MathsStudy Notes

Year 62019/20

Maths Notesweek 1

Mr. Duncan Mallia

Place Value

Place value determines the value of a digit in a number, based on thelocation of the digit.

4 3 2 7 . 8 9 5

4 thousand (4000)3 hundred (300)2 tens (20)7 units/ones (7)8 tenths (0.8)9 hundredths (0.09)5 thousandths (0.005)

The decimal point separates whole numbers from decimals (pieces).

TH H T U t h th

whole numbers

decimals

Rounding numbers to the nearest 10, 100 or 1000

6 5 3 9

Number in words: six thousand, five hundred and thirty-nine.

When rounding to the nearest 10 we must look at the tens digit, whenrounding to the nearest 100 we must look at the hundreds digit whilewhen rounding to the nearest 1000, we must look at the thousandsdigit. Then always remember the easy trick!

So,6539to the nearest 10 is 6540 6539to the nearest 100 is 6500 6539to the nearest 1000 is 7000

TH H T U

Rounding numbers to the nearest whole number, tenth or hundredth

4 . 1 7 3

When rounding to the nearest whole number, always look at the unitsdigit. On the other hand, when rounding to the nearest tenth, always look at the tenths digit and when rounding to the nearest hundredth look at the hundredths digit. Then use the rounding trick again to get your answer.

So,4.173 to the nearest whole number is 44.173 to the nearest tenth is 4.24.173 to the nearest hundredth is 4.17

U t h th

Remember

Any zeros that are after the decimal point at the very end can be removed.

Examples:

5.70 = 5.7

8.30 = 8.3

2.340 = 2.34

6.020 = 6.02

9.0 = 9

28.0 = 28

10.002 = 10.002

Dividing by 10

When you divide by a whole number, the answer is smaller than theoriginal number.

Method 1: When you divide a number by 10, the digits move one placevalue to the right (forwards).

736 ÷ 10 = 73.6

H T U t hhundreds tens units tenths hundreds

7 3 6

7 3 6

Dividing by 10

Method 2: Another method to use when dividing by 10 is to move thedecimal point to the left. If you are dividing by 10, move the decimalpoint one place to the left (backwards).

736 ÷ 10 = 73.6

7 3 6 . = 73.6

Remember that if you have a whole number and you cannot see a decimal point, always put it at the very end.

Dividing by 100

When you divide by a whole number, the answer is smaller than theoriginal number.

Method 1: When you divide a number by 100, the digits move twoplace value to the right (forwards).

736 ÷ 100 = 7.36

H T U t hhundreds tens units tenths hundreds

7 3 6

7 3 6

Dividing by 100

Method 2: Another method to use when dividing by 100 is to movethe decimal point to the left. If you are dividing by 100, move thedecimal point two places to the left (backwards).

736 ÷ 100 = 7.36

7 3 6 . = 7.36

Remember that if you have a whole number and you cannot see a decimal point, always put it at the very end.

Multiplying by 10

When you multiply by a whole number, the answer is bigger than theoriginal number.

Method 1: When you multiply a number by 10, the digits move oneplace value to the left (backwards).

73.6 x 10 = 736

H T U t hhundreds tens units tenths hundreds

7 3 6

7 3 6

Multiplying by 10

Method 2: Another method to use when multiplying by 10 is to movethe decimal point to the right. If you are multiplying by 10, move thedecimal point one place to the right (forwards).

73.6 x 10 = 736

7 3 . 6 = 736

Remember that if you have a whole number and you cannot see a decimal point, always put it at the very end.

Multiplying by 100

When you multiply by a whole number, the answer is bigger than theoriginal number.

Method 1: When you multiply a number by 100, the digits move twoplace value to the left (backwards).

7.36 x 100 = 736

H T U t hhundreds tens units tenths hundreds

7 3 6

7 3 6

Multiplying by 100

Method 2: Another method to use when multiplying by 100 is to movethe decimal point to the right. If you are multiplying by 100, move thedecimal point two places to the right (forwards).

7.36 x 100 = 736

7 . 3 6 = 736

Remember that if you have a whole number and you cannot see a decimal point, always put it at the very end.

Remember

When dividing a multiple of 10, 100, 1000, etc, we cross out the zeros from the back of the number.

Examples:

54Ø ÷ 1Ø = 54

16ØØ ÷ 1ØØ = 16

831Ø ÷ 1Ø = 831

535ØØ ÷ 1ØØ = 535

9ØØØ ÷ 1ØØØ = 9

8760Ø ÷ 1Ø = 8760

400ØØ ÷ 1ØØ = 400

Maths Notesweek 2

Mr. Duncan Mallia

Multiplying and dividing

Remember

Multiplication is the inverse of division and vice versa.

Examples:

5 x 6 = 30 30 ÷ 6 = 5

7 x 8 = 56 56 ÷ 8 = 7

4 x 9 = 36 36 ÷ 4 = 9

8 x 3 = 24 24 ÷ 8 = 3

45 ÷ 9 = 5 9 x 5 = 45

28 ÷ 7 = 4 7 x 4 = 28

36 ÷ 6 = 6 6 x 6 = 36

80 ÷ 10 = 8 10 x 8 = 80

Important: Multiplying and dividing any number by 0 makes 0.15 x 0 = 024 ÷ 0 = 0

Doubling

Different methods for doubling numbers:

double 37

Method 1: 37 + 37 = 74

Method 2: 37 x 2 = 74

Method 3: double 30 = 60double 7 = 1460 + 14 = 74

Method 4: double 35 = 70double 37 = 70 + (double 2) = 74

Doubling

Different methods for doubling numbers:

double 59

Method 1: 59 + 59 = 118

Method 2: 59 x 2 = 118

Method 3: double 50 = 100double 9 = 18100 + 18 = 118

Method 4: double 60 = 120double 59 = 120 - (double 1) = 118

Halving

Different methods for halving numbers:

half of 420

Method 1: 420 ÷ 2 = 210

Method 2: half of 400 = 200half of 20 = 10200 + 10 = 210

Important: Use same methods to double and half numbers with decimal points.Examples: double 2.4 = 4.8 double 4.6 = 9.2

half of 10.6 = 5.3 half of 11.8 = 5.9

2 11.8 5.9

Maths Notesweek 3

Mr. Duncan Mallia

Multiplying

One of the most basic ways to multiply is to add. When we multiplytwo numbers we are simply adding one number to itself a certainnumber of times. For example, if the problem says to multiply 20 x 5,you are adding the number 20 to itself five times.

So, 20 + 20 + 20 + 20 + 20 = 100, or 20 x 5 = 100.

When multiplying larger numbers we can use various methods.

Remember, choose the method that you prefer most!The most important thing is that you get your answers right!

Let’s see some methods together!

Multiplying

37 x 56

Method 1:

Add your answers: 1500 + 350 + 180 + 42 = 2072

30 750 1500 3506 180 42

Multiplying

37 x 56

Method 2:

Add your answers: 1850 + 222 = 2072

3750

376

1850 222

x x

Multiplying

37 x 56

Method 3:

Multiply both sides: 30 x 50 = 150030 x 6 = 1807 x 50 = 3507 x 6 = 42

Add your answers: 1500 + 180 + 350 + 42 = 2072

30 7 50 6

Multiplying by nearly 50, by nearly 100, by 25

Multiplying by nearly 50: 84 x 4984 x 100 = 840084 x 50 = 420084 x 49 = 4200 – 84 = 4116

Multiplying by nearly 100: 67 x 10167 x 100 = 670067 x 101 = 6700 + 67 = 6767

Multiplying by 25: 46 x 2546 x 100 = 460046 x 50 = 230046 x 25 = 1150

halving

halving

Multiplying by doubling and halvingdoubling

halving

x2 x4 x8 x16 x32

2 4 8 16 32

4 8 16 32 64

6 12 24 48 96

8 16 32 64 128

10 20 40 80 160

12 24 48 96 192

14 28 56 112 224

16 32 64 128 256

18 36 72 144 288

20 40 80 160 320

Multiplying by doubling and halvingdoubling

halving

x6 x12 x24

3 x 6 = 18 3 x 12 = 36 3 x 24 = 72

x7 x14 x28

5 x 7 = 35 5 x 14 = 70 5 x 28 = 140

x8 x16 x32

7 x 8 = 56 7 x 16 = 112 7 x 32 = 224

x9 x18 x36

8 x 9 = 72 8 x 18 = 144 8 x 36 = 288

Maths Notesweek 4

Mr. Duncan Mallia

Fractions

Fractions

Fractions

Finding fractions of quantities

¾ of 96

Step 1: Divide the number by the denominator.

4 96

24

Step 2: Multiply the answer by the numerator.

24 x3

72

Therefore,

¾ of 96 = 72

Finding fractions of quantities

⅝ of 280

Step 1: Divide the number by the denominator.

8 280

35

Step 2: Multiply the answer by the numerator.

35 x5

175

Therefore,

⅝ of 280 = 175

Equivalent Fractions by Dividing

Equivalent Fractions by Multiplying

Simplifying a fraction

To simplify a fraction find a common factor of both the numeratorand the denominator and divide the fraction by that factor.

Example: Simplify .

In this case, 5 is a common factor of both 15 and 25.

We can say that is a fraction in its simplest form because there areno common factors for 3 and 5. A fraction in its simplest form is alsocalled a fraction in its lowest terms.

Remember Find the largestpossible number to divide the numerator and the denominator to simplify the fraction at once.

Ordering fractions

Before putting fractions in order, one must first make the fractions with the same denominator so they can be compared.

Example: Put these fractions in order smallest first.

Now that the fractions have a common denominator, one can orderthem starting from the smallest first. Remember in your answer youmust write the fractions given in the question.

Comparing Fractions with different denominators

Watch this:

https://www.youtube.com/watch?v=KNdUJQ_qd4U

Finding the midpoint between two fractions

To find the midpoint between two fractions always start bymultiplying the fraction by 2. If the fraction is a mixed fraction, donot multiply the whole number.

Example: What is the midpoint between ?

Therefore we can say that the midpoint between

Maths Notesweek 5

Mr. Duncan Mallia

Bonds to 10

Adding to make 100 and 1000

100 1000

66 + 34 = 100 660 + 340 = 1000

53 + 47 = 100 530 + 470 = 1000

88 + 12 = 100 880 + 120 = 1000

71 + 29 = 100 710 + 290 = 1000

25 + 75 = 100 250 + 750 = 1000

16 + 84 = 100 160 + 840 = 1000

52 + 48 = 100 520 + 480 = 1000

33 + 67 = 100 330 + 670 = 1000

Remember: When adding to 100, the units (ones) add up to 10 (pair of 10).

Example: 43 + 57 = 100 (3 + 7 = 10)

Adding to make 10 and €10 (with decimal point)

10 €10

3.7 + 6.3 = 10 €3.26 + €6.74 = €10

4.2 + 5.8 = 10 €7.79 + €2.21 = €10

6.7 + 3.3 = 10 €8.81 + €1.19 = €10

8.5 + 1.5 = 10 €5.27 + €4.73 = €10

9.8 + 0.2 = 10 €4.01 + €5.99 = €10

2.9 + 7.1 = 10 €2.55 + €7.45 = €10

1.4 + 8.6 = 10 €1.33 + €8.67 = €10

5.3 + 4.7 = 10 €9.18 + €0.82 = €10

Remember: When adding to 10, the numbers after the point (tenths) add up to 10.

When adding to €10, the numbers after the point (tenths and hundredths) add up to 100.

What should be added to this number to make 100?

64 + ? = 100

Methods:

100 –6436

Counting on…

64

6 + = 36

OR 70 100

30

What should be added to this number to make 10?

1.5 + ? = 10

Methods:

100 –1585

Counting on…

15

5 + = 85

OR 20 100

80

Let’s pretend that 1.5 is 15 and we want to write the pair to 100.

8.5

What should be added to this number to make 10?

9.1 + ? = 10

Methods:

100 –919

Counting on…

91

9

OR 100

Let’s pretend that 9.1 is 91 and we want to write the pair to 100.

0.9

What should be added to this number to make €10?

€6.23 + ? = €10

Methods:

10.00 –6.233.77

Counting on…

6.23

0.77 + = 3.77

OR 7.00 10.00

3.0 3.77

What should be added to this number to make 300?

140 + ? = 300

Methods:

300 –140160

Counting on…

140

60 + = 160

OR 200 300

100 160

What should be added to this number to make 1200?

750 + ? = 1200

Methods:

1200 –750450

Counting on…

750

50 + = 450

OR 800 1200

400 450

Sam the cat weighs 3.54 kg. How many grams must it gain to weigh 4 kg?

3.54 kg + ? = 4 kg

Methods:

4.00 –3.540.46

Counting on…

3.54

0.46

OR 4.0

0.46

14.2cm + ? = 21cm

Methods:

21.0 –14.27.8

Counting on…

14.2

0.8 + = 6.8

OR 15.0 21.0

6.0 6.8cm

The sun flower is 14.2cm tall.How many cm must it grow to be 21cm tall?

We are going to set out a calculation in a column.

We are going to do

35 + 12T U

3 5+ 1 2

The biggest number

always goes on the top!

Now, we’re ready to

add…

First, we add together the units, and write the total in the units column…

7

Next, we add together the tens, then we write the total in the tens column…

4

Let’s get tougher…

This time, we’re going to try H T U!

H T U

3 2 8+ 1 2 3

1541

Let’s add four numbers together.

T U

9 97 32 8

+ 1 4

4122

Take away 99 from 157

H T U

1 5 7- 9 9

85

.1

4. 1

Methods:

Counting on…

99

1 + = 58

100 157

57

OR

€58 - €2.99 = ?

58.00- 2.99

10

.19.

Methods:

Counting on…

2.99

0.01 + = €55.01

3.00 58.00

55.00

OR

.5

7

5

Maths Notesweek 6

Mr. Duncan Mallia

The multiples of anumber are thosenumbers which arefound in the timestable of thatparticular number.

The multiples of 2for example are2, 4, 6, 8, 10,etc. The multiplesare always equalto or larger thanthe number given.

Common MultipleA multiple that 2 or more numbers

have in common.

Example:

Find the first 3 common multiples of 3 and 6.

Common Multiples: 6, 12, 18

3, 6,9,12, 15, 18…3:6, 12, 18, 24, 30,36…6:

Least Common Multiple (LCM)

14, 28, 42 and 70 are allcommon multiples of 2 and 7but the lowest commonmultiple is 14 because it is thesmallest number from all thecommon multiples.To find the common multipleof two or more numbers startby finding the multiples of thebiggest number until you finda number which is a multipleof the other numbers.

Least Common Multiple (LCM)

Properties of Numbers – Divisibility Rules

÷ 2 Is it even? Is the last digit 0,2,4,6,8?÷ 3 Is its digital total a multiple of 3?÷ 4 Do the last two digits make a multiple of 4?÷ 5 Is its units digit 0 or 5?÷ 6 Is it even and its digit total a multiple of 3?÷ 8 Is half of half of it even?÷ 9 Is its digit total a multiple of 9?÷ 10 Is its unit digit 0?÷ 25 Are the last two digits 00, 25, 50 or 75?

Maths Notesweek 7

Mr. Duncan Mallia

Multiplying Decimals

by 10 and 100

Hundreds Tens Units Tenths HundredthsThousands

To multiply by 10 all the digits move one space to the left.

6 8 9

68.9 x 10 = _____

6 8 9

689

Hundreds Tens Units Tenths HundredthsThousands

To multiply by 10 all the digits move one space to the left.

3 7 2

3.72 x 10 = _____

3 7 2

37.2

Hundreds Tens Units Tenths HundredthsThousands

1 9 5

1.95 x 100 = _____

1 9 5

195

To multiply by 100 all the digits move two spaces to the left.

Hundreds Tens Units Tenths HundredthsThousands

4 1 2

41.23 x 100 = _____

4 1 2

4123

To multiply by 100 all the digits move two spaces to the left.

3

3

Dividing Decimals

by 10 and 100

Hundreds Tens Units Tenths HundredthsThousands

To divide by 10 all the digits move one space to the right.

7 5 5

755 ÷ 10 = _____

7 5 5

75.5

Hundreds Tens Units Tenths HundredthsThousands

To divide by 10 all the digits move one space to the right.

2 1 7

21.7 ÷ 10 = _____

2 1 7

2.17

Hundreds Tens Units Tenths HundredthsThousands

4 1 8

41.8 ÷ 100 = ______

4 1 8

0.418

To divide by 100 all the digits move two spaces to the right.

Thousandths

0

Hundreds Tens Units Tenths HundredthsThousands

0 6

0.6 ÷ 100 = ______

0 6

0.006

Thousandths

00

To divide by 100 all the digits move two spaces to the right.

Maths Notesweek 8

Mr. Duncan Mallia

To multiply a 1-digit number by a 2 or 3-digit

number (H/TU x U) we partition the number

into separate HUNDREDS, TENS and UNITS.

Example:

4 x 72 = ?

70 + 2

(4 x 70) + (4 x 2) =280 + 8 = 288

Example:

6 x 39 = ?

30 + 9

(6 x 30) + (6 x 9) =

180 + 54 = 234

Example:

5 x 217 = ?

200 + 10 + 7

(5 x 200) + (5 x 10) + (5 x 7) =

1000 + 50 + 35 = 1085

Apart from using multiplication method by

partition, we can use the vertical method.

Example: 28 x 4 = ?

28x 4

4 x 8 324 x 2080

211

Example: 513 x 5 = ?

513x 5

5 x 3 155 x 10

2500

55

5 x 500

50

6 2

Example: 947 x 7 = ?

947x 7

7 x 7 497 x 40

6300

96

7 x 900

280

2 6

1

Example: 3578 x 3 = ?

3578x 3

3 x 8 243 x 70

900043

3 x 500210

15003 x 3000

710

Example: 6197 x 6 = ?

6197x 6

6 x 7 42 6 x 90

3600028

6 x 100540600

6 x 6000

137

1

Example: 185 x 29 = ?

185x 29

9 x 5 459 x 80

2000

55

9 x 100720

6

2 20 x 5

20 x 80

20 x 100

900100

1600

3

Example: 344 x 64 = ?

344x 64

4 x 4 164 x 40

1800062

4 x 300160

1

1

60 x 4

60 x 40

60 x 300

1200240

2400

0

1

2

1

Long Multiplication

Watch these:

https://www.youtube.com/watch?v=FJ5qLWP3Fqo

https://www.youtube.com/watch?v=RVYwunbpMHA

Factors & Square Numbers

Factors are the numbers you multiply together to get another number.

When you multiply a whole number times itself, the resulting product is called a Square number, or a perfect square or simply "a square.“

So 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 are all square numbers.

Maths Notesweek 9

Mr. Duncan Mallia

Multiplication & Division are inverse operations.

Example:

2 x 4 = 8

So,

8 ÷ 2 = 4

8 ÷ 4 = 2

Use the same three

numbers to find

division facts.

Multiplication:

1.2 x 6 =

Division:

7.2 ÷ 1.2 = 6

7.2 ÷ 6 = 1.2

Use the same three numbers

to find division facts.

1.2

x 66 x 0.26 x 1.0

1.26.07.2

7.2

Multiplication and Division Fact Family Triangles

Multiply the bottom two numbers to give the top number.

8 3x

24

Multiplication and Division Fact Family Triangles

Multiply the bottom two numbers to give the top number.

5 30x

150

7 25

175

x

Multiplication and Division Fact Family Triangles

When one of the bottom numbers is unknown, divide the

top number by the one at the bottom.

6÷120

20

120 ÷ 6 = ?

Now let’s have a look at all the Multiplication and Division Facts

6÷120

20Multiplication:

6 x 20 = 120

20 x 6 = 120

Division:

120 ÷ 6 = 20

120 ÷ 20 = 6

÷

x

Multiplication and Division Fact Family Triangles

When one of the bottom numbers is unknown, divide the

top number by the one at the bottom.

4÷

200

50

200 ÷ 4 = ?

Multiplication and Division Facts

Multiplication:

4 x 50 = 200

50 x 4 = 200

Division:

200 ÷ 4 = 50

200 ÷ 50 = 4

4

200

50÷ ÷

x

What is a Remainder?

• Sometimes when dividing there is something left over.

• It is called the remainder.

Example 1:

There are 7 bones to share with 2 puppies.

But 7 cannot be divided exactly into 2 groups,so each puppy gets 3 bones,and there is 1 left over:

Remainder as a fraction

• We say:

• "7 divided by 2 equals 3 with a remainder of 1"

• And we write:

• 7 ÷ 2 = 3 R 1

It is also possible to cut the remaining bone in half.

7 ÷ 2 = 3 R 1 = 3 1

2bones

1 out of 2

NOTE: The remainder on top,

and the number you are dividing

by at the bottom.

• Work out 19 ÷ 5

• 19 cannot be divided exactly by 5. The closest we can get (without going over) is:

• 3 x 5 = 15 which is 4 less than 19.

Example 2:

= 3 𝟒

𝟓

4 out of 5

Remainder as a fraction should be done with:

• the remainder on top, and• the number you are dividing by at the bottom.

So the answer of 19 ÷ 5 = 3 R 4

• Work out 55 ÷ 4

Example 3:

= 13 𝟑

𝟒

The fraction should be done with:

• the remainder on top, and• the number you are dividing by at the bottom.

55 ÷ 4 = 13 R 3

86 cookies are shared among ten children.

How many cookies do they get each?

Example 4:

= 8 𝟔

𝟏𝟎𝐜𝐨𝐨𝐤𝐢𝐞𝐬86 ÷ 10 = 8 R 6

Remainder as a fraction

Remainder as a decimal

8 𝟔

𝟏𝟎= 8.6

The confectioner has 789 cupcakes. Boxes hold 100 cupcakes.

How many boxes do I need to hold all the cupcakes?

Example 5:

= 7 𝟖𝟗

𝟏𝟎𝟎𝐜𝐮𝐩𝐜𝐚𝐤𝐞𝐬789 ÷ 100 = 7 R 89

Remainder as a fraction

Remainder as a decimal= 7.897 𝟖𝟗

𝟏𝟎𝟎

44 children are going on a picnic, by car. Each car holds 5 children.

How many trips must the car make?

Example 6:

= 8 𝟒

𝟓𝐭𝐫𝐢𝐩𝐬44 ÷ 5 = 8 R 4

Remainder as a fraction

Remainder as a decimal= 8.8= 8 𝟖

𝟏𝟎8 𝟒

𝟓

x2

x2

Converting Fractions to Decimals

Watch these:

https://www.youtube.com/watch?v=WV5VY76Pf5U

https://www.youtube.com/watch?v=do_IbHId2Os

Maths Notesweek 10

Mr. Duncan Mallia

These numbers are called

whole numbers

115

6

3

47

22

These numbers are called

Proper fractions

1

2 3

15

5

6

2

34

7

12

22

These numbers are called

Improper fractions

4

2

25

15 7

6

9

310

7

30

22

An improper fraction has a top number larger than (or equal to) the bottom number.

It is "top-heavy"

When a number includes both whole numbersand fractions.

1

What is it called?

It is called a

mixed number

1

2

Here are some

mixed numbers

32

51

46

8

1

3

4

9

1 2

2 5

Mixed numbers and Improper Fractions

Changing fractions from Mixed to Improper fractions

To change fractions from mixed to improper, get M.A.D.!!

Step 1: Multiply the whole number by the denominator.

= 3 x 4 = 12

Step 2: Add the answer to the numerator. 12 + 1 = 13

Step 3: Denominator remains the same as before. 134

96

Convert mixed number to improper fraction

3

6=

6 x 1 = 6

6 + 3 =

Keep the denominator the

same.

755 x 1 = 5

5 + 2 =2

5=

Keep the denominator the

same.

Convert mixed number to improper fraction

733 x 2 = 6

6 + 1 =

1

3=

Keep the denominator the

same.

Convert mixed number to improper fraction

234

4 x 5 = 20

20 + 3 =3

4=

Keep the denominator the

same.

Convert mixed number to improper fraction

Convert improper fractions to mixed fractions

• Divide the numerator by the denominator.

• Write down the whole number answer.

• Then, write down any remainder above the denominator.

Example:

14 ÷ 5 =𝟏𝟒

𝟓= 2

𝟒

𝟓divide by 5

This is the same as this...

Convert improper fractions to mixed fractions

30 ÷ 8 =𝟑𝟎

𝟖= 3

𝟔

𝟖divide by 8

This is the same as this...

Convert improper fractions to mixed fractions

56 ÷ 10 =𝟓𝟔

𝟏𝟎= 5

𝟔

𝟏𝟎divide by 10

This is the same as this...

Convert improper fractions to mixed fractions

34 ÷ 6 =𝟑𝟒

𝟔= 5

𝟒

𝟔divide by 6

This is the same as this...

Tenths, Hundredths, Thousandths

N22

Year 6

There are:

4 tenths

5 hundredths

6 thousandths

3.174

There are:

1 tenth

7 hundredths

4 thousandths

tenth hundredth thousandth

0.906

There are:

9 tenths

0 hundredths

6 thousandths

tenth hundredth thousandth

Good to know that:

1cm (centimetre) = 10mm

1dm (decimetre) = 10cm = 100mm

1m (meter) = 10dm = 100cm = 1000mm

The smallest unit of measurement

Write the number of metres.

4dm, 5cm and 7mm

Note: The smallest unit of measurement is mm

1m = 1000mm

1dm = 10cm = 100mm4dm = 400mm

So, convert every measurement to mm

5cm = 50mm

7mm

400 + 50 + 7 = 457mm

Answer: 0.457m

dm cm mm

457 ÷ 1000 = 0.457m

Write the number of metres.

6dm, 4cm and 2mm

Note: The smallest unit of measurement is mm

1m = 1000mm

1dm = 10cm = 100mm6dm = 600mm

So, convert every measurement to mm

4cm = 40mm

2mm

600 + 40 + 2 = 642mm

Answer: 0.642m

642 ÷ 1000 = 0.642m

Write the number of metres.

9dm and 1mm

Note: The smallest unit of measurement is mm

1m = 1000mm

1dm = 10cm = 100mm9dm = 900mm

So, convert every measurement to mm

1mm

900 + 1 = 901mm

Answer: 0.901m

901 ÷ 1000 = 0.901m

Write the number of metres.

8cm and 8mm

Note: The smallest unit of measurement is mm

1m = 1000mm

1cm = 10mm8cm = 80mm

So, convert every measurement to mm

8mm

80 + 8 = 88mm

Answer: 0.088m

88 ÷ 1000 = 0.88m

Good to know that:

1cl (centilitre) = 10ml

1dl (decilitre) = 10cl = 100ml

1l (litre) = 10dl = 100cl = 1000ml

re

re

re

re

Write the value of 6.

0.516l

l mldl cl

6ml

6

1000l

or

0.328l

Write the value of 2.

20

1000l =

2

100l

l mldl cl

or

2cl

Write the value of 4.

0.460l 0.842l

l mldl cl l mldl cl

400

1000l =

4

10l 4dlor

40

1000l =

4

100l 4clor

Good to know that:

1kg = 1000g

Write the number of kilograms.

9g = ? kg

9 ÷ 1000 = 0.009 kg

63g = ? kg

63 ÷ 1000 = 0.063 kg

Write the number of kilograms.

280g = ? kg

280 ÷ 1000 = 0.280 kg

56g = ? kg

56 ÷ 1000 = 0.056 kg

Ordering decimalsWhen we order decimals we are going to compare digit

by digit.

0. 4 5 6 7

0. 4 8 1

For example, let’s compare these two numbers:

First we compare the whole number in front of

the decimal

These are the same, so we move on.We now compare the first number after the

decimal point.

These are again the same, so we move on to the

second decimal place.

We now have one decimal bigger than the other

so we can say that the whole number is bigger

We can say that 0.481 is larger than 0.4567

We now have one decimal bigger than the other

so we can say that the whole number is bigger.

Order these from the smallest to the largest?

0.0456 0.123

0.04569 0.4

0.0456, 0.04569, 0.123, 0.4

In order from smallest to largest

Order these from the smallest to the largest?

5.5 5.455.504

5.45, 5.5, 5.504

In order from smallest to largest

Order these from the smallest to the largest?

3.91 3.193.091

3.091, 3.19, 3.91

In order from smallest to largest

Order these from the smallest to the largest?

1.003 1.031.303

1.003, 1.03, 1.303

In order from smallest to largest

Ordering Decimals

Watch these:

https://www.youtube.com/watch?v=lnHvuxp5Q9c

https://www.youtube.com/watch?v=cw9RCCx9Rs8

Maths Notesweek 11

Mr. Duncan Mallia

Measuring Length

When we measure length, we find the size of the dimensions ofobjects. Some tools which measure length include ruler and measuringtape. Use a table to convert from one unit of length to another.

Remember well:

¼km = 0.25km = 250m

½km = 0.5km = 500m

¾km = 0.75km = 750m

¼m = 0.25m = 25cm

½m = 0.5m = 50cm

¾m = 0.75m = 75cm

½cm = 0.5cm = 5mm

Converting Units of Measure

Multiply or divide the measurements accordingly to convert from one unit to the other.

Measuring Weight

When we measure weight, we find out how heavy an object is. A scaleis used to measure weight. Use a table to convert from one unit ofweight to the other.

Remember well:

¼kg = 0.25kg = 250g

½kg = 0.5kg = 500g

¾kg = 0.75kg = 750g

Converting Units of Measure

Multiply or divide the measurements accordingly to convertfrom one unit to the other.

Maths Notesweek 12

Mr. Duncan Mallia

Bar-line graphs

A bar-line graph is similar to a bar graph but the bars are replacedby lines instead. Bar-line graphs are used to represent different data.

bar-line graphbar graph

Favourite sport Eye colour

sports colours

Bar-line graphs

Bar graphs and bar-line graphs (or charts) must have a title. The titleoffers a short explanation of what is in your graph. This helps thereader identify what they are about to look at. It can be creative orsimple as long as it tells what is in the graph.

Bar graphs and bar-line graphs have anx-axis and a y-axis. In most bar graphs,the x-axis runs horizontally, whereas,the y-axis runs vertically. The y-axisusually starts counting at 0 and can bedivided into as many equal parts as youwant to.

Line graphs

A line chart or line graph is a type of chart which displaysinformation as a series of data points called 'markers' connected bystraight line segments. It is a basic type of chart common in manyfields. It is used to show change over time.

A line graph always has a title. Ithas two different axes; onehorizontal and one vertical. It isimportant to read what informationis on each axis. There must be alabel on each axis that describesthe information.Much like a bar graph, firstly welook at the horizontal axis, to plotour points.

Line graphs

Learn to draw and interpret line graphs.

Miles traveled during the week

0

10

20

30

40

50

60

70

Mon. Tues. Wed. Thurs. Fri. Sat. Sun.

Days

Mile

s T

rav

ele

d

Conversion graphs

A conversion graph can be used to change one quantity to anotherwhen the units are changed. Here are some examples:

Converts pounds and francs Converts inches and centimetres

Convert 50% to Marks

A conversion graph

010

20304050

607080

90100

0 5 10 15 20 25 30 35 40 45 50 55 60

Marks

Pe

rce

nta

ge

Step 1: Find 50% on the vertical axis and draw a straight line to meet the conversion line on the graph.

Step 2: Draw a straight line to the other axis. This position gives the conversion value.

A conversion graph

010

20304050

607080

90100

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40

Inches

Ce

nti

me

tre

s

Convert: 12 inch = ____ centimetres

24 inch = ____ centimetres

32 inch = ____ centimetres

30

60

80

A conversion graph

0

10

20

30

40

50

60

70

80

0 5 10 15 20 25 30 35 40 45 50

Miles

Kilo

me

tre

s

Convert: 35 miles = ____ kilometres

25 miles = ____ kilometres

50 miles = ____ kilometres

55

40

80

The Mean

The mean is the average of the numbers. It is easy to calculate:

Step 1: add up all the numbersStep 2: divide by how many numbers there are

In other words, it is the sum divided by the count.

The Mean

The mean is the average of the numbers. It is easy to calculate:

Step 1: add up all the numbersStep 2: divide by how many numbers there are

In other words, it is the sum divided by the count.

Maths Notesweek 13

Mr. Duncan Mallia

Multi-Digit Addition and Subtraction

Watch these:

https://www.youtube.com/watch?v=mAvuom42NyY

https://www.youtube.com/watch?v=Y6M89-6106I

Maths Notesweek 14

Mr. Duncan Mallia

Area of rectangles

Area is a measure of how much space there is inside ashape. Calculating the area of a shape or surface can be useful ineveryday life, for example you may need to know how much paint tobuy to cover a wall.

Calculating Area Using the Grid MethodWhen a shape is drawn on a scaled grid you can find the area bycounting the number of grid squares inside the shape.

In this example there are 10 grid squaresinside the rectangle. (5 x 2 = 10cm2)

Area of rectangles

In order to find an area value, using the grid method, we need to knowthe size that a grid square represents.The example below uses centimetres but the same method applies forany unit of length or distance. You could, for example be using inches,metres, miles, feet etc. In this example each grid square has a widthof 1cm and a height of 1cm. In other words each grid square is one'square centimetre'. Count the grid squares inside the large square tofind its area.

There are 16 small squares so the area of the largesquare is 16 square centimetres.We abbreviate 'square centimetres' to cm2.The 2 means ‘squared’.

The area of the large square is 16cm2 (4 x 4 = 16cm2)

Areas of squares and rectangles

The simplest (and most commonly used) area calculations are forsquares and rectangles.To find the area of a rectangle multiply its height by its width.

For a square you only need to find the length of one of the sides (aseach side is the same length) and then multiply this by itself to findthe area.

Length x Breadth/Width

Calculating area of Irregular shapes

Often, in real life, shapes can be more complex. For example, imagineyou want to find the area of a floor, so that you can order the rightamount of carpet. A typical floor-plan of a room may not consist of asimple rectangle or square:

In this example, the trick is to split the shape into several rectangles (or squares). Find the area of each shape and then find the total area.

It doesn’t matter how you split the shape - any of the three solutions will result in the same answer.

Finding the length / breadth of a missing side involving Area

If you are given the breadth and the area of a rectangle and areasked to find its length, you should use the formula

Length = Area ÷ Breadth

If you are given the length and the area of a rectangle and are askedto find its breadth, you should use the formula

Breadth = Area ÷ Length

area =72cm2

Length = Area ÷ Breadth

breadth = 8cmarea = 72cm2

length = ?

72cm2 ÷ 8 = 9cm

area =100cm2

Breadth = Area ÷ Length

length = 25cmarea = 100cm2

breadth = ?

100cm2 ÷ 25 = 4cm

25cm

8cm

Area of triangles

It can be useful to think of a triangle as half of a square orparallelogram.

The area of a triangle is (Length × Breadth) ÷ 2.In other words you can work out the area of a triangle in the same way as the area for a square or parallelogram just divide your answer by 2.

Area of triangle3cm x 3cm = 9cm2

9cm2 ÷ 2 = 4.5cm2

The area of this triangle is 4.5cm2.

Finding the Length of a missing side in a triangle

Remember:The area of a rectangle is doublethe area of a triangle.

Area of rectangle = double 33cm2

66cm2

Length = Area ÷ Breadth

66cm2 ÷ 5cm = 16cm

Therefore, the length of side Xof the triangle in 16cm

area =33cm25cm

X

Area

Watch these:

https://www.youtube.com/watch?v=xCdxURXMdFY

https://www.youtube.com/watch?v=d_6XOBpoDz4

Finding the Perimeter of Regular Polygons

A polygon is a plane shape (two-dimensional) with straight sides.Examples include triangles, quadrilaterals, pentagons, hexagons and soon. A regular polygon has sides of equal length, and all its interiorangles are of equal size.

To find the perimeter of a regular polygon simply multiply the value of one of its sides by the total number of sides.

For example:If one of the sides of a regular pentagon is 6cm. To find its perimeter work out: 6cm x 5 (no. of sides) = 30cm

Finding the length of a missing side involving Perimeter

perimeter =48cm

Two lengths = 19cm + 19cm = 38cmTwo breadths = 48cm (perim) – 38cm = 10cmOne breadth = 10cm ÷ 2 = 5cm

Therefore the missing side is 5cm long

19cm

perimeter =80cm

Two breadths = 14cm + 14cm = 28cmTwo lengths = 80cm (perim) – 28cm = 52cmOne breadth = 52cm ÷ 2 = 26cm

Therefore the missing side is 26cm long

14

cm

Perimeter

Watch these:

https://www.youtube.com/watch?v=AAY1bsazcgM

https://www.youtube.com/watch?v=JAy_CETEyUM

Maths Notesweek 15

Mr. Duncan Mallia

Reflective Symmetry

When you draw a line of symmetry of a shape, you divide the shapeinto two identical parts which when folded cover each other in anexact way.

Symmetrical Shapes

Symmetrical shapes can have one or more lines of symmetry.

Reflective Symmetry

Important:

To complete the pattern without any mistakes, it is best to tackle the given shaded boxes one by one.

Reflective Symmetry

Important:

To complete the pattern without any mistakes, it is best to tackle the given shaded boxes one by one.

Reflective Symmetry

Important:

To complete the pattern without any mistakes, it is best to tackle the given shaded boxes one by one.

Symmetry

Watch these:

https://www.youtube.com/watch?v=YFzktJNmnPU

https://www.youtube.com/watch?v=MtqtIiJsfiE&t=110s

Maths Notesweek 16 - 18

Mr. Duncan Mallia

Maths Notesweek 19

Mr. Duncan Mallia

Maths Notesweek 20

Mr. Duncan Mallia

Division

Watch these:

Basic division:https://www.youtube.com/watch?v=KGMf314LUc0&t=68s

Long division:https://www.youtube.com/watch?v=LGqBQrUYua4

Long division with 2-digit divisors:https://www.youtube.com/watch?v=HdU_rf7eMTI

Maths Notesweek 21

Mr. Duncan Mallia

Capacity

Capacity is the total amount of fluid / liquid that can be contained in acontainer. It is the word we use when we are measuring liquids. Sometools which measure capacity include a measuring jug and cups.

1 litre = 1000 millilitres¼ litre = 250 millilitres½ litre = 500 millilitres¾ litre = 750 millilitres

Multiply or divide the capacity accordingly to convert from one

unit to the other

Converting between litres and centilitres

Remember:

To convert litres to centilitres multiply by 100.To convert centilitres to litres divide by 100.

1 litre = 100 centilitres2 litres = 200 centilitres3 litres = 300 centilitres

4.5 litres = 450 centilitres

Capacity

Watch these:

https://www.youtube.com/watch?v=VpRa5OY-Q-k

https://www.youtube.com/watch?v=TPFQd7ZjErU

Maths Notesweek 22

Mr. Duncan Mallia

Units of time• Weeks in a year: 1 year = 52 weeks

• Days in a normal year: 1 year = 365 days

• Days in a week: 1 week = 7 days

• Days in a fortnight: 2 weeks = 14 days

• Hours in a day: 1 day = 24 hours

• Minutes in an hour: 1 hour = 60 minutes

• Seconds in a minute: 1 minute = 60 seconds

Other terms you should know:

• Days in a leap year: 1 leap year = 366 days

• Days in a fortnight: 1 fortnight = 2 weeks = 14 days

Leap Year

Once in every four years, instead of 28 days,February has 29 days. Hence, it is a Leap Year.

How can I find if a year is a leap year?Divide the year by 4. If you get a remainder, then the year is not aleap year. If the number is exactly divisible by 4 and you get noremainder, then that year is a leap year.

Since in the first one we have a remainder, we can say that 1998 wasnot a leap year. On the other hand, when we divided the year 2000 by4, we did not get any remainder. Therefore we can say that it was aleap year.

4 1998499r2

4 2000500

Days in the months

30 Days has September,April, June and November.

All the rest have 31,Except for February, it’s the one,

Which only has 28 days clear,And 29 in each Leap Year.

Count the months on your knuckles and the grooves between yourknuckles. Leave out your thumb knuckle. Every month that lands on aknuckle is 31 days, every month that lands on a groove betweenknuckles is 30 days (or 28 for February).

Maths Notesweek 23

Mr. Duncan Mallia

12-hour and 24-hour clock times

There are two ways of telling the time: 12-hour or 24-hour clock times.

• In 12-hour clock times we use a.m. and p.m. to distinguish betweenmorning and evening. It runs from 1am to 12 noon and then from1pm to 12 midnight.

• In 24-hour clock times we use 00 to 11 hour digits to show morningtimes (instead of a.m.) and 12 to 23 hour digits to show eveningtimes (instead of p.m.) It uses the numbers 00:00 to 23:59(midnight is 00:00).

Examples: 10:30 a.m. = 10:303:50 p.m. = 15:50 (3 + 12 = 15)

6:15 p.m. = 18:15 (6 + 12 = 18)

9:00 p.m. = 21:00 (9 + 12 = 21)

clock times are converted to 24-hour by adding

12:00 to the time.

24-hour clock 12-hour clock

00:00 12:00 midnight

01:00 1:00 a.m.

02:00 2:00 a.m.

03:00 3:00 a.m.

04:00 4:00 a.m.

05:00 5:00 a.m.

06:00 6:00 a.m.

07:00 7:00 a.m.

08:00 8:00 a.m.

09:00 9:00 a.m.

10:00 10:00 a.m.

11:00 11:00 a.m.

12:00 12:00 noon

24-hour clock 12-hour clock

13:00 1:00 p.m.

14:00 2:00 p.m.

15:00 3:00 p.m.

16:00 4:00 p.m.

17:00 5:00 p.m.

18:00 6:00 p.m.

19:00 7:00 p.m.

20:00 8:00 p.m.

21:00 9:00 p.m.

22:00 10:00 p.m.

23:00 11:00 p.m.

24:00 12:00 midnight• a.m. (ante meridiem, meaning before midday) • p.m. (post meridiem, meaning past midday)

Timetables

A timetable is a detailed plan showing when events or activities willhappen; a list of the times when buses, trains, and planes leave andarrive; a list of the times when lessons in school happen, etc.

Maths Notesweek 24

Mr. Duncan Mallia

Different Types of Angles

Angles are measured in degrees ex. 24º, 90º, 115º, 180º, 230º, 360º.

The following are the six types of angles you should be familiar with:

• Acute angle (from 1º to 89º)

• Right angle (90º)

• Obtuse angle (91º to 179º)

• Angle on a straight line (180º)

• Reflex angle (181º to 359º)

• One complete turn / one whole revolution (360º)

Acute Angles (from 1º to 89º)

Right Angles (90º)

Obtuse Angles (from 91º to 179º)

Angles on a straight line (180º)

There are 2 right angles in an angle on a straight line90º + 90º = 180º

Reflex Angles (from 181º to 359º)

One complete turn / one whole revolution (360º)

There are 4 right angles in one complete turn90º + 90º + 90º + 90º = 360º

Angles in Compass Points

Remember:

When you move clockwise you move in the direction in which the handsof the clock move. When you move anticlockwise you move in theopposite direction in which the hands of the clock move.

From N, clockwise to E = 90ºFrom N, clockwise to S = 180ºFrom N, clockwise to W = 270ºFrom N, clockwise to N = 360ºFrom W clockwise to NE = 135ºFrom NW, anticlockwise to NE = 270ºFrom S, clockwise to NE = 225ºFrom NE, anticlockwise to NW = 90º

anticlockwise anticlockwise

Angles in Clocks

In one hour the minute hand (the big hand of the clock) turned anangle of 360º while the hour hand (the small hand of the clock) turnedan angle of 30º. Remember that the angle between any twoconsecutive numbers in a clock is 30º.

Therefore, the angle between 3 and 4 is 30º. The angle between 3and 5 is 60º. The angle between 3 and 6 is 90º (right angle).

1 hour passed

Angles in Clocks⅓ right angle

30º

⅔ right angle60º

Remember:

1 right angle = 90º

More about Angles

Study these:

⅓ right angle 30º

½ right angle 45º

⅔ right angle 60º

1 right angle 90º

2 right angles 180º

3 right angles 270º

4 right angles 360º

Angles

Finding Missing Angles in different Triangles

To find the missing angle:

Add 90º to the given angle.90º + 27º = 117º

Subtract the answer from 180º.180º - 117º = 63º

The missing angle is 63º.

Right-angled Triangle• 1 angle is a right

angle (90º)

?

Finding Missing Angles in different Triangles

To find the missing angle:

Add the two given angles together.80º + 60º = 140º

Subtract the answer from 180º.180º - 140º = 40º

The missing angle is 40º.

Scalene triangle• no equal sides• no equal angles

? 80º

60º

Finding Missing Angles in different Triangles

To find the missing angles:

Subtract the given angle from 180º.180º - 30º = 150º

Half the answer.150º ÷ 2 = 75º

The missing angles are 75º.

Isosceles Triangle• 2 equal sides• 2 equal angles

? ?

30º

Finding Missing Angles in different Triangles

To find the missing angle:

Double the given angle.70º x 2 = 140º

Subtract the answer from 180º.180º - 140º = 40º

The missing angle is 40º.

Isosceles Triangle• 2 equal sides• 2 equal angles

?

70º

Finding Missing Angles in Right Angles

1 right angle = 90ºTherefore the missing angle is90º - 62º = 28º

Finding Missing Angles on a Straight Line

angles on a straight line = 180ºTherefore the missing angle is180º - 77º = 103º

Finding Missing Angles in One Whole Revolution

one whole revolution = 360ºTherefore the missing angle is360º - 230º = 130º

one whole revolution = 360º63º + 50º + 75º + 110º = 298ºTherefore the missing angle is360º - 298º = 62º

Finding Missing Angles in a Perfect Cross

one whole revolution = 360º

Use the RULE: Opposite angles are equal!So I can say that angle x is 55º.

Since angles x and y are on a straight line, I can find angle y.180º - 55º = 125º

Once again use the rule that opposite angles are equal for angle z.I can say that angle z is 125º because it is opposite angle y.

Using a protractor

If you are asked to calculate the angle, you should find the angle bymeans of sums and calculations. But if you are asked to measure theangle you should use a protractor.

Make sure you know how to place the protractor on the angles, so as to measure and draw angles.

50º135º

Maths Notesweek 25

Mr. Duncan Mallia

Percentages (%)

In percentages the whole number is always 100%.

For example, 40% means 40 out of 100 =

You can change 40% into a fraction in its simplest form = =

10% = 80% = = 25% =

15% = 20% = = 50% =

60% = = 7% = 75% =

410

25

110

320

610

35

810

45

210

15

7100

14

12

34

Finding Percentages

To find a percentage of a given number:Example: 25% of 260

Write the percentage as a fraction over 100.Simplify the fraction.

25% = = ¼

Then use the fraction to find the answer

¼ of 260 = 654 260

65 x 1 = 65

When finding fractions of quantities,

remember to dividethe number by the denominator and

multiply the answer by the numerator.

Finding 10% of an amount

Examples:

• 10% of 650 650 ÷ 10 = 65

• 10% of 1150 1150 ÷ 10 = 115

• 10% of 880 880 ÷ 10 = 88

• 10% of 790 790 ÷ 10 = 79

• 10% of 100 100 ÷ 10 = 10

• 10% of 70 70 ÷ 10 = 7

To find 10% of an amount divide the

amount by 10

10100

110

Reduction in Prices including Percentages

Reduction in price = 60% of €20

of €2Ø = €12

The new price is €20 - €12 = €8

Reduction in price = 20% of €70

of €7Ø = €14

The new price is €70 - €14 = €56

6Ø1ØØ

2Ø1ØØ

SALE 20% OFF

SALE 60% OFF

Reduction in Prices including Percentages

Reduction in price = 50% of €40

of €4Ø = €20

The new price is €40 - €20 = €20

Reduction in price = 25% of €60

of €6Ø = €15

The new price is €60 - €15 = €45

5Ø1ØØ

2510Ø

SALE 50% OFF

SALE 25% OFF

Fractions Decimals Percentages

0.5 50%

0.25 25%

0.75 75%= 0.20 20%

0.10 10%

0.70 70%

It is important to study

these fractions

and decimals by heart.

Maths Notesweek 26

Mr. Duncan Mallia

Square Numbers

A square number is the answer of a number multiplied by itself.Examples: 4 x 4 = 16

9 x 9 = 81 Therefore, 16 and 81 are called square numbers.

The square numbers which we use frequently are: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.

More examples: The square number of 15 (152) is 225The square number of 30 (302) is 900

If the square number of 16 (162) is 256, what is 16 x 17?

• 16 x 16 = 256• 16 x 17 = 256 + 16 = 272• Therefore, 16 x 17 = 272

Remember 1 is the smallest square number

Sequences

31, 33, ____ , ____ , 39, ____ , 43, ____ .

Before you start writing the missing numbers, you should check if thenumbers are increasing (+) or decreasing (-) and by how much. In thiscase the numbers are increasing by 2.

31, 33, 35, 37, 39, 41, 43, 45.

3, 4, 6, ____ , 13, ____ , ____ , 31.

Pay attention because sometimes there is a pattern as to how numbersin sequences are increasing or decreasing. In the next sequence youhave a pattern. Once you realise what the pattern is, you can fill in themissing numbers.

3, 4, 6, 9, 13, 18, 24, 31.+1 +2 +3 +4 +5 +6 +7

+2 +2 +2 +2 +2 +2 +2

Even and Odd Numbers

An even number is a number that is divisible by 2 (÷2).

Ex. 2, 4, 6, 8, 10, 12, 14, etc.

If a number is not divisible by 2 it is an odd number.

Ex. 3, 5, 7, 9, 11, 13, 15, etc.

To find if a big number is odd or even look at the last digit. If thelast digit is even, the whole number is even and if the last digit is odd,the whole number is odd.

Ex. 19,356 is even because the last digit 6 is even.24,483 is odd because the last digit 3 is odd.67,272 is even because the last digit 2 is even.

even

odd

Maths Notesweek 27

Mr. Duncan Mallia

Proportions as Fractions

When writing proportions as fractions, the fractions should be in theirsimplest form.

red =

blue =

Proportion of the gridwhich is not coloured

yellow

green

1025

1025

25

25

2025

45

=

coloured

525

15

=

not coloured

610

35

=

410

25

=

Proportions - Problems

Problem 1: Chris worked for 10 days in the month of June, whatproportion of the same month did he work for?

Problem 2: Kurt has 40 sweets. 15 are chocolates. What proportionare not chocolate?

40 – 15 = 25 not chocolate

Problem 3: During the months of July and August, Maria had 18 pianolessons. What proportion of the same months she didn’t have lessons?

62 – 18 = 44 no lessons

1030

13

=

2540

=58

=

4462

=2231

=

Problem 4: Colour the proportions shown in the following grid:

orange

purple

pink

orange (colour 1 box orange)

purple (colour 4 boxes purple)

pink (colour 15 boxes pink)

120

15

34

Since in the grid there are 20 small squares, before colouring in the proportions you should

change the fractions to having a denominator 20.

120

15

420

=x4

x4

34

1520

=x5

x5

Maths Notesweek 28

Mr. Duncan Mallia

Three-Dimensional Shapes

Three-dimensional shapes are also known as solid shapes. The followingare the three-dimensional shapes (3-d) which you should be familiar to:

cylinder cube sphere prism

pyramid cuboid cone

Pyramids and Prisms

What is the difference between a pyramid and a prism?• In a pyramid the sides coming out of the base meet at a point (like

the pyramids of Egypt).• In a prism the base and the lid have the same shape.

pyramid prism

Different prisms and pyramids are given names according to the shapeof their base:

Faces, Edges and Vertices

Three-dimensional shapes have faces, edges and vertices.

Remember:

2 vertices, but 1 vertex

Faces, Edges and Vertices

Shapes Names Faces Edges Vertices

cube 6 12 8

cuboid 6 12 8

cylinder 3 2 0

cone 2 1 1

pyramid 5 8 5

sphere 1 0 0

Faces, Edges and Vertices – Pyramids and Prisms

Shape Faces Edges Vertices

Square Pyramid 5 8 5

Triangular Pyramid 4 6 4

Rectangular Pyramid 5 8 5

Pentagonal Pyramid 6 10 6

Hexagonal Pyramid 7 12 7

Octagonal Pyramid 9 16 9

Triangular Prism 5 9 6

Pentagonal Prism 7 15 10

Hexagonal Prism 8 18 12

Octagonal Prism 10 24 16

Nets of Three-Dimensional Shapes

When a three-dimensional shape is ‘opened’ the shape formed by itsfaces is called the net of that particular shape.

cube cuboidcone

cylinder

pyramid prism

Maths Notesweek 29

Mr. Duncan Mallia

Types of Triangles

Maths Notesweek 30

Mr. Duncan Mallia