Transcript File - Hodnet Primary School

Maths ‘Fractions’ Workshop
Can you help me with my maths homework
please?
The question that all parents
dread…
In this workshop:
 Aims for the new curriculum and our school
 Progression of fractions
 Teaching methods and resources
 Ways in which you can help your child with their day to
day maths
Aims of the New National
Curriculum
The national curriculum for mathematics aims to ensure that all
pupils:
• become fluent in the fundamentals of mathematics, including
through varied and frequent practice with increasingly
complex problems over time, so that pupils develop conceptual
understanding and the ability to recall and apply knowledge
rapidly and accurately.
• reason mathematically by following a line of enquiry,
conjecturing relationships and generalisations, and developing
an argument, justification or proof using mathematical
language
• can solve problems by applying their mathematics to a variety
of routine and non-routine problems with increasing
sophistication, including breaking down problems into a series
of simpler steps and persevering in seeking solutions.
In other words…
• Fluent recall of mental maths facts: e.g. times
tables, number bonds etc.
• To reason mathematically: children need to be able
to explain the mathematical concepts with number
sense; they must explain how they got the answer
and why they are correct.
• Problem solving: applying their skills to real-life
contexts.
Good practice in mathematics
• All children need to learn maths in a real life context.
As well as knowing 7x7=49. Children need to be able to
do the following:
There are 7 fields, each field has 7 sheep in them. How
many sheep are there in total?
• Children need to be able to explain how they have
calculated or solved a problem.
• In the new curriculum, written calculations are taught
at an earlier age. The mental methods are essential for
supporting pupils understanding of these written
calculations.
Key Differences in the
new Maths Curriculum:
 Simple fractions (1/4 and 1/2) are taught from Year
1, and by the end of primary school, children should be
able to convert decimal fractions to simple fractions and
multiply and divide fractions.
 By the age of nine, children are expected to know times
tables up to 12×12 (used to be 10×10 by the end of
primary school).
 End of KS2 mental maths test has been replaced
with an arithmetic test.
Expectations
Expectations in the new curriculum for each year
group are now much higher.
I have created a progression map with the
statutory requirements for fractions for each year
group (see hand-out).
FDP Progression in School
Halving of sets of objects begins as early as
EYFS and Year 1.
It is vital that children know
halves and quarters must be
equal in size.
In Year 2 children begin to use ½, ¼, ¾ and 1/3;
find simple fractions of amounts (½ of £12 =
£6) and know equivalence of 1/2 = 2/4.
http://resources.hwb.wales.gov.uk/VTC/ngfl/ngfl-flash/fractions/fractions.html
http://www.topmarks.co.uk/Flash.aspx?f=WhatFractionv3
Progression in KS2
In Year 3 we begin to use the terms denominator and
numerator in writing proper fractions.
We recognise, find and write fractions of objects and
begin to compare and order them.
We begin to add and subtract them and solve problems.
In year 4 we read and write fractions, ordering them
and recognising equivalent fractions. Children also
find fractions of amounts (3/5 of 25Kg = 15Kg).
They continue to add and subtract fractions and
understand more equivalences, including tenths and
hundredths.
Progression
By Year 5 children simplify fractions. They
relate fractions to decimals and percentages,
and begin to multiply them.
They begin to convert mixed numbers to
improper and vice versa.
In Year 6 we ask children to find common
factors in numerators and denominators.
They multiply and divide fractions and associate
them with division. Children convert between
F, D and P and recall and use equivalences.
So what is a fraction?
What do you think?
Fractions - what are they?
1
2
Part of a whole.
When an object or number
is divided into a number of
equal parts, then each part
is called a fraction.
A fraction is a part of a whole
Slice a pizza, and you will have fractions:

1/
2
1/
4
3/
8
(One-Half) (One-Quarter) (Three-Eighths)
The top number tells how many slices you have
The bottom number tells how many slices the
pizza was cut into.
Parts of a Fraction
2
5
Numerator
How many parts you have.
Denominator
The number of parts the whole is divided into (total).
The denominator is downstairs!
Language of fractions
• Language of fractions is used all around children,
“here’s my half.” “I’ll cut this cake into equal parts for
the four of us.” etc
• We would encourage correct use of terms from early
on - not “My half is bigger than yours!”
Improper
Fraction
Equivalent fractions
• Equivalent fractions are fractions that
are equal in size but have different
denominators or numerators.
1
2
=
2
4
There are many more!
We use a fraction wall as well
as lots of images and models
to aid understanding.

Some fractions may look different, but are
really the same, for example:
 4 /8
=
(Four-Eighths)
2/
4
(Two-Quarters)
=
1/
2
(One-Half)
It is usually best to show an answer using the
simplest fraction ( 1/2 in this case ). That is
called Simplifying the Fraction.
Ordering fractions
To order fractions of the same denominator
we need to look at the numerator.
6
1 4 3 7
8
8 8 8 8
Smallest to largest
1 3 4 6
7
8
8 8 8 8
Ordering Fractions
When fractions are to be ordered or added
than they need to have the same denominator
(which links to equivalent fractions) e.g.
3 1 1 1 7 becomes 6 1 4
4 8 2 4 8
8 8 8
2
8
7
8
Changing fractions to a common denominator
1) Change all the fractions to the same denominator. (Find a common multiple)
2) In this case we will use 12 because 2, 4, 6, and 3 all go into it.
3) Whatever you do to the top (numerator), you do to the bottom (denominator).
1x6
2x6
1x3
4 x3
5 x2
6 x2
2 x4
3 x4
4) Your fractions will now be:
6
12
3
12
10
12
8
12
5) Now put your fractions in order (smallest to biggest.)
3
6
8
10
12
12
12
12
6) Change them back, keeping them in order.
1
1
2
4
2
3
5
6
Ordering fractions
When ordering fractions with different
denominators, it’s best to convert all
fractions to the lowest common
denominator.
3 1 1 1 7 becomes 6 1 4 2 7
4 8 2 4 8
8 8 8 8 8
Smallest to largest
1 2 4 6 7 and then 1 1 1 3 7
8 8 8 8 8
8 4 2 4 8


Method 1
Try dividing both the top and bottom of the
fraction until you can't go any further (try
dividing by 2,3,5,7,... etc).
Example: Simplify the fraction 24/108 :
÷2
÷2
÷3
24
12
6
2
108
54
27
9


Method 2
Divide both the top and bottom of the
fraction by the Greatest Common Factor, (you
have to work it out first!).


Example: Simplify the fraction 8/12 :
1) The largest number that goes exactly into
both 8 and 12 is 4, so the Greatest Common
Factor is 4.
2) Divide both the top and bottom by 4:
÷4
8
12
2
3
Finding fractions of an amount or quantities.
As a fraction is a division of a whole then
we use division to find a fraction of
amounts.
Finding a half means dividing by 2 to make
two equal sized groups.
1 ÷4
1 ÷ 3
1 ÷ 5
4
3
5
When it is a unit fraction (1 as the
numerator) just divide by the denominator.
Finding fractions of an amount or quantities.
To find fractions such as 2/3 then we need to
find a third and then multiply this by 2.
2 of 12 = 1 (12 ÷ 3) = 4 then x 2 = 8
3
3
Divide by denominator and then multiply this
by the numerator.
“Divide by the bottom and times by the top!”
Mixed Numbers & Improper fractions
An improper fraction has a numerator that is bigger
than its denominator, for example 10
7
9 is also an improper fraction. It means nine
4
quarters. If you think of this as cakes, nine
quarters are more than two whole cakes. It is 2 1/4
cakes.
Mixed Numbers & Improper fractions
Mixed numbers and improper fractions show
values where there are more than one whole
being shown.
Two and a half pieces is the same as 5 halves.
2
1
2
=
5
2
Changing a mixed number to an improper
fraction
Multiply the whole number by the denominator.
Add the numerator.
This gives you the new numerator; the
denominator stays the same.
e.g. 2 ¼ = 2 x 4 = 8. 8 + 1 = 9 = 9/4

You can add fractions easily if the bottom
number (the denominator) is the same:
 1 /4
+
1/
4
=
2/
=
4
1/
2

(One-Quarter) (One-Quarter) (Two-Quarters) (One-Half)

Another example:
 5 /8
+
1/
8
=
6/
8
=
3/
4

But what if the denominators are not the
same? As in this example:
 3 /8

1/
+
= ?
4
You must somehow make the denominators
the same.


So find an equivalent fraction so they have
they same denominator.
In this case it is easy, because we know that
1/ is the same as 2/ :
4
8
 3 /8
+
2/
8
=
5/
8
This is the same when subtracting fractions. Make
sure they have the same denominator, then just
subtract one numerator from the other.
22 x 1= 8x1= 8=12=11
3
2 3 2 6
6
3
2 2 ÷ 1 = 8 x 2 = 16 = 5 1
3 2 3 1
3
3
Dividing Fractions
To divide fractions you need to use KFC!
Keep the first fraction.
Flip the second one.
Change it to a multiplication.
E.g. 1 ÷ 1
2 6
=
1x6
2 1
=
6
2
Simplify the fraction if you can so
6=3
2
Decimals, Fractions and Percentages are just
different ways of showing the same value:




A Half can be written...
As a fraction: 1/2
As a decimal: 0.5
As a percentage: 50%




A Quarter can be written...
As a fraction: 1/4
As a decimal: 0.25
As a percentage: 25%
Why do children find fractions difficult?
Difficulties with fractions often stem from the fact that
they are different from natural numbers in that they are
relative rather than a fixed amount - the same fraction
might refer to different quantities and different fractions
may be equivalent (Nunes, 2006).
Would you rather have one quarter of £20 or half of £5?
The fact that a half is the bigger fraction does not
necessarily mean that the amount you end up with will be
bigger. The question should always be, 'fraction of what?';
'what is the whole?'. Fractions can refer to objects,
quantities or shapes, thus extending their complexity.
Fraction terminology
Numerator: the number on the top of a fraction showing the number of equal
parts in the fraction eg 3/4
Denominator: the number on the bottom of the fraction showing the total
number of equal parts in the whole eg 3/4
Proper fraction: the numerator is less than the denominator eg 2/3
Improper fraction: the numerator is larger than the denominator indicating that
the parts come from more than one whole (top-heavy fractions) eg 9/5
Mixed fraction: has a whole number and a fraction eg 8 ½
Equivalent fraction: the same fraction written in different ways so each one
gives the same answer in a calculation, even though they look different eg ½
and 3/6
Common denominator: a number that can be divided by the denominators of all
of the fractions eg 2/3 5/8 7/12 all the denominators divide into 24 so
2/ becomes 16/ , 5/
15/ , 7/
14/ .
3
24
8 becomes
24
12 becomes
24
So 24 is the lowest common denominator as this is the smallest number that 3,
8 and 12 will divide into.
Place Value
• In the number 327:
• the "7" is in the Ones position, meaning
just 7 (or 7 "1"s),
• the "2" is in the Tens position meaning 2
tens (or twenty),
• and the "3" is in the Hundreds position,
meaning 3 hundreds
• As we move right, each position is 10
times smaller. From Hundreds, to Tens, to
Ones
• But what if we continue past Ones?
• What is 10 times smaller than One?
1/
10
ths (Tenths) are!
Decimals
• But we must first write a decimal point,
so we know exactly where the Ones
position is: "three hundred and twenty
seven and four tenths“
• but we usually just say "three hundred and
• The decimal point is the most important
part of a Decimal Number. It is exactly to
the right of the Ones position. Without it,
we would be lost ... and not know what
each position meant.
• Now we can continue with smaller and
smaller values, from tenths, to
hundredths, and so on, like in this
example:
Percentages
• When you say "Percent" you are really
saying "per 100"
• So 50% means 50 per 100
(50% of this box is green)
And 25% means 25 per 100
(25% of this box is green)
How can you help?
Number facts need to be
learned!
• Children should learn number
bonds/facts to 10, 20 and beyond.
• They need regular rehearsal to retain
these so practise, practise, practise!
Number facts
They need be able to rapidly recall:
4 + 6 = 10
5+2=7
8 + 2 = 10
10 – 3 = 7
10 + 90 = 100
12 + 8 = 20
35 + 65 = 100
Times Tables are also vital!
• Nothing has changed... Children need to
learn their times tables.
• Without their times tables they may
know the method for multiplication, but
still arrive at a wrong answer.
• Children need to know them upside
down, inside out and back to front!
Times Tables
Counting in multiples
e.g. 3, 6, 9, 12…
is not good enough.
They must be able to rapidly recall:
4x6
8x2
11 x 9
5x7
12 x 3
45 ÷ 9
Applying Times Tables
4 x 3 = 12
3 x 4 = 12
12 ÷ 3 = 12
12 ÷ 4 = 12
40 x 3 = 120
4 x 300 = 1200
40 x 30 = 1200
0.4 x 3 = 1.2
Everyday ways to improve your child’s
(and your) maths
• A prominent clock in the kitchen – ideally analogue and digital.
• Display a traditional calendar.
• Board games that involve dice and spinners – helps not only with
counting but with the concepts of chance.
• Traditional playing cards – simple games such as snap are a
natural way of learning about sorting and chance.
• Dominoes – to help with number combinations.
• A calculator.
• Measuring with scales/kitchen scales – weight, length and
capacity.
• Tape measure and ruler – involve your child in ‘real life’
situations.
• Dried pasta…or Smarties! – useful for counting large collections
to investigate remainders etc.
• An indoor/outdoor thermometer.
• Money – coin recognition, working out how much you’ll
need/change/saving.
• Please don’t…
– Tell them that they are doing ‘sums’
‘Sum’ is a mathematical word that means ‘addition’,
everything else is a ‘calculation’ or ‘number
sentence’.
– Teach your children that to multiply by 10 you ‘just
add a zero’
You ‘move the digits to the left and add zero as a
place holder’.
e.g. 2.34 x 10 = 2.340 is incorrect
– Tell them that you can move the decimal point
You can’t. You can only move the digits to the left
or to the right.
Useful Websites
• I have printed a list of websites that
may be useful. I will put it onto the
website so you can click on the hyperlink
to get direct access if needed.
• Remember, if you are ever unsure or
need any help/advice, ask your child’s
teacher – we’ll try not to bite!
•
•
•
•
Remember what is important in maths!
A focus on mental calculations.
The ability to estimate.
To use maths in a real life context.
To ask children to explain how they have
calculated something using a method that
suits them.
• Teach children written calculations, but
only when children are ready.