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| − | '''Scope of this document'''
| |
| − | <br>
| |
| − | <br>
| |
| | | | |
| | + | = Introduction = |
| | | | |
| − | The following is a | + | The following is a background literature for teachers. It |
| − | background literature for teachers. It summarises the things to be | + | summarises the various concepts, approaches to be known to a teacher |
| − | known to a teacher to teach this topic more effectively . This | + | to teach this topic effectively . This literature is meant to be a |
| − | literature is meant to be a ready reference for the teacher to | + | ready reference for the teacher to develop the concepts, inculcate |
| − | develop the concepts, inculcate necessary skills, and impart | + | necessary skills, and impart knowledge in fractions from Class 6 to |
| − | knowledge in fractions from Class 6 to Class 10. | + | Class 10. |
| | | | |
| | | | |
| − | It is a well known fact | + | It is a well known fact that teaching and learning fractions is a |
| − | that teaching and learning fractions is a complicated process in | + | complicated process in primary and middle school. Although much of |
| − | primary and middle school. Although much of fractions is covered in | + | fractions is covered in the middle school, if the foundation is not |
| − | the middle school, if the foundation is not holistic and conceptual, | + | holistic and conceptual, then topics in high school mathematics |
| − | then topics in high school mathematics become very tough to grasp. | + | become very tough to grasp. Hence this documents is meant to |
| − | Hence this documents is meant to understand the research that has | + | understand the research that has been done towards simplifying and |
| − | been done towards simplifying and conceptually understanding topics | + | conceptually understanding topics of fractions. |
| − | of fractions. | + | |
| | + | |
| | + | |
| | + | = Mind Map = |
| | + | |
| | + | [[Image:KOER%20Fractions_html_m700917.png]] |
| | | | |
| | | | |
| − | It is also very common
| + | = Different Models for interpreting and teaching-learning fractions = |
| − | 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.
| |
| | | | |
| | + | == Introduction == |
| | | | |
| − | Also commonly fractions
| + | Commonly fractions are always approached by teaching it through |
| − | are always approached by teaching it through one model or | + | one model or interpretation namely the '''part-whole '''model |
| − | interpretation namely the '''part-whole '''model | |
| | where the '''whole '''is | | where the '''whole '''is |
| | divided into equal parts and the fraction represents one or more | | divided into equal parts and the fraction represents one or more |
| | of the parts. The limitations of this method, especially in | | of the parts. The limitations of this method, especially in |
| | explaining mixed fractions, multiplication and division of fractions | | explaining mixed fractions, multiplication and division of fractions |
| − | be fractions has led to educators using other interpretations such as
| + | has led to educators using other interpretations such as '''equal |
| − | '''equal share''' and | + | share''' and '''measure'''. |
| − | '''measure'''. These | + | These approaches to fraction teaching are discussed here. |
| − | approaches to fraction teaching are discussed. | |
| | | | |
| | | | |
| − | Also
| + | == Objectives == |
| − | a brief understanding of the common errors that children make when it
| + | |
| − | comes to fractions are addressed to enable teachers to understand the
| + | The objective of this section is to |
| − | child's levels of conceptual understanding to address the
| + | enable teachers to visualise and interpret fractions in different |
| − | misconceptions.
| + | ways in order to clarify the concepts of fractions using multiple |
| | + | methods. The idea is for teachers to be able to select the |
| | + | appropriate method depending on the context, children and class they |
| | + | are teaching to effectively understand fractions. |
| | | | |
| | | | |
| − | <br>
| + | == Part-whole == |
| − | <br>
| + | |
| − | | |
| | | | |
| − | = Syllabus =
| + | The |
| − |
| + | most commonly used model is the part whole model where where the |
| − | {| border="1"
| + | '''whole '''is |
| − | |-
| + | divided into <u>equal</u> |
| − | |
| + | parts and the fraction represents one or more of the parts. |
| − | '''Class 6''' | |
| | | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_78a5005.gif]] |
| − | '''Class 7'''
| |
| | | | |
| | | | |
| − | |-
| + | Half |
| − | |
| + | (½) : The whole is divided into '''two |
| − | Fractions:
| + | equal '''parts. |
| | | | |
| | | | |
| − | Revision of what a fraction is, Fraction as a
| + | One part is coloured, this part |
| − | part of whole, Representation of fractions (pictorially and on
| + | represents the fraction ½. |
| − | number line), fraction as a division, proper, improper & mixed
| |
| − | fractions, equivalent fractions, comparison of fractions, addition
| |
| − | and subtraction of
| |
| | | | |
| | | | |
| − | fractions
| + | [[Image:KOER%20Fractions_html_6fbd7fa5.gif]] |
| | | | |
| | | | |
| − | <br>
| + | One-Fourth |
| − | <br>
| + | (1/4) : The whole is divided into '''four |
| | + | equal '''parts. |
| | | | |
| | | | |
| − | Review of the idea of a decimal fraction, place
| + | One part is coloured, this part represents the fraction ¼. |
| − | 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.
| |
| | | | |
| | | | |
| − | <br>
| |
| − | <br>
| |
| | | | |
| − |
| |
| − | Word problems involving addition and
| |
| − | subtraction of decimals (two operations together on money,mass,
| |
| − | length, temperature and time)
| |
| | | | |
| | | | |
| − | <br>
| + | [[Image:KOER%20Fractions_html_43b75d3a.gif]] |
| | | | |
| | | | |
| − | |
| + | One |
| − | '''Fractions and rational numbers: ''' | + | (2/2 or 1) : The whole is divided into '''two |
| | + | equal '''parts. |
| | | | |
| | | | |
| − | <br>
| + | '''Two''' |
| | + | part are coloured, this part represents the fraction 2/2 |
| | | | |
| | | | |
| − | Multiplication of fractions ,Fraction as an operator
| + | which is equal to the whole or 1. |
| − | ,Reciprocal of a fraction
| |
| | | | |
| | | | |
| − | Division of fractions ,Word problems involving mixed fractions
| + | [[Image:KOER%20Fractions_html_2faaf16a.gif]] |
| | | | |
| | | | |
| − | Introduction to rational numbers (with representation on number
| + | Two |
| − | line)
| + | Fifth (2/5) : The whole is divided into '''five |
| | + | equal '''parts. |
| | | | |
| | | | |
| − | Operations on rational numbers (all operations)
| + | '''Two''' |
| | + | part are coloured, this part represents the fraction 2/5. |
| | | | |
| | | | |
| − | 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)
| + | [[Image:KOER%20Fractions_html_9e5c77.gif]]Three |
| | + | Seventh (3/7) : The whole is divided into '''seven |
| | + | equal '''parts. |
| | | | |
| | | | |
| − | <br>
| + | '''Three''' |
| | + | part are coloured, this part represents the fraction 3/7. |
| | | | |
| | | | |
| − | '''Percentage-'''
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | An introduction w.r.t life situation.
| |
| | | | |
| − |
| |
| − | '''Understanding percentage as a fraction with denominator 100'''
| |
| | | | |
| | | | |
| − | Converting fractions and decimals into percentage and
| + | [[Image:KOER%20Fractions_html_m30791851.gif]] |
| − | vice-versa.
| |
| | | | |
| | | | |
| − | Application to profit & loss (single transaction only)
| |
| | | | |
| − |
| |
| − | Application to simple interest (time period
| |
| | | | |
| | | | |
| − | in complete years)
| + | Seven |
| | + | tenth (7/10) : The whole is divided into '''ten |
| | + | equal '''parts. |
| | | | |
| | | | |
| − | |}
| + | '''Seven''' |
| − | <br>
| + | part are coloured, this part represents the fraction 7/10 . |
| − | <br>
| |
| | | | |
| | | | |
| − | = Concept Map =
| |
| − |
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m8e7238e.jpg]]<br>
| |
| − | <br>
| |
| | | | |
| − |
| |
| − | = Theme Plan =
| |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| | | | |
| − |
| |
| − | {| border="1"
| |
| − | |-
| |
| − | |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | |
| + | |
| − | '''THEME PLAN FOR THE TOPIC
| |
| − | FRACTIONS'''
| |
| | | | |
| | | | |
| − | |
| + | '''Terms Numerator |
| − | <br>
| + | and Denominator and their meaning''' |
| | | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_3bf1fc6d.gif]] |
| − | <br>
| |
| | | | |
| | | | |
| − | |-
| + | Three |
| − | |
| + | Eight (3/8) The whole is divided into '''eight |
| − | '''CLASS''' | + | equal '''parts. |
| | | | |
| | | | |
| − | |
| |
| − | '''SUBTOPIC'''
| |
| | | | |
| − |
| |
| − | |
| |
| − | '''CONCEPT <br>
| |
| − | DEVELOPMENT'''
| |
| | | | |
| | | | |
| − | |
| + | '''Three''' |
| − | '''KNOWLEDGE''' | + | part are coloured, this part represents the fraction 3/8 . |
| | | | |
| | | | |
| − | |
| |
| − | '''SKILL'''
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| − |
| |
| − | |
| |
| − | '''ACTIVITY'''
| |
| | | | |
| | | | |
| − | |-
| + | 3/8 is also written as |
| − | |
| + | numerator/denominator. Here the number above the line- numerator |
| − | 6
| + | tells us '''HOW MANY PARTS''' are involved. It 'enumerates' or |
| | + | counts the coloured parts. |
| | | | |
| | | | |
| − | |
| + | The number '''BELOW''' the line tells – denominator tells us |
| − | Introduction to Fractions
| + | '''WHAT KIND OF PARTS''' the whole is divided into. It 'denominates' |
| | + | or names the parts. |
| | | | |
| | | | |
| − | |
| + | |
| − | 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
| |
| | | | |
| | | | |
| − | |
| + | The important factor to note here is '''WHAT IS THE WHOLE . '''In |
| − | Terms - Numerator and Denominator.
| + | both the figures below the fraction quantity is 1/4. In fig 1 one |
| | + | circle is the whole and in fig 2, 4 circles is the whole. |
| | | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_1683ac7.gif]] |
| − | To be able to Identify/specify
| |
| − | fraction quantities from any whole unit that has been divided.
| |
| − | Locate a fraction on a number line.
| |
| | | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_729297ef.gif]][[Image:KOER%20Fractions_html_4282c1e5.gif]][[Image:KOER%20Fractions_html_6fbd7fa5.gif]] |
| − | ACTIVITY1
| + | |
| | + | |
| | + | |
| | + | |
| | + | |
| | | | |
| | | | |
| − | |-
| + | == Equal Share == |
| − | |
| + | |
| − | 6
| + | 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 |
| | + | |
| | + | |
| | + | [[Image:KOER%20Fractions_html_3176e16a.gif]] |
| | | | |
| | | | |
| − | |
| + | |
| − | Proper and Improper Fractions
| + | |
| | | | |
| | | | |
| − | |
| + | The relations 3/6 = 1⁄2 and 2/6 = 1/3 also follow from the |
| − | The difference between Proper and | + | process of distribution. Another way of distributing the rotis would |
| − | Improper. Know that a fraction can be represented as an Improper
| + | be to divide the first roti into 6 equal pieces give one piece each |
| − | or mixed but have the same value.
| + | 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 |
| | | | |
| | | | |
| − | |
| + | |
| − | Terms – proper, improper or mixed
| + | |
| − | fractions
| |
| | | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_m39388388.gif]] |
| − | Differentiate between proper and
| |
| − | improper fraction. Method to convert fractions from improper to
| |
| − | mixed representation
| |
| | | | |
| | | | |
| − | |
| |
| − | ACTIVITY2
| |
| | | | |
| − |
| + | |
| − | |-
| |
| − | |
| |
| − | 6
| |
| | | | |
| | | | |
| − | |
| + | It is important to note here that the fraction symbols on both |
| − | Comparing Fractions
| + | 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 |
| − | Why do we need the concept of LCM
| + | automatic. 6/5 is the share that one child gets when 6 rotis are |
| − | for comparing fractions
| + | 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 |
| | | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_m799c1107.gif]] |
| − | Terms to learn – Like and Unlike
| |
| − | Fractions
| |
| | | | |
| | | | |
| − | |
| + | Share interpretation does not provide a direct method to answer |
| − | Recognize/identify like /unlike
| + | the question ‘how much is the given unknown quantity’. To say |
| − | fractions. Method/Algorithm to enable comparing fractions
| + | 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. ''''' |
| | | | |
| | | | |
| − | |
| + | |
| − | ACTIVITY3
| + | |
| | | | |
| | | | |
| − | |-
| + | To understand the |
| − | |
| + | equal share model better, use the GeoGebra file explaining the equal |
| − | 6
| + | share model available on [[http://rmsa.karnatakaeducation.org]]. |
| | + | See figure below. Move the sliders m and n and see how the equal |
| | + | share model is interpreted. |
| | | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_17655b73.png]] |
| − | Equivalent Fractions
| + | |
| | | | |
| | | | |
| − | |
| |
| − | Why are fractions equivalent and not
| |
| − | equal
| |
| | | | |
| | + | |
| | + | |
| | + | |
| | + | == Measure Model == |
| | | | |
| − | |
| + | Measure interpretation defines the unit fraction ''1/n ''as the |
| − | Know the term Equivalent Fraction
| + | 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 |
| − | Method/Algorithm to enable comparing
| + | gives a direct approach to answer the ‘how much’ question and the |
| − | fractions
| + | 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. |
| | | | |
| | | | |
| − | |
| |
| − | ACTIVITY4
| |
| | | | |
| − |
| + | |
| − | |-
| |
| − | |
| |
| − | 6
| |
| | | | |
| | | | |
| − | |
| + | === Introducing Fractions Using Share and Measure Interpretations === |
| − | Addition of Fractions
| |
| − | | |
| | | | |
| − | |
| + | One of the major difficulties a child faces with fractions is |
| − | Why do we need LCM to add fractions.
| + | making sense of the symbol ''m/n''. In order to facilitate |
| − | Understand Commutative law w.r.t. Fraction addition
| + | 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 |
| − | Fraction addition Algorithm
| + | 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 |
| − | Applying the Algorithm and adding
| + | rectangles, from our experience we feel circular objects are more |
| − | fractions. Solving simple word problems
| + | 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. |
| | | | |
| | | | |
| − | |
| |
| − | ACTIVITY5
| |
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| − |
| |
| − | |-
| |
| − | |
| |
| − | 6
| |
| | | | |
| − |
| |
| − | |
| |
| − | Subtraction of Fractions
| |
| | | | |
| | | | |
| − | |
| + | == Activities == |
| − | Why we need LCM to subtract
| |
| − | fractions.
| |
| − | | |
| | | | |
| − | |
| + | === Activity1: Introduction to fractions === |
| − | Fraction subtraction Algorithm
| |
| − | | |
| | | | |
| − | |
| + | '''''Learning |
| − | Applying the Algorithm and adding
| + | Objectives ''''' |
| − | fractions. Solving simple word problems
| |
| | | | |
| | | | |
| − | |
| + | Introduce |
| − | ACTIVITY6
| + | fractions using the part-whole method |
| | | | |
| | | | |
| − | |-
| + | '''''Materials and |
| − | |
| + | resources required ''''' |
| − | 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
| |
| | | | |
| | | | |
| − | |
| + | # Write the Number Name and the number of the picture like the example [[Image:KOER%20Fractions_html_m1d9c88a9.gif]]Number Name = One third Number: [[Image:KOER%20Fractions_html_52332ca.gif]] |
| − | Represent decimal numbers on the
| |
| − | number line. How to convert simple decimal numbers into fractions
| |
| − | and vice versa
| |
| − | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_2625e655.gif]][[Image:KOER%20Fractions_html_m685ab2.gif]][[Image:KOER%20Fractions_html_55c6e68e.gif]][[Image:KOER%20Fractions_html_mfefecc5.gif]][[Image:KOER%20Fractions_html_m12e15e63.gif]] |
| − | ACTIVITY7
| |
| | | | |
| − |
| + | |
| − | |-
| + | Question: |
| − | |
| + | What is the value of the numerator and denominator in the last figure |
| − | 6
| + | , the answer is [[Image:KOER%20Fractions_html_m2dc8c779.gif]] |
| | | | |
| | | | |
| − | |
| |
| − | (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
| |
| | | | |
| | | | |
| − | |
| + | # Colour the correct amount that represents the fractions |
| − | Transition from Additive Thinking to
| |
| − | Multiplicative Thinking
| |
| − | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_19408cb.gif]][[Image:KOER%20Fractions_html_m12e15e63.gif]][[Image:KOER%20Fractions_html_m6b49c523.gif]][[Image:KOER%20Fractions_html_m6f2fcb04.gif]] |
| − | ACTIVITY8
| |
| | | | |
| − |
| |
| − | |-
| |
| − | |
| |
| − | 7
| |
| | | | |
| − |
| + | |
| − | |
| + | 7/10 3/8 |
| − | Multiplication of Fractions
| + | 1/5 4/7 |
| | | | |
| | | | |
| − | |
| + | Question: |
| − | Visualise the quantities when a
| + | Before colouring count the number of parts in each figure. What does |
| − | fraction is multiplied 1) whole number 2) fraction. Where is
| + | it represent. Answer: Denominator |
| − | multiplication of fractions used
| |
| | | | |
| | | | |
| − | |
| |
| − | “of” Operator means
| |
| − | multiplication. Know the fraction multiplication algorithm
| |
| | | | |
| − |
| |
| − | |
| |
| − | Apply the algorithm to multiply
| |
| − | fraction by fraction
| |
| | | | |
| − |
| |
| − | |
| |
| − | ACTIVITY9
| |
| | | | |
| | | | |
| − | |-
| + | # Divide the circle into fractions and colour the right amount to show the fraction |
| − | |
| |
| − | 7
| |
| − | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]] |
| − | 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
| + | |
| | + | 3/5 |
| | + | 6/7 1/3 5/8 2/5 |
| | | | |
| | | | |
| − | |
| |
| − | The denominator of a fraction is
| |
| − | always 100.
| |
| | | | |
| − |
| + | |
| − | |
| |
| − | Convert from fraction to percentage
| |
| − | and vice versa
| |
| | | | |
| | | | |
| − | |
| + | # Draw the Fraction and observe which is the greater fraction – observe that the parts are equal for each pair |
| − | Convert percentage
| |
| − | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]] |
| − | ACTIVITY11
| |
| | | | |
| | | | |
| − | |-
| + | [[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]] |
| − | |
| |
| − | 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
| |
| | | | |
| | | | |
| − | |
| + | 1/3 2/3 4/5 2/5 |
| − | ACTIVITY12
| + | 3/7 4/7 |
| | | | |
| | | | |
| − | |-
| |
| − | |
| |
| − | 8
| |
| | | | |
| − |
| |
| − | |
| |
| − | (Linking
| |
| − | to Fraction Topic)
| |
| | | | |
| | | | |
| − | Rational & Irrational Numbers
| |
| | | | |
| | + | |
| | + | |
| | + | # Draw the Fraction and observe which is the greater fraction – Observe that the parts are different sizes for each pair. |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]] |
| − | The number line is fully populated
| |
| − | with natural numbers, integers and irrational and rational numbers
| |
| | | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]] |
| − | 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
| |
| | | | |
| | | | |
| − | |}
| |
| − | <br>
| |
| − | <br>
| |
| | | | |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| | | | |
| | | | |
| − | = Curricular Objectives =
| |
| − |
| |
| − | # Conceptualise and understand algorithms for basic operations (addition, subtraction, multiplication and division) on fractions.
| |
| − | # Apply the understanding of fractions as simple mathematics models.
| |
| − | # Understand the different mathematical terms associated with fractions.
| |
| − | # 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 <u>equal</u>
| |
| − | parts and the fraction represents one or more of the parts.
| |
| | | | |
| − |
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_78a5005.gif]]<br>
| |
| | | | |
| | | | |
| − | Half
| + | 1/3 |
| − | (½) : The whole is divided into '''two
| + | 1/4 1/5 1/8 |
| − | equal '''parts.
| + | 1/6 1/2 |
| | | | |
| | | | |
| − | One part is coloured, this part represents the fraction ½.
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| − |
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_6fbd7fa5.gif]]<br>
| |
| | | | |
| | | | |
| − | One-Fourth
| + | # Solve these word problems by drawing |
| − | (1/4) : The whole is divided into '''four
| + | ## 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? |
| − | equal '''parts.
| + | ## 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. |
| − | | + | ## 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? |
| | + | # |
| | + | # The circles in the box represent the whole; colour the right amount to show the fraction [[Image:KOER%20Fractions_html_m78f3688a.gif]]''Hint: Half is 2 circles'' [[Image:KOER%20Fractions_html_m867c5c2.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]] |
| | | | |
| − | One part is coloured, this part represents the fraction ¼.
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_43b75d3a.gif]]<br> | + | [[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]] |
| | | | |
| − |
| |
| − | 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.
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_2faaf16a.gif]]Two
| |
| − | Fifth (2/5) : The whole is divided into '''five
| |
| − | equal '''parts.
| |
| | | | |
| | | | |
| − | '''Two''' | + | '''''Pre-requisites/ |
| − | part are coloured, this part represents the fraction 2/5.
| + | Instructions Method ''''' |
| | | | |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | 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. |
| | + | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_9e5c77.gif]]Three
| + | After |
| − | Seventh (3/7) : The whole is divided into '''seven
| + | the activity sheet is completed, please use the evaluation questions |
| − | equal '''parts.
| + | to see if the child has understood the concept of fractions |
| | | | |
| | | | |
| − | '''Three''' | + | '''''Evaluation''''' |
| − | part are coloured, this part represents the fraction 3/7.
| |
| | | | |
| | + | |
| | + | === Activity 2: Proper and Improper Fractions === |
| | | | |
| − | <br>
| + | '''''Learning |
| | + | Objectives''''' |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m30791851.gif]]<br>
| + | Proper and Improper Fractions |
| | | | |
| | | | |
| − | Seven
| + | '''''Materials |
| − | tenth (7/10) : The whole is divided into '''ten
| + | and resources required ''''' |
| − | equal '''parts.
| |
| | | | |
| | | | |
| − | '''Seven''' | + | # [[Image:KOER%20Fractions_html_5518d221.jpg]]If 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.''' |
| − | part are coloured, this part represents the fraction 7/10 .
| + | |
| | + | [[Image:KOER%20Fractions_html_5e906d5b.jpg]][[Image:KOER%20Fractions_html_5518d221.jpg]] |
| | | | |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | [[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]] |
| | | | |
| − |
| |
| − | '''Terms Numerator and Denominator and their meaning'''
| |
| | | | |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_3bf1fc6d.gif]]Three
| |
| − | Eight (3/8) The whole is divided into '''eight
| |
| − | equal '''parts.
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | '''Three'''
| + | [[Image:KOER%20Fractions_html_5518d221.jpg]][[Image:KOER%20Fractions_html_5e906d5b.jpg]] |
| − | part are coloured, this part represents the fraction 3/8 .
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | 3/8
| + | [[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]] |
| − | 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.
| |
| | | | |
| | | | |
| − | <br>
| + | [[Image:KOER%20Fractions_html_5518d221.jpg]] |
| | + | |
| | | | |
| | | | |
| − | == Equal Share ==
| + | [[Image:KOER%20Fractions_html_55f65a3d.gif]] |
| | + | |
| | + | |
| | | | |
| − | <br>
| + | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | 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
| |
| | | | |
| − |
| + | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_3176e16a.gif]]
| |
| | | | |
| | | | |
| − | <br>
| + | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | The relations 3/6 = 1⁄2
| + | # 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 |
| − | 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
| |
| | | | |
| | | | |
| − | <br>
| + | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m39388388.gif]]
| + | |
| | + | |
| | | | |
| | | | |
| − | <br>
| + | [[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_5e906d5b.jpg]][[Image:KOER%20Fractions_html_55f65a3d.gif]] |
| − | <br>
| + | |
| | | | |
| | | | |
| − | It is important to note
| + | [[Image:KOER%20Fractions_html_5518d221.jpg]] |
| − | 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
| |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m799c1107.gif]] | + | [[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_55f65a3d.gif]][[Image:KOER%20Fractions_html_5518d221.jpg]] |
| | + | |
| | | | |
| | | | |
| − | Share interpretation
| + | [[Image:KOER%20Fractions_html_55f65a3d.gif]] |
| − | 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. '''''
| |
| | | | |
| | | | |
| − | <br>
| |
| − | <br>
| |
| | | | |
| − |
| |
| − | '''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
| + | '''Pre-requisites/ |
| − | (like cake) do not come in the same size and can be divided into n
| + | Instructions Method ''' |
| − | 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
| + | Examples of Proper and improper |
| − | children to the use of both circles and rectangles, from our
| + | fractions are given. The round disks in the figure represent rotis |
| − | experience we feel circular objects are more useful when use the
| + | and the children figures represent children. Cut each roti and each |
| − | share interpretation as children can draw as many small circles as
| + | child figure and make the children fold, tear and equally divide the |
| − | they need and since the emphasis not so much on the size as in the
| + | roits so that each child figure gets equal share of roti. |
| − | 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
| + | '''''Evaluation''''' |
| − | 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: ==
| + | # What happens when the numerator and denominator are the same, why ? |
| | + | # What happens when the numerator is greater than the denominator why ? How can we represent this in two ways ? |
| | | | |
| − | <br>
| + | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | # 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.
| + | === Activity 3: Comparing Fractions === |
| − | # In [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m2988e86b.gif]], 5 is called the '''numerator''' and 7 is called the '''denominator'''.
| |
| − | # Fractions can be shown on a number line. Every fraction has a point associated with it on the number line.
| |
| − | # 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'''.
| |
| − | # An improper fraction can be written as a combination of a whole and a part, and such fraction then called '''mixed fractions'''.
| |
| − | # 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.
| |
| − | # 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 : ==
| + | '''''Learning |
| | + | Objectives''''' |
| | + | |
| | | | |
| − | <br>
| + | Comparing-Fractions |
| − | <br>
| |
| | | | |
| | | | |
| − | # [[http://vimeo.com/22238434]] Video on teaching fractions using the equal share method made by Eklavya an NGO based in Madhya Pradesh, India
| + | '''''Materials |
| − | # [[http://mathedu.hbcse.tifr.res.in/]] Mathematics resources from Homi Baba Centre for Science Education
| + | and resources required ''''' |
| | + | |
| | | | |
| − | <br>
| + | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | <br>
| + | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | = Errors with fractions =
| + | [[http://www.superteacherworksheets.com/fractions/comparing-fractions.pdf]] |
| | + | |
| | | | |
| − | When
| + | [[http://www.superteacherworksheets.com/fractions/comparing-fractions2.pdf]] |
| − | 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'''.
| |
| | | | |
| | | | |
| − | <br>
| + | '''Pre-requisites/ |
| | + | Instructions Method''' |
| | | | |
| | | | |
| − | The
| + | Print the |
| − | five levels of resistance to N-Distracters that a child develops are:
| + | document and work out the |
| | + | activity sheet |
| | | | |
| | | | |
| − | <br>
| + | '''''Evaluation''''' |
| | | | |
| | | | |
| − | # '''''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. | + | # Does the child know the symbols '''>, <''' and '''=''' |
| − | # '''''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.
| + | # What happens to the size of the part when the denominator is different ? |
| − | # '''''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.
| + | # Does it decrease or increase when the denominator becomes larger ? |
| − | # '''''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. | + | # Can we compare quantities when the parts are different sizes ? |
| − | # '''''Resistance to N-Distracter: '''''The student is completely free (conceptually and algorithmically) of N-Distracter errors. | + | # What should we do to make the sizes of the parts the same ? |
| | | | |
| − | <br>
| |
| | | | |
| | + | |
| | + | |
| | + | |
| | + | === Activity 4: Equivalent Fractions === |
| | | | |
| − | <br>
| + | '''''Learning |
| | + | Objectives''''' |
| | | | |
| | | | |
| − | == Key vocabulary: ==
| + | To understand Equivalent Fractions |
| | + | |
| | | | |
| − | <br>
| + | '''''Materials |
| − | <br>
| + | and resources required''''' |
| | | | |
| | | | |
| − | # '''N-Distractor''': as defined above.
| + | [[http://www.superteacherworksheets.com/fractions/fractions-matching-game.pdf]] |
| | + | |
| | | | |
| − | == Additional resources: ==
| + | |
| | + | |
| | + | |
| | | | |
| − | <br>
| + | '''Pre-requisites/ |
| − | <br>
| + | Instructions Method''' |
| | | | |
| | | | |
| − | # [[www.merga.net.au/publications/counter.php?pub=pub_conf&id=1410]] A PDF Research paper titled Probing Whole Number Dominance with Fractions.
| + | Print 10 copies |
| − | # [[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 ”
| + | of the document from pages 2 to 5 fractions-matching-game |
| − | # [[http://books.google.com/books?id=Y5Skj-EA2_AC&pg=PA251&lpg=PA251&dq=streefland+fractions&source=bl&ots=aabKa]][[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
| + | Cut the each fraction part. Play memory game as described in |
| − |
| + | the document in groups of 4 children. |
| − | = Operations on Fractions =
| |
| − |
| |
| − | == Addition and Subtraction ==
| |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | Adding and subtracting like fractions is simple. It must be
| + | '''''Evaluation''''' |
| − | 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.
| |
| | | | |
| | | | |
| − | <br>
| + | # What is reducing a fraction to the simplest form ? |
| − | | + | # What is GCF – Greatest Common Factor ? |
| | + | # Use the document [[simplifying-fractions.pdf]] |
| | + | # Why are fractions called equivalent and not equal. |
| | | | |
| − | Adding and subtracting unlike fractions requires the child to
| + | == Evaluation == |
| − | 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.
| |
| − | | |
| | | | |
| − | <br>
| + | == Self-Evaluation == |
| − | | |
| | | | |
| − | <br>
| + | == Further Exploration == |
| − | | + | |
| | + | == Enrichment Activities == |
| | | | |
| − | == Multiplication == | + | = Errors with fractions = |
| | | | |
| − | <br>
| + | == Introduction == |
| − | | |
| | | | |
| − | Multiplying a fraction by a whole number: Here the repeated addition
| + | A brief |
| − | logic of multiplying whole numbers is still valid. 1/6 multiplied by
| + | understanding of the common errors that children make when it comes |
| − | 4 is 4 times 1/6 which is equal to 4/6.
| + | to fractions are addressed to enable teachers to understand the |
| | + | child's levels of conceptual understanding to address the |
| | + | misconceptions. |
| | | | |
| | | | |
| − | <br>
| + | == Objectives == |
| − | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_714bce28.gif]]
| + | 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-Distractor. 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-distractors and a slow-down of learning when |
| | + | moving from the '''concrete level to the abstract level'''. |
| | | | |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | 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.
| |
| | | | |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | 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.
| |
| | | | |
| | | | |
| − | <br>
| + | == N-Distractors == |
| − | | |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m66ce78ea.gif]]<br>
| |
| | | | |
| | | | |
| − | <br>
| + | The five levels of resistance to |
| | + | N-Distractors that a child develops are: |
| | | | |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | # '''''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. |
| − | | + | # '''''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. |
| | + | # '''''Spontaneous refutation of N-Distractor errors:''''' The student may still make N-Distractor 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. |
| | + | # '''''Free of N-Distractor: '''''The written work is free of N-Distractors. This could mean a thorough understanding of the methods/algorithms of manipulating fractions. |
| | + | # '''''Resistance to N-Distractor: '''''The student is completely free (conceptually and algorithmically) of N-Distractor errors. |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | == Activities == |
| − | | |
| | | | |
| − | We can apply that idea to fractions, too.
| + | == Evaluation == |
| − | | |
| | | | |
| − | * The one side of the rectangle is 1 unit (in terms of length).
| + | == Self-Evaluation == |
| − | * The other side is 1 unit also.
| |
| − | * The whole rectangle also is ''1 square unit'', in terms of area.
| |
| | | | |
| − | <br>
| + | == Further Exploration == |
| − | | + | |
| | + | # [[www.merga.net.au/publications/counter.php?pub=pub_conf&id=1410]] A PDF Research paper titled Probing Whole Number Dominance with Fractions. |
| | + | # [[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 ” |
| | + | # [[http://books.google.com/books?id=Y5Skj-EA2_AC&pg=PA251&lpg=PA251&dq=streefland+fractions&source=bl&ots=aabKaciwrA&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 |
| | + | |
| | + | == Enrichment Activities == |
| | | | |
| − | See figure below to see how the following multiplication can be
| |
| − | shown.
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| − |
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m6c9f1742.gif]]
| |
| | | | |
| | | | |
| − | <br>
| + | = Operations on Fractions = |
| − | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_753005a4.gif]]
| + | == Introduction == |
| − | | |
| | | | |
| − | <br>
| + | This topic introduces the different operations on fractions. When |
| | + | learners move from whole numbers to fractions, many of the operations |
| | + | are counter intuitive. This section aims to clarify the concepts |
| | + | behind each of the operations. |
| | | | |
| | | | |
| − | <br>
| + | == Objectives == |
| − | | |
| | | | |
| − | <br>
| + | The aim of this section is to visualise and conceptually |
| | + | understand each of the operations on fractions. |
| | | | |
| | | | |
| − | <br>
| + | == Addition and Subtraction == |
| − | | |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | 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. |
| | | | |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | 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. |
| | | | |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | '''Remember:
| |
| − | '''The two fractions to multiply
| |
| − | represent the length of the sides, and the answer fraction represents
| |
| − | area.
| |
| | | | |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | == Division == | + | == Multiplication == |
| | | | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | Dividing
| + | Multiplying a |
| − | a fraction by a whole number can be demonstrated just like division
| + | fraction by a whole number: Here the repeated addition logic of |
| − | of whole numbers. When we divide 3/4 by 2 we can visualise it as
| + | multiplying whole numbers is still valid. 1/6 multiplied by 4 is 4 |
| − | dividing 3 parts of a whole roti among 4 people.
| + | times 1/6 which is equal to 4/6. |
| | | | |
| | | | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_1f617ac8.gif]] | + | [[Image:KOER%20Fractions_html_714bce28.gif]] |
| | | | |
| | | | |
| − | Here
| + | |
| − | 3/4 is divided between two people. One fourth piece is split into
| |
| − | two.<br>
| |
| − | Each person gets 1/4 and 1/8.
| |
| | | | |
| | | | |
| − | <br>
| + | 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. |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m5f26c0a.gif]]
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | OR
| + | 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. |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m25efcc2e.gif]]<br>
| |
| | | | |
| − |
| |
| − | 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.
| |
| | | | |
| − |
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_7ed8164a.gif]]
| |
| | | | |
| | | | |
| − | <br>
| + | [[Image:KOER%20Fractions_html_m66ce78ea.gif]] |
| | | | |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | When
| |
| − | dividing a fraction by a fraction, we use the measure interpretation.
| |
| | | | |
| − |
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m3192e02b.gif]]<br>
| |
| | | | |
| | | | |
| − | When
| + | |
| − | we divide 2 by ¼ we ask how many times does ¼ fit into 2
| |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m257a1863.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m257a1863.gif]]<br>
| + | |
| | | | |
| | | | |
| − | <br>
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | It
| + | |
| − | fits in 4 times in each roti, so totally 8 times.
| |
| | | | |
| | | | |
| − | <br>
| + | We can apply that |
| | + | idea to fractions, too. |
| | | | |
| | | | |
| − | We
| + | * The one side of the rectangle is 1 unit (in terms of length). |
| − | write it as
| + | * The other side is 1 unit also. |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m390fcce6.gif]]
| + | * The whole rectangle also is ''1 square unit'', in terms of area. |
| | + | |
| | + | |
| | | | |
| | | | |
| − | <br>
| + | See figure below |
| − | <br>
| + | to see how the following multiplication can be shown. |
| | | | |
| | | | |
| − | <br>
| + | |
| − | <br>
| |
| | | | |
| | | | |
| − | == Key vocabulary: ==
| + | [[Image:KOER%20Fractions_html_m6c9f1742.gif]] |
| | + | |
| | | | |
| − | <br>
| + | |
| − | <br>
| |
| | | | |
| | | | |
| − | # '''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.
| + | [[Image:KOER%20Fractions_html_753005a4.gif]] |
| | + | |
| | | | |
| − | # '''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: ==
| + | |
| | + | |
| | | | |
| − | <br>
| + | |
| − | <br>
| |
| | | | |
| | | | |
| − | # [[http://www.youtube.com/watch?v=41FYaniy5f8]] detailed conceptual understanding of division by fractions
| + | |
| − | # [[http://www.homeschoolmath.net/teaching/f/understanding_fractions.php]] understanding fractions
| + | |
| − | # [[http://www.geogebra.com]] Understand how to use Geogebra a mathematical computer aided tool
| |
| − | # [[http://www.superteacherworksheets.com]] Worksheets in mathematics for teachers to use
| |
| | | | |
| − | = Linking Fractions to other Topics =
| + | '''Remember: '''The |
| | + | two fractions to multiply represent the length of the sides, and the |
| | + | answer fraction represents area. |
| | + | |
| | | | |
| − | == 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.
| |
| | | | |
| | + | |
| | + | |
| | + | == Division == |
| | | | |
| − | 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.
| |
| | | | |
| | | | |
| − | <br>
| + | Dividing a fraction by a whole number |
| − | <br>
| + | 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. |
| | | | |
| | | | |
| − | The
| + | [[Image:KOER%20Fractions_html_1f617ac8.gif]] |
| − | 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.
| |
| | | | |
| | | | |
| − | <br>
| + | Here 3/4 is divided between two |
| − | <br>
| + | people. One fourth piece is split into two. Each person gets |
| | + | 1/4 and 1/8. |
| | | | |
| | | | |
| − | The
| + | |
| − | number line between 0 and 1 is divided into ten parts. Each of these
| |
| − | ten parts is '''1/10''', a '''tenth'''.
| |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_3d7b669f.gif]]<br> | + | [[Image:KOER%20Fractions_html_m5f26c0a.gif]] |
| − | <br>
| |
| | | | |
| | | | |
| − | 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.
| + | OR |
| | | | |
| | | | |
| − | We
| + | [[Image:KOER%20Fractions_html_m25efcc2e.gif]] |
| − | 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
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m7f1d448c.gif]] | |
| | | | |
| | | | |
| − | Note:
| + | Another way of solving the same |
| − | A common error one sees is 0.7 is written as 1 /7. It is seven
| + | problem is to split each fourth piece into 2. |
| − | 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
| + | This means we change the 3/4 |
| − | coloured portion represents 0.6 or 6/10 and the whole block
| + | into 6/8. |
| − | represents 1.
| |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_1cf72869.gif]] <br> | + | [[Image:KOER%20Fractions_html_7ed8164a.gif]] |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | |
| − | <br>
| |
| | | | |
| | | | |
| − | == Percentages ==
| + | When dividing a fraction by a fraction, |
| − |
| + | we use the measure interpretation. |
| − | 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
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m1369c56e.gif]]
| |
| − | where '''10 is the numerator''' and the '''denominator is '''<u>'''always'''</u>'''
| |
| − | 100'''. In this case 10 % of the cost of the book is
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m50e22a06.gif]].
| |
| − | So you can buy the book for 200 – 20 = 180 rupees.
| |
| | | | |
| | | | |
| − | <br>
| + | [[Image:KOER%20Fractions_html_m3192e02b.gif]] |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | When we divide 2 by ¼ we ask how many |
| − | <br>
| + | times does ¼ |
| | | | |
| | | | |
| − | There
| |
| − | are a number of common ones that are useful to learn. Here is a table
| |
| − | showing you the ones that you should learn.
| |
| | | | |
| − |
| |
| − | {| border="1"
| |
| − | |-
| |
| − | |
| |
| − | Percentage
| |
| | | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_m257a1863.gif]][[Image:KOER%20Fractions_html_m257a1863.gif]] |
| − | Fraction
| + | |
| | + | |
| | + | '''fit into 2'''. |
| | | | |
| | | | |
| − | |-
| + | |
| − | |
| |
| − | 100%
| |
| | | | |
| | | | |
| − | |
| + | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m15ed765d.gif]]
| |
| | | | |
| | | | |
| − | |-
| + | It fits in 4 times in each roti, so |
| − | |
| + | totally 8 times. |
| − | 50%
| |
| | | | |
| | | | |
| − | |
| + | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_df52f71.gif]]
| |
| | | | |
| | | | |
| − | |-
| + | We write it as |
| − | |
| + | [[Image:KOER%20Fractions_html_m390fcce6.gif]] |
| − | 25%
| |
| | | | |
| | | | |
| − | |
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m6c97abb.gif]]
| |
| | | | |
| | + | |
| | + | |
| | + | |
| | + | == Activities == |
| | + | |
| | + | === Activity 1 Addition of Fractions === |
| | | | |
| − | |-
| + | '''''Learning |
| − | |
| + | Objectives''''' |
| − | 75%
| |
| | | | |
| | | | |
| − | |
| + | Understand Addition of Fractions |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m6cb13da4.gif]]
| |
| | | | |
| | | | |
| − | |-
| + | '''''Materials |
| − | |
| + | and resources required ''''' |
| − | 10%
| + | |
| | + | |
| | + | [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Addition.html]] |
| | | | |
| | | | |
| − | |
| + | '''''Pre-requisites/ |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_26bc75d0.gif]]
| + | Instructions Method ''''' |
| | | | |
| | | | |
| − | |-
| + | Open link |
| − | |
| + | [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Addition.html]] |
| − | 20%
| |
| | | | |
| | | | |
| − | |
| + | [[Image:KOER%20Fractions_html_m3dd8c669.png]] |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m73e98509.gif]] | + | |
| | | | |
| | | | |
| − | |-
| + | Move the sliders |
| − | |
| + | Numerator1 and Denominator1 to set Fraction 1 |
| − | 40%
| |
| | | | |
| | | | |
| − | |
| + | Move the sliders |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m2dd64d0b.gif]]
| + | Numerator2 and Denominator2 to set Fraction 2 |
| | | | |
| | | | |
| − | |}
| + | See the last bar |
| − | <br>
| + | to see the result of adding fraction 1 and fraction 2 |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | When you move |
| − | <br>
| + | the sliders ask children to |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m60c76c68.gif]]To
| + | Observe and |
| − | see 40 % visually see the figure :
| + | describe what happens when the denominator is changed. |
| | | | |
| | | | |
| − | You
| + | Observe and |
| − | can see that if the shape is divided into 5 equal parts, then 2 of
| + | describe what happens when denominator changes |
| − | those parts are shaded.
| |
| | | | |
| | | | |
| − | If
| + | Observe and |
| − | the shape is divided into 100 equal parts, then 40 parts are shaded. | + | describe the values of the numerator and denominator and relate it to |
| | + | the third result fraction. |
| | | | |
| | | | |
| − | These
| + | Discuss LCM and |
| − | are equivalent fractions as in both cases the same amount has been
| + | GCF |
| − | shaded.
| + | |
| | + | |
| | + | '''''Evaluation''''' |
| | | | |
| | + | |
| | + | === Activity 2 Fraction Subtraction === |
| | | | |
| − | <br>
| + | '''''Learning |
| − | <br>
| + | Objectives ''''' |
| | | | |
| | | | |
| − | == Ratio and Proportion ==
| + | Understand Fraction Subtraction |
| | + | |
| | | | |
| − | It
| + | '''''Materials and |
| − | is important to understand that fractions also can be interpreted as
| + | resources required''''' |
| − | 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.
| |
| | | | |
| | | | |
| − | <br>
| + | [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Subtraction.html]] |
| − | <br>
| + | |
| | + | |
| | + | '''''Pre-requisites/ |
| | + | Instructions Method ''''' |
| | | | |
| | | | |
| − | '''What
| + | Open link |
| − | is ratio?'''
| + | [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Subtraction.html]] |
| | | | |
| | | | |
| − | 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
| + | [[Image:KOER%20Fractions_html_481d8c4.png]] |
| − | * Use 1 shovel (bucket) of cement to 3 shovels (buckets) of sand
| + | |
| − | * Use 3 parts blue paint to 1 part white
| + | |
| | | | |
| − | Ratio
| + | Move the sliders Numerator1 and Denominator1 to set Fraction 1 |
| − | is the number of '''parts''' to a mix. The paint mix is 4
| |
| − | parts, with 3 parts blue and 1 part white.
| |
| | | | |
| | | | |
| − | The
| + | Move the sliders Numerator2 and Denominator2 to set Fraction 2 |
| − | 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
| + | See the last bar to see the result of subtracting fraction 1 and |
| − | paint in the ratio 3:1 (3 parts blue paint to 1 part white paint)
| + | fraction 2 |
| − | means 3 + 1 = 4 parts in all.
| |
| | | | |
| | | | |
| − | 3
| + | |
| − | parts blue paint to 1 part white paint = is ¾ blue paint to ¼ white
| + | |
| − | paint.
| |
| | | | |
| | | | |
| − | <br>
| + | When you move the sliders ask children to |
| − | <br>
| |
| | | | |
| | | | |
| − | Cost
| + | observe and describe what happens when the denominator is |
| − | of a pen is Rs 10 and cost of a pencil is Rs 2. How many times the
| + | changed. |
| − | cost of a pencil is the cost of a pen? Obviously it is five times.
| |
| − | This can be written as
| |
| | | | |
| | | | |
| − | <br>
| + | observe and describe what happens when denominator changes |
| − | <br>
| |
| | | | |
| | | | |
| − | The
| + | observe and describe the values of the numerator and denominator |
| − | ratio of the cost of a pen to the cost of a pencil =
| + | and relate it to the third result fraction. |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m762fb047.gif]]
| |
| | | | |
| | | | |
| − | <br>
| + | Discuss LCM and GCF |
| − | <br>
| |
| | | | |
| | | | |
| − | What
| + | '''''Evaluation''''' |
| − | 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
| + | === Activity 3 Multiplication of fractions === |
| − | pots in a ratio of 3:1
| + | |
| | + | '''''Learning |
| | + | Objectives ''''' |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m22cda036.gif]]<br>
| + | Understand Multiplication of fractions |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | '''''Materials and |
| − | <br>
| + | resources required''''' |
| | | | |
| | | | |
| − | <br>
| + | [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]] |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | '''''Pre-requisites/ |
| − | <br>
| + | Instructions Method ''''' |
| | | | |
| | | | |
| − | <br>
| + | Open link |
| − | <br>
| + | [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]] |
| | | | |
| | | | |
| − | <br>
| + | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | But
| + | [[Image:KOER%20Fractions_html_12818756.png]] |
| − | 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
| + | Move the sliders Numerator1 and Denominator1 to set Fraction 1 |
| − | we double the amount of blue paint we need 6 pots.
| |
| | | | |
| | | | |
| − | If
| + | Move the sliders Numerator2 and Denominator2 to set Fraction 2 |
| − | we double the amount of white paint we need 2 pots.
| |
| | | | |
| | | | |
| − | Six
| + | On the right hand side see the result of multiplying fraction 1 |
| − | paint pots in a ratio of 3:1
| + | and fraction 2 |
| | | | |
| | | | |
| − | <br>
| + | '''Material/Activity Sheet''' |
| − | <br>
| |
| | | | |
| | | | |
| − | The
| + | Please open |
| − | amount of blue and white paint we need increase in direct proportion
| + | [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]] |
| − | to each other. Look at the table to see how as you use more blue
| + | in Firefox and follow the process |
| − | paint you need more white paint:
| |
| | | | |
| | | | |
| − | Pots
| + | When you move the sliders ask children to |
| − | of blue paint 3 6 9 12
| |
| | | | |
| | | | |
| − | Pots
| + | observe and describe what happens when the denominator is |
| − | of white paint 1 2 3 4
| + | changed. |
| | | | |
| | | | |
| − | <br>
| + | observe and describe what happens when denominator changes |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | One unit will be the large square border-in blue solid lines |
| − | <br>
| |
| | | | |
| | | | |
| − | Two
| + | A sub-unit is in dashed lines within one square unit. |
| − | quantities which are in direct proportion will always produce a graph
| |
| − | where all the points can be joined to form a straight line.
| |
| | | | |
| | | | |
| − | <br>
| + | The thick red lines represent the fraction 1 and 2 and also the |
| − | <br>
| + | side of the quadrilateral |
| | | | |
| | | | |
| − | '''What
| + | The product represents the area of the the quadrilateral |
| − | is Inverse Proportion ?'''
| |
| | | | |
| | | | |
| − | Two
| + | '''''Evaluation''''' |
| − | 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
| + | When |
| − | the table below, observe that as the speed increases time taken to | + | two fractions are multiplied |
| − | cover the distance decreases
| + | is the product larger or smaller that the multiplicands – why ? |
| | | | |
| | | | |
| − | <br>
| |
| − | <br>
| |
| | | | |
| − |
| + | |
| − | {| border="1"
| |
| − | |-
| |
| − | |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | |
| + | |
| − | Walk
| + | |
| | | | |
| | | | |
| − | |
| + | === Activity 4 Division by Fractions === |
| − | Run
| + | |
| | + | '''''Learning |
| | + | Objectives ''''' |
| | | | |
| | | | |
| − | |
| + | Understand Division by Fractions |
| − | Cycle
| |
| | | | |
| | | | |
| − | |
| + | '''''Materials and |
| − | Bus
| + | resources required''''' |
| | | | |
| | | | |
| − | |-
| + | [[http://www.superteacherworksheets.com/fractions/fractionstrips_TWQWF.pdf]] |
| − | |
| |
| − | Speed
| |
| − | Km/Hr
| |
| | | | |
| | | | |
| − | |
| + | Crayons/ colour |
| − | 3
| + | pencils, Scissors, glue |
| | | | |
| | | | |
| − | |
| + | '''''Pre-requisites/ |
| − | 6
| + | Instructions Method ''''' |
| − | (walk speed *2)
| |
| | | | |
| | | | |
| − | |
| + | Print out the pdf |
| − | 9
| + | [[http://www.superteacherworksheets.com/fractions/fractionstrips_TWQWF.pdf]] |
| − | (walk speed *3)
| |
| | | | |
| | | | |
| − | |
| + | Colour each of the unit fractions in different colours. Keep the |
| − | 45
| + | whole unit (1) white. |
| − | (walk speed *15)
| |
| | | | |
| | | | |
| − | |-
| + | Cut out each unit fraction piece. |
| − | |
| + | |
| − | Time
| + | |
| − | Taken (minutes)
| + | Give examples |
| | + | [[Image:KOER%20Fractions_html_m282c9b3f.gif]] |
| | | | |
| | | | |
| − | |
| + | For example if we try the first one, |
| − | 30
| + | [[Image:KOER%20Fractions_html_21ce4d27.gif]] |
| | + | See how many |
| | + | [[Image:KOER%20Fractions_html_m31bd6afb.gif]]strips |
| | + | will fit exactly onto whole unit strip. |
| | | | |
| | | | |
| − | |
| + | |
| − | 15
| + | |
| − | (walk Time * ½)
| |
| | | | |
| | | | |
| − | |
| + | '''''Evaluation''''' |
| − | 10
| |
| − | (walk Time * 1/3)
| |
| | | | |
| | | | |
| − | |
| + | When |
| − | 2
| + | we divide by a fraction is the result larger or smaller why ? |
| − | (walk Time * 1/15)
| |
| | | | |
| | | | |
| − | |}
| + | |
| − | <br>
| + | |
| − | <br>
| |
| | | | |
| | | | |
| − | As
| + | == Evaluation == |
| − | Zaheeda doubles her speed by running, time reduces to half. As she
| + | |
| − | increases her speed to three times by cycling, time decreases to one
| + | == Self-Evaluation == |
| − | third. Similarly, as she increases her speed to 15 times, time
| + | |
| − | decreases to one fifteenth. (Or, in other words the ratio by which
| + | == Further Exploration == |
| − | time decreases is inverse of the ratio by which the corresponding
| + | |
| − | speed increases). We can say that speed and time change inversely in
| + | # [[http://www.youtube.com/watch?v=41FYaniy5f8]] detailed conceptual understanding of division by fractions |
| − | proportion.
| + | # [[http://www.homeschoolmath.net/teaching/f/understanding_fractions.php]] understanding fractions |
| − | | + | # [[http://www.superteacherworksheets.com]] Worksheets in mathematics for teachers to use |
| | + | |
| | + | = Linking Fractions to other Topics = |
| | | | |
| − | === Moving from Additive Thinking to Multiplicative Thinking === | + | == Introduction == |
| | | | |
| − | Avinash
| + | It is also very common for the school system to treat themes in a |
| − | thinks that if you use 5 spoons of sugar to make 6 cups of tea, then
| + | separate manner. Fractions are taught as stand alone chapters. In |
| − | you would need 7 spoons of sugar to make 8 cups of tea just as sweet
| + | this resource book an attempt to connect it to other middle school |
| − | as the cups before. Avinash would be using an '''''additive
| + | topics such as Ratio Proportion, Percentage and high school topics |
| − | transformation''''''''; '''he thinks that since we added 2 more
| + | such as rational and irrational numbers, inverse proportions are |
| − | cups of tea from 6 to 8. To keep it just as sweet he would need to
| + | made. These other topics are not discussed in detail themselves, but |
| − | add to more spoons of sugar. What he does not know is that for it to
| + | used to show how to link these other topics with the already |
| − | taste just as sweet he would need to preserve the ratio of sugar to
| + | understood concepts of fractions. |
| − | 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.
| |
| | | | |
| − |
| + | |
| − | <br>
| |
| | | | |
| | | | |
| − | The
| + | == Objectives == |
| − | 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.
| |
| − | | |
| | | | |
| − | <br>
| + | Explicitly link the other |
| − | <br>
| + | topics in school mathematics that use fractions. |
| | | | |
| | | | |
| − | == Rational & Irrational Numbers == | + | == Decimal Numbers == |
| | | | |
| − | After
| + | “Decimal” |
| − | the number line was populated with natural numbers, zero and the | + | comes from the Latin root '''''decem''''', |
| − | negative integers, we discovered that it was full of gaps. We
| + | which simply means ten. The number system we use is called the |
| − | discovered that there were numbers in between the whole numbers -
| + | decimal number system, because the place value units go in tens: you |
| − | fractions we called them.
| + | have ones, tens, hundreds, thousands, and so on, each unit being 10 |
| | + | times the previous one. |
| | | | |
| | | | |
| − | But,
| + | In |
| − | soon we discovered numbers that could not be expressed as a fraction.
| + | common language, the word “decimal number” has come to mean |
| − | These numbers could not be represented as a simple fraction. These
| + | numbers which have digits after the decimal point, such as 5.8 or |
| − | were called irrational numbers. The ones that can be represented by a
| + | 9.302. But in reality, any number within the decimal number system |
| − | simple fraction are called rational numbers. They h ad a very
| + | could be termed a decimal number, including whole numbers such as 12 |
| − | definite place in the number line but all that could be said was that
| + | or 381. |
| − | 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
| + | The |
| − | is another subset called transcendental numbers which have now been | + | simplest way to link or connect fractions to the decimal number |
| − | discovered. These numbers cannot be expressed as the solution of an
| + | system is with the number line representation. Any scale that a |
| − | algebraic polynomial. "pi" and "e" are such
| + | child uses is also very good for this purpose, as seen in the figure |
| − | numbers.
| + | below. |
| | | | |
| | | | |
| − | == Vocabulary ==
| + | |
| − |
| + | |
| − | Decimal
| |
| − | Numbers, Percentages, Ratio, Direct Proportion, Inverse Proportion,
| |
| − | Rational Numbers, Irrational Numbers
| |
| | | | |
| | | | |
| − | <br>
| + | The |
| − | <br>
| + | number line between 0 and 1 is divided into ten parts. Each of these |
| | + | ten parts is '''1/10''', a '''tenth'''. |
| | | | |
| | | | |
| − | == Additional Resources ==
| + | [[Image:KOER%20Fractions_html_3d7b669f.gif]] |
| − |
| |
| − | [[http://www.bbc.co.uk/skillswise/numbers/wholenumbers/]] | |
| | | | |
| − |
| |
| − | [[http://en.wikipedia.org/wiki/Koch_snowflake]]
| |
| | | | |
| | | | |
| − | [[http://www.realmagick.com/5552/bringing-it-down-to-earth-a-fractal-approach/]]
| + | 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. |
| | | | |
| | | | |
| − | = Activities : =
| + | We |
| − |
| + | can write any fraction with '''tenths (denominator 10) '''using the |
| − | == Activity1: Introduction to fractions ==
| + | 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 |
| − | === Objective: ===
| + | tenths or |
| − |
| + | [[Image:KOER%20Fractions_html_m7f1d448c.gif]] |
| − | Introduce
| |
| − | fractions using the part-whole method
| |
| | | | |
| | | | |
| − | === Procedure: ===
| + | 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. |
| | + | |
| | | | |
| − | Do
| + | The |
| − | the six different sections given in the activity sheet. For each | + | coloured portion represents 0.6 or 6/10 and the whole block |
| − | section there is a discussion point or question for a teacher to ask
| + | represents 1. |
| − | children.
| + | |
| | + | |
| | + | [[Image:KOER%20Fractions_html_1cf72869.gif]] |
| | + | |
| | | | |
| | + | |
| | + | == Percentages == |
| | | | |
| − | After
| + | Fractions and percentages are different ways of writing the same |
| − | the activity sheet is completed, please use the evaluation questions | + | thing. When we say that a book costs Rs. 200 and the shopkeeper is |
| − | to see if the child has understood the concept of fractions
| + | giving a 10 % discount. Then we can represent the 10% as a fraction |
| | + | as |
| | + | [[Image:KOER%20Fractions_html_m1369c56e.gif]] |
| | + | where '''10 is the numerator''' and the '''denominator is '''<u>'''always'''</u>''' |
| | + | 100'''. In this case 10 % of the cost of the book is |
| | + | [[Image:KOER%20Fractions_html_m50e22a06.gif]]. |
| | + | So you can buy the book for 200 – 20 = 180 rupees. |
| | | | |
| | | | |
| − | <br>
| |
| − | <br>
| |
| | | | |
| − |
| + | |
| − | '''Material/Activity
| |
| − | Sheet'''
| |
| | | | |
| | | | |
| − | # Write the Number Name and the number of the picture like the example [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m1d9c88a9.gif]]Number Name = One third Number: [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_52332ca.gif]]
| |
| − |
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_2625e655.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m685ab2.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55c6e68e.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_mfefecc5.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m12e15e63.gif]]
| |
| | | | |
| − |
| + | |
| − | Question:
| |
| − | What is the value of the numerator and denominator in the last figure
| |
| − | , the answer is [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m2dc8c779.gif]]
| |
| | | | |
| | | | |
| − | # Colour the correct amount that represents the fractions
| + | There |
| − |
| + | are a number of common ones that are useful to learn. Here is a table |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_19408cb.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m12e15e63.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m6b49c523.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m6f2fcb04.gif]]<br>
| + | showing you the ones that you should learn. |
| − | <br>
| |
| | | | |
| − |
| + | |
| − | 7/10 3/8
| + | {| border="1" |
| − | 1/5 4/7
| + | |- |
| | + | | |
| | + | Percentage |
| | | | |
| | | | |
| − | Question:
| + | | |
| − | Before colouring count the number of parts in each figure. What does
| + | Fraction |
| − | it represent. Answer: Denominator
| |
| | | | |
| | | | |
| − | <br>
| + | |- |
| − | <br>
| + | | |
| | + | 100% |
| | | | |
| | | | |
| − | # Divide the circle into fractions and colour the right amount to show the fraction
| + | | |
| − |
| + | [[Image:KOER%20Fractions_html_m15ed765d.gif]] |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
| |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | |- |
| − | <br>
| + | | |
| | + | 50% |
| | | | |
| | | | |
| − | <br>
| + | | |
| − | <br>
| + | [[Image:KOER%20Fractions_html_df52f71.gif]] |
| − | | |
| − |
| |
| − | 3/5
| |
| − | 6/7 1/3 5/8 2/5
| |
| | | | |
| | | | |
| − | <br>
| + | |- |
| − | <br>
| + | | |
| | + | 25% |
| | | | |
| | | | |
| − | # Draw the Fraction and observe which is the greater fraction – observe that the parts are equal for each pair
| + | | |
| − |
| + | [[Image:KOER%20Fractions_html_m6c97abb.gif]] |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
| |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
| + | |- |
| | + | | |
| | + | 75% |
| | | | |
| | | | |
| − | <br>
| + | | |
| | + | [[Image:KOER%20Fractions_html_m6cb13da4.gif]] |
| | | | |
| | | | |
| − | <br>
| + | |- |
| | + | | |
| | + | 10% |
| | | | |
| | | | |
| − | 1/3 2/3 4/5 2/5
| + | | |
| − | 3/7 4/7
| + | [[Image:KOER%20Fractions_html_26bc75d0.gif]] |
| | | | |
| | | | |
| − | <br>
| + | |- |
| | + | | |
| | + | 20% |
| | | | |
| | | | |
| − | <br>
| + | | |
| | + | [[Image:KOER%20Fractions_html_m73e98509.gif]] |
| | | | |
| | | | |
| − | # Draw the Fraction and observe which is the greater fraction – Observe that the parts are different sizes for each pair.
| + | |- |
| | + | | |
| | + | 40% |
| | + | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br> | + | | |
| | + | [[Image:KOER%20Fractions_html_m2dd64d0b.gif]] |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
| + | |} |
| | + | |
| | + | |
| | | | |
| | | | |
| − | <br>
| + | |
| | + | |
| | | | |
| | | | |
| − | <br>
| + | [[Image:KOER%20Fractions_html_m60c76c68.gif]]To |
| | + | see 40 % visually see the figure : |
| | | | |
| | | | |
| − | <br>
| + | You |
| | + | can see that if the shape is divided into 5 equal parts, then 2 of |
| | + | those parts are shaded. |
| | | | |
| | | | |
| − | <br>
| + | If |
| | + | the shape is divided into 100 equal parts, then 40 parts are shaded. |
| | | | |
| | | | |
| − | 1/3 1/4 1/5 1/8
| + | These |
| − | 1/6 1/2
| + | are equivalent fractions as in both cases the same amount has been |
| | + | shaded. |
| | | | |
| | | | |
| − | <br>
| + | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | # Solve these word problems by drawing
| + | == Ratio and Proportion == |
| − | ## 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?
| + | |
| − | ## 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.
| + | It |
| − | ## 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?
| + | 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 |
| − | === Evaluation Questions ===
| + | be established between Ratio and Proportion and the fraction |
| − |
| + | representation. Connecting multiplication of fractions is key to |
| − | == Activity 2: Proper and Improper Fractions ==
| + | understanding ratio and proportion. |
| − |
| |
| − | === 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
| |
| | | | |
| − |
| |
| − | # [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5518d221.jpg]]If 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.'''
| |
| − |
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5e906d5b.jpg]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5518d221.jpg]]<br>
| |
| − | <br>
| |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
| + | '''What |
| − | <br>
| + | is ratio?''' |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5e906d5b.jpg]]<br>
| + | Ratio |
| − | <br>
| + | is a way of comparing amounts of something. It shows how much bigger |
| | + | one thing is than another. For example: |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5518d221.jpg]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5518d221.jpg]]<br>
| + | * Use 1 measure detergent (soap) to 10 measures water |
| − | <br>
| + | * Use 1 shovel (bucket) of cement to 3 shovels (buckets) of sand |
| − | | + | * Use 3 parts blue paint to 1 part white |
| | | | |
| − | <br>
| + | Ratio |
| − | <br>
| + | is the number of '''parts''' to a mix. The paint mix is 4 |
| | + | parts, with 3 parts blue and 1 part white. |
| | | | |
| | | | |
| − | <br>
| + | The |
| − | <br>
| + | 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. |
| | | | |
| | | | |
| − | # 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
| + | Mixing |
| | + | paint in the ratio 3:1 (3 parts blue paint to 1 part white paint) |
| | + | means 3 + 1 = 4 parts in all. |
| | + | |
| | | | |
| − | <br>
| + | 3 |
| − | <br>
| + | parts blue paint to 1 part white paint = is ¾ blue paint to ¼ white |
| | + | paint. |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5e906d5b.jpg]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5518d221.jpg]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
| + | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_5518d221.jpg]]<br>
| + | Cost |
| − | <br>
| + | 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 |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]][[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_55f65a3d.gif]]<br>
| + | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | <br>
| + | The |
| − | <br>
| + | ratio of the cost of a pen to the cost of a pencil = |
| | + | [[Image:KOER%20Fractions_html_m762fb047.gif]] |
| | | | |
| | | | |
| − | <br>
| + | |
| − | <br>
| + | |
| | | | |
| | | | |
| − | === Evaluation Question ===
| + | What |
| | + | is Direct Proportion ? |
| | + | |
| | | | |
| − | # What happens when the numerator and denominator are the same, why ?
| + | Two |
| − | # What happens when the numerator is greater than the denominator why ? How can we represent this in two ways ?
| + | 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. |
| | + | |
| | | | |
| − | <br>
| + | Paint |
| − | <br>
| + | pots in a ratio of 3:1 |
| | | | |
| | | | |
| − | == Activity 3: Comparing Fractions ==
| + | [[Image:KOER%20Fractions_html_m22cda036.gif]] |
| − |
| + | |
| − | === Objective: ===
| |
| − |
| |
| − | Comparing-Fractions
| |
| | | | |
| | | | |
| − | === Procedure: ===
| |
| − |
| |
| − | Print
| |
| − | the document '''Comparing-Fractions.pdf ''' and'''
| |
| − | Comparing-Fractions2 a'''nd work
| |
| − | out the activity sheet
| |
| | | | |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| | | | |
| − |
| |
| − | '''Material/
| |
| − | Activity Sheet'''
| |
| | | | |
| | | | |
| − | [[Comparing-Fractions.pdf]]
| |
| | | | |
| − |
| |
| − | [[Comparing-Fractions2.pdf]]
| |
| | | | |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| | | | |
| | | | |
| − | === Evaluation Question ===
| |
| − |
| |
| − | # Does the child know the symbols '''>, <''' and '''='''
| |
| − | # What happens to the size of the part when the denominator is different ?
| |
| − | # Does it decrease or increase when the denominator becomes larger ?
| |
| − | # Can we compare quantities when the parts are different sizes ?
| |
| − | # What should we do to make the sizes of the parts the same ?
| |
| − |
| |
| − | == Activity 4: Equivalent Fractions ==
| |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| | | | |
| − |
| |
| − | === 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 ===
| + | 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. |
| | + | |
| | | | |
| − | # What is reducing a fraction to the simplest form ?
| + | If |
| − | # What is GCF – Greatest Common Factor ?
| + | we double the amount of blue paint we need 6 pots. |
| − | # Use the document [[simplifying-fractions.pdf]]
| |
| − | # Why are fractions called equivalent and not equal.
| |
| − |
| |
| − | == Activity 5: Fraction Addition ==
| |
| − |
| |
| − | === Objective: ===
| |
| − |
| |
| − | Understand Addition of
| |
| − | Fractions
| |
| | | | |
| | | | |
| − | === Procedure: ===
| + | If |
| − |
| + | we double the amount of white paint we need 2 pots. |
| − | <br>
| |
| − | <br>
| |
| | | | |
| | | | |
| − | Open
| + | Six |
| − | Geogebra applications
| + | paint pots in a ratio of 3:1 |
| | | | |
| | | | |
| − | Open
| |
| − | link
| |
| − | [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Addition.html]]
| |
| | | | |
| − |
| |
| − | Move
| |
| − | the sliders Numerator1 and Denominator1 to set Fraction 1
| |
| | | | |
| − |
| |
| − | Move
| |
| − | the sliders Numerator2 and Denominator2 to set Fraction 2
| |
| | | | |
| | | | |
| − | See
| + | The |
| − | the last bar to see the result of adding fraction 1 and fraction 2 | + | 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: |
| | | | |
| | | | |
| − | '''Activity
| + | Pots |
| − | Sheet'''
| + | of blue paint 3 6 9 12 |
| | | | |
| | | | |
| − | Please
| + | Pots |
| − | open
| + | of white paint 1 2 3 4 |
| − | [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Addition.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
| |
| | | | |
| | | | |
| − | Observe
| |
| − | and describe the values of the numerator and denominator and relate
| |
| − | it to the third result fraction. Discuss LCM and GCF
| |
| | | | |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| | | | |
| − |
| |
| − | === Evaluation Question ===
| |
| − |
| |
| − | == Activity 6: Fraction Subtraction ==
| |
| − |
| |
| − | === Objective: ===
| |
| − |
| |
| − | Understand Fraction
| |
| − | Subtraction
| |
| | | | |
| | | | |
| − | === Procedure: ===
| + | Two |
| − |
| + | quantities which are in direct proportion will always produce a graph |
| − | Open Geogebra
| + | where all the points can be joined to form a straight line. |
| − | applications
| |
| | | | |
| | | | |
| − | Open link
| |
| − | [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Subtraction.html]]
| |
| | | | |
| − |
| |
| − | 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
| + | '''What |
| − | the result of subtracting fraction 1 and fraction 2
| + | is Inverse Proportion ?''' |
| | | | |
| | | | |
| − | <br>
| + | Two |
| − | <br>
| + | 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 |
| − | '''Material/Activity
| + | workers increase. Similarly when the speed increases the time to |
| − | Sheet'''
| + | cover a distance decreases. Zaheeda can go to school in 4 different |
| | + | ways. She can walk, run, cycle or go by bus. |
| | | | |
| | | | |
| − | Please open link
| + | Study |
| − | [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_Subtraction.html]]
| + | the table below, observe that as the speed increases time taken to |
| − | in Firefox and follow the process
| + | cover the distance decreases |
| | | | |
| | | | |
| − | 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
| + | {| border="1" |
| − | the values of the numerator and denominator and relate it to the
| + | |- |
| − | third result fraction. Discuss LCM and GCF
| + | | |
| | | | |
| − |
| |
| − | === Evaluation Question ===
| |
| − |
| |
| − | == Activity 7: Linking to Decimals ==
| |
| − |
| |
| − | === Objective: ===
| |
| − |
| |
| − | Fractions
| |
| − | representation of decimal numbers
| |
| | | | |
| | | | |
| − | === Procedure: ===
| + | | |
| − |
| + | Walk |
| − | <br>
| |
| − | <br>
| |
| | | | |
| | | | |
| − | Make copies of the
| + | | |
| − | worksheets decimal-tenths-squares.pdf and
| + | Run |
| − | decimal-hundreths-tenths.pdf
| |
| | | | |
| | | | |
| − | <br>
| + | | |
| − | <br>
| + | Cycle |
| | | | |
| | | | |
| − | '''Activity Sheet'''
| + | | |
| | + | Bus |
| | | | |
| | | | |
| − | decimal-tenths-squares.pdf
| + | |- |
| | + | | |
| | + | Speed |
| | + | Km/Hr |
| | | | |
| | | | |
| − | decimal-hundreths-tenths.pdf
| + | | |
| | + | 3 |
| | | | |
| | | | |
| − | <br>
| + | | |
| − | <br>
| + | 6 |
| | + | (walk speed *2) |
| | | | |
| | | | |
| − | === Evaluation Question ===
| + | | |
| − |
| + | 9 |
| − | # 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
| + | (walk speed *3) |
| − | # 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: ===
| + | | |
| − |
| + | 45 |
| − | Use
| + | (walk speed *15) |
| − | the NCERT Class 6 mathematics textbook chapter 12 and work out
| |
| − | Exercise 12.1
| |
| | | | |
| | | | |
| − | <br>
| + | |- |
| − | <br>
| + | | |
| | + | Time |
| | + | Taken (minutes) |
| | | | |
| | | | |
| − | '''Activity Sheet'''
| + | | |
| | + | 30 |
| | | | |
| | | | |
| − | NCERT [[Class6 Chapter 12 RatioProportion.pdf]] Exercise 12.1
| + | | |
| | + | 15 |
| | + | (walk Time * ½) |
| | | | |
| | | | |
| − | <br>
| + | | |
| − | <br>
| + | 10 |
| | + | (walk Time * 1/3) |
| | | | |
| | | | |
| − | === Evaluation Question ===
| + | | |
| | + | 2 |
| | + | (walk Time * 1/15) |
| | + | |
| | | | |
| − | # Explain what the numerator means in the word problem
| + | |} |
| − | # Explain what the denominator means
| + | |
| − | # Finally describe the whole fraction in words in terms of ratio and proportion.
| + | |
| | + | |
| | | | |
| − | <br>
| + | As |
| − | <br>
| + | 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. |
| | | | |
| | | | |
| − | == Activity 9: Fraction Multiplication ==
| + | |
| | + | |
| | + | |
| | | | |
| − | === Objective: ===
| + | '''Moving from Additive Thinking to |
| | + | Multiplicative Thinking ''' |
| | + | |
| | | | |
| − | Understand
| |
| − | Multiplication of fractions
| |
| | | | |
| − |
| |
| − | === Procedure: ===
| |
| − |
| |
| − | Open Geogebra
| |
| − | applications
| |
| | | | |
| | | | |
| − | Open link
| + | Avinash |
| − | [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/Fraction_MultiplyArea.html]]
| + | 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. |
| | | | |
| | | | |
| − | Move the sliders
| + | === Proportional Reasoning === |
| − | Numerator1 and Denominator1 to set Fraction 1
| |
| − | | |
| | | | |
| − | Move the sliders
| + | '''''Proportional |
| − | Numerator2 and Denominator2 to set Fraction 2
| + | 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 |
| − | On the right hand side
| + | quantities. It involves quantitative thinking as well as qualitative |
| − | see the result of multiplying fraction 1 and fraction 2
| + | 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. |
| | | | |
| | | | |
| − | '''Material/Activity
| |
| − | Sheet'''
| |
| | | | |
| − |
| |
| − | Please open
| |
| − | [[http://rmsa.karnatakaeducation.org.in/sites/rmsa.karnatakaeducation.org.in/files/documents/F]][[raction_MultiplyArea.html]]
| |
| − | in Firefox and follow the process
| |
| | | | |
| | | | |
| − | When you move the
| + | The |
| − | sliders ask children to
| + | 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. |
| | | | |
| | | | |
| − | 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
| + | == Rational & Irrational Numbers == |
| − | dashed lines within one square unit.
| |
| − | | |
| | | | |
| − | The thick red lines
| + | After |
| − | represent the fraction 1 and 2 and also the side of the quadrilateral
| + | 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. |
| | | | |
| | | | |
| − | The product represents | + | But, |
| − | the area of the the quadrilateral | + | 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. |
| | | | |
| | | | |
| − | === Evaluation Question ===
| + | How |
| − |
| + | can we be sure that an irrational number cannot be expressed as a |
| − | When
| + | fraction? This can be proven algebraic manipulation. Once these |
| − | two fractions are multiplied is the product larger or smaller that
| + | "irrational numbers" came to be identified, the numbers |
| − | the multiplicands – why ?
| + | that can be expressed of the form p/q where defined as rational |
| | + | numbers. |
| | | | |
| | | | |
| − | <br>
| + | There |
| − | <br>
| + | 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. |
| | | | |
| | | | |
| − | == Activity 10: Division of fractions == | + | == Activities == |
| | | | |
| − | === Objective: === | + | === Activity 1 Fractions representation of decimal numbers === |
| | | | |
| − | Understand Diviion by
| + | '''''Learning |
| − | Fractions
| + | Objectives ''''' |
| | | | |
| | | | |
| − | === Procedure: ===
| + | Fractions representation of decimal |
| | + | numbers |
| | + | |
| | | | |
| − | <br>
| + | '''''Materials and |
| − | <br>
| + | resources required''''' |
| | | | |
| | | | |
| − | Print out the
| + | [[http://www.superteacherworksheets.com/decimals/decimal-tenths-squares_TWDWQ.pdf]] |
| − | [[fractionsStrips.pdf]] | |
| | | | |
| | | | |
| − | Colour each of the unit
| + | [[http://www.superteacherworksheets.com/decimals/decimal-hundredths-tenths.pdf]] |
| − | fractions in different colours. Keep the whole unit (1) white.
| |
| | | | |
| | | | |
| − | Cut out each unit
| + | |
| − | fraction piece.
| + | |
| | | | |
| | | | |
| − | Give examples
| + | '''''Pre-requisites/ |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m282c9b3f.gif]]
| + | Instructions Method ''''' |
| | | | |
| | | | |
| − | For example if we try
| + | Make copies of the worksheets |
| − | the first one, | + | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_21ce4d27.gif]]
| + | |
| − | See how many
| + | [[http://www.superteacherworksheets.com/decimals/decimal-tenths-squares_TWDWQ.pdf]] |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m31bd6afb.gif]]strips | |
| − | will fit exactly onto whole unit strip.
| |
| | | | |
| | | | |
| − | <br>
| + | [[http://www.superteacherworksheets.com/decimals/decimal-hundredths-tenths.pdf]] |
| − | <br>
| |
| | | | |
| | | | |
| − | '''Material /Activity
| |
| − | Sheet'''
| |
| | | | |
| − |
| + | |
| − | [[fractionsStrips.pdf]]
| |
| − | , Crayons, Scissors, glue
| |
| | | | |
| | | | |
| − | <br>
| + | '''''Evaluation''''' |
| − | <br>
| |
| | | | |
| | | | |
| − | === Evaluation Question ===
| + | # Draw a number line and name the fraction and decimal numbers on the number line. Take a print of the document [[http://www.superteacherworksheets.com/decimals/decimal-number-lines-tenths.pdf]] ''' . '''Ask students to place any fraction and decimal numbers between between 0 and 10 on the number line |
| | + | # Write 0.45, 0.68, 0.05 in fraction form and represent as a fraction 100 square. |
| | | | |
| − | When
| + | |
| − | we divide by a fraction is the result larger or smaller why ?
| + | |
| | | | |
| | | | |
| − | == Activity 11: Percentages ==
| + | |
| | + | |
| | + | |
| | | | |
| − | === Objective: === | + | === Activity 2 Fraction representation and percentages === |
| | | | |
| − | Understand fraction
| + | '''''Learning |
| − | representation and percentages
| + | Objectives ''''' |
| | | | |
| | | | |
| − | <br>
| + | Understand fraction representation and percentages |
| − | <br>
| |
| | | | |
| | | | |
| − | === Procedure: ===
| + | |
| − |
| + | |
| − | <br>
| |
| − | <br>
| |
| | | | |
| | | | |
| − | Please print copies of the 2 activity sheets [[percentage-basics-1.pdf]]
| + | '''''Materials and |
| − | and [[percentage-basics-2.pdf]]
| + | resources required''''' |
| − | and discuss the various percentage quantities with the various | |
| − | shapes.
| |
| | | | |
| | | | |
| − | Then print a copy each of [[spider-percentages.pdf]]
| + | [[http://www.superteacherworksheets.com/percents/converting-fractions-decimals-percents_EASIE.pdf]] |
| − | and make the children do this activity
| |
| | | | |
| | | | |
| − | <br>
| + | [[http://www.superteacherworksheets.com/percents/basic-percentages1.pdf]] |
| − | <br>
| |
| | | | |
| | | | |
| − | '''Activity Sheet''' | + | '''''Pre-requisites/ |
| | + | Instructions Method ''''' |
| | | | |
| | | | |
| − | Print
| + | Please print |
| − | out [[spider-percentages.pdf]]
| + | copies of the 2 activity sheets |
| | + | [[http://www.superteacherworksheets.com/percents/converting-fractions-decimals-percents_EASIE.pdf]] and |
| | + | [[http://www.superteacherworksheets.com/percents/basic-percentages1.pdf]] and discuss the various percentage quantities with |
| | + | the various shapes. |
| | | | |
| | | | |
| − | <br>
| |
| − | <br>
| |
| | | | |
| − |
| |
| − | === Evaluation Question ===
| |
| − |
| |
| − | What
| |
| − | value is the denominator when we represent percentage as fraction ?
| |
| | | | |
| | | | |
| − | What
| + | Then print a copy |
| − | does the numerator represent ?
| + | each of [[spider-percentages.pdf]] |
| | + | and make the children do this activity |
| | | | |
| | | | |
| − | What
| |
| − | does the whole fraction represent ?
| |
| | | | |
| − |
| |
| − | What
| |
| − | other way can we represent a fraction whoose denominator is 100.
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | <br>
| + | '''''Evaluation''''' |
| − | <br>
| |
| | | | |
| | | | |
| − | == Activity 12: Inverse Proportion ==
| + | # 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 whose denominator is 100. |
| | | | |
| − | === Objective: ===
| |
| − |
| |
| − | Understand fraction
| |
| − | representation and Inverse Proportion.
| |
| | | | |
| − |
| |
| − | === Procedure: ===
| |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| | | | |
| | | | |
| − | Use
| |
| − | the NCERT Class 8 mathematics textbook chapter 13 and work out
| |
| − | Exercise 13.1
| |
| | | | |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| | | | |
| − |
| |
| − | '''Activity Sheet'''
| |
| | | | |
| | | | |
| − | [[NCERT Class 8 Chapter 13 InverseProportion.pdf]] Exercise 13.1
| + | === Activity 3 === |
| − | | |
| | | | |
| − | <br>
| + | '''''Learning |
| − | <br>
| + | Objectives ''''' |
| | | | |
| | | | |
| − | '''Evaluation Question'''
| + | Understand fraction representation and rational and irrational |
| | + | numbers |
| | | | |
| | | | |
| − | 1. Given a set of
| + | '''''Materials and |
| − | fractions are they directly proportional or inversely proportional ?
| + | resources required''''' |
| | | | |
| | | | |
| − | 2.
| + | Thread |
| − | In the word problem, identify the numerator, identify the denominator
| + | of a certain length. |
| − | and explain what the fraction means in terms of Inverse proportions
| |
| − | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| − | == Activity 13: Rational and Irrational Numbers ==
| + | '''''Pre-requisites/ |
| − |
| + | Instructions Method ''''' |
| − | === Objective: ===
| |
| − |
| |
| − | Understand fraction
| |
| − | representation and rational and irrational numbers
| |
| − | | |
| − |
| |
| − | === Procedure: ===
| |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| | | | |
| | | | |
| Line 2,680: |
Line 2,114: |
| | | | |
| | | | |
| − | <br>
| + | |
| | | | |
| | | | |
| Line 2,688: |
Line 2,122: |
| | | | |
| | | | |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_m1a6bd0d0.gif]]<br> | + | [[Image:KOER%20Fractions_html_m1a6bd0d0.gif]] |
| | | | |
| | | | |
| Line 2,696: |
Line 2,130: |
| | | | |
| | | | |
| − | <br>
| |
| − |
| |
| − |
| |
| − | <br>
| |
| − |
| |
| − |
| |
| − | <br>
| |
| | | | |
| − |
| |
| − | <br>
| |
| | | | |
| | | | |
| | Identify | | Identify |
| − | the various places where pi, "e" and the golden ratio | + | the various places where pi, "e" and the golden ratio |
| | occur | | occur |
| | | | |
| | | | |
| − | '''Material'''
| |
| | | | |
| − |
| |
| − | Thread
| |
| − | of a certain length.
| |
| | | | |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| | | | |
| | | | |
| − | === 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 ?
| |
| | | | |
| | | | |
| − | <br>
| + | '''''Evaluation''''' |
| − | <br>
| |
| | | | |
| | | | |
| − | = Interesting Facts =
| + | # 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 ? |
| | | | |
| − | 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.
| |
| | | | |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| − |
| |
| − |
| |
| − | 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
| |
| | | | |
| − |
| |
| − | [[Image:Fractions-Resource_Material_Subject_Teacher_Forum_September_2011_html_636d55c5.gif]]<br>
| |
| − | <br>
| |
| | | | |
| | | | |
| − | , appears, which
| + | == Evaluation == |
| − | 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.
| |
| − | | |
| | | | |
| − | <br>
| + | == Self-Evaluation == |
| − | <br>
| |
| − | | |
| | | | |
| − | In early Egyptian and
| + | == Further Exploration == |
| − | 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.
| |
| − | | |
| | | | |
| − | <br>
| + | # Percentage and Fractions, [[http://www.bbc.co.uk/skillswise/numbers/wholenumbers/]] |
| − | <br>
| + | # A mathematical curve Koch snowflake, [[http://en.wikipedia.org/wiki/Koch_snowflake]] |
| − | | + | # Bringing it Down to Earth: A Fractal Approach, [[http://www.realmagick.com/5552/bringing-it-down-to-earth-a-fractal-approach/]] |
| | | | |
| − | The Romans used a system
| + | = See Also = |
| − | 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 :
| |
| − | | |
| | | | |
| − | <br>
| + | # Video on teaching fractions using the equal share method made by Eklavya an NGO based in Madhya Pradesh, India [[http://vimeo.com/22238434]] |
| − | <br>
| + | # Mathematics resources from Homi Baba Centre for Science Education , [[http://mathedu.hbcse.tifr.res.in/]] |
| − | | + | # [[http://www.geogebra.com]] Understand how to use Geogebra a mathematical computer aided tool |
| − |
| |
| − | 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
| |
| − | | |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| − | | |
| − |
| |
| − | 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.
| |
| − | | |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| − | | |
| − |
| |
| − | 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.
| |
| − | | |
| − |
| |
| − | <br>
| |
| − | <br>
| |
| − | | |
| − |
| |
| − | 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
| + | = Teachers Corner = |
| − | | |
| | | | |
| − | fractionstrips.pdf
| + | = Books = |
| − | | |
| | | | |
| − | NCERT Class6 Chapter 12
| + | # [[http://books.google.com/books?id=Y5Skj-EA2_AC&pg=PA251&lpg=PA251&dq=streefland+fractions&source=bl&ots=aabKaciwrA&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 |
| − | RatioProportion.pdf
| |
| − | | |
| | | | |
| − | NCERT Class8 Chapter 13
| + | = References = |
| − | DirectInverseProportion.pdf
| |
| − | | |
| | | | |
| − | percentage-basics-1.pdf
| |
| | | | |
| − |
| |
| − | percentage-basics-2.pdf
| |
| − |
| |
| − |
| |
| − | simplifying-fractions.pdf
| |
| | | | |
| | | | |
| − | spider-percentages.pdf
| + | # Introducing Fractions Using Share and Measure Interpretations: A Report from Classroom Trials . Jayasree Subramanian, Eklavya, Hoshangabad, India and Brijesh Verma,Muskan, Bhopal, India |
| | + | # Streefland, L. (1993). “Fractions: A Realistic Approach.” In Carpenter, T.P., Fennema, E., Romberg, T.A. (Eds.), Rational Numbers: an Integration of Research. (289-325). New Jersey: Lawrence, Erlbaum Associates, Publishers. |
