maths methods taught in key stage 2 -...

Learning Together for

Life

Maths methods

taught in Key Stage 2

Mathematics – LKS2 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 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


Life

Addition

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

2. 57 + 35

3. 98 + 57

4. 365 + 478

= 79

= 843

= 155

= 92

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?


Life

Subtraction

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

Actual answer: 68

Subtract 20

Subtract 3

Try these:

1. 85 - 36

2. 52 - 25

3. 96 - 58

4. 124 - 89

= 49

= 35

= 38

= 27

Column method - Exchanging & cancelling.

832–279


Life

Multiplication

In Year 3

• Understand how to use partioning and arrays to solve

TU x U

eg 13 x 3

10 x 3 = 30 3 x 3 = 9

Calculate 23 × 8

Split the large number up:

20 × 8

3 × 8

Add the answers together as you have a total of 23 eights.

Answer: 184

= 160

= 24

This is 2 × 8 with a zero on

the end

Have a go at these:

1. 17 × 6

2. 26 × 4

3. 43 × 7

4. 64 × 9

= 102

= 104

= 301

= 576

In Year 3

x 10 3

3 30 9

30 + 9 = 39

Grid Method 255 x 5 = ?

x

200

50

5

200 x 5

50 x 5

5 x 5

5

Finally, add the three numbers together to get your answer.

1000

+

250

+

25

=

1275

So 255 x 5 = 1 275

1000 25 250

Then, multiply each number in the column by each number in the row. Have a look at the grid below.

x

200

50

5

20

5

Add up each column, then add the resulting numbers together.

255 x 25 = 6 375

Grid Method – part 2

255 x 25 = ?

4000

250 1000

100 1000

25

4000 + 1000 + 100 = 5100

1000 + 250 + 25 = 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 = ?

2 1

x 1 3 3 x 1 = 3 write down the 3.

3 6 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

1 0 2

2 7 3

Traditional method

45 x 34 = ?

4 5

x 3 4 4 x 5 = 20 write down the 0, carry the 2.

0 8 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

5 0 3

5 3 0

1

1

1

1

2

2 5

5 1

0 5

2

2 x 5 =

10

5 x 5 = 25

Lattice method – part 1

25 x 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

2 5

5 1

0

5 1

5

2

2

Add along the

diagonal 2 + 0 = 2


25 x 5 = ?

3 6

8 2

4 8

4

3 x 8

=24

6 x 8 = 48


36 x 8 = ?


as shown

2. Multiply each number

above a column by the

numbers in every row




each triangle

3 6

8 2

4

8 2

8

4

8

Add along the

diagonal line 4 + 4 = 8


36 x 8 = ?

3 6

1

3

3 6

1 x 3 = 3

1 x 6 = 6


36 x 13 = ?


as shown

2. Multiply each number

above a column by the

numbers in every row




each triangle

9 8 1

3 x 6 = 18

3 x 3 = 9

0 0

0

3 6

1

3

3 6


36 x 13 = ?

9 8

4 6 8

1

Add along the diagonal

line

6 + 1 + 9 = 16

1

0 0

0

Have a go at calculating 21 × 32

Do it in as many ways possible…

Which of these did you use to calculate 21 × 32?

20 1

30

2

600 30

40 2

Are there any other methods?

32 × 21

32 + 640

Answer: 672 Answer: 672

2 1

3

2

0 6

0 3

0 4

0 2

2 7

6

0

Answer: 672

Choose a method to calculate 43 × 17?

40 3

10

7

400 30

280 21

Are there any other methods?

43 × 17 301

+ 430 Answer: 731 Answer: 731

4 3

1

7

0 4

0 3

2 8

2 1

1 3

7

0

Answer: 731

1

Multiply these using whichever method you like (no calculators!):

1. 26 × 14

2. 74 × 39

3. 124 × 16

4. 249 × 179

= 364

= 2886

= 1984

= 44571

643 x 27

6 4 3

2

7

1 2

0 8

0 6

2 1

2 8

4 2

1 6 1

3

1

7

1

= 17,361

23.6 x 3.2

2 3 6

3

2

0 6

0 9

1 8

1 2

0 6

0 4

2 5 1

5

1

7

= 75.52

Using decimals

Real Life Problem (6a)

1. Amy bought 48 teddy bears at £9.55 each.

Work out total amount she paid.

£458.40

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.

Real Life Problem (6a)

855.4 kg


Life

Division

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 Divisible by 2?

Divisible by 3?

Divisible by 4?

Divisible by 5?

Divisible by 6?

26

120

975

12,528

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

The sum of its digits is a multiple of 3

The number made by the last 2 digits is a multiple of 4

It ends in a 0 or 5

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

2. 320 ÷ 4

3. 33,000 ÷ 3

4. 42,000,000 ÷ 7

= 400

= 80

= 11,000

= 6,000,000

Division: Learning to show the remainder

58 ÷ 4 = 14 r2

5 8 4

1 1

4 r 2

Division: Learning to show the remainder For each question show the three ways of

showing the remainder.

1 - 28 ÷ 5

2 - 37 ÷ 2

3 - 49 ÷ 4

4 - 39 ÷ 5

5 - 73 ÷ 2

6 - 58 ÷ 4

7 - 87 ÷ 4

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!

Draw the bus stop:

Divide each value by 3, not forgetting to carry any

remainders.

The answer is on top!

Answer: 45

0

3

Calculate 135 ÷ 3

Bus Stop Multiples

We need to do the following calculations:

10 × 3 = 30

10 × 3 = 30

10 × 3 = 30

10 × 3 = 30

5 × 3 = 15

How many 3s in total?

Answer: 45

1 3 5

4 5 1 1 Total so far: 30

Total so far: 60

Total so far: 90

Total so far: 120

Total so far: 135

Divisor

Draw the bus stop:

Divide each value by 14, not forgetting to carry any

remainders.

The answer is on top!

Answer: 23

0

14

What about 322 ÷ 14

Bus Stop Multiples

We need to do the following calculations:

10 × 14 = 140

10 × 14 = 140

3 × 14 = 42

How many 14s in total?

Answer: 23

3 2 2

2 3 3 4 Total so far: 140

Total so far: 280

Divisor

Total so far: 322

Calculate these, without a calculator!

1. 168 ÷ 7

2. 222 ÷ 6

3. 384 ÷ 12

4. 952 ÷ 17

= 24

= 37

= 32

= 56

Long Division Methods

Method 1

We are going to try to solve

839 ÷ 27

839 ÷ 27

Method 1

8 3 9 - 2 7 0 (27 x 10 = 270)

839 ÷ 27

Method 1

8 3 9 - 2 7 0 (27 x 10 = 270) 5 6 9

839 ÷ 27

Method 1

8 3 9 - 2 7 0 (27 x 10 = 270)

- 2 7 0 (27 x 10 = 270) 5 6 9

2 9 9

839 ÷ 27

Method 1

8 3 9 - 2 7 0 (27 x 10 = 270)

- 2 7 0 (27 x 10 = 270)

- 2 7 0 (27 x 10 = 270)

5 6 9

2 9 9

2 9

839 ÷ 27

Method 1

8 3 9 - 2 7 0 (27 x 10 = 270)

- 2 7 0 (27 x 10 = 270)

- 2 7 0 (27 x 10 = 270)

5 6 9

2 9 9

2 9 - 2 7 (27 x 1 = 27)

2

839 ÷ 27

Method 1

8 3 9 - 2 7 0 (27 x 10 = 270)

- 2 7 0 (27 x 10 = 270)

- 2 7 0 (27 x 10 = 270)

5 6 9

2 9 9

2 9 - 2 7 (27 x 1 = 27)

2 10 + 10 + 10 + 1 = 31

839 ÷ 27

Method 1

= 3 1 r 2

Or 31 2

27


Method 2


839 ÷ 27

839 ÷ 27

Method 2

becomes

27 ) 8 3 9

839 ÷ 27

Method 2

27 ) 8 3 9

27 x 10 = 270

Miles off!

839 ÷ 27

Method 2

27 ) 8 3 9

27 x 10 = 270

27 x 20 = 540

Getting better!

839 ÷ 27

Method 2

27 ) 8 3 9

27 x 10 = 270

27 x 20 = 540

27 x 30 = 810

Much better!

839 ÷ 27

Method 2

27 ) 8 3 9

27 x 30 = 810

Put the 810 underneath.

- 8 1 0

2 9

27 x 30 = 810

839 ÷ 27

Method 2

27 ) 8 3 9

- 8 1 0

2 9

27 x 30 = 810

27 x 1 = 27 - 2 7 2

839 ÷ 27

Method 2

27 ) 8 3 9

- 8 1 0

2 9

27 x 30 = 810

27 x 1 = 27

- 2 7 2

30 + 1 = 31

3 1

839 ÷ 27

Method 2

27 ) 8 3 9

- 8 1 0

2 9 Don’t forget the remainder! - 2 7

2

3 1 r 2

839 ÷ 27

Method 2

= 3 1 r 2

Or 31 2

27


Method 3


839 ÷ 27

839 ÷ 27

Method 3

becomes

27 ) 8 3 9

839 ÷ 27

Method 3

27 ) 8 3 9

Calculate 8 ÷ 27

839 ÷ 27

Method 3

27 ) 8 3 9

We can’t do it, so we write the answer 0 here

0

839 ÷ 27

Method 3

27 ) 8 3 9

So we next look at 83 ÷ 27

0

Use you repeated subtraction here if this helps

839 ÷ 27

Method 3

27 ) 8 3 9

0

2 x 27 = 54

3 x 27 = 81 3

839 ÷ 27

Method 3

27 ) 8 3 9

0

3 x 27 = 81

3

- 8 1

2

We need to take off 81 from the 83 to get the remainder

839 ÷ 27

Method 3

27 ) 8 3 9

0 3

- 8 1

2 9

Drop the next digit next to the 2

839 ÷ 27

Method 3

27 ) 8 3 9

0 3 1

- 8 1

2 9

Now we are going to do 29 ÷ 27 and put the answer here

839 ÷ 27

Method 3

27 ) 8 3 9

0 3 1

- 8 1

2 9

Now we are going to do 29 - 27 to get the remainder

2 7

839 ÷ 27

Method 3

27 ) 8 3 9

0 3 1 r 2

- 8 1

2 9

2 7 2

839 ÷ 27

Method 3

= 3 1 r 2

or 31 2 27

Long division Again use the method that gives you the correct answer !!

Question : 2987 23

23 46 69 92 115 138161 184207230

23 times table

2 9 8 7 23 29

1 6

68

2 22

227

9 20

Answer : 129 r 20

Now try 1254 17 and check on your calculator – Why is the remainder different?

How would this be calculated....

• How many 2s in 4........

• How many 2s in 6........

• How many 2s in 8.........

2 4 6 8

1) Work out the following:-

(a) (b) (c) 2742 ÷ 3 7364 ÷ 7 3231 ÷ 9

2 7 4 2 3

Answer = 914

1 2

0 9 1 4

Answer = 1052

7 3 6 4 7 3

1 0 5 1

2 3 2 3 1 9 3

0 3 5

5 8

9

Answer = 359

Fractions Review

Level 1 2 3 4 5 6

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.

Use & count in tenths. Recognise, find & write fractions. Recognise some equivalent fractions for ¼ ½ ¾ Add/subtract fractions with the same denominator within one whole. Order fractions with common denominator

Identify equivalent fractions. Add & subtract fractions with common denominators. Recognise common equivalent fractions including decimal equivalents for ¼ ½ ¾ and tenths and hundredths.

Compare & order fractions. Add & subtract fractions with different denominators, with mixed numbers. Multiply and divide fractions by units. Write decimals as fractions. Link percentages to fractions & decimals Multiply fractions and mixed numbers

Compare & simplify fractions. Use equivalents to add fractions. Multiply simple fractions. Divide fractions by whole numbers. Multiply and divide fractions with different denominators, writing the answer in its simplest form.

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.

0

12 =

Does not exist!

Fractions

A fraction with a numerator of 0 equals 0.

156

0 = 0

4

0 = 0

Fractions

• If the numerator is larger than the denominator, it is called an improper fraction.

Find the improper fraction

11

7

12

10

73

56

9

4

8

10

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

8 9 8 x 9 72

4 7 7 x 4 28 = = x

Multiplication - Let’s Try It!

9 2

1 7 = x

5 3

1 7 = x

7 11

9 4 = x

4 14

7 30 = x

18

7

15

7

77

36

56

210

56

210 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

72

28 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

72

28 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

72 ÷

÷ 4

4

=

= 7

18 So

72

28 =

7

18

Simplification - Let’s Try It!

3

1

7

2

3

1

63

18 =

5

2

114

78 =

7

2

19

13

9

3 =

84

24 =

21

7 =

15

6 =

Division

Just like multiplication with one more step:- • Turn the ÷ into a x symbol. • Invert / flip the second fraction and multiply

8 2

1 3 = ÷

8 1

2 3 x =

8

6

Division - Let’s Try It!

9 2

1 7 = ÷

= 5 3

1 7 ÷

7 11

9 4 = ÷

= 4 10

7 20 ÷

9

14

5

21

63

44

7

50

Addition

• To add two fractions, you must make sure they have a Common Denominator

8

3

16

5 +

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?

8

3

16

5 +

The smallest number that has both of these as factors is 16

8 goes into 16 two times 16 goes into 16 one time

16

6

8

3

2

2 x = 16

5

16

5

1

1 x =

16

6

16

5 + =

16

11 Once you have a common denominator, add the numerators.

Addition - Let’s Try It!

4 2

1 1 = +

= 8 3

2 6 +

16 8

2 4 = +

= 16 4

3 13 +

4

3

12

17

2

1

16

25

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?

8

3

16

5 -

The smallest number that has both of these as factors is 16

8 goes into 16 two times 16 goes into 16 one time

16

6 = 8

3

2

2 x 16

5 = 16

5

1

1 x

16 6

16 5

- = 16 1

Once you have a common denominator, subtract the numerators.

Subtraction - Let’s Try It!

= 8 2

1 7 -

8 2

1 6 = -

= 16 8

3 9 -

4 16

7 5 = -

8

3

4

1

16

3

16

13

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

maths methods taught in key stage 2 -...

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