mathematics study guide - gogsat · 2010-07-09 · roman numerals ... pie chart/circle graph ... 1...

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Mathematics Study Guide


Table of Contents

NUMBER ........................................................................................................................................................................................... 4

Place Value .................................................................................................................................................................................. 4

Face Value .................................................................................................................................................................................... 4

Fractions ...................................................................................................................................................................................... 5

What are Fractions? ................................................................................................................................................................ 5

Fraction Terms ........................................................................................................................................................................ 6

Proper Fractions ...................................................................................................................................................................... 6

Improper Fractions.................................................................................................................................................................. 6

Mixed Numbers ....................................................................................................................................................................... 7

Equivalent Fractions ............................................................................................................................................................... 7

Adding Fractions With Like Denominators ............................................................................................................................ 8

Subtracting Fractions With Like Denominators ..................................................................................................................... 8

Reducing Fractions.................................................................................................................................................................. 9

Decimals ..................................................................................................................................................................................... 10

Addition ..................................................................................................................................................................................... 11

Important Addition Terms .................................................................................................................................................... 11

How to Write Addition Problems ......................................................................................................................................... 11

Commutative Property of Addition ..................................................................................................................................... 11

Associative Property of Addition ......................................................................................................................................... 12

Zero Property of Addition (Additive Identity Property) ..................................................................................................... 12

Distributive Property of Addition......................................................................................................................................... 12

Division ...................................................................................................................................................................................... 13

Quotient, Dividend and Divisor ............................................................................................................................................ 13

Rounding ................................................................................................................................................................................... 14

Rounding Numbers to the Nearest Ten ............................................................................................................................... 14

Rounding Numbers to the Nearest Hundred ....................................................................................................................... 14

Roman Numerals ....................................................................................................................................................................... 15

Uses of Roman Numerals ...................................................................................................................................................... 15

Exponents .................................................................................................................................................................................. 16

Whole Numbers ......................................................................................................................................................................... 17

Counting Numbers .................................................................................................................................................................... 17

Prime Numbers .......................................................................................................................................................................... 17

Composite Numbers .................................................................................................................................................................. 17

Odd Numbers ............................................................................................................................................................................ 17

MEASUREMENT .............................................................................................................................................................................. 18

Measuring Length ..................................................................................................................................................................... 18

Measuring Mass ......................................................................................................................................................................... 18

Measuring Capacity ................................................................................................................................................................... 19

Measuring Time ......................................................................................................................................................................... 19

GEOMETRY ..................................................................................................................................................................................... 20


Angles ........................................................................................................................................................................................ 20

Acute Angles ......................................................................................................................................................................... 20

Obtuse Angles....................................................................................................................................................................... 20

Right Angles ......................................................................................................................................................................... 21

Complementary Angles ........................................................................................................................................................ 21

Supplementary Angles ......................................................................................................................................................... 21

Polygons .................................................................................................................................................................................... 22

Triangle ................................................................................................................................................................................. 22

Quadrilateral ........................................................................................................................................................................ 22

Rectangle .............................................................................................................................................................................. 22

Square ................................................................................................................................................................................... 23

Parallelogram ....................................................................................................................................................................... 23

Rhombus ............................................................................................................................................................................... 23

Trapezoid .............................................................................................................................................................................. 24

Pentagon............................................................................................................................................................................... 24

Hexagon ................................................................................................................................................................................ 24

Heptagon .............................................................................................................................................................................. 25

Octagon ................................................................................................................................................................................ 25

Nonagon ............................................................................................................................................................................... 25

Decagon ................................................................................................................................................................................ 26

Parts of a Circle .......................................................................................................................................................................... 26

STATISTICS ...................................................................................................................................................................................... 27

Bar Graph................................................................................................................................................................................... 27

Parts of the Bar Graph .......................................................................................................................................................... 27

Line Graph ................................................................................................................................................................................. 29

Parts of the Line Graph ........................................................................................................................................................ 29

Pictograph/Picture Graph .......................................................................................................................................................... 30

Parts of the Pictograph ......................................................................................................................................................... 30

Pie Chart/Circle Graph ............................................................................................................................................................... 31

Parts of the Pie Chart ............................................................................................................................................................ 31

Sample vs Population ................................................................................................................................................................ 32

Mean .......................................................................................................................................................................................... 32

Mode .......................................................................................................................................................................................... 32

Median ....................................................................................................................................................................................... 32

Sets ............................................................................................................................................................................................. 34

Empty Sets ............................................................................................................................................................................. 34

Intersection of Sets ............................................................................................................................................................... 35

Unions ................................................................................................................................................................................... 36

Finite Sets .............................................................................................................................................................................. 37

Infinite Sets ........................................................................................................................................................................... 37


NUMBER

Place Value

Numbers are written using the following ten digits.

0 1 2 3 4 5 6 7 8 9

The number 456 has 3 digits, each in a different place. Each place has a name that tells it value.

Hundreds Tens Ones 4 5 6

The digit 6 is in the ones place. This means the digit has a value of 6 ones.

The digit 5 is in the tens place. This means the digit has a value of 5 tens.

The digit 4 is in the hundreds place. It has a value of 4 hundreds.

Face Value

Face value is the actual value of the digit. In the number 456, the "5" has a face

value of 5.


Fractions

What are Fractions?

A fraction may represent a whole, part of a whole or more than a whole.

Whole Part of a whole More than a whole

6 4 10 6 6 6

A fraction may show a part of one whole

This circle is divided into 6 equal parts. One part of the circle is gone. You can use a fraction to show how much of the circle is gone.

1 part of the circle is gone. 6 parts make up the whole circle.


A fraction may also show a part of one whole group.

Here is a whole group of squares. Two of the squares are blue. You can use a fraction to show the number of blue squares in the group.

2 squares are blue. 5 squares are in the group.

Fraction Terms

Fractions are made up of a numerator and a denominator.

2 Numerator (this is the number of parts you are talking about ) 5 Denominator (this is the number of equal parts in the whole)

Proper Fractions

This is a fraction where the numerator is smaller than the denominator. All proper

fractions have a value of less than one.

2 TWO IS SMALLER THAN 5 5

Improper Fractions

In an improper fraction, the numerator is equal to or larger than the denominator.

10 TEN IS LARGER THAN 5. 5


Mixed Numbers

A mixed number is made up of a whole number (other than zero) and a fraction.

(Whole Number) 2 1 Fraction 2

Equivalent Fractions

Some fractions have the same value even though they use different numbers. All the fractions below have equal value.

1 2 3

2

4

6


Adding Fractions With Like Denominators

When adding fractions with like denominators, add the numerators and leave the denominators.

3 + 1 = 3+1 = 4 5 5 5 5

Subtracting Fractions With Like Denominators

When subtracting fractions with like denominators, subtract the numerators and keep the denominators the same.

7 - 5 = 7-5 = 2 8 8 8 8

Write your answer in its simplest form.

2 ÷ 2 = 1 8 2 4


Reducing Fractions

A fraction is in its simplest form if both the numerator and denominator cannot be divided by any number except one.

Reduce 9/18 to its simplest form.

Step 1: List the factors of 9: 1,3,9

Step 2: List the factors of 18: 1,2,3,6,9,18.

(1, 3 and 9 are factors of both numbers, 9 is the greatest common factor (GCF))

Step 3: Divide the numerator and denominator by the GCF.

9 ÷ 9 1 18 9 2

Step 4: Write the fraction in its lowest terms.

9 = 1 18 2


Decimals

A decimal number is written using a decimal point.

Let’s look at the number 86.76

86 . 76 whole number part decimal point decimal part

We use place value with decimals.

Tens Ones Decimal Point Tenths Hundredths

8 6 . 7 6

In the number 86.76 the digit 7 has a value of 7 tenths or 0.7


Addition

You add numbers together to find out how many you have in all.

Important Addition Terms

Addends – These are numbers that are added.

Sums – Answers to addition.

5 Addend + 2 Addend

7 Sum

How to Write Addition Problems

Addition problems can be written in a line or column.

Line 5 + 2 = 7

5 Column + 2 7

Commutative Property of Addition

Commutative property: When you add two numbers, the sum is the same regardless of the order of the addends.

For example 6 + 2 = 2 + 6


Associative Property of Addition

Associative Property: When you add three or more numbers, the sum is the same regardless of the grouping of the addends.

For example (6 + 3) + 4 = 6 + (3 + 4)

Zero Property of Addition (Additive Identity Property)

Additive Identity Property: When you add zero to any number the sum is the original number.

For example 6 + 0 = 6.

Distributive Property of Addition

Distributive property: The sum of two numbers times a third number is equal to the sum of each addend times the third number.

For example 4 * (6 + 3) = 4*6 + 4*3


Division

Division breaks things into equal parts.

Quotient, Dividend and Divisor

In mathematics, a quotient is the result of a division.

For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend and 3 the divisor.

The quotient can also be expressed as the number of times the divisor divides into the dividend.


Rounding

We round numbers (approximate them) to make them easier to work with in your head.

Rounding Numbers to the Nearest Ten

Numbers ending in 1 through 4 should be rounded down to the next lower number that ends in 0.

For example 73 rounded to the nearest 10 would be 70.

Numbers ending in 5 or more should be rounded up to the next even ten.

For example, the number 89 rounded to the nearest ten would be 90.

Rounding Numbers to the Nearest Hundred

Numbers ending in 1 through 49 should be rounded down to the next lower number that ends in 00.

For example 424 rounded to the nearest hundred would be 400.

Numbers ending in 50 or more should be rounded up to the nearest hundred.

The number 989 rounded to the nearest hundred would be 1000.


Roman Numerals

Roman numerals are a numeral system of ancient Rome based on letters of the alphabet, which are combined to signify the sum (or in some cases, the difference) of their values.

I = 1 V = 5 X = 10 L = 50 C = 100 D = 500 M = 1000

Uses of Roman Numerals

Roman numerals are used to mark the hours on clock faces.

Roman numerals are used to number pages in the prefaces of books. Roman numerals are used to express copyright dates. Roman numerals are used to count items in a series.


Exponents

Exponential notation is useful in situations where the same number is multiplied repeatedly. The number being multiplied is called the base and the exponent tells how many times the base is multiplied by itself.

Example: 2 2 2 written in exponential notation is 23

We refer to this as two to the third power, or two to the power of three.

The base in this example is 2 and the exponent is 3.

A number written with an exponent of 1 is the same as the given number. For example 21 is the same as 2.


Whole Numbers

Whole Numbers are the numbers 0, 1, 2, 3, 4, 5, … (and so on)

Counting Numbers

Counting Numbers are Whole Numbers without the zero such as 1, 2, 3, 4, 5, … (and so on).

Prime Numbers

A prime is a positive whole number (greater than 1) which can only be divided evenly by itself and 1. Examples of prime numbers include:

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67

Composite Numbers

A composite number is a positive integer which has a positive divisor other than one or itself.

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 128, 129, 130, 132, 133, 134, 135, 136, 138, 140.

Odd Numbers Odd numbers leave a remainder of 1 if divided by 2. Additionally, all odd numbers end in 1, 3, 5, 7 or 9. Examples of odd number include 71, 123 and 23.


MEASUREMENT

There are different units of measurement for length, mass and capacity.

Measuring Length

Length is a measure of how long or wide something is. Rulers and tape measures can be used to measure length.

Length is measured in millimetres (mm), centimetres (cm), metres (m) or kilometres (km). These are known as metric units of length. 1 cm = 10 mm 1 m = 100 cm 1 km = 1000 m

Measuring Mass

Mass is a measure of how heavy something is. Scales can be used to measure mass.


Measuring Capacity

Capacity or volume is a measure of how much space something takes up. Measuring spoons or measuring jugs can be used to measure capacity.

Measuring Time

1 minute = 60 seconds 1 hour = 60 minutes 1 day = 24 hours 1 week = 7 days 1 fortnight = 14 days 1 year = 12 months = 52 weeks = 365 days 1 leap year = 366 days


GEOMETRY

Angles

Acute Angles

An acute angle is an angle measuring between 0 and 90 degrees.

Obtuse Angles

An obtuse angle is an angle measuring between 90 and 180 degrees.


Right Angles

A right angle is an angle measuring 90 degrees.

Complementary Angles

Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees.

Supplementary Angles

Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees.


Polygons

A polygon is a closed figure made by joining line segments, where each line segment intersects exactly two others.

Triangle

A three-sided polygon. The sum of the angles of a triangle is 180 degrees.

Quadrilateral A four-sided polygon. The sum of the angles of a quadrilateral is 360 degrees.

Rectangle

The term rectangle normally refers to a quadrilateral with four right angles.


Square

A four-sided polygon having equal-length sides meeting at right angles. The sum of the angles of a square is 360 degrees.

Parallelogram

A Parallelogram is a four-sided polygon with two pairs of parallel sides. The sum of the angles of a parallelogram is 360 degrees.

Rhombus

In geometry, a rhombus is a quadrilateral whose four sides all have the same length. The rhombus is often called a diamond. The sum of the angles of a rhombus is 360 degrees.


Trapezoid

A four-sided polygon having exactly one pair of parallel sides. The two sides that are parallel are called the bases of the trapezoid. The sum of the angles of a trapezoid is 360 degrees.

Pentagon

A five-sided polygon. The sum of the angles of a pentagon is 540 degrees.

Hexagon

A six-sided polygon. The sum of the angles of a hexagon is 720 degrees


Heptagon

A seven-sided polygon. The sum of the angles of a heptagon is 900 degrees.

Octagon

An eight-sided polygon. The sum of the angles of an octagon is 1080 degrees

Nonagon

A nine-sided polygon. The sum of the angles of a nonagon is 1260 degrees.


Decagon

A ten-sided polygon. The sum of the angles of a decagon is 1440 degrees.

Parts of a Circle


STATISTICS

Bar Graph A bar graph can be used to give a visual representation of the relationship of data that has been collected.

Parts of the Bar Graph

A bar graph includes:

1. Two axis with labels 2. Scale 3. Title 4. Bars


Line Graph

A line graph uses points connected by a line to show data.

Parts of the Line Graph


Pictograph/Picture Graph

A pictograph uses pictures or symbols to show data.

Parts of the Pictograph


Pie Chart/Circle Graph

Pie chart diagrams (also called circle graphs) are useful for displaying information about the percentages or parts of a whole.

Parts of the Pie Chart

Boys 40%

Girls 60%

Composition of Mrs. Jones Class


Sample vs Population

A sample is a group of units selected from a larger group (the population). We study a sample to come to some conclusion about the group as a whole. The population for a study of students sitting the Jamaican GSAT is all students sitting the GSAT in all schools in Jamaica. A sample might be all students sitting the GSAT at your school.

Mean

In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all of the list divided by the number of items in the list.

Find the mean of the following numbers: 5, 4, 6, 2, 5 Step 1: Add the numbers (5+4+6+2+5 = 22) Step 2: Divide 22 by 5 (the 'Mean' (Average) is 4.4)

Mode

In statistics, the mode is the value that occurs the most frequently in a data set. In the following set (5, 4, 6, 2, 5) the mode is 5.

Median

The median can be described as the middle value. It is technically a numeric value separating the higher half of a sample from the lower half.

Example: Find the Median number in the following set (5, 4, 6, 2, 5). Step 1: Line up your numbers (smallest to largest): 2,4,5,5,6 Step 2: The Median (The number in the middle) is: 5


Example: Find the Median number in the following set (8, 3, 44, 17, 12, 6).

Step 1: Line up your numbers (smallest to larges): 3, 6, 8, 12, 17, 44 Step 2: Add the 2 middles numbers and divide by 2: (8 +12 = 20) (20 ÷ 2) = 10 The Median (The number in the middle) is 10


Sets

Empty Sets

In set theory, the empty set is the unique set having no elements. The symbol for the empty set is shown below.


Intersection of Sets

Intersection of two sets A ∩ B

The intersection (denoted as ∩) of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements.

For example: The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.

http://upload.wikimedia.org/wikipedia/commons/9/99/Venn0001.svg


Unions

In set theory, the union (denoted as ∪) of a collection of sets is the set of all distinct elements in the collection.

For example: A = {1,2,3,4} B = {5,6,7,8} A ∪ B = {1,2,3,4,5,6,7,8}

Union of two sets: A ∪ B

http://upload.wikimedia.org/wikipedia/commons/3/30/Venn0111.svg


Finite Sets In mathematics, a finite set is a set that has a finite number of elements.

Infinite Sets

In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.

Some examples are:

The set of all integers, {..., -1, 0, 1, 2, ...}, is a countable infinite set.

The set of all real numbers is an uncountable infinite set.

mathematics study guide - gogsat · 2010-07-09 · roman numerals ... pie chart/circle graph ... 1...

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