Friday, 2 May 2014

Teaching Fractions


Fractions ‘The Problem’.                                       
“Children often have trouble comprehending the values the fractional numerals represent as they are not given materials or encouraged to visualise the materials. Many procedures for computing fractions appear complex, are often taught abstractly and before children understand what a fraction is,  and are inconsistent with the rules used for operating on whole numbers.  (When you multiply a quantity by a proper fraction, you end up with a number which is less than the original number.  Conversely, dividing a number by a fraction generates a result which is more than the starting number.) This does not fit with children’s exiting knowledge about multiplication and division.                    When fractions are explored as a way of expressing the relationships between various objects the concept, rather than the language, is the focus of a child’s experience.  In an experiential context, the notion of part-whole relations is not difficult for even quite young children to grasp.  On the other hand, couching fraction concepts in symbols causes a great deal of confusion.  In order to learn about fractions, children need extensive experiences exploring them as relationships between objects.  They must also develop their own ways of representing fractions before symbols and operations are introduced.  Fractions are only difficult to understand when they are seen as symbols to be manipulated, rather than ideas to be conceptualised.                                                   
           Children gain an understanding of fractions by using materials to explore part-whole relationships.  Giving children access to their own sets of materials is important.  It enables  them to form and check ideas that are relevant to them.  In the beginning it is advisable to use materials that are easy to manipulate, wooden blocks, foam shapes... As the mathematical ideas become established, and as motor skills develop, children will be able to work with more compact shapes (Card and paper....)”             
Adapted from Fractions M3 Mathematics

Ratio and Fractions                                           
There are two ways to describe a set of 5 counters where 2 are blue and 3 are yellow. 
    By giving the two parts (the ratio)  e.g. two blue to three yellow.   Comparing the two parts to each other is the most common way for learners to see such situations.           
   By giving one part as a fraction of the total e.g. two blue in every five.  The most commonly used way to describe them in mathematics is by comparing each part to the whole (the fraction)

Students need to understand fractions as regions and as sets, and later to see how they relate to decimals and percentages, and ratios

Knowledge students need                                   
• The English names for fractions before the symbols; thus half, quarter etc
• Symbols for unit fractions halves and fourths                       
• Symbols for thirds and fifths                                   
• Doubles and corresponding halves 6 + 6  1/2 of 12 is 6                   
• Symbols for tenths and fractions greater than 1
• Orders fractions with same denominators  1/4 and 3/4                   
• Orders fractions, halves, quarters,thirds,fifths,tenths                   
• Forward and backward word sequence for halves, quarters,thirds,fifths, and tenths                              • Can identify and fraction including tenths, hundredths,thousandths and those greater than one            • Say the number one thousandth, one hundredth, one tenth, one ,ten etc before and after                      • Equivalent proportions for halves, thirds,, with numbers to one hundred and one thousand  e.g. 1 in 4 is equivalent to 25 in 100 250 in 1000

Strategy development                                       
• Find a fraction of a number by sharing out the objects equally e.g. 1/4 of 12 is       
• Find a fraction of a number by skip counting or adding the same number over and over again.  4 + 4 + 4  = 12 so 1/3 of 12 = 4                   
• Use multiplication and addition facts to find answers to fraction problems for example 3/4 of 20 is 1/2 of 20 and 1/4 of 20: 1/6 of 18 is found by finding 1/3 of 18 and halving   
• Use a variety of strategies for finding 1/3 of 27:   3/8 of 28 is the same as 1/2 of 3/4 of 28                   • Make connections between decimals, fractions and percentages

• Circular fractions kit/ Strip Fraction Kit/region fraction kit  us these to connect physical models with symbols.  Include fractions above one               
• Investigate the effect of increasing the bottom number of a fraction  while keeping the top constant                                           
• Investigate with materials equivalence of fractions  2/4 = 1/2               
• Cut food and other objects into fractions share food (cookies, sweets) you have 8 cookies and 2 children and need to halve, How would you halve the cookies?  extend to get children imaging what would have to be done                                   
• Use playdough, clay for exploring fractions                            
• Share a piece of string between 4 what fraction would each get               
• Unifix/multilink blocks for showing fractions of a number                   
• Use pattern block shapes to show fraction relationships of a hexagon       
• Fractions of a 100’s board                                   
• Turns as fractions                                       
• Hands of clock as fractions                                   
• Double Number Lines for showing 1/2 of 12, 1/4 of 12                   
• Ratio Tables:  I have 12 lollies you can have 1/4 of them. How many is that?   
                     Fraction:        1        1/2        1/4                                   
                                            12        6        3                   
• Sharing counters and using ratio tables                           
• Sets of counters  2 out of 3 repeated 1 set is 2/3   2 sets is 4/6  3 sets is 6/9 etc               
• Estimation cards for estimating fractions, and equivalent decimals...           
• Show each domino as a fraction                                                                                       
Possible Resources                                       
• Fraction Fun through Cooperative Learning, Laurie Robertson                
• Family Math Book (Fraction Kits and activities)                       
• OHP Fraction regions                                         
• Fractions M3 Mathematics Helen Pengelly                           
• The BEANZ Cookbook Mathemaction                               
• Multilink Fraction Activities Bob Stone et al                           
• Ideas from the Arithmetic Teacher NCTM                           
• The Mathworks Greenes et al                                   
• Commercial Fraction Sets                                   
• Fraction games from ORDA or similar                               
• Fraction Dominoes                                        
•   Interactive fraction site                       
• Lesson plans and worksheets/games etc

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