MATHS Plane Shapes Plane shapes are two dimensional and closed. In other words, they are shapes that lies on flat surface. There are thirteen types of plane shapes. Quadrilateral, triangle, pentagon, hexagon, octagon, circle, oval, square, rectangle, kite, trapezium and rhombus. Polygons = 2D and only contain straight lines.

Regular = All sides/angles the same.

Irregular = All sides/angles different.

Diagonal = A line which goes from corner to corner. Question!

Give me two plane shapes that have two diagonal. Answer:

Any type of quadrilateral.

Triangles There are three sides and three angles. There are different types of triangles according to their sizes. There are six different types of triangles. Scalene, isosceles, equilateral, acute-angled, right-angled and obtuse-angled. Scalene = No sides/angles equal

Isosceles = Two sides equal

= Angles opposite equal sides are equal

Equilateral = All sides equal

= Three 60 degree angles

Acute-angled = All angles are acute

Right-angled = One angle is a right angle

Obtuse-angled = One angle is obtuse Question!

If two of the angles are 60 degree and 70 degree, what type of angles is the last one? Answer: acute angle Triangle = angle a* + b* + c* = 180*

Quadrilateral = angle a* + b* + c* + d* = 360*

Isosceles Triangle = angle a* = b*

Equilateral Triangle = angle a* = b* = c*

Straight Angle = a* + b* = 180*

Angles at a point = a* + b* + c* = 360* Size of an Angle You can find the angle sum of a polygon by starting from one corner and divide the polygon into triangles. The angles in each triangle add up to 180*. Polygons that have all sides and angles equal are called regular polygons. Angle Sum of a Polygon Solid shapes have thickness as well as length and width. They are called three-dimensional (3D). Solids Many shapes are symmetrical. This means that they seemed well balanced and in the right proportion. In maths, there are two types of symmetry. Line symmetry and rotational symmetry. Symmetry Solids shapes can be made from plane shapes. This is done by drawing the net of the solid on a piece of paper. The net shows how the faces of the solid are joined to each other. When these faces are folded along their edges, the solid is formed. Some nets are quite easy to make, while others are quite difficult. Nets of Solids In many technical situations a solid is represented by drawing it from different views. It is most often drawn looking at it from the front, the top and the sides. When used together these drawings can be used to "describe" the solid. Looking at Solids from Different Views The square, rectangle, parallelogram and rhombus are special types of quadrilaterals. They have many properties that are related to their sides, angles and diagonals. Quadrilaterals These are the summary of the properties of the six different quadrilaterals. Summary of the properties of quadrilateral. Trapezium = one pair of opposite sides parallel.

Parallelogram = two pairs of parallel sides, opposite sides/angles are equal and diagonals bisect one another.

Rhombus = all properties of parallelogram and all sides are equal, diagonals bisect each other at right angles, diagonals bisect the angles though which they pass.

Rectangle = all properties of parallelogram and all angles are right angles, diagonals are equal.

Square = all properties of rhombus and rectangle, four sides equal and have four right angles.

Kite = two pairs of adjacent sides equal, diagonals are perpendicular and one diagonal is an axis of symmetry. Question!

Which quadrilaterals have all sides equal? Answer: rhombus and square RULES Question!

If I have an equilateral triangle with two 60* angles. What value would the third angle be? Answer: 60* Quadrilaterals pentagon Number of sides Number of triangles 5 2 3 4 (cc) image by anemoneprojectors on Flickr hexagon Angle Sum of polygon 6 4 heptagon 7 5 octagon 8 6 2 180* = 360* 3 180* = 540* 4 180* = 720* 5 180* = 900* 6 180* = 1080* Polygon Question!

What is the angle sum of a decagon? Answer: 1440* Parts of a solid. Face: a surface of the solid. Edge: a line where two faces meet. Vertex: a corner where three or more faces meet. There are two main families of solids. They are the prisms and the pyramids. Line Symmetry

A shape have line symmetry if it can be divided by a line into two identical mirror image. The dividing line is called the axis of symmetry. Rotational Symmetry

A shape have rotational symmetry if it can be spun about a point so that it repeats its shape more than once in a rotational. If it repeats its shape after half a turn, it is said to have point symmetry. The point about which the shape spins is called the center of symmetry. Prisms

All prisms have a special pair of parallel faces. These faces are the only two faces that need not be rectangular in shape.

If a prism is "sliced" parallel to these faces, the same shape always results. This shape is called the cross-section. Pyramids

All pyramids have one face that need not be triangular. This face is used to name the pyramid. All the other faces of a pyramid are triangular. Pyramids cannot be sliced like prisms so that identical shapes always result. Other Solids

Some solids are neither prisms nor pyramids. The most common of these are the cylinder, cone and sphere. Question!

Which solid shapes have no vertices? Answer: cylinder and sphere Net of a cube Question!

Which of these nets is the net of a pentagonal prism? Question!

How many axis of symmetry does a square have? Answer: four A B C Answer: B Front Right side Top Left side Question!

Name the solid above. Answer: sphere BY: Elina Lee Shapes Number: Its Order and Structure Grouping symbols are often used to tell us which operations to perform first. Three commonly used grouping symbols are = parentheses ( ) , brackets [ ] & braces { } Order of Operations When doing order of operations, you can use a solution way called BIDMAS. You have to first do the work inside a bracket, then indices. Finally do division or multiplication and addition or subtraction. Question!

What is the answer for 28+(2 3) -7= ? Answer: 27 Property Using Number Properties Example 1. Multiplying any number by one leaves it unchanged. 2. Multiplying any number by zero gives the answer zero. 3. Adding zero to any number leaves it unchanged. 4. When adding two numbers, the order does not change the answer. 5. When multiplying two numbers, the order does not change the answer. 6. When adding more than two numbers, we may add them in any order. 7. When multiplying more than two numbers, we may multiply them in any order. 6583 1 = 6583 749 0 = 0 96 + 0 = 96 14 = 9 = 9 + 14 9 7 = 7 9 41+154+59+6 = (41+59)+(154+6) = 260 4 836 25 = (4 25) 836 = 83600 Question!

(18+6)+3 = 18+(6+3)

True or False Answer: True Abbreviations used in mathematics

% = per cent

= therefore

eg = for example

ie = that is

K = thousands

am = before noon

pm = after noon

BC = before Christ

AD = Anno Domini Language & Symbols used in Maths Name of Group Special Sets of Whole Number Symbol used in mathematics

= means is equal to

= = is not equal to

= is approximately equal to

< = is less than

< = is less than or equal to

< = is not less than

> = is greater than

> = is greater than or equal to

= the square root of 9 or the number that is multiplied by itself to give 9. Eg 3x3=9 = 3

= the cube root of 8 or the number used in a product 3 times to give 8. Eg 2x2x2=8 = 2 QUESTION!

To the quotient of 63 and 9, add 34. Answer: 41 Pattern Diagram/ Explanation Cardinal Numbers 0,1,2,3,... 0+counting number Counting Numbers 1,2,3,4,... Even Number Odd Number 2,4,6,8,... 1,3,5,7,... Square Numbers 1,4,9,16,... Triangular Numbers 1,3,6,10,15,... Hexagonal Numbers 1,7,19,37,... Fibonacci Numbers 1,1,2,3,5,8,... Palindromic Numbers Examples:

929, 7337 QUESTION!

What is the difference between a cardinal number and a counting number? Answer: Cardinal Number counted zero. Counting Number don't. A factor of a counting number divides it exactly.

A multiple of a counting number is found when you multiply it by another counting number. Eg 3x8=24

Here3 and 8 are factors of 24, and 24 is a multiple of both 3 and 8. Factors and Multiples QUESTION!

List all the factors of the number 24 and 121. Answer: 24 = 1,2,3,4,6,8,12,24

121 = 1,11,121 A prime number is a counting number that has exactly two factors, itself and 1. For example, 3 has only two factors, 3 and 1, so 3 is a prime number. Prime and Composite A composite number has more than two factors. For example, 25 has three factors, 25, 1 and 5, so 25 is a composite. The number 1 has only one factor, 1, so 1 is neither prime nor composite. QUESTION!

Is 2 a prime or composite number? Answer: prime Divisor Divisibility Tests Divisibility Test Example 2 3 4 The number must be even, ie it must end in 0,2,4,6 or 8 The sum of the digits is divisible by 3 The number formed by the last two digits must be divisible by 4 5 The last digit must be 5 or 0 6 The number must be divisible by both 2 and 3 8 The number formed by the last three digits is divisible by 8 9 The sum of the digits is divisible by 9 10 The last digit must be 0 11 The sum of the digits in odd-numbered places will be equal to the sum of the digits in even-numbered places, or will differ by a multiple of 11 25 The last two digits will be 00, 25, 50 or 75 100 The last two digits will be 00 4136 is divisible by 2 as it is even. 30012 is divisible by 3 as (3+0+0+1+2) is divisible by 3 76112 is divisible by 4 as 12 is divisible by 4 11225 is divisible by 5 as it ends with a 5 40002 is even and its digit sum is divisible by 3 963216 is divisible by 8 as 216 is divisible by 8 142128 is divisible by 9 as (1+4+2+1+2+8) is divisible by 9 814710 is divisible by 10 as it ends with 0 7081426 is divisible by 11 as (7+8+4+6)-(0+1+2) is 22, which is divisible by 11 80925 is divisible by 25 as it ends in 25 81700 is divisible by 100 as it ends in 00 1. Which of these numbers are divisible by 2? QUESTION! 571, 3842, 5816, 2221, 887, 9000, 374555, 8774, 8166, 7008 2. Which of these numbers are divisible by 4? 1004, 67814, 7118, 2222 6124, 8156, 98, 61852, 934 Answer: 1. 3842, 5816, 9000, 8774, 8166, 7008

2. 1004, 6124, 8156, 61852 If the square of 15 is 225, then the square root of 225 is 15 Square and Cube Roots If the cube of 8 is 512, then the cube root of 512 is 8 QUESTION!

What is the square root of 324?

What is the cube root of 729? Answer: square root: 18

Cube root: 9 It's amazing how often we read measurements on scale or digital readout. Measurements Here are some examples: Measuring Instruments weighing scales speedometer protractor thermometer digital watch clocks microwave oven measuring jug In the metric system, the basic unit of length is the metre. The prefixes kilo, centi and milli are then added to give the other most common units. Units of Length 10mm = 1cm

100cm = 1m

1000mm = 1m

1000m = 1km To convert small units into large units, you divide.

To convert large units to small units, you multiply. You only need to know how to multiply a decimal by 10, 100 or 1000 An interval is part of a line with a definite length.

A line goes on forever in both directions. QUESTION!

Convert each of these measurements.

1. 6cm = ...mm 2. 1200cm = ...m

3. 8.3km = ...m 4. 54000m = ...km Answer: 1. 60mm 2. 12m

3. 8300m 4. 54km To measure a length we need an instrument that will do the job. Some of them are shown below. Measuring Length Rulers and tape measures would be marked in centimetres or millimetres to measure shorter lengths accurately.

Trundle wheels might only be used to measure a length to the nearest metre, and sometimes may have a counter attached to count the metres.

Odometers on a car usually measures distances in tenths of a kilometre. The end digit (often in red) measures the tenths, so the remaining digits indicate the number of kilometres the car has traveled. QUESTION!

Which of the instrument below will you use to measure...

1. the height of a person

2. the length of a soccer field Answer: 1. ruler/tape 2. trundle wheel Trundle Wheel Ruler People often use very general ways of estimating lengths or distances, and with practice it is possible to become quite good at it. Estimating Length QUESTION!

Estimate the length below to the nearest cm. 1 cm 1. 2. Answer: 1. 4cm 2. 2cm The perimeter of a figure is the sum of the lengths of the sides of the figure. It is the distance around the figure. Perimeter 5 For a square, if 5 stands for the length of one side, then a formula for perimeter would be : P = 4x5 7 3 For a rectangle with a length of 3 and a breadth of 7, the perimeter would be : P = 2x3+2x7 QUESTION!

Calculate the perimeter of these figures. 1. 2. Answer: 1. 28m 2. 70mm People have always been concerned with time and its measurement. Our whole existence is locked into the passage of time. Calendar and Dates In order to measure time, we must refer to something that is constant, ie doesn't change. Hence our main reference points are astronomical bodies such as the sun and stars. 1 year = 365 days

1 leap year = 366 days When in a leap year, there will be another day in February. QUESTION!

1. How many days in a fortnight?

2. How many years in a century? Answer: 1. 14 2. 100 For convenience, each day is broken up into smaller units of time, so that we know what 'time of the day' it is. Without this division it would be difficult to coordinate our activities with other people. Clocks and Times 60 seconds = 1 minute

60 minute = 1 hour

24 hour = 1 day 24-hour time is always given as a 4-digit number, the first two indicating the hour after midnight and the second two indicating the minutes past the hour. Eg 0815

Digital time does not use zero as a first figure, eg 8:15. One disadvantage is that if am or pm is not used we would not know whether 10:30 referred to 24-hour time or digital time.

am = between 12 midnight and 12 noon

pm = between 12 noon and 12 midnight QUESTION!

Change this to 24-hour time.

11:45pm

Change this to 12-hour time.

0523 Answer: 1. 2345 2. 5:23am When operating with time, keep the hours and minutes in separate columns to not get confused. Operating with Time EST = Eastern Standard Time Central time is 1/2 an hour behind EST.

Western time is 2 hours behind EST. QUESTION!

2 h 20 min + 1 h 50 m = ? Answer: 4 h 10 min Timetables are part of our life. We use them to predict tides, catch trains and many more. Timetables To graph a number on the number line, place a large dot at the position of that number on the number line. Directed Number ps. please ignore the www.laosworld.net background, thank you. =] The number line can be very useful. It can help us to perform additions, subtractions, multiplications and even divisions. The number line can also be used to graph sets of numbers. Graphing Points on the Number Line On a street directory, a letter and a number are given. Come down from the letter and across from the number. For example, J5 Reading a street directory On the street directory, a letter and a number were used to name a position. On the number plane, two numbers are used. These are called coordinates, and are written in parentheses and separated by a comma. The two axes used are called the x-axis (horizontal) and the y-axis (vertical). The x-coordinate are always written before the y-coordinate. The Number Plane QUESTION!

In what coordinate is the black dot at? Answer: 2,-4 Many of the numbers we use represent situations that have direction as well as size. We call these numbers directed numbers. Once a direction is chosen to be positive (+), the opposite direction is taken to be negative (-). A directed whole number without a decimal is called an integer. Directed Numbers Chosen Direction for numbers

If north is positive, then south is negative

If east is positive, then west is negative

If above zero degree is positive, then below zero degree is negative

If to the right is positive, then to the left is negative QUESTION!

Arrange these directed numbers in order from smallest to largest. 1. 2,-3,-2,3 2. 0,5,-6,1 Answer: 1. -3,-2,2,3 2. -6,0,1,5 A negative number can represent a loss; a positive number can represent a gain. Adding the negative of a number is the same as subtracting the positive number, eg -2 + -3 = -2-3. A number added to its opposite gives zero, eg (-4) + (4) = 0 Addition & Subtraction of Directed Numbers Rules for multiplication by extending number patterns Multiplication of Directed Numbers If you take one less away, the answer increases by one. Two minus signs become a plus. To take away -3, add its opposite. The same methods can be used with decimals and fractions. Subtracting a negative number For multiplication & division

two like signs give a plus

two unlike signs give a minus Division of Directed Numbers The directed numbers ...,-4,-3,-2,-1,0,1,2,3,4,... are called integers. Using Directed Numbers The key on a calculator Directed Numbers and the Calculator The size of a fraction is unchanged if both the numerator and the denominator are multiplied or divided by the same number Eg 3/8 = 3x5/8x5 = 15/40

To compare fractions, make their denominators the same. Then compare the numerators. Eg 3/8 < 4/8

An improper fraction has a numerator that is greater than its denominator. Eg 5/2, 13/10

A mixed number is one that has a whole number part and a fraction part. Eg 2 1/2, 1 3/10

When fractions have the same denominator, we can add them by adding numerators. Eg 3/10 + 4/10 = 7/10. We can subtract them by subtracting numerators. Eg 7/8 - 2/8 = 5/8

5 x 3/4 means '5 lot of 3/4' or '5x3/4 = 15/4 or 3 3/4

To find 7/8 of a number, find 1/8 of the number and then multiply by 7. (to find 1/8 of a number, divide it by 8).

Fractions, Percentages & Probability QUESTION!

1. -7 + 8 =?

2. 6 - 9 =? Answer: 1. 1 2. -3 QUESTION!

1. -5--3 =?

2. -4 + -7 =? Answer: 1. -2 2. 11 Multiplying two unlike signs gives a minus. Eg -8x7 = -56 (plus x minus=minus) (minus x plus = minus)

Multiplying two like signs gives a plus. Eg -8x-7 = 56 (minus x minus = plus) QUESTION!

1. 4x-7=? 2. -6x-8=? Answer: 1. -28 2. 48 If -6 x -5 = 30 then 30 -5 = -6 QUESTION!

1. -7 -7 =?

2. 18 -3 =? Answer: 1. 1 2. -6 (-3 ) = -3 x -3 = 9 The fraction bar groups both the top and bottom. Integers are often called the set J. QUESTION!

(-6 ) = ? Answer: 36 Modern calculator allow you to enter negative numbers using the key. The is pressed after the second part of the number. QUESTION! (use a calculator)

-2160 16 = ? Answer: -135 Any numbers that can be written as a fraction is called a rational number. Comparing Fractions 2/0 is not real. You cannot have two of zero equal parts. Summary of important ideas Review of Fractions To multiply two fraction, multiply the numerators and multiply the denominators.

Answer = numerator 1 x numerator 2

denominator 1 x denominator 2 Multiplication of Fractions Probability is involved in making decisions. We often used probability: 'Where is the most likely place to find my watch?', 'How can I best avoid an accident?' Probability Before adding or subtracting fractions, we must express each fraction with the same denominator. This is called a common denominator. We always try to use the lowest common denominator (the LCD). Addition & Subtraction of Fractions Some Notes... Addition & Subtraction of Mixed Numbers Notes... Fractions of Quantities When fractions have the same denominator, you can divide the numerator to get the answer. Eg 2/4 5/8 = 6/8 5/8 = 6 5 Division involving Fractions To find a percentage of a quantity, write the percentage as a decimal/fraction and multiply by the quantity. Eg 11% of 800 = 0.11 x 800 = 88 Finding a Percentage of a Quantity Summary of important ideas... Review of Percentages Notes... Changing Fractions & Decimals to Percentages To write one quantity as a percentage of another... One Quantity as a Percentage of Another Eg 4/5 = 12/15, 2/3 = 10/15 therefore 4/5 + 2/3 = 12/15 + 10/15 = 22/15 = 1 7/15 QUESTION!

1. 7/10 + 2/10 =?

2. 8/10 + 3/5 =? Answer: 1. 9/10

2. 14/10 = 1 4/10 = 1 2/5 20/20 = 1

1 - 17/20 = 1/1 - 17/20 = 20/20 - 17/20 = 3/20

1 - 3/10 = 7/10 Subtract the whole number first. QUESTION!

1. 1 1/2 + 5 =?

2. 1 - 2/10 =? Answer: 1. 6 1/2 2. 8/10 = 4/5 Changing to smaller units makes the working easier. They must be changed to the same units. QUESTION!

1/4 of one hour Answer: 15 minutes make sure that the two quantities are in the same units

write the first as a fraction of the second

change the fraction to a percentage by multiplying by 100/1 QUESTION!

12 of my 30 fishes are goldfish. What percentage are not goldfish? Answer: 60% We describe the chance of something happening using either words or numbers. Words: impossible, unlikely, fifty-fifty, likely, certain

Numbers: 0, between 0 & 1/2, 50%, 1/2/0.5, between 1/2 & 1, 1 QUESTION!

If you toss a coin, you will get a head. What is the probability of getting a head? Answer: fifty-fifty/50% QUESTION!

5% of 660m Answer: 33m to express a fraction as a percentage, multiply by 100/1 %

to express a decimal as a percentage, multiply by 100%

Eg 62/100 = 0.62 = 62% QUESTION!

1. Change fraction to percentage.

17/20 = ?

2. Change decimal to percentage.

0.02 = ? Answer: 1. 85% 2. 2% A percentage is a convenient way of writing fractions that have a denominator of 100. 'Per cent', written %, means 'per 100' or 'for every hundred'. Eg 7% = 7/100

To write a percentage as a fraction or mixed number, first write it as a fraction with denominator 100, then simplify. Eg 125% = 125/100 = 1 1/4

To change fractions to percentages, first change the denominator of the fraction to 100. Eg 3/20 = 15/100 = 15%

To change a percentage to a decimal, we can write it first as hundredths and then as a decimal. Eg 93% = 93/100 = 0.93

To change a decimal to a percentage, we can write it as a fraction first, then change it to a decimal, or we can multiply the decimal by 100%. Eg 0.93 = o.93 x 100% = 93% QUESTION!

Change to fraction.

1. 99% 2. 8% Answer: 1. 99/100 2. 8/100 To divide by a fraction, invert the fraction and multiply. Eg 3/4 2/5 = 3/4 x 5/2 Invert means turn upside down. QUESTION!

5/8 1/2 = ? Answer: 1 1/4 Simplifying the work when multiplying fractions

If there is a number that will divide exactly into both a numerator and a denominator, this division should be done. It is easier to do it before multiplying rather than when we reduce the answer. To multiply mixed numerals, write them in improper fractions first. If two numbers multiply to give 1, they are called reciprocals of one another. 3/10 x 3/10 = 9/100 = 0.09 OR 3/10 x 3/10 = 0.3 x 0.3 therefore 0.3 x 0.3 = 0.09 QUESTION!

1/2 x 2/5 = ? Answer: 1/5 Common means 'belonging to both' Fractions like 6/1 are called whole numbers. (The 1 in the bottom is what makes it special) Remember to reduce answers to lowest terms. QUESTION!

1. Write the improper fraction as a mixed number. 21/6 =?

2. Write the mixed number as an improper fraction. 42 2/3 =? Answer: 1. 3 3/6 = 3 1/2

2. 128/3

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# Math Revision Sheet =)

It is to help people with the Math test.