Year 3 - 4 MATHS NOTEBOOK

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Compiled by E. C. Wood 0

Number Some Important Basics Number Bonds Number bonds are pairs of numbers that combine to make a third. Number bonds help show that every whole number larger than one is made up of other whole numbers. 1 & 4, 2 & 3 both make five. They are the number bonds of five.

Number Bonds means addition sums that make particular totals and their corresponding subtraction facts (see example below). Number Bonds of 10 10 – 10 = 0 10 – 9 = 1 10 – 8 = 2 10 – 7 = 3 10 – 6 = 4 10 – 5 = 5 10 – 4 = 6 10 – 3 = 7 10 – 2 = 8 10 – 1 = 9 10 – 0 = 10

0 + 10 = 10 1 + 9 = 10 2 + 8 = 10 3 + 7 = 10 4 + 6 = 10 5 + 5 = 10 6 + 4 = 10 7 + 3 = 10 8 + 2 = 10 9 + 1 = 10 10 + 0 = 10

Number bonds are vital for mental arithmetic. When you ‘partition’, or ‘decompose’ a number you get number bonds.. For instance, knowing that can be 15 can be partitioned to 10 and 5 helps you perform calculations with 15. You can calculate with the ’10′ part, and then with the ’5′ part, and combine their answers. 5 and 10 are number bonds of 15. Knowing and remembering the number bonds of 100 makes it easier to handle money, or give change. If I know that 70 and 30 are number bonds of 100, then I know instantly how much change to expect from a pound when I buy a 70p packet of sweets. Number bonds can help you add. Knowing that 5 and 2 are number bonds of 7 makes Place Value adding 7 much faster. five to five get 10, andathen add indicates the remaining two Pupils needtotofive understand thatAdd the position of to a digit within number its value. to example, get 12! And this is just scratching the 4surface. For they need to know that the in 24 is 4 units but the 4 in 47 is 4 tens or 40.

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Place Value It is important to understand that the position of a digit within a number indicates its value. For example, the 4 in 24 is 4 units but the 4 in 47 is 4 tens or 40. Partitioning A skill vital to many of the calculations in this booklet is that of partitioning, which is related to the concept of place value. Partitioning means splitting a number into its digits and giving each its place value, as in the examples below.

27 20

638 7

600

30

45·6 8 40

5

0·6

Relationship between Calculation Types It is important to understand the relationship between addition, subtraction, multiplication and division. They will learn that: Addition and subtraction are inverse operations – that is, the opposite of one another. Adding 3 is the opposite of subtracting 3, so that if 6 + 3 = 9 then 9 – 3 = 6. Multiplication and division are also inverse operations. One way of viewing multiplication is that it is repeated addition: so, for example 5 x 3 (we can say ‘five threes’) is the same as adding five threes together: 5 x 3 = 3 + 3 + 3 + 3 + 3 = 15. Division can also be viewed as repeated addition. If I divide 16 by 5, I can simply start from zero and count up in groups of 5 until I have less than 5 left, therefore 16 ÷ 5 = 3 r 1. Commutative Law The commutative law says that when adding or multiplying numbers together, the order of the numbers does not matter. So: 4 + 6 is the same as 6 + 4 20 x 3 is the same as 3 x 20 4 + 9 + 16 + 11 is the same as 16 + 4 + 11 + 9 From this we also find that: If 9 – 6 = 3 then 9 – 3 = 6 If 20 ÷ 5 = 4 then 20 ÷ 4 = 5. Distributive Law The distributive law says that when multiplying or dividing a number, it is possible to split the number, multiply or divide each part, then recombine it. For example: 4 x 26 is the same as (4 x 20) + (4 x 6) 28 ÷ 2 is the same as (20 ÷ 2) + (8 ÷ 2) 2

Addition

increase

add

more

altogether

sum

total

make

addition

greater

Partitioning - splitting numbers down into hundreds, tens and units

47 40

+

1. Split 47 and 76 into tens and units.

76

7

70

6

2. Add the tens together and then the units together. 40 + 70 = 110 7 + 6 = 13 3. Finally add the unit and tens together 110 + 13 = 123

With partitioning, some people will find it easier to only split one of the numbers up. 47 + 76 = (47 + 70) + 6 = 123 Rearranging the numbers so as to add from the largest number 47 + 76 = 76 + 40 + 7 Using a number line Using a combination of partitioning and a number line can aid addition.

Example 1 76 + 47 = +10 76

+10 86

+10 96

+10 106

1. Split the smaller of the two numbers into hundreds tens units etc. 47 = 4 tens and 7 units

+7 116 123

47 + 76 = 123 Example 2 3

2. Starting from the larger number use the number line and count up in tens (either as individual tens or all together) and then the units.

47 + 76 =

+40

47

In this example the same method is used. However, instead of counting up steps of ten, the tens are just added in one whole step.

+7 116

123

47 + 76 = 123 Adding the nearest 10 and Adjusting (for example when adding 19, 29 etc) 36 + 29 36 + 30 – 1

This method works when adding numbers which are one away from the nearest ten, for example 19, 29 etc.

36 + 30 = 66

In this example view 29 as 30 - 1

66 – 1 = 65

Add 36 and 30 = 66 Finally subtract 1 from 66 = 65

Standard written method Example 1

135 + 54 = H T U 1 3 5+ 5 4 1 8 9

Start by rewriting the sum down the page so that the units, tens, hundreds etc are lined up in the same column. Add up each column, starting with the units, writing in the answers as you go. Units: 5 + 4 =9 Tens: 3 + 5 = 8 Hundreds: 1 + 0 = 1

Example 2

147 + 7 = 14 7+ 76 223 1 1

In this example the columns add up to 10 or more, so it is necessary to carry across into the next column. Units: 7 + 6 = 13; write down the 3 and carry the one to the tens column. Tens: 4 + 7 + 1 = 12; write down the 2 and carry the one to the hundreds column. Hundreds: 1 + 1 = 2; write down the 2 4

Subtraction

minus less

difference subtract

between

leave

takeaway

Using a number line Using a Number Line Draw a line from 0 to the higher number (the number you are subtracting from). Partition the number to be taken away 48 = 40 + 8. Take away first the ten, then the units.

Example 1 86 - 48 = -2 -6 0 38 40

-10 46

-10 56

-10 66

-10 76

86

86 - 48 = 38

Example 2 86 - 48 =

-2

In this example the same method is used. However, instead of counting down in steps of ten, the tens are subtracted in just one step.

-40

-6

38 40

46

86

86 - 48 = 38 Example 3 103 - 92 = 21

+8 92

+10 90

In this example the number line is used to find the difference by counting up from the smaller of the two numbers.

+3 100

103

Subtracting the nearest 10 and Adjusting (for example when subtracting 19, 29 etc) 5

86 - 29

This method works when subtracting numbers which are one away from the nearest ten, for example 19, 29 etc.

86 – 30 + 1

In this example view 29 as 30 =+ 1 86 - 30 = 56 Subtract 30 from 86 = 56 56 + 1 = 57 Finally 56 + 1 = 57

Expanded Methods These involve partitioning both numbers before subtracting. Partitioning means separating the digits of the numbers e.g. 43 is 40 plus 3. Example 1 - with no decomposition. Split both of the numbers into tens and units. Subtract the ten( 40 -20) and the units (6-2) Add the two answers together (20 +4) to give you a final answer. 46 - 22 = 24

46 - 22

40 + 6 - 20 + 2 20 + 4 = 24

Example 2 - with decomposition. 73 - 47

60 13

70 + 3 40 + 7 20 + 6 = 26

-

In this example 7 units cannot be taken away from 3 units so decompositionis used. The number 73 is changed from 70 + 3 to 60 + 13 by moving ten from the tens column to the units column. Subtract the ten( 60 - 40) and the units (137). Add the two answers together (20 +6) to give you a final answer. 73 - 47 = 26

Example 3 - in this example decomposition is used twice. 714 - 286

100 600

-

110

14

700 + 10 + 4 200 + 80 + 6 400 + 20 + 8 + 428 6

Standard written method Example 1

987 – 76 = 9 8 776 911

Start by rewriting the sum down the page so that the units, tens, hundreds etc are lined up in the same column. Remember to put the bigger number on the top. Subtract the bottom number from the top number in each column, starting with the units, writing in the answers as you go. Units: 7 - 6 = 1 Tens: 8 - 7 = 1 Hundreds: 9 - 0 = 9

Example 2.

138 – 29 = 2

1

1 3 82 9 1 0 9

In this example the number on the bottom of a column is larger than the number on the top, so it is necessary to use decomposition (to borrow from the next column to the left). Units: because 9 is bigger than 8 a ten is taken from the tens column of the top number to make it 18 - 9 = 9 Tens: because a ten has been moved to the units column the 3 is crossed out and replaced by a 2. Therefore 2 -2 = 0. Hundreds: 1 - 0 = 1; write down the 2

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Multiplication

multiply

lots of

multiples of

groups of

times

repeated addition product

of

multiplied by

Mental Methods Multiplication Tables. There’s no escaping them – these are still vital! By the end of Year4 all facts up to 10 x 10 should be secure. However, knowing tables facts is not just a matter of chanting tables: it is important to know that 6x8 is 48 or that 48 ÷ 8 is 6 without having to recite the rest of the 8 times table in order to reach the answer. A good knowledge of tables facts enables pupils to concentrate more fully on grasping new methods while still producing accurate work. Using memorised tables facts Combining knowledge of tables with multiplying by powers of 10 so, for example: 4 x 8 = 32 so 4 x 80 = 320 Repeated addition. Multiplication can be seen as repeated addition. 2+ 2 + 2 + 2 + 2 = 10 is the same as 5 lots of 2 = 10 or 5 x 2 = 10. Number lines

0

4

A number line can be used to support repeated addition.

8

4 x 3 can be done by counting up groups of 4.

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Arrays. These are a systematic arrangement of objects, usually in rows and columns. 5x3= X X X X X X X X X X = 15 X X X X X 8

Partitioning numbers and multiplying each part of the number So, for example: 64 x 2= 64 60 + x2

120

x

1. Split 64 into tens and units: 60 + 4

2

2. Multiply both of the numbers by 2: 60 x 2 = 120 4x2=8

4 x2

+

8

= 128

3. Finally add the unit and tens together: 120 + 8 = 128

64 x 2 = 128

Written Method 1) The Grid method. The grid method builds on the use of partitioning to multiply two-digit by one-digit numbers and then three-digit by one-digit numbers. Example 1

In example 1 the 45 has been partitioned into 40 and 5 Each part is then multiplied by 3. 40 x 3 =120 and 5 x 3 =15 The products are then added back together to give the answer. 120 + 15 = 135

45 x 3 = X

40

5

3

120

15

135

45 x 3 = 135 Example 2

In example 2 a three-digit number multiplied in the same way. The 263 has been partitioned into 200, 60 and 3. Each part is then multiplied by 4. 200 x 4 =800, 60 x 4 =240 and 3 x4 =12 The products are then added back together to give the answer. 800 + 240 + 12 = 1052

263 x 4 = X

200

60

3

4

800

240

12

1052

263 x 4 = 1052

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2) The Standard Compact Method In example 1 a two digit number is multiplied a single digit Starting with the units, multiply each of the numbers on the top row by 4. 9 x 4 = 36 - write in the 6 (unit) and carry the 3 (tens) across to the tens column. 1x4=4+3=7

Example 1

19 x 4 = 1 9x 4 7 6 3

19 x 4 = 76

Example 2

In example 2 a three digit number is multiplied a single digit Starting with the units, multiply each of the numbers on the top row by 4. 5 x 4 = 20 - write in the 0 (unit) and carry the 2 (tens) across to the tens column. 4 x 4 = 16 + 2 = 18 - write in the 8 (tens) and carry the 1 (tens) across to the hundreds column. 3 x 4 = 12 + 1 = 13 - write in the 3 (hundreds) and carry the 1 (thousand) across to the thousands column.

345 x 4 = 3 4 5x 4 13 8 0 1 1 2

345 x 4 = 1380

Both the Grid Method and the Standard Compact Method can be used in later years for

long multiplication. e.g. 345 x 23

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Division

divided by

share equally share

divide

division

equal groups of

divided into

Mental Methods Using memorised tables facts: from knowing that 6 x 4 = 24 you can deduce that 24 ÷ 4 = 6 and 24 ÷ 6 = 4 Grouping Using jottings to support mental calculation. Drawing pictures to represent the calculation can be useful. 15 ÷ 5 means that 15 needs to be split into 5 equal groups. 15 ÷ 5 = 3 15 ÷ 5 = 3 Number lines +5 0

+5 5

The use of the number line depends on understanding that division is related to repeated addition; so to divide 15 by 5 you can add in groups of 5 repeatedly until you reach 15. Count how many groups of 5 you have to find the answer.

+5 10 15

15 ÷ 5 = 3

Chunking 52 ÷ 4 = 13 In this example repeated addition is still used, however, the first ten groups have been chunked together. There are 13 groups of 4 in 52, therefore 52 ÷ 4 =13

10 lots of 4 +4 0

40

+4 44

+4 48

52

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Expanded Methods Short division

As you look at this division sum, ask: ‘How many twos in twenty four?’ or ‘What is 24 divided by 2?

Example 1.

24

2

2 )2 4 1 2

or

Rewrite the question, either way works.

1 2 2)2 4

Divide each digit of 24 by 2. 2÷2=1 4÷2=2 so 24 ÷ 2 = 12

Example 2.

665

7

7 )6 66 35 0 9 5

or

0 9 5 1 7) 6 6635 7

In example 2 it is necessary to use remainders during the calculation At the very start 7 does not fit into 6, so we say it goes in zero times remainder 6. The remainder (6) is placed next to the following number to make 66. The process the process continues, but this time it is how many times 7 goes into 66, which is 9 times (because 7 goes into 63) remainder 3. Continue this process until you reach the end of the sum.

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Times Tables There is no getting away from it, learning times table is very important. Below are the times tables that you should try to know by the end of Year 4 and some helpful hints that might make learning them easier.

The Best Trick Remember that every multiplication has a twin, which may be easier to remember. For example if you forget 8×5, you might remember 5×8. This way, you only have to remember half the table.

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The 2 times table. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20....... Remember, all numbers in the 2 times table are even. All even numbers can be divided by 2.

The 3 times table. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30....... In the 3 times table, the digits will add up to a multiple of 3. For example: 3 x 6 = 18. The digits of 18 are 1 and 8. 1+8=9 9 is a multiple of 3. (3 x 3 = 9) Another example: 3 x 7 = 21 The digits of 21 are 2 and 1. 2 + 1 = 3. 3 is a multiple of 3. (3 x 1 = 3) This rule can help you with division too. If you add the digits of a number together and the answer is a multiple of 3 (in the 3 times table), the number can be divided by 3. E.g. 351

=3+5+1=9

So 351 can be divided by 3 The 4 times table. 4, 8, 12, 16, 20, 24, 28, 32, 36, 40...... Remember, all numbers in the 4 times table are even. Double it and double again! Look at 4 as 2 x 2, and since 2’s are doubles, just double the number then double that answer. For example 4x6 6 x 2 = 12 or 6 + 6 = 12 12 x 2 = 24 12 + 12 = 24 4 x 6 = 24 14

The 5 times table. 5, 10, 15, 20, 25, 30, 35, 40, 45, 50...... Any number multiplied by 5 will always end 0 or 5 The 6 times table. 6, 12, 18, 24, 30, 36, 42, 48, 54, 60...... Every number in the 6 times table is even and if you add the digits together they will always give you a multiple of 3. 6 x 7 = 42 4+2=6

6 is a multiple of 3 (in the 3 times table).

If you multiply 6 by an even number, they both end in the same digit. Example: 6 × 2 = 12, 6 × 4 = 24, 6 × 6 = 36, etc The 7 times table. 7, 14, 21, 28, 35, 42, 49, 56, 63, 70...... The 7 times table is one of the harder ones to learn, but this little trick below can help.

0

1

2

2

3

4

4

5

6

1. Start with a 3 x 3 grid. 2. Fill in the numbers as shown. 3. Starting from the top right hand corner, put in the numbers 1 -9 going down the page. 4. You are now left with the 7 times table.

07 14 21 28 35 42 49 56 63

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The 8 times table. 8, 16, 24, 32, 40, 48, 56, 64, 72, 80...... The numbers in the 8 × table are always even. This means they can be divided by 2 without remainder. If it’s an odd number then it is not in the 8 × table! Have a look at the 8 × table again. The unit digits have a regular pattern - they go down in 2s. 8 × 4 is the same as 4 × 8 (= 32) so you can use the 4 × table if you know it better.

The 9 times table. 9, 18, 27, 36, 45, 54, 63, 72, 81, 90...... Notice how the "units" place goes down: 9,8,7,6, ...? And at the same time, the "tens" place goes up: 1,2,3,...? If you add the answer's digits together, you get 9. Example: 9×5=45 and 4+5=9. (But not with 9×11=99) 1. Hold your hands in front of you with your fingers spread out. 2. For 9 X 3 bend your third finger down. (9 X 4 would be the fourth finger etc.) 3. You have 2 fingers in front of the bent finger and 7 after the bent finger. 4. Thus the answer must be 27. 5. This technique works for the 9 times tables up to 10. The 10 times table. 10, 20, 30, 40, 50, 60, 70, 80, 90, 100...... Any number multiplied by 10 will always end with a 0

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