. However, at least in the beginning we need to instruct children how to divide a circle into three or five equal parts and if we use the circular model for measure interpretation, we would need ready made teaching aids such as the circular fraction kit for repeated use.

Rectangular objects (like cake) do not come in the same size and can be divided into n equal parts in more than one way. Therefore we need to address the issues (i) that the size of the whole should be fixed (ii) that all 1⁄2’s are equal– something that children do not see readily. The advantage of rectangular objects is that we could use paper models and fold or cut them into equal parts in different ways and hence it easy to demonstrate for example that 3/5 = 6/10 using the measure interpretation .

Though we expose children to the use of both circles and rectangles, from our experience we feel circular objects are more useful when use the share interpretation as children can draw as many small circles as they need and since the emphasis not so much on the size as in the share, it does not matter if the drawings are not exact. Similarly rectangular objects would be more suited for measure interpretation for, in some sense one has in mind activities such as measuring the length or area for which a student has to make repeated use of the unit scale or its subunits.

Measure Model

Measure interpretation defines the unit fraction 1/n as the measure of one part when one whole is divided into n equal parts. The composite fraction m/n is as the measure of m such parts. Thus 5/6 is made of 5 piece units of size 1/5 each and 6/5 is made of 6 piece units of size 1/5 each. Since 5 piece units of size make a whole, we get the relation 6/5 = 1 + 1/5.

Significance of measure interpretation lies in the fact that it gives a direct approach to answer the ‘how much’ question and the real task therefore is to figure out the appropriate n so that finitely many pieces of size will be equal to a given quantity. In a sense then, the measure interpretation already pushes one to think in terms of infinitesimal quantities. Measure interpretation is different from the part whole interpretation in the sense that for measure interpretation we fix a certain unit of measurement which is the whole and the unit fractions are sub-units of this whole. The unit of measurement could be, in principle, external to the object being measured.

Key vocabulary:

1. 1. (a) A fraction is a number representing a part of a whole. The whole may be a single object or a group of objects. (b) When expressing a situation of counting parts to write a fraction, it must be ensured that all parts are equal.
2. In File:Fractions-Resource Material Subject Teacher Forum September 2011 html m2988e86b.gif, 5 is called the numerator and 7 is called the denominator.
3. Fractions can be shown on a number line. Every fraction has a point associated with it on the number line.
4. In a proper fraction, the numerator is less than the denominator. The fractions, where the numerator is greater than the denominator are called improper fractions.
5. An improper fraction can be written as a combination of a whole and a part, and such fraction then called mixed fractions.
6. Each proper or improper fraction has many equivalent fractions. To find an equivalent fraction of a given fraction, we may multiply or divide both the numerator and the denominator of the given fraction by the same number.
7. A fraction is said to be in the simplest (or lowest) form if its numerator and the denominator have no common factor except 1.

Additional resources :

1. [[1]] Video on teaching fractions using the equal share method made by Eklavya an NGO based in Madhya Pradesh, India
2. [[2]] Mathematics resources from Homi Baba Centre for Science Education

Errors with fractions

When fractions are operated erroneously like natural numbers, i.e. treating the numerator and the denominators separately and not considering the relationship between the numerator and the denominator is termed as N-Distracter. For example 1/3 + ¼ are added to result in 2/7. Here 2 units of the numerator are added and 3 & four units of the denominator are added. This completely ignores the relationship between the numerator and denominator of each of the fractions. Streefland (1993) noted this challenge as N-distrators and a slow-down of learning when moving from the concrete level to the abstract level.

The five levels of resistance to N-Distracters that a child develops are:

1. Absence of cognitive conflict: The child is unable to recognize the error even when she sees the same operation performed resulting in a correct answer. The child thinks both the answers are the same in spite of different results. Eg. ½ + ½ she erroneously calculated as 2/4. But when the child by some other method, say, through manipulatives (concrete) sees ½ + ½ = 1 does not recognize the conflict.
2. Cognitive conflict takes place: The student sees a conflict when she encounters the situation described in level 1 and rejects the ½+1/2 = 2/4 solution and recognizes it as incorrect. She might still not have a method to arrive at the correct solution.
3. Spontaneous refutation of N-Distracter errors: The student may still make N-Distracter errors, but is able to detect the error for herself. This detection of the error may be followed by just rejection or explaining the rejection or even by a correct solution.
4. Free of N-Distracter: The written work is free of N-Distracters. This could mean a thorough understanding of the methods/algorithms of manipulating fractions.
5. Resistance to N-Distracter: The student is completely free (conceptually and algorithmically) of N-Distracter errors.

Key vocabulary:

1. N-Distractor: as defined above.

Additional resources:

1. www.merga.net.au/publications/counter.php?pub=pub_conf&id=1410 A PDF Research paper titled Probing Whole Number Dominance with Fractions.
2. www.merga.net.au/documents/RP512004.pdf A PDF research paper titled “Why You Have to Probe to Discover What Year 8 Students Really Think About Fractions ”
3. [[3]]ciwrA&sig=DcM0mi7r1GJlTUbZVq9J0l53Lrc&hl=en&ei=0xJBTvLjJ8bRrQfC3JmyBw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBUQ6AEwAA#v=onepage&q&f=false A google book Fractions in realistic mathematics education: a paradigm of developmental By Leen Streefland

Operations on Fractions

Addition and Subtraction

Adding and subtracting like fractions is simple. It must be emphasised thought even during this process that the parts are equal in size or quantity because the denominator is the same and hence for the result we keep the common denominator and add the numerators.

Adding and subtracting unlike fractions requires the child to visually understand that the parts of each of the fractions are differing in size and therefore we need to find a way of dividing the whole into equal parts so that the parts of all of the fractions look equal. Once this concept is established, the terms LCM and the methods of determining them may be introduced.

Multiplication

Multiplying a fraction by a whole number: Here the repeated addition logic of multiplying whole numbers is still valid. 1/6 multiplied by 4 is 4 times 1/6 which is equal to 4/6.

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Multiplying a fraction by a fraction: In this case the child is confused as repeated addition does not make sense. To make a child understand the of operator we can use the language and demonstrate it using the measure model and the area of a rectangle.

The area of a rectangle is found by multiplying side length by side length. For example, in the rectangle below, the sides are 3 units and 9 units, and the area is 27 square units.

File:Fractions-Resource Material Subject Teacher Forum September 2011 html m66ce78ea.gif

We can apply that idea to fractions, too.

• The one side of the rectangle is 1 unit (in terms of length).
• The other side is 1 unit also.
• The whole rectangle also is 1 square unit, in terms of area.

See figure below to see how the following multiplication can be shown.

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File:Fractions-Resource Material Subject Teacher Forum September 2011 html 753005a4.gif

Remember: The two fractions to multiply represent the length of the sides, and the answer fraction represents area.

Division

Dividing a fraction by a whole number can be demonstrated just like division of whole numbers. When we divide 3/4 by 2 we can visualise it as dividing 3 parts of a whole roti among 4 people.

File:Fractions-Resource Material Subject Teacher Forum September 2011 html 1f617ac8.gif

Here 3/4 is divided between two people. One fourth piece is split into two.
Each person gets 1/4 and 1/8.

File:Fractions-Resource Material Subject Teacher Forum September 2011 html m5f26c0a.gif

OR

File:Fractions-Resource Material Subject Teacher Forum September 2011 html m25efcc2e.gif

Another way of solving the same problem is to split each fourth piece into 2.

This means we change the 3/4 into 6/8.

File:Fractions-Resource Material Subject Teacher Forum September 2011 html 7ed8164a.gif

When dividing a fraction by a fraction, we use the measure interpretation.

File:Fractions-Resource Material Subject Teacher Forum September 2011 html m3192e02b.gif

When we divide 2 by ¼ we ask how many times does ¼ fit into 2

File:Fractions-Resource Material Subject Teacher Forum September 2011 html m257a1863.gifFile:Fractions-Resource Material Subject Teacher Forum September 2011 html m257a1863.gif

It fits in 4 times in each roti, so totally 8 times.

We write it as File:Fractions-Resource Material Subject Teacher Forum September 2011 html m390fcce6.gif

Key vocabulary:

1. Least Common Multiple: In arithmetic and number theory, the least common multiple (also called the lowest common multiple or smallest common multiple) of two integers a and b, usually denoted by LCM(a, b), is the smallest positive integer that is a multiple of both a and b. It is familiar from grade-school arithmetic as the "lowest common denominator" that must be determined before two fractions can be added.

1. Greatest Common Divisor: In mathematics, the greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder. For example, the GCD of 8 and 12 is 4.

Additional resources:

1. [[4]] detailed conceptual understanding of division by fractions
2. [[5]] understanding fractions
3. [[6]] Understand how to use Geogebra a mathematical computer aided tool
4. [[7]] Worksheets in mathematics for teachers to use

Linking Fractions to other Topics

Decimal Numbers

“Decimal” comes from the Latin root decem, which simply means ten. The number system we use is called the decimal number system, because the place value units go in tens: you have ones, tens, hundreds, thousands, and so on, each unit being 10 times the previous one.

In common language, the word “decimal number” has come to mean numbers which have digits after the decimal point, such as 5.8 or 9.302. But in reality, any number within the decimal number system could be termed a decimal number, including whole numbers such as 12 or 381.

The simplest way to link or connect fractions to the decimal number system is with the number line representation. Any scale that a child uses is also very good for this purpose, as seen in the figure below.

The number line between 0 and 1 is divided into ten parts. Each of these ten parts is 1/10, a tenth.

File:Fractions-Resource Material Subject Teacher Forum September 2011 html 3d7b669f.gif

Under the tick marks you see decimal numbers such as 0.1, 0.2, 0.3, and so on. These are the same numbers as the fractions 1/10, 2/10, 3/10 and so on.

We can write any fraction with tenths (denominator 10) using the decimal point. Simply write after the decimal point how many tenths the number has. 0.6 means six tenths or 1/6. 1.5 means 1 whole and 5 tenths or File:Fractions-Resource Material Subject Teacher Forum September 2011 html m7f1d448c.gif

Note: A common error one sees is 0.7 is written as 1 /7. It is seven tenths and not one seventh. That the denominator is always 10 has to be stressed. To reinforce this one can use a simple rectangle divided into 10 parts , the same that was used to understand place value in whole numbers.

The coloured portion represents 0.6 or 6/10 and the whole block represents 1.

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Percentages

Fractions and percentages are different ways of writing the same thing. When we say that a book costs Rs. 200 and the shopkeeper is giving a 10 % discount. Then we can represent the 10% as a fraction as File:Fractions-Resource Material Subject Teacher Forum September 2011 html m1369c56e.gif where 10 is the numerator and the denominator is always 100. In this case 10 % of the cost of the book is File:Fractions-Resource Material Subject Teacher Forum September 2011 html m50e22a06.gif. So you can buy the book for 200 – 20 = 180 rupees.

There are a number of common ones that are useful to learn. Here is a table showing you the ones that you should learn.

Percentage	Fraction
100%
50%
25%
75%
10%
20%
40%

File:Fractions-Resource Material Subject Teacher Forum September 2011 html m60c76c68.gifTo see 40 % visually see the figure :

You can see that if the shape is divided into 5 equal parts, then 2 of those parts are shaded.

If the shape is divided into 100 equal parts, then 40 parts are shaded.

These are equivalent fractions as in both cases the same amount has been shaded.

Ratio and Proportion

It is important to understand that fractions also can be interpreted as ratio's. Stressing that a fraction can be interpreted in many ways is of vital importance. Here briefly I describe the linkages that must be established between Ratio and Proportion and the fraction representation. Connecting multiplication of fractions is key to understanding ratio and proportion.

What is ratio?

Ratio is a way of comparing amounts of something. It shows how much bigger one thing is than another. For example:

• Use 1 measure detergent (soap) to 10 measures water
• Use 1 shovel (bucket) of cement to 3 shovels (buckets) of sand
• Use 3 parts blue paint to 1 part white

Ratio is the number of parts to a mix. The paint mix is 4 parts, with 3 parts blue and 1 part white.

The order in which a ratio is stated is important. For example, the ratio of soap to water is 1:10. This means for every 1 measure of soap there are 10 measures of water.

Mixing paint in the ratio 3:1 (3 parts blue paint to 1 part white paint) means 3 + 1 = 4 parts in all.

3 parts blue paint to 1 part white paint = is ¾ blue paint to ¼ white paint.

Cost of a pen is Rs 10 and cost of a pencil is Rs 2. How many times the cost of a pencil is the cost of a pen? Obviously it is five times. This can be written as

The ratio of the cost of a pen to the cost of a pencil = File:Fractions-Resource Material Subject Teacher Forum September 2011 html m762fb047.gif

What is Direct Proportion ?

Two quantities are in direct proportion when they increase or decrease in the same ratio. For example you could increase something by doubling it or decrease it by halving. If we look at the example of mixing paint the ratio is 3 pots blue to 1 pot white, or 3:1.

Paint pots in a ratio of 3:1

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But this amount of paint will only decorate two walls of a room. What if you wanted to decorate the whole room, four walls? You have to double the amount of paint and increase it in the same ratio.

If we double the amount of blue paint we need 6 pots.

If we double the amount of white paint we need 2 pots.

Six paint pots in a ratio of 3:1

The amount of blue and white paint we need increase in direct proportion to each other. Look at the table to see how as you use more blue paint you need more white paint:

Pots of blue paint 3 6 9 12

Pots of white paint 1 2 3 4

Two quantities which are in direct proportion will always produce a graph where all the points can be joined to form a straight line.

What is Inverse Proportion ?

Two quantities may change in such a manner that if one quantity increases the the quantity decreases and vice-versa. For example if we are building a room, the time taken to finish decreases as the number of workers increase. Similarly when the speed increases the time to cover a distance decreases. Zaheeda can go to school in 4 different ways. She can walk, run, cycle or go by bus.

Study the table below, observe that as the speed increases time taken to cover the distance decreases

	Walk	Run	Cycle	Bus
Speed Km/Hr	3	6 (walk speed *2)	9 (walk speed *3)	45 (walk speed *15)
Time Taken (minutes)	30	15 (walk Time * ½)	10 (walk Time * 1/3)	2 (walk Time * 1/15)

As Zaheeda doubles her speed by running, time reduces to half. As she increases her speed to three times by cycling, time decreases to one third. Similarly, as she increases her speed to 15 times, time decreases to one fifteenth. (Or, in other words the ratio by which time decreases is inverse of the ratio by which the corresponding speed increases). We can say that speed and time change inversely in proportion.

Moving from Additive Thinking to Multiplicative Thinking

Avinash thinks that if you use 5 spoons of sugar to make 6 cups of tea, then you would need 7 spoons of sugar to make 8 cups of tea just as sweet as the cups before. Avinash would be using an additive transformation'''; he thinks that since we added 2 more cups of tea from 6 to 8. To keep it just as sweet he would need to add to more spoons of sugar. What he does not know is that for it to taste just as sweet he would need to preserve the ratio of sugar to tea cup and use multiplicative thinking. He is unable to detect the ratio.

Proportional Reasoning

Proportional thinking involves the ability to understand and compare ratios, and to predict and produce equivalent ratios. It requires comparisons between quantities and also the relationships between quantities. It involves quantitative thinking as well as qualitative thinking. A feature of proportional thinking is the multiplicative relationship among the quantities and being able to recognize this relationship. The relationship may be direct (divide), i.e. when one quantity increases, the other also increases. The relationship is inverse (multiply), when an increase in one quantity implies a decrease in the other, in both cases the ratio or the rate of change remains a constant.

The process of adding involved situations such as adding, joining, subtracting, removing actions which involves the just the two quantities that are being joined, while proportional thinking is associated with shrinking, enlarging, scaling , fair sharing etc. The process involves multiplication. To be able to recognize, analyse and reason these concepts is multiplicative thinking/reasoning. Here the student must be able to understand the third quantity which is the ratio of the two quantities. The preservation of the ratio is important in the multiplicative transformation.

Rational & Irrational Numbers

After the number line was populated with natural numbers, zero and the negative integers, we discovered that it was full of gaps. We discovered that there were numbers in between the whole numbers - fractions we called them.

But, soon we discovered numbers that could not be expressed as a fraction. These numbers could not be represented as a simple fraction. These were called irrational numbers. The ones that can be represented by a simple fraction are called rational numbers. They h ad a very definite place in the number line but all that could be said was that square root of 2 is between 1.414 and 1.415. These numbers were very common. If you constructed a square, the diagonal was an irrational number. The idea of an irrational number caused a lot of agony to the Greeks. Legend has it that Pythagoras was deeply troubled by this discovery made by a fellow scholar and had him killed because this discovery went against the Greek idea that numbers were perfect.

How can we be sure that an irrational number cannot be expressed as a fraction? This can be proven algebraic manipulation. Once these "irrational numbers" came to be identified, the numbers that can be expressed of the form p/q where defined as rational numbers.

There is another subset called transcendental numbers which have now been discovered. These numbers cannot be expressed as the solution of an algebraic polynomial. "pi" and "e" are such numbers.

Vocabulary

Decimal Numbers, Percentages, Ratio, Direct Proportion, Inverse Proportion, Rational Numbers, Irrational Numbers

Additional Resources

[[8]]

[[9]]

[[10]]

Activities :

Activity1: Introduction to fractions

Objective:

Introduce fractions using the part-whole method

Procedure:

Do the six different sections given in the activity sheet. For each section there is a discussion point or question for a teacher to ask children.

After the activity sheet is completed, please use the evaluation questions to see if the child has understood the concept of fractions

Material/Activity Sheet

1. Write the Number Name and the number of the picture like the example File:Fractions-Resource Material Subject Teacher Forum September 2011 html m1d9c88a9.gifNumber Name = One third Number: File:Fractions-Resource Material Subject Teacher Forum September 2011 html 52332ca.gif

Question: What is the value of the numerator and denominator in the last figure , the answer is File:Fractions-Resource Material Subject Teacher Forum September 2011 html m2dc8c779.gif

1. Colour the correct amount that represents the fractions

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7/10 3/8 1/5 4/7

Question: Before colouring count the number of parts in each figure. What does it represent. Answer: Denominator

1. Divide the circle into fractions and colour the right amount to show the fraction

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3/5 6/7 1/3 5/8 2/5

1. Draw the Fraction and observe which is the greater fraction – observe that the parts are equal for each pair

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1/3 2/3 4/5 2/5 3/7 4/7

1. Draw the Fraction and observe which is the greater fraction – Observe that the parts are different sizes for each pair.

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1/3 1/4 1/5 1/8 1/6 1/2

1. Solve these word problems by drawing
   1. Amar divided an apple into 8 equal pieces. He ate 5 pieces. He put the a other 3 in a box. What fraction did Amar eat?
   2. There are ten biscuits in the box. 3 are cream biscuits. 2 are salt biscuits. 4 are chocolate biscuits. 1 is a sugar biscuit. What fraction of the biscuits in the box are salt biscuits.
   3. Radha has 6 pencils. She gives one to Anil and he gives one to Anita. She keeps the rest. What fraction of her pencils did she give away?

Activity 2: Proper and Improper Fractions

Objective:

Proper and Improper Fractions

Procedure:

Examples of Proper and improper fractions are given. The round disks in the figure represent rotis and the children figures represent children. Cut each roti and each child figure and make the children fold, tear and equally divide the roits so that each child figure gets equal share of roti.

Material/Activity Sheet

1. File:Fractions-Resource Material Subject Teacher Forum September 2011 html 55f65a3d.gifFile:Fractions-Resource Material Subject Teacher Forum September 2011 html 55f65a3d.gifFile:Fractions-Resource Material Subject Teacher Forum September 2011 html 5518d221.jpgIf you want to understand proper fraction , example 5/6. In the equal share model , 5/6 represents the share that each child gets when 5 rotis are divided among 6 children equally.

File:Fractions-Resource Material Subject Teacher Forum September 2011 html 5e906d5b.jpgFile:Fractions-Resource Material Subject Teacher Forum September 2011 html 5518d221.jpg

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File:Fractions-Resource Material Subject Teacher Forum September 2011 html 55f65a3d.gifFile:Fractions-Resource Material Subject Teacher Forum September 2011 html 55f65a3d.gifFile:Fractions-Resource Material Subject Teacher Forum September 2011 html 5e906d5b.jpg

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1. If you want to understand improper fraction , example 8/3. In the equal share model , 8/3 represents the share that each child gets when 8 rotis are divided among 3 children equally. The child in this case will usually distribute 2 full rotis to each child and then try to divide the remaining rotis. At this point you can show the mixed fraction representation as 2 2/3

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File:Fractions-Resource Material Subject Teacher Forum September 2011 html 5518d221.jpg

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Evaluation Question

1. What happens when the numerator and denominator are the same, why ?
2. What happens when the numerator is greater than the denominator why ? How can we represent this in two ways ?

Activity 3: Comparing Fractions

Objective:

Comparing-Fractions

Procedure:

Print the document Comparing-Fractions.pdf and Comparing-Fractions2 and work out the activity sheet

Material/ Activity Sheet

Comparing-Fractions.pdf

Comparing-Fractions2.pdf

Evaluation Question

1. Does the child know the symbols >, < and =
2. What happens to the size of the part when the denominator is different ?
3. Does it decrease or increase when the denominator becomes larger ?
4. Can we compare quantities when the parts are different sizes ?
5. What should we do to make the sizes of the parts the same ?

Activity 4: Equivalent Fractions

Objective:

To understand Equivalent Fractions

Procedure:

Print 10 copies of the document from pages 2 to 5 fractions-matching-game.pdf

Cut the each fraction part

Play memory game as described in the document in groups of 4 children.

Activity Sheet

fractions-matching-game.pdf

Evaluation Question

1. What is reducing a fraction to the simplest form ?
2. What is GCF – Greatest Common Factor ?
3. Use the document simplifying-fractions.pdf
4. Why are fractions called equivalent and not equal.

Activity 5: Fraction Addition

Objective:

Understand Addition of Fractions

Procedure:

Open Geogebra applications

Open link [[11]]

Move the sliders Numerator1 and Denominator1 to set Fraction 1

Move the sliders Numerator2 and Denominator2 to set Fraction 2

See the last bar to see the result of adding fraction 1 and fraction 2

Activity Sheet

Please open [[12]] in Firefox and follow the process

When you move the sliders ask children to

Observe and describe what happens when the denominator is changed.

Observe and describe what happens when denominator changes

Observe and describe the values of the numerator and denominator and relate it to the third result fraction. Discuss LCM and GCF

Activity 6: Fraction Subtraction

Objective:

Understand Fraction Subtraction

Procedure:

Open Geogebra applications

Open link [[13]]

Move the sliders Numerator1 and Denominator1 to set Fraction 1

Move the sliders Numerator2 and Denominator2 to set Fraction 2

See the last bar to see the result of subtracting fraction 1 and fraction 2

Material/Activity Sheet

Please open link [[14]] in Firefox and follow the process

When you move the sliders ask children to

observe and describe what happens when the denominator is changed.

observe and describe what happens when denominator changes

observe and describe the values of the numerator and denominator and relate it to the third result fraction. Discuss LCM and GCF

Activity 7: Linking to Decimals

Objective:

Fractions representation of decimal numbers

Procedure:

Make copies of the worksheets decimal-tenths-squares.pdf and decimal-hundreths-tenths.pdf

Activity Sheet

decimal-tenths-squares.pdf

decimal-hundreths-tenths.pdf

Evaluation Question

1. Draw a number line and name the fraction and decimal numbers on the number line. Take a print of the document decimal-number-lines-1.pdf . Ask students to place any fraction and decimal numbers between between 0 and 10 on the number line
2. Write 0.45, 0.68, 0.05 in fraction form and represent as a fraction 100 square.

Activity 8: Ratio and Proportion

Objective:

Linking fractional representation and Ratio and Proportion

Procedure:

Use the NCERT Class 6 mathematics textbook chapter 12 and work out Exercise 12.1

Activity Sheet

NCERT Class6 Chapter 12 RatioProportion.pdf Exercise 12.1

Evaluation Question

1. Explain what the numerator means in the word problem
2. Explain what the denominator means
3. Finally describe the whole fraction in words in terms of ratio and proportion.

Activity 9: Fraction Multiplication

Objective:

Understand Multiplication of fractions

Procedure:

Open Geogebra applications

Open link [[15]]

Move the sliders Numerator1 and Denominator1 to set Fraction 1

Move the sliders Numerator2 and Denominator2 to set Fraction 2

On the right hand side see the result of multiplying fraction 1 and fraction 2

Material/Activity Sheet

Please open [[16]]raction_MultiplyArea.html in Firefox and follow the process

When you move the sliders ask children to

observe and describe what happens when the denominator is changed.

observe and describe what happens when denominator changes

One unit will be the large square border-in blue solid lines

A sub-unit is in dashed lines within one square unit.

The thick red lines represent the fraction 1 and 2 and also the side of the quadrilateral

The product represents the area of the the quadrilateral

Evaluation Question

When two fractions are multiplied is the product larger or smaller that the multiplicands – why ?

Activity 10: Division of fractions

Objective:

Understand Diviion by Fractions

Procedure:

Print out the fractionsStrips.pdf

Colour each of the unit fractions in different colours. Keep the whole unit (1) white.

Cut out each unit fraction piece.

Give examples File:Fractions-Resource Material Subject Teacher Forum September 2011 html m282c9b3f.gif

For example if we try the first one, File:Fractions-Resource Material Subject Teacher Forum September 2011 html 21ce4d27.gif See how many File:Fractions-Resource Material Subject Teacher Forum September 2011 html m31bd6afb.gifstrips will fit exactly onto whole unit strip.

Material /Activity Sheet

fractionsStrips.pdf , Crayons, Scissors, glue

Evaluation Question

When we divide by a fraction is the result larger or smaller why ?

Activity 11: Percentages

Objective:

Understand fraction representation and percentages

Procedure:

Please print copies of the 2 activity sheets percentage-basics-1.pdf and percentage-basics-2.pdf and discuss the various percentage quantities with the various shapes.

Then print a copy each of spider-percentages.pdf and make the children do this activity

Activity Sheet

Print out spider-percentages.pdf

Evaluation Question

What value is the denominator when we represent percentage as fraction ?

What does the numerator represent ?

What does the whole fraction represent ?

What other way can we represent a fraction whoose denominator is 100.

Activity 12: Inverse Proportion

Objective:

Understand fraction representation and Inverse Proportion.

Procedure:

Use the NCERT Class 8 mathematics textbook chapter 13 and work out Exercise 13.1

Activity Sheet

NCERT Class 8 Chapter 13 InverseProportion.pdf Exercise 13.1

Evaluation Question

1. Given a set of fractions are they directly proportional or inversely proportional ?

2. In the word problem, identify the numerator, identify the denominator and explain what the fraction means in terms of Inverse proportions

Activity 13: Rational and Irrational Numbers

Objective:

Understand fraction representation and rational and irrational numbers

Procedure:

Construct Koch's snowflakes .

Start with a thread of a certain length (perimeter) and using the same thread construct the following shapes (see Figure).

File:Fractions-Resource Material Subject Teacher Forum September 2011 html m1a6bd0d0.gif

See how the shapes can continue to emerge but cannot be identified definitely with the same perimeter (length of the thread).

Identify the various places where pi, "e" and the golden ratio occur

Material

Thread of a certain length.

Evaluation Question

How many numbers can I represent on a number line between 1 and 2.

What is the difference between a rational and irrational number, give an example ?

What is Pi ? Why is it a special number ?

Interesting Facts

In this article we will look into the history of the fractions, and we’ll find out what the heck that line in a fraction is called anyway.

Nearly everybody uses, or has used, fractions for some reason or another. But most people have no idea of the origin, and almost none of them have any idea what that line is even called. Most know ways to express verbally that it is present (e.g. “x over y-3,” or “x divided by y-3″), but frankly, it HAS to have a name. To figure out the name, we must also investigate the history of fractions.

The concept of fractions can be traced back to the Babylonians, who used a place-value, or positional, system to indicate fractions. On an ancient Babylonian tablet, the number

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, appears, which indicates the square root of two. The symbols are 1, 24, 51, and 10. Because the Babylonians used a base 60, or sexagesimal, system, this number is (1 * 60 0 ) + (24 * 60-1 ) + (51 * 60-2 ) + (10 * 60-3 ), or about 1.414222. A fairly complex figure for what is now indicated by √2.

In early Egyptian and Greek mathematics, unit fractions were generally the only ones present. This meant that the only numerator they could use was the number 1. The notation was a mark above or to the right of a number to indicate that it was the denominator of the number 1.

The Romans used a system of words indicating parts of a whole. A unit of weight in ancient Rome was the as, which was made of 12 uncias. It was from this that the Romans derived a fraction system based on the number 12. For example, 1/12 was uncia, and thus 11/12 was indicated by deunx (for de uncia) or 1/12 taken away. Other fractions were indicated as :

10/12 dextans (for de sextans),

3/12 quadrans (for quadran as)

9/12 dodrans (for de quadrans),

2/12 or 1/6 sextans (for sextan as)

8/12 bes (for bi as) also duae partes (2/3)

1/24 semuncia (for semi uncia)

7/12 septunx (for septem unciae)

1/48 sicilicus

6/12 or 1/2 semis (for semi as)

1/72 scriptulum

5/12 quincunx (for quinque unciae)

1/144 scripulum

4/12 or 1/3 triens (for trien as)

1/288 scrupulum

This system was quite cumbersome, yet effective in indicating fractions beyond mere unit fractions.

The Hindus are believed to be the first group to indicate fractions with numbers rather than words. Brahmagupta (c. 628) and Bhaskara (c. 1150) were early Hindu mathematicians who wrote fractions as we do today, but without the bar. They wrote one number above the other to indicate a fraction.

The next step in the evolution of fraction notation was the addition of the horizontal fraction bar. This is generally credited to the Arabs who used the Hindu notation, then improved on it by inserting this bar in between the numerator and denominator. It was at this point that it gained a name, vinculum. Later on, Fibonacci (c.1175-1250), the first European mathematician to use the fraction bar as it is used today, chose the Latin word virga for the bar.

The most recent addition to fraction notation, the diagonal fraction bar, was introduced in the 1700s. This was solely due to the fact that, typographically, the horizontal bar was difficult to use, being as it took three lines of text to be properly represented. This was a mess to deal with at a printing press, and so came, what was originally a short-hand, the diagonal fraction bar. The earliest known usage of a diagonal fraction bar occurs in a hand-written document. This document is Thomas Twining’s Ledger of 1718, where quantities of tea and coffee transactions are listed (e.g. 1/4 pound green tea). The earliest known printed instance of a diagonal fraction bar was in 1784, when a curved line resembling the sign of integration was used in the Gazetas de Mexico by Manuel Antonio Valdes.

When the diagonal fraction bar became popularly used, it was given two names : virgule, derived from Fibonacci’s virga; and solidus, which originated from the Roman gold coin of the same name (the ancestor of the shilling, of the French sol or sou, etc.). But these are not the only names for this diagonal fraction bar.

According to the Austin Public Library’s website, “The oblique stroke (/) is called a separatrix, slant, slash, solidus, virgule, shilling, or diagonal.” Thus, it has multiple names.

A related symbol, commonly used, but for the most part nameless to the general public, is the “division symbol,” or ÷ . This symbol is called an obelus. Though this symbol is generally not used in print or writing to indicate fractions, it is familiar to most people due to the use of it on calculators to indicate division and/or fractions.

Fractions are now commonly used in recipes, carpentry, clothing manufacture, and multiple other places, including mathematics study; and the notation is simple. Most people begin learning fractions as young as 1st or 2nd grade. The grand majority of them don’t even realize that fractions could have possibly been as complicated as they used to be, and thus, don’t really appreciate them for their current simplicity.

ANNEXURE A – List of activity sheets attached

comparing-fractions.pdf

comparing-fractions2.pdf

fractions-matching-game.pdf

fractionstrips.pdf

NCERT Class6 Chapter 12 RatioProportion.pdf

NCERT Class8 Chapter 13 DirectInverseProportion.pdf

percentage-basics-1.pdf

percentage-basics-2.pdf

simplifying-fractions.pdf

spider-percentages.pdf