Year 6 Maths Workshop Presentation

Report
Maths Workshop for Year 6
Parents and Carers
12 January 2015
Mrs Claire Searle – Maths
Leader
What is a fraction?
Talk to someone else – what do you think?
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.
What do Year 6 pupils need to know and do with
fractions?
Numerators and Denominators
•A fraction is made up of 2 numbers. The top number is called the
NUMERATOR and the bottom number is called the DENOMINATOR.
In the fraction ¾, 3 is the numerator and 4 is the denominator.
•DENOMINATOR
This number shows how many equal ‘pieces’ something has been divided
into. In the fraction ¾, 4 is the denominator showing that there are 4
equal pieces making up the whole.
•NUMERATOR
This number shows how many of those pieces there are. In the fraction
¾ there are 3 pieces out of the total of 4.
Numerators and Denominators
For example, if a pizza is cut into 4 equal slices there will be 4 pieces on the
plate. This makes a fraction of 4/4 (1 whole).
If I eat one of those pieces, ( ¼) then there are 3 pieces
left. ( ¾ ). The denominator stays the same, there are
still 4 parts that made up the whole pizza, but the
numerator has changed, as there are only 3 parts of the
pizza left.
Simplifying fractions
Some fractions can be made simpler by finding the highest common
factor . (The highest number that will go into both parts of the
fraction.)
Eg for 8/10 both the numerator and denominator can be divided by 2
to give 4/5.
16/24 Both the numerator and denominator can be divided by 2, 4
and 8. The highest common factor (HCF) is 8, so this fraction can be
simplified to give 2/3.
Try this!
Simplify 16/36
These can be divided by 2 and 4. The HCF is 4, so
the answer is 4/9
Exploring equivalence using a tangram
What fraction is each part of
the whole?
What other fractions can
you make?
What equivalences can you
find?
Equivalent fractions
½ = 2/4 = 3/6 = 4/8 = 5/10 = 6/12 = ...
¼ = 2/8 = 3/12 = 4/16 = 5/20 = ...
1/3 = 2/6 = 3/9 = 4/12 = 5/15 = ...
Make fraction strips showing quarters, thirds, sixths, eighths, tenths
Fraction strips
Use your strips of paper to:
Make some different fraction
strips.
What fractions can you find
that are equivalent to 1/3?
Which is larger, 5/8 or ¾?
Fraction strips
How can fraction strips help children make sense of problems like this?
Comparing and ordering fractions
Putting fractions in order of size can be difficult. It’s easiest to convert them
(temporarily) to fractions with the same denominator if you are unsure.
Try putting these fractions in order:
3/4
9/10
1/3
4/5
15/8
5/16
1/3
¾
4/5
9/10
1¼
5/16
1¼
15/8
Addition and Subtraction
Addition and subtraction need to be done with common denominators.
Addition and subtraction
Add or subtract these fractions. Remember to convert them into fractions with the
same denominator first.
•Look for the smallest number that the denominators will all go into. Eg for 3/7 +
2/5 the smallest number that both 7 and 5 will go into is 35.
•For 3/7, there are 5 lots of 7 in 35, so multiply both parts of 3/7 by 5 = 15/35.
•For 2/5, there are 7 lots of 5 in 35, so multiply both parts of 2/5 by 7 = 14/35.
• Now you can add the fractions easily. 15/35 + 14/35 = 29/35.
½+¾=
2/4 + ¾ = 5/4 = 1 ¼
2/3 + 1/6 =
4/6 + 1/6 = 5/6
¾ - 2/3 =
9/12 – 8/12 = 1/12
9/10 – 3/5 =
9/10 – 6/10 = 3/10
3/8 + 5/6 + ¾ =
9/24 + 20/24 + 18/24 = 47/24 = 1 23/24
Multiplication by a whole number
½x3=
To multiply a fraction by a whole number, first convert the whole number to
an improper fraction. ½ x 3/1 =
• Now multiply both numerators together and then both denominators
giving 3/2.
•Finally divide the numerator by the denominator, giving a mixed fraction 1 ½
•So the answer to ½ x 3 is 1 ½ .
•You can also think of it as ½ + ½ + ½ also giving 1 ½
Try this: 2/3 x 6 =
2/3 x 6/1 = 12/3 = 4
Multiplication by a fraction
½x¾=
•It is useful to imagine the multiplication sign means ‘of’ so this
calculation can be expressed as ‘what is ½ of ¾?’ and ‘what is ¾ of
½?’
• Multiply the numerators together and the denominators together.
½ x ¾ = 3/8
This answer is the same for both calculations above,
as multiplications can always be done either way round and will give
the same answer.
Try this:
¾ of 5/8 = 15/32
Division
Children need to be able to divide proper fractions by whole numbers,
Eg ¼ ÷ 2 = 1/8 .
To do this, turn the whole number into a fraction : 2/1
Then turn the fraction upside down: 1/2
Then multiply it by the first fraction
¼ x 1/2 = 1/8
2=
The denominator
has been doubled,
so the value has
been halved.
Try this!
1/3 ÷ 4 = ?
1/3 ÷ 4/1
1/3 x ¼ = 1/12
Decimal fractions
Finding decimal fractions
What is 1/5 as a decimal?
To convert a fraction to a decimal, simply divide the denominator (bottom
part) into the numerator (top part).
So to find 1/5 as a decimal, divide 1 by 5 which gives 0.2
1/5 = 1 ÷ 5 = 0.2
¾ = 3 ÷ 4 = 0.75
Try this!
What is 4/5 as a decimal?
4/5 = 4 ÷ 5 = 0.8
Converting decimals to fractions
First make the fraction’s denominator (its bottom part) 10, 100, 1000 and so
on for every digit after the decimal point.
0.75
Decimal number
with 2 places
after the decimal
point
75
3
100
4
Count the
decimal places;
if there is 1 digit, the
denominator is 10,
if there are 2 then it
is 100. The numerator
is the number after
the decimal point.
Now divide both
numbers by the
highest number
that goes into both 25.
.
Have a go!
Change 0.6 into a
fraction.
0.6
6
10
3
5
Equivalences between fractions, decimals and
percentages
•Converting between decimals and percentages is easy if the decimal
number is below 1. Percentage just means out of 100.
•So 0.8 is 80% which is 8 tenths or 80 hundredths.
•
•0.65 is 65% which is 65 hundredths.
•Children need to be sure about place value in decimals to be able to do this
conversion easily.
They also need to be able to know and use equivalences between fractions
decimals and percentages.
Which of these fractions are the same?
70%
4/5
80%
3/4
0.55
0.45
8/10
34%
Finding percentages of whole numbers
•To find 10% of any number, divide by 10. 10% of 86 = 8.6
•To find 5% of any number, divide by 10 and then halve that number.
5% of 86 - halve 8.6 to give 4.3
•To find 15% of any number, add 10% and 5% together.
So for 86, add 8.6 and 4.3 = 12.9
•To find 1%, divide by 100. 1% of 18 is 0.18
•Using these it is possible to find any percentage of a number.
See how quickly you can do these:
30% of 60
15% of 20
7% of 50
110% of 75
Price reduced by 20%! Was £15, now ______
25% off! Now £60! What was the price before
it was reduced?
Example SATs questions
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 number of parts examined, shown on the top, is
less than the whole eg 2/3
•Improper fraction – the larger numerator indicates that the parts come
from more than one whole (also called 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 2/3 becomes 16/24, 5/8 becomes 15/24,
7/12 becomes 14/24. So 24 is the lowest common denominator as this is
the smallest number that 3, 8 and 12 will divide into.
Ratio and Proportion
Ratio and Proportion
Ratio compares the size of quantities.
Proportion compares the relationship between 2 sets of quantities.
Ratios show how much bigger one thing is than another.
Two things are in proportion when a change in one causes a related
change in the other.
A fruit bowl with a ratio of 6 apples to 2 bananas can be written
like this
6:2
This can be divided by 2 and simplified to
there is 1 banana.
3:1 meaning that for every 3 apples
How many apples would there be if there were 6 bananas in the fruit
bowl?
Tomato soup!
6 tomatoes make 1 bowl of soup.
How would you write the ratio?
6:1
How many tomatoes do you need to make 2 bowls of soup?
3 bowls of soup?
6 bowls of soup?
What operation did you use to find the answers?
multiplying by 6
How many bowls of soup could you make with 48 tomatoes?
What about with 120 tomatoes?
6 million tomatoes?
What operation did you use this time?
12
18
36
8
20
1 million!
Dividing by 6
The tomatoes and the bowls of soup are in direct proportion. The ratio
between them is always the same.
Proportion
What proportion of the stick is blue?
Proportion means ‘fraction’ or ‘percentage’.
6/10 or 3/5 or 60% of the stick is blue.
For every 6 blue cubes there are 4 yellow cubes.
If the stick had 9 blue cubes, how many yellow cubes would there be?
If the stick had 60 blue cubes how many yellow cubes would there be?
What is the ratio of blue cubes to yellow cubes?
3:2
Fish Pie
Omar makes fish pie for 3 people.
How many grams of fish should he use?
Mary used 2kg of potato to make a fish pie.
How many people did her fish pie feed?
How much butter was in her fish pie?
How much fish was in her fish pie?
From the Nrich website
Example SATs questions
Useful websites
Fractions
http://www.bbc.co.uk/skillswise/topic/fractions
http://www.bbc.co.uk/skillswise/factsheet/ma17frac-l1-f-fraction-wall
http://www.bbc.co.uk/bitesize/ks2/maths/number/fractions/read/1/
http://primarygamesarena.com/fractions
Ratio and Proportion
http://www.bbc.co.uk/education/topics/zsq7hyc
http://www.11plusforparents.co.uk/Maths/ratio1.html
http://nrich.maths.org/8959
http://resources.woodlands-junior.kent.sch.uk/maths/fractions/

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