Download Calculations Power

Maths methods
taught in Key Stage 2
Learning
Together for
Life
Mathematics – LKS2
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No ratio required in LKS2
Written division moved to UKS2
No calculator skills included
Carroll / Venn diagrams no longer required
Y3: Formal written methods for + & —
Y3: Compare, order & + & — easy fractions
Y3: Vocabulary of angles & lines
Y3: Time including 24h clock & Roman numerals
Y4: Recognise equivalent fractions/decimals
Y4: Solve fractions & decimals problems
Y4: Perimeter/area of compound shapes
Y4: Know multiplication tables to 12 x 12
Mathematics – UKS2
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No calculator skills included
No probability included
Data handling greatly reduced content
Y5: Use decimals to 3dp, including problems
Y5: Use standard multiplication & division methods
Y5: Add/subtract fractions with same denominator
Y5: Multiply fractions by whole numbers
Y6: Long division
Y6: Calculate decimal equivalent of fractions
Y6: Use formula for area & volume of shapes
Y6: Calculate area of triangles & parallelograms
Y6: Introductory algebra & equation-solving
Addition
Learning
Together for
Life
Split the numbers up!
The art of this is to split the numbers up into
units, tens, hundreds etc.
Add each “bit” separately then join them
together at the end.
If you are asked to find the “sum” of some
numbers, add them.
An example of this:
Calculate 34 + 55
Split it up into tens and units:
Tens: 30 + 50 = 80
Units: 4 + 5 = 9
Final answer: 89
Another example of this:
Calculate 187 + 264
Split it up into tens and units:
Hundreds: 100 + 200 = 300
Tens: 80 + 60 = 140
Units: 7 + 4 = 11
Final answer: 451
Try these:
1. 26 + 53 = 79
2. 57 + 35 = 92
3. 98 + 57 = 155
4. 365 + 478 = 843
Using a standard method. Column Addition
386+542
+
Now you try…
• Use a method you haven’t used before – a
number line, expanded method or standard
column addition.
587 + 348 =
When you have your answer, compare your
answer and method with the person next to
you.
Do you have the same answer?
Did you find it in a different way?
Subtraction
Learning
Together for
Life
The key to subtracting is…
…to estimate an answer first.
Perform the subtraction in stages.
If you are asked to find the “difference” between
two numbers, subtract them.
What does this mean?
Calculate 72 – 38
Make it easier:
72 – 30
Answer: 42
42 - 8
Answer: 34
Firstly take
subtract 30
Then
subtract 8
A different method:
Calculate 72 – 38
Make it easier:
72 – 40
Answer: 32
Actual answer: 34
+2 to the
number
you’re
subtracting
+2 to your
answer
Another example:
Calculate 91 – 23
Make it easier:
91 – 20
Answer: 71
71 - 3
Subtract 20
Subtract 3
Actual answer: 68
Try these:
1. 85 - 36 = 49
2. 52 - 25 = 27
3. 96 - 58 = 38
4. 124 - 89 = 35
Column method - Exchanging &
cancelling.
832–279
Multiplication
Learning
Together for
Life
In Year 3
• Understand how to use partioning and arrays to solve
TU x U
eg 13 x 3
10 x 3 = 30
3x3=9
Calculate 23 × 8
Split the large number up:
This is 2 × 8
with a zero on
the end
20 × 8 = 160
3 × 8 = 24
Add the answers together as you have a total of 23
eights.
Answer: 184
Have a go at these:
1. 17 × 6
= 102
2. 26 × 4
= 104
3. 43 × 7
= 301
4. 64 × 9
= 576
In Year 3
x
10
3
3
30
9
30 + 9 = 39
Grid Method
255 x 5 = ?
200
50
5
200 x 5
50 x 5
5x5
x
5
1000
250
25
Finally, add the three numbers together to get your answer.
1000
+ 250
So 255 x 5 = 1 275
+ 25
= 1275
Grid Method – part 2
255 x 25 = ?
x
200
50
5
20
4000
1000
100
5
1000
250
25
Add up each column, then add the resulting numbers together.
4000 + 1000 + 100
1000 + 250 + 25
=
=
5100
1275
6375
Over to you!
Have a go at solving these multiplications using
the grid method.
65 x 8
74 x 45
92 x 53
Traditional method
21 x 13 = ?
x
2
1
1
3
3 x 1 = 3 write down the 3.
6
3
2
1
0
2
7
3
3 x 2 = 6 write down the 6
10 x 1 = 10 write down the 10
1 x 2 = 2 write down the 2
Add the numbers
Traditional method
45 x 34 = ?
1
2
x
4
5
3
4
4 x 5 = 20 write down the 0, carry the 2.
1
8
0
1
3
5
0
1
5
3
0
4 x 4 = 16, add 2 write down the 18
30 x 5 = 150 write down the 50, carry the one
3 x 4 =12, add the 1, write down the 13
Add the numbers
Lattice method – part 1
25 x 5 = ?
5 x 5 = 25
2x5=
10
2
5
1
2
0
5
5
1. Make the lattice (grid)
as shown
2. Multiply each
number above a
column by the numbers
in every row
3. Write the answers in
the lattice. Making sure
you have only 1 digit in
each triangle
Lattice method – part 2
25 x 5 = ?
2
5
1
2
0
1
2
2+0=2
5
5
5
Add along the
diagonal
Lattice method – part 1
36 x 8 = ?
6 x 8 = 48
3x8
=24
3
6
2
4
4
8
8
1. Make the lattice (grid)
as shown
2. Multiply each number
above a column by the
numbers in every row
3. Write the answers in
the lattice. Making sure
you have only 1 digit in
each triangle
Lattice method – part 2
36 x 8 = ?
3
6
2
4
4
2
8
4+4=8
8
8
8
Add along the
diagonal line
Lattice method – part 1
36 x 13 = ?
1x6=6
3
1x3=3
6
0
0
3
3x3=9
0
6
1
9
8
1
1. Make the lattice (grid)
as shown
3
2. Multiply each number
above a column by the
numbers in every row
3 x 6 = 18
3. Write the answers in
the lattice. Making sure
you have only 1 digit in
each triangle
Lattice method – part 2
36 x 13 = ?
3
0
6
3
0
1
4
0
1
9
6
6 + 1 + 9 = 16
6
8
8
1
3
Add along the diagonal
line
Have a go at calculating 21 × 32
Do it in as many ways possible…
Which of these did you use to
calculate 21 × 32?
30
2
20
1
600
30
40
2
Answer: 672
32
× 21
32
+ 640
Answer: 672
2
0
0
0
6
1
6
4
7
0
0
3
3
2
2
2
Answer: 672
Are there any other methods?
Choose a method to calculate 43 × 17?
10
7
40
3
400
30
280
21
Answer: 731
43
× 17
301
+ 430
Answer: 731
4
0
0
2
7
3
4
8
13
0
2
3
1
1
7
1
Answer: 731
Are there any other methods?
Multiply these using whichever
method you like (no calculators!):
1. 26 × 14
= 364
2. 74 × 39
= 2886
3. 124 × 16
= 1984
4. 249 × 179 = 44571
643 x 27 = 17,361
6
1
4
0
3
0
2
1
2
6
8
2
2
14
7
2
8
1
1
7
6
1
3
Using decimals
23.6 x 3.2 = 75.52
2
0
3
0
6
1
3
6
8
9
0
1
10
2
4
6
2
1
7
5
2
5
Real Life Problem (6a)
1. Amy bought 48 teddy bears at £9.55 each.
Work out total amount she paid.
£458.40
Real Life Problem (6a)
2. Nick takes 26 boxes out of his van.
The weight of each box is 32.9kg.
Work out the total weight of the 26 boxes.
855.4 kg
Division
Learning
Together for
Life
Starter
For each number in the table, put a tick if it is divisible by 2, 3, 4,
5, or 6. How can you work these out without actually working out
the division?
Number
26
120
975
12,528
Divisible
by 2?
Divisible
by 3?
Divisible
by 4?
Divisible
by 5?
Divisible
by 6?
Multiples Investigation
Do you know any of the rules for checking divisibility?
A number can be divided by 2 if: It ends in a 0, 2, 4, 6 or 8
A number can be divided by 3 if: The sum of its digits is a multiple of 3
A number can be divided by 4 if: The number made by the last 2 digits is a multiple of 4
A number can be divided by 5 if: It ends in a 0 or 5
A number can be divided by 6 if: It can be divided by both 2 and 3 (ends in an even number
and is a multiple of 3)
Divisibility Rules (8)
• A number is divisible
by 8 if
• the number made by
the last three digits
will be divisible by 8
Divisibility Rules (9)
• A number is divisible
by 9 if
• the sum of all the
digits will add to 9
Multiplication is the inverse of division.
Multiplication and division are inverse operations; this means they are the opposites of each other. By
knowing the answer to one problem you can work out all the others.
Example
20
÷
5
÷
×
4
We can use our tables – how?
300 ÷ 6 is easy because we already know 30 ÷ 6.
30 ÷ 6 = 5
So 300 ÷ 6 =50
2800 ÷ 4 is easy because we know 28 ÷ 4.
28 ÷ 4 = 7
So 2800 ÷ 4 = 700.
Have a go at these:
1. 2000 ÷ 5 = 400
2. 320 ÷ 4 = 80
3. 33,000 ÷ 3 = 11,000
4. 42,000,000 ÷ 7 = 6,000,000
Division: Learning to show the remainder
58 ÷ 4
= 14 r2
14
1
4 58
r
2
Division: Learning to show the remainder
For each question show the three ways of
showing the remainder.
1 - 28 ÷ 5
5 - 73 ÷ 2
2 - 37 ÷ 2
6 - 58 ÷ 4
3 - 49 ÷ 4
7 - 87 ÷ 4
4 - 39 ÷ 5
8 - 67 ÷ 5
The “bus stop” method
You do need to know your tables!
The most common method is called “the bus
stop”, because it looks like the numbers are
waiting in a bus stop.
The divisor goes outside the “bus stop”.
You can also add multiples of your divisor, but
this will take you longer!
Calculate 135 ÷ 3
Bus Stop
Draw the bus stop:
Divisor
0
3
1
4
1
3
5
1
5
Divide each value by 3, not
forgetting to carry any
remainders.
The answer is on top!
Answer: 45
Multiples
We need to do the following
calculations:
10 × 3 = 30
Total so far: 30
10 × 3 = 30
Total so far: 60
10 × 3 = 30
Total so far: 90
10 × 3 = 30
Total so far: 120
5 × 3 = 15
Total so far: 135
How many 3s in total?
Answer: 45
What about 322 ÷ 14
Bus Stop
Draw the bus stop:
Divisor
0
14
3
2
3
2
3
4
2
Divide each value by 14, not
forgetting to carry any
remainders.
The answer is on top!
Answer: 23
Multiples
We need to do the following
calculations:
10 × 14 = 140
Total so far: 140
10 × 14 = 140
Total so far: 280
3 × 14 = 42
Total so far: 322
How many 14s in total?
Answer: 23
Calculate these, without a calculator!
1. 168 ÷ 7
= 24
2. 222 ÷ 6
= 37
3. 384 ÷ 12 = 32
4. 952 ÷ 17 = 56
Long Division Methods
Method 1
We are going to try to solve
839 ÷
27
Method 1
839
-270
839 ÷
27
(27 x 10 = 270)
Method 1
839
-270
569
839 ÷
27
(27 x 10 = 270)
Method 1
839
-270
569
-270
299
839 ÷
27
(27 x 10 = 270)
(27 x 10 = 270)
Method 1
839
-270
569
-270
299
-270
29
839 ÷
27
(27 x 10 = 270)
(27 x 10 = 270)
(27 x 10 = 270)
839 ÷
27
Method 1
839
-270
569
-270
299
-270
29
-27
2
(27 x 10 = 270)
(27 x 10 = 270)
(27 x 10 = 270)
(27 x 1
= 27)
839 ÷
27
Method 1
839
-270
569
-270
299
-270
29
-27
2
(27 x 10 = 270)
(27 x 10 = 270)
(27 x 10 = 270)
(27 x 1
= 27)
10 + 10 + 10 + 1 = 31
Method 1
839 ÷
27
=31 r 2
Or 31 2
27
Long Division Methods
Method 2
We are going to try to solve
839 ÷
27
Method 2
839 ÷
27
becomes
27 ) 8 3 9
Method 2
839 ÷
27
27 ) 8 3 9
27 x 10 = 270
Miles off!
Method 2
839 ÷
27
27 ) 8 3 9
27 x 10 = 270
27 x 20 = 540
Getting
better!
Method 2
839 ÷
27
27 ) 8 3 9
27 x 10 = 270
27 x 20 = 540
27 x 30 = 810
Much better!
Method 2
27 x 30 = 810
839 ÷
27
Put the 810
underneath.
27 ) 8 3 9
-810
29
27 x 30 = 810
Method 2
839 ÷
27
27 ) 8 3 9
-810
29
-27
2
27 x 30 = 810
27 x 1 = 27
Method 2
839 ÷
27
31
27 ) 8 3 9
-810
29
-27
2
27 x 30 = 810
27 x 1 = 27
30 + 1 = 31
Method 2
839 ÷
27
31
27 ) 8 3 9
-810
29
-27
2
r 2
Don’t forget
the remainder!
Method 2
839 ÷
27
=31 r 2
Or 31 2
27
Long Division Methods
Method 3
We are going to try to solve
839 ÷
27
Method 3
839 ÷
27
becomes
27 ) 8 3 9
Method 3
839 ÷
27
Calculate 8 ÷ 27
27 ) 8 3 9
Method 3
839 ÷
27
We can’t do it, so
we write the
answer 0 here
0
27 ) 8 3 9
Method 3
839 ÷
27
So we next look at
83 ÷ 27
0
27 ) 8 3 9
Use you repeated subtraction
here if this helps
Method 3
2 x 27 = 54
839 ÷
27
03
27 ) 8 3 9
3 x 27 = 81
Method 3
3 x 27 = 81
839 ÷
27
03
27 ) 8 3 9
-81
2
We need to
take off 81
from the
83 to get
the
remainder
Method 3
839 ÷
27
03
27 ) 8 3 9
-81
29
Drop the
next digit
next to the
2
Method 3
839 ÷
27
Now we are
going to do
29 ÷ 27 and
put the
answer
here
031
27 ) 8 3 9
-81
29
Method 3
839 ÷
27
Now we are going
to do 29 - 27 to
get the remainder
031
27 ) 8 3 9
-81
29
27
Method 3
839 ÷
27
031
27 ) 8 3 9
-81
29
27
2
r 2
Method 3
839 ÷
27
or
= 31
31 2
27
r 2
Again use the method that gives
you the correct answer !!
Long division
Question : 2987  23
12 9
23 times
table
23
46
69
92
115
138
161
184
207
230
29
68
23
6
22
2987
20
Answer : 129 r 20
227
Now try 1254  17 and check on
your calculator – Why is the
remainder different?
How would this be calculated....
2 468
• How many 2s in 4........
• How many 2s in 6........
• How many 2s in 8.........
1)Work out the following:-
2742 ÷ 3
0 9 1 4
3 2 27 4 12
(a)
Answer = 914
7364 ÷ 7
1 0 5 2
7 7 3 36 14
(b)
Answer = 1052
3231 ÷ 9
0 3 5 9
9 3 32 53 81
(c)
Answer = 359
Fractions Review
Level
1
2
Fractions
Recognise
& use 1⁄2
& 1⁄4
Find and write
simple
fractions.
Understand
equivalence of
e.g. 2/4 = 1/2
recognise,
find, name
and write
fractions 1/3 ¼
2/4 and ¾ of
a length,
shape, set of
objects or
quantity.
3
Use & count in
tenths.
Recognise, find &
write fractions.
Recognise some
equivalent
fractions for
¼½¾
Add/subtract
fractions with the
same denominator
within one whole.
Order fractions
with common
denominator
4
Identify
equivalent
fractions.
Add & subtract
fractions with
common
denominators.
Recognise
common
equivalent
fractions
including
decimal
equivalents for
¼ ½ ¾ and
tenths and
hundredths.
5
6
Compare & order
fractions.
Compare & simplify
fractions.
Add & subtract
fractions with
different
denominators, with
mixed numbers.
Use equivalents to
add fractions.
Multiply and divide
fractions by units.
Divide fractions by
whole numbers.
Write decimals as
fractions.
Multiply and divide
fractions with
different
denominators,
writing the answer
in its simplest form.
Link percentages to
fractions & decimals
Multiply simple
fractions.
Multiply fractions
and mixed numbers
Today we are learning
I am starting the lesson on level _____________________
By the end of this lesson I want to be able to _____________________
Fractions
A number in the form
Numerator
Denominator
Or
N
D
Fractions
The denominator can never be equal to 0.
12
0
Does not
=
exist!
Fractions
A fraction with a numerator of 0 equals 0.
0
4
= 0
0
156
= 0
Fractions
• If the numerator is larger than the denominator, it
is called an improper fraction.
Find the improper fraction
7
10
4
56
10
11
8
9
73
12
Maths with Fractions
Four basic functions
• Multiply
• Divide
• Add
• Subtract
Multiplication
• Multiply the numerators and put in
the numerator of the result
• Multiply the denominators and put
in the denominator of the result
7
8
x
4
9
=
7x4
8x9
=
28
72
Multiplication - Let’s Try It!
7
9
7
5
x
x
1
2
1
3
=
=
7
4
18
7
7
30
15
4
x
x
9
11
7
14
=
=
36
77
210
56
210
56
These numbers get
pretty big!
What if we needed
to multiply again?
Let’s make the fraction more simple,
so it will be easier to use in the future.
Simplification
• Divide by the Greatest Common Factor
28
72
But what is a
Common Factor?
Factors
• A factor is a number that can be divided
into another number with no remainder
– 8’s factors are:
•
•
•
•
1 (8/1 =8)
2 (8/2 = 4)
4 (8/4 = 2)
8 (8/8 = 1)
– 3 is NOT a factor of 8, because 8 is not evenly
divisible by 3 (8/3 = 2 with R=3)
Common Factors
• A common factor is a factor that
two numbers have in common
– For example, 7 is a factor of both 21 and 105,
so it is a common factor of the two.
– The greatest common factor is the largest
factor that the two number share
So let’s go back to our simplification
problem from before…
Simplification
• Divide both numerator and denominator by
the Greatest Common Factor
28
72
Factors are 1, 2, 4, 7, 14, 28
Factors are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 48, 72
Greatest Common Factor is 4
28 ÷ 4 = 7
72 ÷ 4 = 18
28
7
So
=
72
18
Simplification - Let’s Try It!
3
1
=
9
3
7
1
=
21
3
18
2
=
63
7
24
2
=
84
7
6
2
=
15
5
78
13
=
114 19
Division
Just like multiplication with one more step:• Turn the ÷ into a x symbol.
• Invert / flip the second fraction and multiply
3
8
÷
1
2
=
3
8
x
2
1
=
6
8
Division - Let’s Try It!
7
1
÷
9
2
7
5
÷
1
3
14
=
9
=
4
9
44
÷
=
7
11
63
21
20
5
4
÷
7
10
=
50
7
Addition
• To add two fractions, you must make
sure they have a Common Denominator
3
8
+
5
16
What is a
Common
Denominator?
Common Denominator
• A common denominator is a number with which both
of the denominators share at least one factor that is
not the number 1
– For example, if the denominators are 4 and 7, then a
common denominator is 28.
– 28 shares the factors 1, 2 and 4 with the number 4,
and the factors 1 and 7 with the number 7.
So let’s go back to our simplification problem
from before…
Addition
• To add two fractions, you must make sure
they have a Common Denominator
• What can you multiply each fraction by to give the smallest
common denominator?
3
5
+
8
16
8 goes into 16 two times
3
8
x
The smallest number that has
both of these as factors is 16
Once you have a common
denominator, add the numerators.
2
2
=
6
16
16 goes into 16 one time
5
16
x
1
1
=
6
5
11
+
=
16
16
16
5
16
Addition - Let’s Try It!
1
1
+
4
2
6
8
+
2
3
=
=
3
4
4
2
1
+
=
16
8
2
17
13
12
16
+
3
4
=
25
16
Subtraction
•
•
To subtract two fractions, they also must
have a Common Denominator
What can you multiply each fraction by to
give the smallest common denominator?
3 - 5
8
16
8 goes into 16 two times
3
2
6
8 x 2 = 16
The smallest number that has
both of these as factors is 16
Once you have a common denominator,
subtract the numerators.
16 goes into 16 one time
5
1
5
16 x 1 = 16
6
5
1
=
16
16
16
Subtraction - Let’s Try It!
7
8
6
8
-
-
1
2
1
2
=
=
3
9
8
16
1
5
4
4
-
-
3
8
7
16
=
=
3
16
13
16
Review
• A fraction has a numerator and a denominator
• The denominator can never be 0
• You can multiply, divide, add and subtract fractions
• A common factor is a number that both denominators
are evenly divisible by
• A common denominator is a number that both
denominators share a factor with