Transcript Multiplication and Division

Maths Workshop for Year 4 Parents and
Carers
16 March 2015
Mrs Claire Searle – Maths Leader
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New National Curriculum introduced
September 2014, for all year groups (except
Years 2 and 6 – they are from September
2015).
Expectations have been increased, and some
of what was in the Year 5 curriculum is now
in Year 4.
All multiplication tables up to 12 x 12 need
to be learnt by the end of Year 4.
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add and subtract numbers with up to 4 digits
using the formal written methods of
columnar addition and subtraction where
appropriate
Steps:
Th H T U
5 6 8 2
•Recognise ones, tens, hundreds and thousands digits
and know their values.
• Know how to set out each column in an addition
question.
•Know to start adding from the right side of the
addition (ones.)
Children in Year 4 often still need to use the
column headings to help them set calculations
out correctly.
7853 + 451=
They need to remember to use 0 as a place-holder!
Adding and subtracting decimals - money
What do
children need
to know to
work these
out?
How would
you set these
calculations
out?
•Need to know that numbers after the decimal point are
smaller than 1.
•Need to know that numbers must be lined up in
columns including the decimal points.
1.
1.
2.
2
Multiplication and Division
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What do you understand by:
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Multiplication?
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Division?
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What vocabulary can you think of that applies to
each of them?
What methods do you use to multiply and divide?
Are they mental or written?
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X
repeated addition eg 5 x 3 is the same as
(equals) 3 + 3 + 3 + 3 + 3
times
lots of
groups of
multiplied by
multiply
times tables
double
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Repeated subtraction
eg 20 ÷5 = 20 – 5 – 5 – 5 - 5
Divide
Divided by
Share
Share equally
Groups
Lots
Halve
Recall multiplication and division facts for
multiplication tables up to 12 × 12
By the end of the summer term in Year 4, all children
should know all multiplication tables off by heart.
This includes multiplication facts out of order eg what
is 8 times 3?
Also corresponding division facts for all multiplication
tables, eg how many 7s in 42?
They need to know these instantly! No fingers or
chanting through the tables.
Practising objectives on Numeracy Passports will help. If
they have completed North America by the end of Year 4
then they have met this requirement. All Numeracy
Passports and parent helpsheets are on the Longfield
School website.
Knowing tables and their multiplication and division facts is
really important as they are a vital part of being able to
problem-solve and reason.
How would you do?
This is one way we test times tables in school –
have a go! You will have 30 seconds to work out
the answers to 10 questions.
Use place value, known and derived facts to multiply
and divide mentally, including: multiplying by 0 and
1; dividing by 1; multiplying together three numbers
Children need to be able to:
•derive and recall multiplication tables (including
multiplying by 0 and 1)
•derive and recall division facts
•use known facts to derive new facts, for example: 3 x 5
=15 so 3 x 50 =150
Try this!
:
What other facts can you derive from 4 x 6 = 24?
40 x 60 = 2400
40 x 6 = 240
400 x 6 = 2400
4 x 60 = 240
4 x 600 = 2400
Multiply two-digit and three-digit numbers by 10
When multiplying by 10, 100 or 1000, we teach
children to ‘move all the digits to the left’ according to
the number of zeros.
So to multiply a whole number by 10, move all the
digits one place to the left, and then place a zero in
the space left as a placeholder.
To multiply by 100, move all the digits 2 places to the
left and place 2 zeros as placeholders..
To multiply by 1000, move all the digits 3 places to
the left and place 3 zeros as placeholders.
632 x 10 =
zero as a
placeholder
The decimal point does not move – we never teach
children to ‘move the decimal point’.
54.15 x 10
zero not needed
as placeholder
With enough practice and visualising, children will
learn to ‘see’ place value boards in their minds
and be able to move the digits to the left, to
multiply, or to the right, to divide, using zero as a
placeholder when necessary.
Try this!
For each number and calculation, said, say
whether the digits need to move the left or to
the right, and then write the final number.
Multiply three numbers together; use my knowledge
of number facts and regrouping to make calculating
easier, for example: (2 x 6) x 4 = 2 x (6 x 4)
Multiplications can be done in any order, so it is
possible to regroup numbers in a calculation to make
the numbers easier to calculate.
So 9 x 10 x 7 can be regrouped as (9 x 7) x 10.
Times tables should mean that a child can work
out 9 x 7 = 63, and knowledge of place value
means that 63 can be easily multiplied by 10
mentally: 63 x 10 = 630
Try this!
Regroup these numbers to make them easier to
multiply.
6x7x5
(6 x 5) x 7
6 x 5 = 30
30 x 7 = 7 x 30
7 x 3 is 21, so 7 x 30 is 210
Recognise and use factor pairs and commutativity
in mental calculations
Children need to understand that factors are
whole numbers you can multiply together to get
another number, for example: 2 and 3 are factors
of 6 (because 2 x 3 = 6).
Understand that factors come in pairs.
Try this!
What numbers can you multiply together to get 24?
1 x 24
2 x 12
3x8
4x6
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Addition and multiplication are commutative operations.
This means that for these operations the numbers can be
added or multiplied in any order and the answer will still be
the same. (You can think of it like the word ‘commuter’ –
like people the numbers can go back and forth or change
place and still be the same!)
So 8 + 4 = 12 is the same as 4 + 8 = 12
And 8 x 5 gives the same answer as 5 x 8.
But subtraction and division are not commutative.
8 – 4 (= 4) is not the same as 4 – 8. (=-4)
And 40 ÷ 5 (= 8) is not the same as 5 ÷ 40 (= 0.125)
Children need to be taught this – they don’t automatically
know or recognise it, and often won’t notice or realise that
they have written the calculation the wrong way round.
Use factor pairs and commutativity to multiply
mentally
for example: 15 x 6 = 15 x 2 x 3
15 x 2 = 30 and 30 x 3 = 90 so 15 x 6 = 90
Your turn!
Try working out this calculation
mentally using a factor pair: 18 x 5
18 x 5 = 9 x 2 x 5
2 x 5 = 10
10 x 9 = 90
Use factor pairs and commutativity to divide mentally
for example: 90 ÷ 6 = 90 ÷ (3 x 2)
90 ÷ 3= 30 and 30 ÷ 2 = 15 so 90 ÷ 6= 15
Try this! Use factor pairs to divide this mentally.
120 ÷ 8 =
120 ÷ (4 x 2)
120 ÷ 4 = 30
30 ÷ 2 = 15
So 120 ÷ 8 = 15
Multiply two-digit and three-digit numbers
by a one-digit number using formal written
layout
Understand that the phrase 'formal written layout',
probably does not include the 'grid method' but
refers to a column method for multiplication.
What is the grid method?
The grid method is a method of multiplying using
partitioning.
2-digits (or more) are split up and each number
multiplied separately,
before being
added together
again.
It is a step on
the way to
short and long
multiplication.
Try the grid method!
84 x 7 =
Short multiplication
Multiplication of a 2-digit (or more) number by a
single-digit number.
Children need to
understand when
they need to carry
tens and
hundreds, and to
add them on once
carried.
HTU
Numbers
need to be
lined up in
columns
according to
place value.
Place value
headings
should be
used at first.
Short multiplication
Your turn!
58 x 8
272 x 5
Need to
remember to
use zero as a
place-holder,
and to carry
both tens and
hundreds.
This
calculation
goes into the
thousands
column.
Division
Chunking
85÷ 5
85
-50
35
-25
10
-10
0
10 x 5
5x5
2x5
17
85 ÷ 5 = 17
Chunking involves
repeatedly taking away
‘chunks’ of easily
calculated numbers
from the number being
divided (dividend – here
the dividend is 85).
They need to be chunks
of the divisor (here the
divisor is 5).
Once zero is reached,
the number of chunks
added together gives
the answer.
Your turn – try chunking!
245 ÷ 5
245
- 50
195
- 50
145
- 50
95
- 50
45
- 45
0
So 245 ÷ 5 = 49
10 x 5
10 x 5
10 x 5
10 x 5
9x5
49
In this
calculation,
children need to
be confident
taking away
chunks of 50
that go over the
hundreds
barriers.
More confident
children will be
able to take
away groups of
20 x 5 to carry
out the
calculation
more quickly.
Solve problems involving multiplying and adding,
including using the distributive law to multiply 2 digit
numbers by 1 digit, integer scaling problems and
harder correspondence problems such as n objects are
connected to m objects.
Children need to know that if you are scaling up you
will be using multiplication for example: to make 7 kg
of concrete you need 1kg of cement, 2kg of sand and
4kg of gravel. So how much cement, sand and gravel
would you need to make 21 kg of concrete?
Can you work it out? What do you need to do?
There are 3 lots of 7 in 21, so you need to multiply all
the amounts by 3. So 3kg cement, 6kg sand, 12kg
gravel. All those added together make 21kg.
Know that if you are scaling down you will be
using division for example: how can you
share 12 sweets between 4 children so each
child has the same number of sweets?
So 12 divided by 4 = 3. Each child has 3
sweets.
Use the relationship between
multiplication and division to solve
problems, for example:
 X 5 = 45
60÷  = 5
Use the inverse
operation – division. So
45 ÷ 5 = 9
9 x5 = 45
Divide 60 by 5 = 12
60 ÷ 12 = 5
What needs to happen here to solve these
problems?
Solve multiplication and division problems in context
including measuring and scaling contexts, for
example: 4 times as high, 8 times as long, 12 sweets
shared between 4 children.
Apple pudding – serves 4
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400g cooking apples
50g soft brown sugar
1 egg
Grated rind of 1 lemon
80g self-raising flour
90g caster sugar
75g butter
100ml milk
If this recipe makes
enough for 4, how
much flour would you
need to make enough
for 12 people?
How much sugar?
Make use of opportunities to give your
child practical experience of mathematics
in the home and everyday life, such as:
• following recipes and changing them
for different numbers of people;
• working out quantities of different
parts to make diluted drinks, colours of
paint or cement;
• comparing prices for single and multibuy packs to decide which is better value.
Example questions
Calculate 453 × 8.
Calculate 942 ÷ 6
Fractions
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.
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.
Exploring equivalence using a
tangram
What fraction is
each part of the
whole?
What other
fractions can you
make?
What equivalences
can you find?
Make some different
fraction
strips.
Use your strips of paper
to:
What fractions can
you find that are
equivalent to 1/4?
Which is larger, 5/8
or ¾?
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.
How can fraction strips help children make sense of
problems like this?
Your turn!
Write these fractions as decimals
4/
10
4/
10 = 0.4
27/
100
27/
100
= 0.27
Write these decimals as fractions
0.7
0.86
0.7 = 7/10 0.86 =
86/
100
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Some fractions with the same denominator will
add to a total of 1, eg 1/3 + 2/3 = 1
If fractions have the same denominator then you
just need to add the numerators together.
Some answers will be less than 1 eg
3/
4
7
10 + /10 = /10
Some answers will be greater
than one. The answers can be
given as an improper (top-heavy) fraction or a mixed
number.
8/
9
17/
7
10 + /10 =
10 = 1 /10
Remember not to add the denominators.
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Children need to know that one tenth or one
hundredth is one of ten or one hundred equal parts
of an object or quantity
They need to record tenths as fractions 1/10, 2/10 etc
They need to understand that 0.1 is the same as
one tenth, 0.2 = two tenths, etc
They need to record hundredths as fractions
1/100, 2/100, etc
They need to know that 0.01 is the same as one
hundredth, 0.02 = two hundredths, etc.
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Subtracting fractions with the same
denominators is very similar
9/
10
- 2/10 = 7/10
If the first number in the subtraction is an improper
(top-heavy) fraction, just subtract in the same way.
20
6
14
6
1
/8 – /8 = /8 =
/8
Your turn!
Add or subtract these fractions. Give the
answers as mixed numbers
7/
10
20/
7
+ 7/10 + 9/10 =
– 9 /7 =
11/
7
23/
10
= 1 4 /7
= 2 3/10
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Go back to the fraction strips
Label the fraction strips with their decimal
equivalents – for ¼ (0.25) ½ (0.5) and ¾ (0.75)
Children need to remember and recall these
decimal equivalents to fractions.
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Need to remember that numbers that end in
1, 2, 3 or 4 round down.
Numbers that end in 5,6, 7, 8 or 9 round up.
So 76, rounded to the nearest 10 is 80.
Likewise 7.6 rounded to the nearest whole
number is 8. The 0.6 rounds up.
7.4 would round down, to 7.
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http://www.bbc.co.uk/bitesize/ks2/maths/number/multip
lication_division/read/1/
http://resources.woodlandsjunior.kent.sch.uk/maths/timestable/interactive.htm
http://www.theschoolrun.com/times-tables
http://www.topmarks.co.uk/maths-games/7-11years/multiplication-and-division
http://www.topmarks.co.uk/maths-games/7-11years/fractions-and-decimals
http://www.bbc.co.uk/bitesize/ks2/maths/number/fractio
ns_basic/play/
http://www.bgfl.org/custom/resources_ftp/client_ftp/ks2/
maths/fractions/index.htm