Fractions

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Fractions

• By Robin Hare

Material can be found at http//www.kegga.com (und
er the Bits and Bobs menu) Reference Mathemati
cs for dyslexics A teaching handbook Chinn
Ashcroft
2
Fractions - Why this topic?

• From experience in the classroom the observed
  use of fractions has been poor and often confused
• Pupils seem scared of fractions
• A strong preference is shown for decimals
• even though they get equally confused with place
  value
• Strong links exist with other Mathematical
  topics
• Percentages and decimals
• Probability
• SSM
• Number e.g HCM and LCM
• Can be (should be?) addressed separately from
  decimals and percentages because rules for
  equivalence, addition/subtraction and
  multiplication/division are completely different

3
Fractions in the National Curriculum
(The boring bit)

• KS2
• Understand unit fractions (e.g. 1/3 and 1/8)
• then fractions of several parts of a whole
  (e.g. 2/3 or 5/8)
• Find them on number lines and find fractions of
  shapes
• Understand equivalent fractions
• Simplify fractions by canceling common factors
• Compare and order by converting them to a common
  denominator
• KS3
• Use fractional notation and understand
  equivalent fractions
• Simplify fractions by canceling common
  denominator
• Order fractions by rewriting with a common
  denominator
• Calculate given fraction of a quantity (answer
  as a fraction)
• Express a number as a fraction of another number
• Add and subtract fractions (by using common
  denominator)
• Perform short division to convert a fraction to
  a decimal
• Use of unit fractions and multiplicative
  inverses
• Multiply and divide a fraction by an integer,
  unit fraction or given fraction

4
Fractions - A way of teaching them

• A consistent image should be kept throughout all
  strands of the topic
• The strands/main topic areas are
• Definition / understanding of what a fraction is
• Equivalence
• Simplifying
• Comparing
• Adding and subtracting
• Multiplying and dividing
• These strands would not be taught together (i.e.
  linearly), but the same image should remain for
  continuity of learning
• Lots of kinaesthetic learning can be
  incorporated
• folding, cutting, sticking, colouring
• Approach concentrates on strong link between the
  folding and the writing down

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Fractions - Basics

• Example lessons aimed at Year 7
• Can safely assume no great expertise from KS2
• Any expertise will be gratefully received and
  may require some differentiation in the classroom
• Start at the beginning (i.e. the bits done in
  KS2)
• So we can use a consistent image
• To bring a more formal / grown up approach to
  learning
• Start to with the whole class together
• Terminology
• Introduce where appropriate, not up-front
• Can slide from pupils own words to correct
  terminology
• Book recommends do not use words numerator and
  denominator early on? since pupils never
  remember what they mean anyway!
  Comments?

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Fractions - First lesson(s)

• What is a fraction?
• Objective definition, terminology and
  visualisation
• Starter A splurge diagram of words they think
  are to do with fractions
• this will be a good way to lead into the lesson
• very good for assessment of their current
  knowledge from KS2
• Main activity
• Arm yourself with lots of squares of paper
• Explain a whole thing segments and parts
  with images
• Unit fractions e.g. 1/5 one part of a whole
  thing made of 5 segments
• On to e.g. 3/5 three parts of a whole thing
  made of 5 segments
• Show e.g. 5/5 five parts of 5 segments a
  whole thing 1
• same for all completely shaded areas
• Activity worksheet 1
• Plenary
• More than a whole thing

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Draw this
Paper divided into segments (like an orange) Talk
about parts of a whole thing
Unit fractions
then
and
5 /5 a whole 1
Plenary
A whole 3 /5
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Fractions - Second lesson(s)

• Objective equivalence and simplifying
• whether this is too much for one lesson will
  depend on the group
• Starter
• Three intersecting circles of multiples of 2, 3,
  5
• Class put numbers in right place
• Main activity
• Equivalence
• Give out or get class to make 3/4 fractions
  (write down what they have)
• Fold it in half (write down what fraction they
  now have)
• Write down 3/4 3 x 2 over 4 x 2 6/8 more
  examples if required
• Go through above example Worksheet 2a
• Simplifying fractions
• The inverse of the operation above
• Go through example 4/10 -gt 2/5 Worksheet 2b
• Plenary
• Links percentages by dividing into 100 squares
  of 10 x 10

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Starter - put in numbers 1 to (class size)
Multiples of 2
Multiples of 3
Multiples of 3 and 5
NOT A MULTIPLE OF ANY
Multiples of 5
Equivalence
Simplification
3 /4
6 / 8
4 / 10 ---------------gt 2 / 5
Write as 3 / 4 (3 2) / (4 2) 6 / 8
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Fractions - Third lesson(s)

• Objective Making segment sizes the same to
  compare fractions
• Starter
• Pentominos link with shaded area concept
• Main activity
• Necessary step towards adding and subtracting
• Method follows philosophy of approach from
  previous lessons
• Look at segments
• folding it has had (for both)
• folding it needs so they have had same folding
• relates to prime factors really doing the LCM
• Comparing follows on naturally from this
  activity
• Worksheet 3
• Plenary
• Converting mixed-fractions to top heavy
  fractions
• Converting top-heavy fractions to mixed
  fractions

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Making segment sizes the same
Thirding
3/4
7/8 3/4
9/12
Halving
Halving
2/3
Halving
(4/6)
8/12
Folding it has had Folding it
needs 7/8 Halving, Halving, Halving - 3/4
Halving, Halving Halving
Folding it has had Folding
it needs 3/4 Halving, Halving
Thirding 2/3 Thirding
Halving, Halving
3/4 3 x 3 / 4 x 4 9/12 2/3
2x2/3x2 4/6 4x2/6x2 8/12
7/8
7/8 3/4 3 x 2 / 4 x 2
6/8
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Plenary
Paper
Written/spoken 2 1/4 2 x 1 1/4 2 x 4
quarters 1 quarter 8 quarters 1
quarter 9 quarters 9/4
Top heavy to mixed is the same images reversed
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Fractions - Fourth lesson(s)

• Objective Adding and subtracting fractions
• Starter
• Revision of making segment sizes the same and
  comparing
• use a Which is bigger? type game.
• Main activity
• Start it simple fractions with same segments
  (denominator)
• Adding fractions with different segment sizes
  e.g. 1/2 2/5
• other examples 7/8 3/4 and 3/4 2/3
• uses familiar working from last lesson
• both involve making top heavy fractions into
  mixed in the final stage
• Adding more than two fractions
• Adding mixed fractions
• No reason why subtraction cannot be done
  simultaneously
• Can be supported by kinaesthetic folding,
  cutting and sticking exercises
• Worksheet 4
• Plenary
• Subtracting mixed fractions where where bigger
  fraction part is subtracted from smaller fraction
  part

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Adding (same segments)
Adding (different segments)
Written Spoken
halfing
1/5 one-fifth 3/5 plus
three-fifths 4/5 four-fifths
fifthing


Formally
2/5 2x2/5x2 4/10 1/2 1x5/2x5
5/10 4/10 5/10 9/10
Rule number of segments always stays the same
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Fractions - Differences between /- and x
This leads to a lot of confusion between the
different operations
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Fractions - Fifth lesson(s)

• Want to cover Multiplying fractions
• Starter
• May be some time later something that revises
  below would be useful
• converting mixed-fractions to top heavy
  fractions
• converting top-heavy fractions to mixed
  fractions
• Main activity
• Preliminaries
• Language of means multiply 3/4 of 8 3/4 x
  8
• Fraction contains hidden divide sign 3/4 3
  4
• Estimation 3/4 lt1 so answer will be smaller
  9/5 gt 1 answer bigger
• Multiply a fraction by a whole number
• Multiply 2 regular fractions together
• Multiplication with mixed fractions make
  improper first
• Worksheet 5
• Plenary
• Multiplying 3 or more fractions together

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Multiply by integer
Fraction times Fraction
4 x 2/5
e.g. 1/3 of 1/2
1/6
x
Overlay the right fraction with the left fraction
8/5
1 3/5
e.g. 3/4 of 2/3
6/12 1/2
x
With hidden sign 4 x 2/5 4 x 2 5 8/5
2/5 of 4 2/5 x 4 5 x 2/5 (commutative
relationship) or 2 5 x 4 (use BODMAS)
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Fraction times Fraction Mixed Fractions
e.g. 1/2 of 1 1/3 1/2 of 4/3
4/6
2/3
x

x
WITH MIXED FRACTIONS A BIT OF STRETCHING IS
NEEDED
Overlay the right fraction with the left fraction
Definitely 4 parts, but segments are 6 not
8 because there are 6 (2 x 3) segments in the
whole
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Fractions - Sixth lesson(s)

• Want to cover Dividing fractions
• Starter
• Something fun
• Main activity
• Can use same image
• Division by making segments the same size
• Division with mixed fractions
• Worksheet 6
• Plenary
• Dividing fractions by inverse multiplication

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Dividing
Paper/image Written Spoken
Three-quarters divided by one- eighth
six-eighths divided by one-eighth (divide
into eight bits) six there are 6 eighths in
6/8 (3/4)
3/4 1/8
Make segment size same as that being divided into
6/8 1/8
Break up eighths
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Dividing by 3/8 is there are 2 lots of 3
(eighths) in 3/4