MATH 7

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Chapter 1:

Number Sense Lesson 1.1 Basic Ideas of Sets

A set may be thought of as a collection of objects. These objects are called elements or members of the set. A set with no element is an empty set. The symbol for empty set is or { }. If a set contains many elements, we often use three dots, …, called an ellipsis, to indicate there are elements in the set that have not been written down. The following are some example of sets where we list some elements and then use an ellipsis to indicate that the pattern is to be continued indefinitely.

N = { 1, 2, 3, 4, 5, …} Three ways to describe a set: 1. The Rooster Notation or Listing Method This is a method describing a set by listing each element of the set inside the symbol { }. In listing the elements of the set, each distinct element is listed once and the order of the elements does not matter.

Example: A = { 1, 2, 3, 4} 2. The Verbal Description Method It is the method of describing a set in words. We can describe the sets named in no. 1 as follows.

Example: Set A is the set of counting numbers less than 5.

3. The Set Builder Notation It is a method that lists the rules that determine whether an object is an element of the set rather than the actual elements. We can describe the sets in no. 1 in set builder notation as: A = { xlx is a counting number less than 5} read as “ the set of all x’s such that x is a counting number less than 5”. The vertical bar after the x is translated as “ such that”.

Equal Sets Two sets that contain exactly the same elements are said to be equal sets. If we are given A = { a, e, i, o, u} and B = { e, o, I, u, a}, then we say that A = B. These two sets contain exactly the same elements and, therefore, are equal.

Equivalent Sets Two sets that contain exactly the same number of elements are equivalent sets. If we are given A = { 1, 2, 3, 4} and B = {m, a, t, h} we say that A is equivalent to B ( A B ). Both sets contain four elements, hence, they are equivalent.

Subsets Set A is a subset of set B written as A , if and only if, every element in A is also an element of B. There are cases where two or more sets contain some, but not all of the same elements. Consider the set of positive even numbers, A = {2, 4, 6, 8, …}, and the set of positive whole numbers, B = {1, 2, 3, 4, …}. We can see that 2 similarly, we note that 8 In fact, every element that is in A is also contained in set B. Therefore, we can say that set A is contained in Set B, or, in symbols, we can write A B. when a set is contained in another set B, we say that A is a subset of B. Every set is a subset of itself. A subset of a given set that is not the set itself is called a proper subset. If set A is a proper subset of set B, then two conditions must be satisfied: first, A must be a subset YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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of B; second set B must contain at least one element that is not found in set A. if A is a proper subset of B, then we say that A is properly contained in B, and we write Consider the sets A = {a,b,c} and B = {a,b,c,d}, we can say that A B since each element in A is also an element in B, and there is at least one element in B not contained in A. We cannot say that B A. because d , but d Hence, B but A B. Universal Set The universal set denoted by U, is a set of all possible elements of any set used in the problem.

The universal set can change from problem to problem, depending on the nature of the set being discussed. For example, the universal set U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 } contains the digit 1 through 9. In a discussion using the universal set, we would only consider those sets whose elements are members of U. for example: A = { 1, 3, 5} might be discussed, but C = { a, b, c} would not be of C are elements of U. Complement of a Set A The complement of a set A, written as A`, is a set of all elements in the universal set (U) that are not in the set A. Example given: U = {1,2,3,4,5}, A = {1,3,5}, B = {1,5}, and C = {} Find: a. A` A` = {2,4} b. B` B = {2,3,4} Operations of Sets A. Intersection of Sets The intersection of sets A and B, written as A B, is a set of elements that are both members of both A and B. Example: Given: U = { 1,2,3,4,5}, A = {1,2,3} and B = { 1,3,4} find (A )’ Solution: (A ) = {1,3} Therefore: (A )’ = {2,4,5 } B. Union of sets The union of sets A and B, written as A U B, is a set of elements that are members of A, or members of B, or members of both A and B. Example: Given: A={2,4,6,…} and B = { 1,3,5,…}, find A B Solution: we have the set of even numbers and the set of odd numbers. Hence, the union of these two sets is the set of counting numbers: A U B = {1,2,3,4,…} Venn Diagram Venn diagrams are very useful in showing the relationship between sets. The Venn diagram which consists of a rectangle represents the universal set and a circle or circles inside the rectangle to represent the set or sets being considered in the discussion. It is understood that the elements in the set are inside the circle that represent the set. Disjoints sets have no elements in common.

Example: Given: U = {a,b,c,d,e}, A = {c,d,e}, and B = {a,b}. use a venn diagram to show A B.

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Solution: U

Set A and B are disjoint sets.

A e

B

c d d ee

a b

Practice Exercise 1.1 Answer the following 1. P = { a,b,c,f,h} and Q = { a,c,d,e,g,i}. Find: a. P b. P U Q 2. Let U = { 1,2,3,4,5,6}, A = {1,3,5}, B = {2,4,5,6} and C = {1,2,4,6} a. A` b. A` U B` c. A’ U C 3. List all possible subsets of each a. {10,20} b. {a,b,c,d,e} c. { 0,5,8,9} 4. Are the following statements true or false? a. {o,n,e}= {n,e,o} b. {a,t,o,m} {t,o,m,a} c. {2,4,6,…,50} {1,3,5,…,49} 5. Tell whether each statement is true or false. a. {a,b,c,d} { a,b,c,d} b. {10, 20} { 1,2,3,4} c. {2,4,6,…} { 1,2,3…} Lesson 1.2 The Set of Real Numbers Name Natural Number Whole Number Integers

Rational Numbers, Q Irrational Numbers

Description N = { 1 ,2,3,4,5,…} These numbers are used for counting.

W = {0,1,2,3,4,…} These numbers are formed by adding 0 to the set of natural numbers.

I = {…,-4,-3,-2,-1,0,1,2,3,4,…} They are formed by adding the negatives of the natural numbers to the set of whole numbers.

The set of rational numbers is the set of all members which can be expressed in the form , where a and b are integers, b . The decimal representation of a rational number either terminates or repeats.

The set of irrational numbers is the set of numbers whose decimal representations are neither terminating nor repeating. These numbers cannot be expressed as a quotient of integers.

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Example 2,3,5,8,17 0,2,3,5,8,17 -19,-8,5,0,3,5,8 -19 =

,

= 0.6 -√ = -1.414 √ = 1.73 = 3.1416 Page 4

Answer the following: 1. Complete the statement using always, sometimes, or never. Explain a. A real number is __________ a rational number. b. An irrational number _________ a real number. c. A negative integer is __________ an irrational number.

2. What is rational number? Use your own words.

3. How is the decimal form of a rational number different from the decimal form of an irrational number?

4. In the definition of irrational number, we may say ”a number that can be put in the form , where a and b are integers and b and b are integers and b ?

”. Why could not we say “ a number of the form , where a

Lesson 1.3 Properties of Real Numbers Let a, b, and c denote real numbers Properties Meaning Closure property of Addition The sum of any real numbers is a real number a + b is a real number

Closure property multiplication

of

The product of any two real number is a real number ab is a real number

Commutative property of addition Associative property of addition

Two real numbers can be added in any order, a+b=b+a If three real numbers are added, it makes no difference which two are added first. (a+b)+c = a+ (b+c)

Associative multiplication

property

Distributive property multiplication addition/subtraction

Examples 2 is a real number and 3.5 is a real number, so 2 + 3.5 or 5.5 is a real number.

12 is a real number and is a real number, so 12. or 4 is a real number.

12 + 7 = 7 + 12

(12+7)+5 = 12+ (7+5) 19 + 5 = 12 + 12 24 = 24 of If three real numbers are multiplied, it makes no (12.7).5 = difference which two are multiplied first.

12.(7.5) (a.b).c = a.(b.c) 84.5 = 12.35 420 = 420 of Multiplication distributes over 7.(5+3)=7.5 + over addition/subtraction.

7.3 a.(b+c) = a.b + a.c = 35 + 21 = 56

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Let a, b, and c denote real numbers Properties Meaning Closure property of Addition The sum of any real numbers is a real number a + b is a real number

Closure property multiplication

of

The product of any two real number is a real number ab is a real number

Commutative property of addition Associative property of addition

Two real numbers can be added in any order, a+b=b+a If three real numbers are added, it makes no difference which two are added first. (a+b)+c = a+ (b+c)

Examples 2 is a real number and 3.5 is a real number, so 2 + 3.5 or 5.5 is a real number.

12 is a real number and is a real number, so 12. or 4 is a real number.

12 + 7 = 7 + 12

(12+7)+5 = 12+ (7+5) 19 + 5 = 12 + 12 24 = 24 Associative property of If three real numbers are multiplied, it makes no (12.7).5 = multiplication difference which two are multiplied first.

12.(7.5) (a.b).c = a.(b.c) 84.5 = 12.35 420 = 420 Distributive property of Multiplication distributes over 7.(5+3)=7.5 + multiplication over addition/subtraction.

7.3 addition/subtraction a.(b+c) = a.b + a.c = 35 + 21 = 56 Identity property of addition Any number added to the identity element 0 will 12 + 0 = 12 remain unchanged. 0 is the identity element for addition a+0 = 0 + a =a Identity property of Any number multiplied to the identity element 1 12 . 1 = 12 multiplication will remain unchanged. 1 is the identity element for multiplication. a.1 = 1. a = a Inverse property of addition The sum of a number and its additive inverse 12 + (-12) = 0 (opposite) is the identity element 0. a and (-a) are additive inverses. a + (-a) = (-a) + a=0 Inverse property of The product of a number and its multiplicative 12. = 1 multiplication inverse (reciprocal) is the identity element 1. a and are multiplicative inverses.

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Answer the following: 1. Explain why division is not communicative.

2. Describe two ways of calculating 8(15+5).

3. Provide an example to illustrate that division is not associative.

4. Is putting your right shoe first before your left shoe communicative? Provide an example of two things that you do that are not communicative.

5. Give examples of two things you do that are communicative.

Lesson 1.4 The set of Integers The set of integers consists of all positive whole numbers, all negative whole numbers, and zero.

I = { …,-3,-2,-1,0,1,2,3,…} The set of natural numbers, N, may also be called the set of positive integers or integers greater than zero.

Integers that are greater than zero are called positive integers. Integers that are less than zero are called negative integers. Zero is neither positive nor negative.

Distance is always a positive value. Absolute value is used to describe distance on a number line.

The absolute value of an integer is equal to its distance from 0. The absolute value of x is written as | |.

Example: The integer 4 is 4 units from 0. The integer -4 is 4 units from 0. So, opposite integers have the same absolute values. The absolute value of 4 is written as | |.

Example: Find the value of each expression. | | | | = 9 + 2 = 11 | | | | = 10 – 5 = 5 An inequality is a mathematical statement that contains the symbol:

.

Two important terms are common in inequality problems: 1. The phrase “at most” means (less than or equal to.

2. The phrase “at least” means (greater than or equal to.

Example: Fill in the appropriate symbol in each translation a. The team (t) must have at least 8 members.

Translation: t 8 b. The height (h) must be at most 6’3” Translation: h 6’3” Practice 1. Anne’s Science teacher told the class that the term paper would have to be at least 10 pages.

Translation: Number of pages of the term paper _____ 10 pages.

2. The publisher told the author he had sold at least 50 000 copies of books.

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Translation: Number of copies sold _____ 50 000.

3. If time is measured in years, and 0 stands for this year (2015), tell what number each year stands for. a. Last year b. Next year c. 5 years before last year d. 2022 e. 1953 4. Which is greater? a. +8 or -7 + 10 or -10 c. 25 or -26 5. List the integers that can replace x to make a true statement. Then graph the integers on a number line. a. | | b. | | c. | |

Lesson 1.5 Adding Integers Rules in Adding Integers Like signs: (+) + (+) or ( - ) + ( - ) Find the sum of their absolute values and use the sign common to both integers.

Unlike signs: (+) + ( - ) or ( - ) + ( + ) Find the difference of their absolute values and use the sign of the integer with the greater absolute value.

Example -9 + (-3) = Solution Find the absolute values: | | = 9; | | Since the signs are the same, add the absolute values: 9 + 3 = 12 The addends are both negative, therefore, the sum is negative: -9 + ( -3) = -12 If opposites are added, the sum is always 0.

Example: 11 + (-11) = 0 -20 + 0 = -20 Practice Add the following integers 1. 85 + (-212) 2. -458 + 384 3. -85 + 31 Find the missing number in each addition sentence 1. 12 + (? ) = 0 2. 0 + (?) = 0 3. 13 + (?) = -17 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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4. -8 + (?) = -15 5. -17 + 4 + (?) + 38 = 14 Lesson 1.6 Subtracting Integers Rule to subtract an integer, add its opposite.

Or For all numbers a and b, a – b = a + ( -b) Example: convert each difference into a sum.

9 – (-3) = 9 + (3) =

change – to +. The opposite of -3 is 3

Example: Subtract 7 – (9) = 7 + (-9) = -2 Practice: subtract. Use the rule for subtracting integers 1. 10 – (-5) 2. -10 – (-12) 3. -12 – 15 4. 42 – (-26) 5. 19 -23

Lesson 1.7 Multiplying Integers Rules in multiplying Integers 1. When you multiply tow numbers with the same sign, the product is positive. (+)(+) = (+) (-)(-) = (+) 2. When you multiply two numbers with different signs, the product is negative. (+)(-) = (-) (-)(+) = 3. Any number multiplied by 0 gives a product of 0.

(0) ( any number) = 0 ( any number)(0) = 0 Example: a. (-2)(-3)(5) = 30 b. (-6)(-2)(-5)(-8) = 480 c. (-1)(-2)(-3)(-4)(-5)= -120

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Practice: Find each product: a. (-7)(-5) b. (-10)(9) c. (-6)(-4)(-2)(-1) d. (-3)(-3)(-3)(-3)(-3) e. (-9)(0)

Lesson 1.8 Dividing Integers Rules in dividing integers 1. When two numbers with the same sign are divided, the quotient is always positive. = (+) 2. When two numbers with different sign are divided, the quotient is always negative. = (-) 3. The rules for dividing zero by a nonzero number and for division by zero still hold.

Example: a. (-40) ÷ (-8)

negative ÷ negative:

=5

b. 72 ÷ (-9)

positive ÷ negative:

= -8

c. -49 ÷ 7

negative ÷ positive:

=7

Practice: 1) (-63) ÷ (9) 2) (-88) ÷ (-8) 3) (-50) ÷ (-10) 4) (125) ÷ (-5) 5) I72I ÷ I12I 6) (56) ÷ (7)

Lesson 1.9 The Set of Rational Numbers A rational number is any number that can be written in the form , where a and b are integers, and where b, the integer in the denominator, is not equal to 0. The set of rational numbers is represented by { | The symbol

} is read “is not equal to”. Thus, b

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means b is not equal to 0.

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A number of the form

means a รท b, where a is the numerator and b is the denominator. Also, if a and

b are both positive, is called a proper fraction if a < b, an improper fraction if a>b, and a whole number if a divides exactly a.

Every rational number can be represented by either a terminating decimal or a repeating decimal.

Every terminating or repeating decimal represents a rational number.

Rule: for all integers a and b and all positive integer c and d: 1. > if and only if ad > bc 2. < if and only if ad < bc Example: which is greater? >

or

the LCD is 12

=

since 9 > 7

>

because (3)(12) > (7)(4) or 36 > 28

Density property for rational Numbers Between every pair of distinct rational numbers, there is another rational number.

Find a number between the pair of number , Rewrite

as similar fractions. Fractions are similar if they have the same denominator.

In and and

, there is no integer between 2 and 1 , 3 is between 2 and 4, hence is in between

Operations with rational Numbers If , then: Addition Subtraction Multiplication Division

=

= =

=

= =

c

Answer the following 1. A real number that can be written as a quotient of two integers is called a/an _____.

2. For all integers a and b and all positive integers c and d, > if and only if _____ > ______.

3. If a and b are rational numbers, and a < b, then the number halfway between a and b is _____.

4. Express the number 54 as a quotient of two integers. _____ 5. Every terminating or repeating decimal represents a/an _____

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Lesson 1.10 Square Roots A perfect square is the square of a whole number. The number 9 is a perfect square because 9 = 3 2. The number 7 is not a perfect square because there is no whole number that can be squared to get 7.

25 is the square of 5 because 52 = 25. You can say that 5 is the square root of 25. -5 is also a square of 25 because (-5)2 = 25. The symbol √ is used to indicate the positive square root and is known as the radical sign. When √ is an integer, the number n is called a perfect square.

Example: Find the square root √ Solution: 112 = (11)(11) = 121; √

= 11

Irrational number is a number that cannot be expressed in the form , where a and b are integers and b is not equal to 0.

Example 6.3 √ √ is a non terminating, non repeating decimal, we say that √

is an irrational number.

Practice: Complete the table. Round each number to three decimal places Square root value Rational or irrational √ √ √ √ √

Lesson 1.11 Scientific Notation The easier way to treat big numbers is to use scientific notation.

In scientific notation 15 000 000 can be written as 1.5 x 10 7 . Write the number from 1 to 10 then multiply by power of 10.

This means that scientific notation is expressing a number as a product of two factors one of which is a power of 10 and the other is a number between 1 and 10.

Examples: YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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1. 5 000 = 5 x 103 2. 18 000 = 1.8 x 104 3. 251 000 = 2.51 x 105 To write the scientific notation to standard form, use multiplication Example: 1. 1.5 x 103 = 1.500 move the decimal point 3 places to the right as a result of multiplying by 1000 = 103 2. 7.394 x 107 = 7.394 x 10 000 000 = 73 940 000 Practice: 1. Write in scientific notation a. 530 b. 7610 c. 890 000 d. 4 600 000 e. 15 700 000 2. Write each in standard form a. 2.7 x 103 b. 5.18 x 105 c. 3.478 x 106 d. 4.326 x 107 e. 8.11 x 109

Chapter 2:

Measurements Lesson 2.1 Measurements

prefix Kilo- Hectosymbol k h value 1 000 100

Dekada 10

Decid or 0.1

CentiC or 0.01

Millim or 0.001

The prefixes deka-, hector-, and kilo- are prefixes that indicate multiplication by 10, 100, 1000. A kilowatt is a unit of measure used in electricity and is equal to 1 000 watts.

The prefixes, micro- and mega- refer to 1 million. A megaton is 1 million tons. A micrometer is onemillionth of a meter. The prefix micro- may remind us of microfilm, the very small film that libraries use to keep copies of printed matter.

Prefix ExaPetaTeraGiga-

Symbol Multiple Equivalent E Quintillion P Quadrillion T Trillion G Billion

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MegaKiloHectoDekaDeciCentiMilliMicroNanoPic0FemtoAttoZeptoYocto-

M K H Da D C M Îź N P F A z y

10

Million Thousand Hundred Ten Tenth Hundredth Thousandth Millionth Billionth Trillionth Quadrillionth Quintillionth Sixtillionth Septillionth

A kilogram is 1 000 grams Hectoliter is 100 liters A milliliter is

of a liter or 0.001 liter

An attometer is one quintillionth of a meter.

Practice: Complete the following 1. 1 kilometer = _____ meter 2. 1 dekameter = _____ meter 3. 1 hectoliter = _____ liter 4. 1 centimeter = _____ meter 5. 1 millimeter = _____ meter Lesson 2.2 Measuring lengths The basic unit of length for metric measure is the meter (m).

Metric units of length word symbol Meaning Kilometer Km 1000 meters Hectometer Hm 100 meters Dekameter Da 10 meters Meter M 1 meter Decimater Dm 0.1 meter Centimeter Cm 0.01 meter Millimeter mm 0.01 meter

These rules to change from one metric to another

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1. Change from a larger unit to a smaller unit ( moving to the right in the diagram). Multiply by a power of ten. Thus, move the decimal point in the given quantity one place to the right for each smaller unit until the decimal unit is r decimal unit reached.

X 10 Km hm da m dm cm mm 2. To change from a smaller unit to a larger unit ( moving to the left in the diagram). Divide by a power of ten. Thus, move the decimal point in the given quantity one place to the left for each larger unit until the decimal unit is reached.

Km hm da m dm cm mm รท 10 Example: Convert 7 km = 7 000 m km = 2 000 m 12.3 cm = 0.000123 The chart shows how customary units of length are related Customary unit: length 12 inches (in.) = 1 foot (ft.)

36 inches = 1 yard (yd) 3 feet = 1 yard (yd) 5 280 feet = 1 mile (mi) Another to convert from unit to another is called dimensional analysis. In this procedure, we use unit fractions.

A unit fraction has two properties: the numerator and denominator contain different units, and the value of the unit fraction is 1.

Example

Example: Convert 8 ft = _____ in.

8 ft = Example The human body has 45 miles of nerves. How long is this in feet?

45 mi x

= 237 600 ft

Practice Complete each 1. 7 m = _____ cm 2. 300 cm = _____ m 3. 5.3 km = _____ m YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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4. 47 mm = _____ cm 5. 53000 cm = _____ km Complete each 1. 15 ft = _____ in 2. 23 mi = _____ ft 3. 63 in. = _____ cm 4. 9 yd = _____ m 5. 42 mm = _____ in Lesson 2.3 Measuring capacity Capacity of a container is the amount of liquid it can hold. Volume and capacity are related in the metric system.

1 L = 1 dm3 = 1 000 cm3 A cubic meter ( m3) is used to measure large volumes. One cubic meter = 1.3 cubic yards A kiloliter is equivalent to 1 000 liters. word symbol Meaning Kiloliter KL 1000 liters Hectoliter HL 100 liters Dekaliter DaL 10 liters liter L 1 liter Deciliterr DL 0.1liter Centiliter CL 0.01liter Milliliter mL 0.001liter

Two rules are used to change metric units.

1. To change from a larger unit to a smaller unit ) moving to the right in the diagram), multiply by a power of ten. Thus, move the decimal point in the given quantity one place to the right for each smaller unit until the decimal unit is reached.

2. To change from a smaller unit to a larger unit ( moving to the right in the diagram), multiply by a power of ten. Thus, move the decimal point in the given quantity one place to the left for each larger unit until the decimal unit is reached.

Example: 0.24 kl = _____L To convert from kiloliter to liter, we start at kiloliters and move three steps to the right to obtain liter X 10 KL hL daL L Hence move the decimal point three places to the right.

0.24 kL = 240 L This is the same as multiplying 0.24 by 1000.

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Thus, 0.24 kL converts to 240 L. Volume in cubic units capacity 1 cubic yard about 200 gallons 1 cubic foot about 7.48 gallons 231 cubic inches about 1 gallon Customary units of capacity 8 fluid ounces (fl. Oz.) = 1 cup ( c ) 2 c = 1 pint (pt ) 2 pt. = 1 quart (qt) 4 qt. = 1 gallon (gal.)

Example: Convert 50 qt. = _____ gal 4 qt = 1 gal.

50 qt x

12.5 gal.

Practice A. Complete each 1. 2 L = _____ mL 2. 0.5 kL = mL 3. 4 cL = daL 4. 4 000 mL = _____ L 5. 2 L = _____ hL B. Complete each 6. 18 qt = _____ pt.

7. 24 gal. = _____ qt.

8. 44 qt. = _____ gal.

9. 100 gal. = _____ pt 10. 16 gal = _____ c Lesson 2.4 Measuring mass The table can be used to find equivalent measures of weight Symbol kg hg Dag g dg cg Word kilogram Hectogram dekagram gram decigram centigram meaning 1 000 grams 100 grams 10 grams 1 gram 0.1 gram 0.01 gram Example Convert 0.3 g = ___ mg YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

mg milligram 0.001 gram

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Start from g and move three places to the right to obtain mg X 10 g dg cg mg 0.2 g = 300 mg Convert each of the following 128 oz. = _____ lb.

1lb. = 16 oz.

128 oz. x

= 8 lb.

Practice A. complete each 1. 30 kg = _____g 2. 12 g = _____ mg 3. 4 732 g = _____ kg 4. 22 dag = _____dg 5. 34 mg = _____ g B. Complete each 1. 52 oz. = _____ lb.

2. 35 lb = _____ oz.

3. 19 000 lb = _____ T 4. 900 kg = _____ lb.

5. 63 oz. = _____ g Lesson 2.5 Measuring Perimeter and Circumference The sum of the lengths of the sides of a polygon is called the perimeter ( P) of the polygon. A Polygon is a closed figure whose sides are segments. A regular polygon is a polygon having all sides congruent and all angles are congruent Formula: Perimeter of a rectangle and regular polygons The perimeter (p) of a rectangle with length (l) and width (w) is given by: P = 2l + 2w or P = 2 (l + w) Equilateral triangle s s s P = 3s Square s s s s P = 4s Regular pentagon s s s s s YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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P = 5s Example Find the perimeter of each polygon a. Regular octagon: side, 9 m An octagon is a polygon with eight sides. Since the decagon is regular, P = 10s.

P = 8s P = 8(9) P = 72 A circle is a set of all points in a plane that are at the same distance from a given point in the plane. The given point is called the center.

The circumference of a circle is the distance around the circle.

Formula for circumference C= Where d is the diameter of the circle and r is the radius of the circle.

Example Find the circumference of a circle whose diameter is 50 ft.

C= write the formula for the circumference C 3.14 (50) Replace 3.14 and d with 50.

C 157 The circumference is approximately equal to 157 ft.

Practice 1. A rectangular playground is 800 feet long and 400 feet wide. If fencing costs P560 per yard, how much would it cost to place fencing around the playground?

2. A square flower bed is 10 feet long on a side. How many plants are needed if they are spaced 6 inches apart around the outside of the bed?

3. One side of a triangular flower garden has a length of 4.5 m, the second side is twice as long as the first side. The length of the third side is 3 m less than that of the second side. What is the perimeter of the garden?

Lesson 2.6 Measuring area The region enclosed by a plane figure is called the area (A) of the figure.

Formula Area of a rectangle YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Area of a rectangle = length x width A = lw Area of a square Area of a square = (length of a side )2 A = s2 Example 1. Find the area of the rectangle whose length is 7 and width is 3 A = lw A = (7)(3) = 21 2. The length of a side of a square is 12 inches. Find its area Solution A = s2 = (12)2 = 144 The area of a square with side 12 inches long is 144 in 2.

Formula of area of a parallelogram Area of parallelogram = base x height A = bh Example Find the area of the parallelogram whose base is 8 in and height is 3 in Solution: A = bh = (8)(3) = 24 The area is 24 sq. in.

Example The area of a parallelogram is 96 cm2 and its base is 8 cm long. Find its height.

Solution: A = bh 96 = 8 h 12 = h Formula of Area of a Triangle Area of a triangle = x base s height = bh

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Example: Find the area of a triangle whose base is 28 in and height is 6 in.

Solution: A= = (28)(6) = 84 The area is 84 sq in.

Formula: Area of a trapezoid = x height (base 1 + base 2) A = h ( b1 + b2) Area of a rhombus = base x height A = bh Example Find the area of a trapezoid whose height is 4 ft., bases are 11 ft. and 5 ft.

A = h (b1 + b2) = (4)(5 + 11) = 32 The area of the trapezoid is 32 ft2.

Find the area of a rhombus whose height is 8 dm and bases is 15 dm.

A = bh = (15)(8) = 120 The area of the rhombus is 12 dm2 A prism is a special type of solid. It has two bases which are congruent polygonal regions lying in parallel planes.

The surface area of a space figure is the sum of the areas of all the faces.

A cube is a prism with six congruent faces. The surface area of a cube is six times the area of the face of 6e2 Formula: Surface area of a cube Surface area = 6 x area of a face SA = 6e2 Example Find the surface area of the cube whose edge is 7 cm Solution SA= 6 e2 = 6 (7)2 = 6 (49) YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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= 294 sq cm.

Surface area of a rectangular solid Surface Area = 2 lw + 2hw + 2hl Example: Find the surface area of the rectangular solid whose length is 11, height is 3, and width is 7 A = 2 lw + 2hw + 2hl = 2 (11)(7) + 2 (3)(7) + 2 (3)(11) = 154 + 42 + 66 = 262 sq cm The surface area of the rectangular solid is 262 sq. cm.

Lesson 2.7 Measuring volume The amount of space a three dimensional figure occupies is called its volume.

Formula of Volume of a Cube The volume (V) of a cube is equal to the cube of (e) which is the length of an edge. e V = e3 Example: Find the volume of a cube with the edge is 10 ft V = e3 V = 103 V = 1 000 ft3 Volume of prism The volume of a prism is equal to the product of the area of the base (B) and its height (h) V = Bh Example Find the volume of the prism with a length is 12, width is 8 ft and height is 6 ft.

V = Bh V = lwh = (12)(8)(6) = 576 The volume is 576 ft3 The volume of a right circular cylinder Formula: V = Bh = Where ( r ) is the radius of the circle at either end and h is the height YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Example: Find volume of a right cylinder whose height is 15 m and diameter is 4 m V = Bh V= h Solution V = 3.14 (2)215 V V Volume of a pyramid Formula: V = Bh Where b is the base and h is the height ( the perpendicular distance from the top to the base._ Example: Find the volume of the pyramid whose height is 9 in and length is 12 in and width is 8 in.

V = (lw) h V = (12 x 8)(9) V = 288 The volume is about 288 in3.

Volume of a cone Formula The volume (V) of a right circular cone of radius ÂŽ equals one third the product of the area of its base (B) and its height (h) V = Bh =

h

Example: Find the volume of the cone whose diameter is 24 and height is 20 cm.

V =

h

V V The volume is about 3 014.4 cm3.

Volume of a sphere The volume of a sphere with radius r is given by the formula V= Example Find the volume of a sphere whose diameter is 6 cm Solution V= V YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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V 904.32 The volume of a sphere is about 904.32 cm3 Practice: Find the volume of each rectangular prism described length width height volume 16 cm 7 cm 12 cm 14 in. 9 in.

11 in.

Find the volume of each rectangular pyramid length width height volume 8 cm 6 cm 17 cm 2.5 in. 2.8 in. 2.9 in.

Find the volume of right circular cone radius height volume 3 ft 12 ft 12 m 14 m Lesson 2.8 Measuring Time Measures of time: Conversion 1 day = 24 hours (hr) 1 hour = 60 minutes (min.)

1 minute = 60 seconds (sec.)

Example: Convert the measurement to the indicated unit.

15 days = _____ hours 15 days x

= 360 hours

Thus, 15 days = 360 hours Find the time or elapse time Between 3:00 P.M. and 2:30 A. M. To get 2:30 AM you need to go another hour and a half past 12:00 midnight. Add 2 hr. 30 min. to 12 hr. to get 14 hr. 30 min.

2: 30 A.M. 14 hr. 30 min.

- 3: 00 P.M. -3 hr. 00 min 11 hr. 30 min.

Practice Complete each 1. 8 min = _____ s.

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2. 6 days = _____ hr.

3. 30 hr. = _____ min.

4. 4 hr. = _____ s.

5. 50 s. = _____ min.

Find the time elapse 1. 2 hr. 43 min. after 7:15 A.M. 2. 11 hr. 4 min. before 10:00 P.M. 3. 5 hr. 25 min after 10:35 A.M. 4. 2 hr. 56 min. before 9:38 P.M. 5. 5 hr. 29 min. after 11:32 A.M. Lesson 2.9 Measuring Temperature Temperature is the hotness or coldness of something. The two most commonly used temperature scales are the Fahrenheit and the Celsius.

The Fahrenheit scale was invented in the early 1700s by Gabriel D. Fahrenheit, a German physicist. A salt-water solution freezed at 00F at sea level. Pure water freezes at 32 0F and boils at 212 0F. The normal temperature of the human body is 98.6 0F. The Celsius scale was developed in 1742 by Anders Celsius, a Swedish astronomer. Pure water freezes at 0 0C and boils at 100 0C. the Celsius scale divides the interval between these two points into 100 equal parts.

The following formulas will help us convert from Fahrenheit to Celsius and vice versa.

Fahrenheit to Celsius Celsius to Fahrenheit

Convert 32 therefore , 32 Convert = Therefore Practice 1. Convert each to Fahrenheit 1. 10 2. 20 3. -15 4. -40 5. 120 2. Convert each to Celsius 1. 95 2. 41 3. 104 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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4. -22 5. -4

Lesson 2.10 Ratio, Rate, and Speed A ratio is a comparison of two similar quantities.

Given two similar quantities, a and b, the ratio of a and b is defined as a:b = , where b

0

a and b are called terms of the ratio Example: There are 10 boys and 30 girls in the drama club. Find the ratio a. The number of boys to the number of girls b. The number of girls to the total number of members in the club.

Solution: a. Ratio of number of boys to the number of girls = 10 : 30 = 1:3 b. Total number of members = 10 + 30 = 40 c. Ratio of the number of girls to the total number of members = 30:40 = 3:4 Equivalent ratios A ratio has no units. It can be expressed as . Because is a fraction, it can have the equivalent fractions; and

for any integer m where m 0. Thus, all ma:mb and

and b can be any rational numbers except b Example Simplify the following ratio

equivalent ratios of a : b where a

.

1: = 1 x 2 : x 2 =2:1 A ratio can be used to represent relationship of more than two quantities. For example, if a = 27, b = 36, and c = 24 then the ratio of a to b to c is: a : b : c = 27 : 36 : 24 = 9 x 3 : 12 x 3 : 8 x 3 = 9: 12 : 8 Rate A rate is a ratio of two measurements having different units of measurements. For example, 300 words in 5 minutes is a rate. When rate is simplified so it has a denominator of 1, it is called a unit rate. To find the unit rate of 300 words in 5 minutes, divide 300 by 5. The result is a unit rate of 60 words in 1 minute or 60 words per minute.

Speed Speed = YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Speed is used for the rate of change of distance travelled. For example, if a man jogs 1 500 meters in 5 minutes, then he jogs at the rate of

or 300 meters per minute, 300 m/min.

If the speed is unchanged during a period of time, it is called a uniform speed or constant speed.

However, it is difficult to maintain a uniform speed for a certain period of time. In real life situation, it is more realistic to use average speed.

Average speed = Example The average speed of a car for the first 3 hours is 70 kph. Its average speed for the next 2 hours is 80 kph.

Find its average speed for the whole trip.

Solution Total distance travelled = 70 (3) + 80 (2) = 210 + 160 = 370 km Total time taken = 3 + 2 = 5 Average speed for the whole trip =

= 74 kph

Practice Express the ratio in simplest form.

1. 75:35 2. 72:66 3. 2 :1 4. 36:144 5. 0.038:44 Find the rate in each case.

1. A 3 liter of orange juice costs P150 2. Rose types 400 words in 8 minutes.

3. A factory manufactures 150 tablets in 4 days 4. The mass of 50 mangoes is 15 kilos 5. Arnold paid P500 for 8 tickets Chapter 3 Algebraic expressions Lesson 3.1 The language of Algebra The branch of mathematics that involves expression with variables is called algebra. The result of combining numbers and variables with ordinary operations of arithmetic is called an algebraic expression or simple expression.

X+7

2 r

b2 â&#x20AC;&#x201C; 4ac

and

English expressions leading to algebraic expressions with addition and subtraction, multiplication and division YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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English phrase The sum of m and 8 10 added to c 5 less than t 18 reduce by n Twice x Ten rimes c The quotient of 8 and m

Algebraic translation m+8 c + 10 t–5 18 – n 2x 10c

The ratio of 7 to a

Practice Translate each into an algebraic expression 1. Nine less than a certain a number.

2. Thirteen decreased by a number.

3. Thrice the sum of x, y and z 4. Six times the number, decrease by 10 5. Eleven more than the sum of two different numbers Lesson 3.2 Classifying algebraic expressions A polynomial is an algebraic expression that represents a sum of one or more terms containing wholenumber exponents on the variables.

A polynomial with one term is called a monomial.

A polynomial with two terms is called binomial.

A polynomial with three terms is called a trinomial.

Examples monomials binomials trinomials 9a2b 5x3 – 2x2 -8a2 + 3a + 4 8x -15x24 + 3 -11x7 – 4x4 – 3x2 Classifying polynomial Degree Example Classification 1 2x + 3y linear polynomial 2 2 3x + 2x -1 quadratic polynomial 3 2 3 4x – 2x + x – 5 cubic polynomial A polynomial containing two or more like terms can be simplified by adding these terms. This process is called combining like terms. Liked terms are two or more terms that contain the same variables and exponents. If the terms differ by at least one variable, they are called unlike terms.

Like terms unlike terms 4x and 7x 4x and 3y YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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4a2b and -5a2b 4a2b and -5ab2 Example Simplify 10x2 + 3 + 12x2 = (10 + 12)x2 + 3 = 22x2 + 3 Practice Simplify by combining like terms 1. 4x + 8 – (5x – 6 ) 2. – ( x – 3) – 3 – x 3. 3a + 4b – 5a + 6b 4. 4x + 5y – 10 + 9x – 11y + 13 5. -3k2g – [ 4kg2 – ( 6k2g – 2kg2)] Lesson 3.3 Evaluating Algebraic expressions When a number is substituted for the variable in a polynomial, the polynomial takes a numerical value.

Finding the value is called evaluating the polynomial.

Steps in evaluating an algebraic expression: 1. Replacing the variable by the given number value and 2. Performing the indicated arithmetic following the order of operations.

- First, simplify expressions within grouping symbols - Then simplify powers - Then simplify products and quotients in order from left to right - Then simplify sums and differences in order from left to right Example Evaluate =

when x = 1 and y = -1 or

Practice Find the value of the following algebraic expressions 1. 9a + 3; a = 5 2. X + 3 (x – 5); x= 25 3. 4a ÷ (13 – 9b); a = 12 and b = 4.

(F – 32) when F = 830 Lesson 3.4 Adding Polynomials

To add two polynomials, write the sum and simply by combining like terms.

Example (3x + 4) + (7x – 8) = ( 3x + 7x ) + ( 4 – 8) = (3 + 7) x + (-4) YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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= 10x – 4 Practice Simplify 1. (-6x -3) + (3x – 9) 2. (3x2 + 5x – 9) + ( -3x + 4) 3. (4x2 + 3x + 6) + ( 6x2 + 3x – 7) 4. (5x2 + 10xy + 4y2) + ( 7x2 + 8xy – 6y2) 5. (2ab2 + 2a3b – 4ab) + ( 4a2b – 3ab – 2a2b) Lesson 3.5 Subtracting Polynomials Subtraction Rule To subtract an expression from another expression, add its negative. That is, a – b = a + (-b) You can subtract a quantity by adding its negative. First change the sign of each term in the subtrahend, then group like terms, and then add.

Example: Subtract (5x + 4) – (7x + 2) = (5x – 7x) + (4 – 2) = - 2x + 2 (9x + 1) – (-2x + 2) = (9x + 1)+ (2x – 2) = (9x + 2x) + (1 – 2) = 11x – 1 Practice 1. (a – 2) – ( a – 9) 2. (b – 7 ) – ( 4b – 2) 3. ( 2x2 – 3y) – ( 3x2 – 5y)

Lesson 3.6 Product of Monomials Laws of exponents 1. Product Rule for Exponents: xm xn = xm+n To multiply powers having the same base, keep the base and add the exponents.

2. Power Rule of Exponents: (xm)n = xmn To find the power of a power of a base, keep the base and multiply the exponents.

3. The power of the product Rule: (xy)m = xmym To find the power of a product, find the power of each factor and then multiply the resulting powers.

Example 1 Simplify each

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=

= 4a26

=

Example 2 Find the volume with the given dimensions: L=x W=4x H=5x Volume = L x w x H V = (x)(4x)(5x) = (4)(5)(x1+1+1) = 20x3 cubic units Exercises: Find the product: 1) 2) 3) Lesson 3.7 Product of Monomials and Polynomials Rule: To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial Example: -4x(x2 – 4x + 11) = -4x(x2) -4x(-4x)-4x(11) = -4x3 + 16x2 -44x Practice: Find the product a) 4(x + 3) b) -3(a – 5) c) d)

m2 (8m5 – 4) m2n(15m + 20n)

e) Lesson 3.8 Product of two or more polynomials Multiplication of two polynomials requires repeated application of the distributive property (4x + 5)( x2 –x + 4) = (4x + 5)(x2) - (4x + 5)(x) + (4x + 5)(4) = (4x3 + 5x2) – ( 4x2 + 5x) + (16x + 20) = 4x3 + 5x2 – 4x2 + 5x + 16x + 20 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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= 4x3 + x2 + 11x + 20 Another method of multiplying two polynomials is to use a vertical format similar to that of multiplication of whole numbers. Let’s use the vertical format to find the product of the previous example. Arrange the polynomials such that like terms are aligned in the same column. More, importantly, the terms must be arranged in increasing or decreasing degrees.

Example: Multiply using the vertical format x2 – x + 3 x–2 2 -2x + 2x – 6 3 x –x2 + 3x____ x3 -3x2 + 5x – 6 FOIL Method Multiplication of two binomials using the foil method F= the product of the first term of each binomial O=the product of the outer term of each binomial I= the product of the inner term of each binomial L = the product of the last term of each binomial Example: Simplify (5x – 6)( 2x + 1) Solution: F O I (5x – 6)( 2x + 1) = (5x)(2x) +(5x)(1) + (-6)(2x) + (-6)(1) = 10x2 + 5x – 12x – 6 = 10x2 – 7x – 6

L

Practice: Multiply each a) ( x + 9) ( x + 4 ) b) (a – 5 ) ( a – 6) c) (5b – 12)( 2b + 9) d) ( 9x – 2y)(4x – 3y) e) (6x + 8y)(8x – 5y) Lesson 3.9 Square of a binomial 2 The expression (a + b) is the square of a binomial (a + b). it can be written as the product (a + b) (a + b).

This product can be expanded using the FOIL method. (a + b) (a + b) = a2 + ab + ab + b2 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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= a2 + 2ab + b2 The square of the difference (a – b)2 can be expanded in the same manner. (a - b) (a - b) = a2 - ab - ab + b2 = a2 - 2ab + b2 Notice after simplifying, the square of a binomial has three terms. The trinomial a 2 + 2ab + b2 and a2 2ab + b2 are called perfect square trinomials.

The Square of a Binomial is a square of the first term plus (or minus) twice the product of the two terms plus the square of the last term.

When you use these special product patterns, remember that x and y can be numbers, variables, or even algebraic expressions.

Trinomials in the form x2 + 2xy + y2 or x2 - 2xy + y2 are called perfect square trinomials because each is the result after squaring a binomial.

Example: (a + 7)2 = a2 + 2 (a) (7) + 72 = a2 14a + 49 Example: A square garden is surrounded by a walk 1 meter wide. If the area of the walk is 96 square meters, what are the dimensions of the garden?

Solution: Let x = the length of the side of the garden Then, x + 2 = the length of the square including the walk The area of the walk is the difference of the areas of the larger square and the smaller square. (x + 2)2 – x2 = 96 the area of the walk is 96 m2 x2 + 4x + 4 - x2 = 96 (x + 2)2 = x2 + 2xy + y2 4x + 4 = 96 simplify 4x = 92 subtract 4 from both sides. x = 23 divide each side by 4 the length of a side of the garden is 23 meters.

Practice: Write each perfect square as a trinomial a) (a + 5)2 b) (8 + c )2 c) (3m + 8)2 d) (5n – 8)2 e) (

)

Lesson 3.10 Cube of a binomial The cube of a binomial ( x+ y) is the sum of four terms where each can be obtained as follows: First term: the cube of x YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Second term: Third term: Fourth term:

three times the product of x2 and y three times the product of x and y2 the cube of y

Cube of a Binomial Pattern For all numbers x and y: (x + y)3 = x3 + 3x2y +3xy2 + y3 (x – y)3 = x3 - 3x2y +3xy2 - y3 Example: Find the product of (2x + 5)3 Solution: First term: the cube of 2x or 8x3 Second term: three times the product of (2x)2 and 5 or 3(2x)2(5) or 60x2 Third term: three times the product of (2x)2 and (5)2 or 3 (2x)25 or 150x Fourth term: the cube of 5 or 125 Example: [a + (2b + 1)]3 = (a)3 + 3(a)2(2b + 1) + 3(a)(2b + 1)2 + (2b + 1)3 = a3 + 6a2b + 3a2 + 3a(4b2 + 4b + 1)2 + 8b3 12b2 + 6b + 1 = a3 + 6a2b + 3a2 + 12ab2 + 12ab + 3a + 8b3 + 12b2 + 6b + 1 Practice Expand the following 1. (a + 3)3 2. ( b – 3)3 3. (5 + c)3 4. (5d + 3)3 5. (3e2 – 4f)3 Lesson 3.11 Products of Sums and differences Multiplication of binomials with special products Using FOIL, it is possible to find a pattern for the product of the sum and difference of two terms and for the square of a binomial.

The expression ( a + b)(a – b ) is the product of the sum and difference of two terms. The first binomial in the expression is a sum; the second is the difference of two terms. The first binomial in the expression is a sum ; the second is a difference. The two terms are a and b. The first term in each binomial is a. The second term in each binomial is b.

The product of the Sum and Difference of Two Terms (a + b) ( a – b) = a2 – ab + ab + b2 = a2 - b2 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Example: Simplify (3x – 2)(3x + 2) Solution: (3x – 2)(3x + 2) = (3x)2 – (2)2 = 9x2 – 4 Practice: a) (3x + 2)(3x – 2) b) (2a + 3)(2a – 3) c) ( d) (7a – 5 ) ( 7a + 5) Lesson 3.12 Squares of Trinomials 2 2 2 2 (a + b + c) = a + b + c + 2ab + 2 ac + 2 bc There are six terms in the square of a trinomial 1) Square of the first term 2) square of the second term 3) square of the third term 4) twice the product of the first and the second term 5) twice the product of the first and the third term 6) twice the product of the second term and the third term Example: (x +2y+ 5z)= x2 + 4y2 + 25z2 + 4xy + 10xz + 20yz Practice: When trinomial x + y + z is squared 1) the first term is the square of _____ 2) the second term is the square of _____ 3) the third term is the square of _____ 4) the fourth term is _____ 5) the fifth term is _____ 6) the sixth term is _____ Find the product: 1) (x + 2y + 4z)2 2) (8m – 2n – 3p)2 3) (10p + 2Q – 3r)2 4) (9s – 2 + 6t)2 Lesson 3.13 Divisions of Monomials Rules of Exponents for Division YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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For every positive integer m and n, and x =

, when m > n;

=

, when m < n;

:

or 1, when m = n Rule 1: If the exponent of the numerator is greater than that of the denominator, find the quotient as follows: =

= x4 ( x

:

Rule 2: If the exponent of the numerator is less than that of the denominator, find the quotient as follows: (x Rule 3: If the exponents of the numerator and denominator are equal, the quotient is 1. ;

=1(x

)

Example: Simplify each. Assume that none of the variables is zero a) b)

=

To divide two monomials, we use the communicative and associative Properties to rearrange the factors. We also use the quotient rules for Exponents to help simplify the quotient. We will assume that variables represent nonzero values so that the expressions are defined.

Rule: Quotient with Monomial Divisor 1) To divide a monomial by a monomial, use the monomial communicative and associative properties to rearrange factors. Simplify using the Quotient rules for Exponents.

2) To divide a polynomial by a monomial, use the Distributive Law. Simplify using the quotient rules for exponents. In symbols, when a, b, and c are real numbers and c is equal to zero then: = + Example: =(

)

= -6x4 =

=

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Practice: Divide each. a) b) c) d) e) Lesson 3.14 Division of Polynomials by another Polynomial To divide a polynomial by another polynomial with more than one term, use a procedure similar to long division in arithmetic.

Example: 3x2 – 14 x – 12 ÷ x – 2 3x – 8 2 X – 2 3x – 14 x – 12 3x2 – 6x -8x – 12 -8x + 16 - 28 The quotient is 3x – 8 The remainder is -28 Rule: 1. Arrange the terms of both the dividend and the divisor in either ascending or descending order.

2. Divide the first term of the dividend by the first term of the divisor, and write the result as the first term of the quotient 3. Multiply the entire divisor by the number obtained in step 2 and subtract the product from the dividend. The remainder will be the new dividend.

4. Divide the new dividend by the first term of the divisor as before, and continue to divide in this way until the divisor is exact or until the remainder is of lower degree than the divisor.

5. If there is a remainder, write it over the divisor and add the fraction to the part of the quotient previously obtained.

Practice Divide and write your answer as a polynomial or mixed expression 1. X2 + 8x + 15 ÷ x + 5 2. X2 – 5x – 36 ÷ x + 4 3. X2 – 3x – 54 x + 6 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Chapter 4 Linear Equations and Inequalities in one variable Lesson 4.1 Introduction to Equations An equation is a statement that the two numbers or two expressions are equal. To solve an equation means to find all of the solutions to the equation. Any value of the variable that makes the equation true statement is a root or solution to the equation. Equations that they solution are equivalent equations.

These equations This equation This equation is This equation is are equivalent is an identity inconsistent conditional x – 5 and x = 11 x + x = 2x X=x+1 2x – 4 =0 Example Write the mathematical equation for each sentence 1. A number by 7 is 25. x + 7 = 25 2. Eleven diminished by a number is 53 11 – n = 53 Practice Write the algebraic equation 1. The quotient of 15 and a number is 5 2. One-third of a number is 30.

3. The product of a number and 15 is 3 Lesson 4.2 Properties of equality Properties equality 1. Addition property of equality (APE) If x = y, then x + z = y + z Equals may be added on both sides of the equation.

2. Subtraction property of equality (SPE) If x = y, then x – z= y – z Equals may be subtracted from both sides of the equation.

3. Multiplication property of equality (MPE) If x = y, then xz = yz Both sides of the equation may be multiplied by equals.

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4. Division property of equality (DPE) If x = y and z 5. Substitution law If x + y = z and x = y, then y + y = z or x + x = z Equals may be substituted for equals 6. Reflexive property symmetric property If x = y, then y = x 7. Transitive property (TPE) 8. If x = y and y = z,then x = z 9. If two quantities are both equal to a third quantity, then they are equal to each other.

Example: Identify the property of equality used to solve each equation a. 5x – 3 = x + 25 5x = x + 28 APE 4x = 28 SPE X=7 DPE b.

+ 5 = 11 =6

2x = 18 X=9

SPE MPE DPE

Practice Identify the property illustrated 1. If k + 1 = 7, then k = 7 -1 2. If p + q = 53 and 53 = r + s, then p + q = r + s 3. If w + 12 = 7, then 5w + 60 = 35 Lesson 4.3 Solving equations using Addition of subtraction An addition equation is an equation involving the sum of a number and a variable. An addition equation can be solved by subtracting the same number from each side of the equation so that the variable is isolated on one side of the equation.

Example: Solve each equation X + 5 = 11 Your goal is to isolate the variable x X + 5 – 5 = 11 – 5 subtract 5 from each side YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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X=6 Subtraction equation is an equation involving the difference between a variable and a number.

You can solve a subtraction equation by adding the same number on each side of the equation.

Example: Solve the equation X – 7 = 10 X – 7 + 7 = 10 + 7 add 7 on both sides X = 17 Practice Solve 1. X + 3 = 6 2. -7 + x = -10 3. X + 12 = 7 4. X – 2.5 = 4 5. -8 + c = -5 Lesson 4.4 Solving simple equations using multiplication or division A multiplication equation is an equation involving a product of a variable and a number. You can solve a multiplication equation by dividing. Again, the goal is to isolate the variable on one side of the equation.

Example 8x = 72 X=9 A division equation is an equation involving a quotient of a variable and a nonzero number. You can solve a division equation by multiplying.

Example

X = 48 Practice Solve the equation 1. 5x = 85 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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2. -44b = 11 3. 16 = 4.

Lesson 4.5 Solving multi-step equations Some equations involve two operations. To solve these two-step equations, undo addition and subtraction first. Then undo multiplication and division.

Example 6x – 5 = 19 Isolate the variable x 6x – 5 + 5 = 19 + 5 6x = 24 X=4 When an equation involves a variable that is multiplied by a fraction, you can use the reciprocal of the fraction to solve the equation.

Example (

)

X = -18 The next example contains variables on both sides. Because variables represent numbers, you may transform an equation by adding a variable expression on each side or by subtracting a variable expression from each side.

Example 6x = 3x – 42 6x – 3x = 3x – 42 – 3x 3x = -42 X = -14 Cross multiplication is equivalent to multiplying both sides of an equation by the least common multiple of the denominators of the two fractions. After cross multiplication, the equation is solved in the usual way by removing brackets and collecting like terms.

Example 12 ( )

(

)

9x = 4(x – 3) YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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9x = 4x – 12 9x – 4x = 4x – 12 – 4x 5x = -12 X= Practice Solve 1. 6 + 4n = 1 2. 11 = -5 + 3.

= 10x + 3

4. 4x + 11 = 3 5. -18 + 4m = 10

Lesson 4.6 Number problems The process involved in solving problem The 3”Rs” and the “ESP” of solving verbal problems R Read the problem thoroughly R Represent the unknown with a variable R Relate the unknown and the values given in the problem.

E Equate form an equation using the facts in the problem S Solve the equation P Prove the answer READING: when we read a problem in mathematics, we must be sure we catch each word.

Also must keep asking ourselves two questions.

1. How would we represent this in mathematical terms?

2. Is the piece of information necessary?

REPRESENTING THE UNKNOWNs The unknown numbers in the problem can be represented in several ways.

For example, if one number is thrice another, we can represent them as x and 3x RELATING THE UNKNOWNs: Look for keywords that translate into equals. Some of these words are: is, are, was, make and equals EQUATION FORMING:As the unknown in the problem are correctly represented, then the meaning of the story can be obtained by expressing these into equation.

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SOLVING AND PROVING: We already know the methods in solving equations and we can check the answers to see if they satisfy the problem.

Example The sum of two numbers is 60. The larger number is four times the smaller. Find the smaller number.

Solution: READ: we know two things about the numbers: a) The sum of the two numbers is 60. b) The larger number is four times the smaller number.

REPRESENT: use the first sentence to represent the unknown numbers Let x = the smaller number RELATE: then, 60 – x = larger number EQUATE: the second sentence gives the equation.

SOLVE: 60 – x = 4x 60 = 4x + x 60 = 5x 12 = x Thus, the smaller number is 12 and the larger number is 4x or 48.

PROVE: show that the numbers satisfy both sentences a. The sum is 60: 12 + 48 = 60 b. The larger number is four times the smaller number: 48 = 4(12) The word consecutive means following in order without interruption, a nonnegative integer refers to a whole number. Hence, consecutive integers are whole numbers which follow a particular order without interruption. Each integer exceeds the integer preceding it by 1.

Even numbers are whole numbers that are divisible by 2, while whole numbers which are not divisible by (2) are odd numbers. An odd number is either one more than or one less than an even number. Consecutive even integers are even numbers in an uninterrupted order which is the same as when you count by two’s such as 2,4,6,8 and 10. Consecutive odd integers are odd numbers in an uninterrupted order, such as 7,9,11,13 and 15.

Consecutive integers: x, x + 1, x + 2, x + 3,…

Consecutive even integers: x, x + 2, x + 4, x + 6,…

Consecutive odd integers: x, x + 2, x + 4, x + 6…

Practice Solve 1. Four times a number increased by 5 is 37. Find the number.

2. Forty-five is equal to twelve more than thrice a number.

3. The difference between eight times a number and 16 is 40. Find the number.

4. Find the number that is 50 greater than its opposite.

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Lesson 4.7 Age problem In dealing with age problems, it is important to keep in mind that the ages of different people change at the same rate. For example, after two years, all the people in the given problem are two years older than they were two years ago. Four years ago, all the people in the problem were four years younger. It is also easier if one makes a table showing the representation for current ages in the problem, “future” ages (a number of years from now) and “past” ages (a number of years ago). If possible, represent the youngest present age by a single letter, then represent the other ages.

Example Michael is one-fourth as old as his father. Next year, their ages will total 42 years. How old is each now?

Solution: Let x = Michael’s age in years now.

Then, 4x = father’s age in years now.

Equation: Next year, their ages will total 42. ( x+ 1) + ( 4x + 1) = 42 X + 1 + 4x + 1 = 42 5x + 2 = 42 5x = 40 X=8 4x = 32 Michael is 8 years old now and his father is 32 years old.

Practice: Solve 1. Bobby is 3 years older than Chris. Last year, the sum of their ages was 39. How old is each now?

2. Beth is 5 years younger than Celeste. Next year, their ages will have a sum equal to 57.

How old is each now?

Lesson 4.8 Solving Work Problems Among the kinds of verbal problems that are solved by fractional equations are work problems.

When several people or different forces are at work on a particular job, it is important to know their work rates to determine how they will work together. If we are going to consider a task as one job, then the rate of doing work can be represented as: where x is the number of time units required to complete the job.

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Another thing that we must know in doing work problems is that the amount of work done is the product of the rate and the time, that is: Amount of the Persons task work Work = Raterate completed by Strategy forthat solving Work Problems person

Time at work Time

x

1. If the job is completed in x days, then the rate is job/day.

2. Construct a table showing rate, time, and work completed 3. The total work completed is the sum of the individual amounts of work completed.

4. If the job is complete, then the total work done is 1 job.

Amount completed by one person/machine

Example State the work rate

+

Amount completed by another person/ machine

Mimie can clean the room in 3 hours.

Mrs. Adoracion can edit a book in 6 weeks.

=

Whole task

job per hour. job per week.

Example Maya takes 12 hours to type a report. Her friend helps her and together they finish the job in 4 hours. How long would it take her friend had she worked alone?

Solution: Let x = the time it would take Mayaâ&#x20AC;&#x2122;s friend to do the job herself.

Then, her work rate is job per hour.

Name Maya Friend

Rate x time = work 4 4

Equation: Part of the job Maya does + part of the job friend does (

=

whole job

)

x + 12 = 3x 2x = 12 X=6 Answer: it would take her friend 6 hours to type the report.

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Practice Solve the problem 1. Carlos can paint a certain fence in 3 hours. If Magno helps him, the job takes only 2 hours. How long would it take Magno to paint the fence by himself.

2. Sandra can encode the whole manuscript in 6 hours. If her daughter helps, they get the job done in

hours. How long would it take her daughter to do the job by herself?

3. A faucet can fill a bath tub in 9 minutes. The drain can empty the full tub in 12 minutes.

With the drain open and the faucet on, how long will it take Marie to do the whole job?

Lesson 4.9 Solving Uniform Motion Problems When the object moves without changing its speed or rate, that object is said to be in uniform motion.

The three relationships 1. Rate x Time = Distance 2.

3.

Example What is the average rate of speed must a car travel to cover a distance of 243 kilometers in 3 hours?

Use the formula R = = R = 81 kph Motion in opposite directions Two buses leave a station, one traveling north at the rate of 60 kph, and the other south at the rate of 65 kph. In how many hours will they be 375 kilometers apart?

Solution Let x = number of hours each bus traveled.

Direction Rate x Time = Distance Northbound 60 x 60x Southbound 65 X 65x The equation is determined by the fact: the distance traveled by the northbound bus plus the distance traveled by the southbound bus equals the total distance

Distance the northbound bus traveled 60x

+ +

Distance the southbound bus 65x traveled =

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Total distance

375 Page 46

Solve the equation 60x + 65x = 375 125x = 375 X = 3 hours Answer: 3 hours Practice: Solve 1. How fast did a car go to overtake a passenger bus in 5 hours if the bus averaged 45 kph and left 4 hours before the car?

2. John climbs a mountain at 10 kph and returns at 5 kph. It it took 1 hour to climb and return, how far did he climb?

3. One Sunday, evening Luisa walked for 2 hours and ran for 30 minutes. If she ran twice as fast as she walked and she covered 12 kilometers altogether, find how fast did she run?

Lesson 4.10 Solving Mixture Problems A solution is a homogeneous mixture of two or more substances which components are uniformly distributed all throughout.

Solutions are usually labeled by the percentages of the solutes. This refers to the strength of the solution. That is, a solution that is 20% salt is twice as strong as a 10% salt solution. We add water to the solution to weaken it; the water that we added is 0% salt. We add salt to the solution to increase its strength. Pure salt is 100% salt.

In solving mixture problems, we label each solution by the percentage of solute in it, rather than by its value. When the amount if the given solution is multiplied by its strength, then we get the amount of solute it contains.

For example, if we 50 grams of a 15% salt solution, then we have 500 x 0.15 = 75 grams of salt.

Steps in solving mixture problem 1. Label each solution with its corresponding percentage.

2. Give the total amount of the combined solution.

3. Form the equation by multiplying vertically.

Example One solution is 80% acid and another one is 20% acid. How much of each solution is needed to make 100 gallons that is 65% acid?

Solution How much is the solution is needed First solution – 80% acid Second solution – 20% acid Final solution – 100 gal.,65% acid Let x = amount of the first solution.

Then, 100 – x = the amount of the second solution YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Solution Amount of solution Percent acid Amount of Acid st 1 solution x 80% 0.8x 2nd solution 100 – x 20% 0.2(100 – x) Final solution 100 65% 0.65(100) Equation: the sum of the amounts of the acid in the two solutions is equal to the amount of acid in the final solution.

0.8x + 0.2 (100 –x ) = 0.65(100) 0.8x + 20 – 0.2x = 65 0.6 x = 45 X = 75 gal. the amount of the first solution needed Thus, 100 – x = 100 – 75 = 25 gal. the amount of the second solution needed Practice 1. How many liters of water should be added to 20 liters of a 30% sulfuric acid solution to reduce it to a 15% solution?

2. How many milliliters of water should be added to 28 milliliters of a 25% hydrochloric acid solution to reduce it to 5% solution?

3. How many gallons of milk containing 4% butterfat must be mixed with 100 gallons of 1% milk to obtain 2% milk?

Lesson 4.11 Business and investment Problems Business formulas: S=C+M Selling price = Cost + Markup S=R–D Sale price = Regular Price – Discount P = br Percentage = base x rate I = Prt Interst = Principal x rate x time In the formula S = C + M, (S) represents the selling price that a business sells a product to a customer, (C) represents the cost which is the price that a business pays for a product, and (M) is the markup which is usually expressed as a percent of the retailer’s cost. Markup is added to a retailer’s cost to cover the expenses of operating a business.

Example Find the cost of shoes for P945 if the markup was 35% of the cost Solution Let x = the cost of the shoes in pesos Then, 0.35x = the markup in pesos and 945 = the selling price in pesos Formula: Cost + Markup = Selling Price Equation: x + 0.35x = 945 1.35x = 945 X = 700 Answer: the cost of the shoes is P700.

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Discount (D) in the formula S = R – D is the amount by which a retailer reduced the regular price of a product for a promotional sale. It is usually expressed as a percent of the regular price or original price.

This percent is called the discount rate.

Example After a discount of 20%, the price of a watch became P2 500. Find the regular price of the watch.

Solution Let x = the regular price of the watch in pesos.

Then, 0.20x = the discount in pesos and P 2 500 = the sale price in pesos Formula: Selling Price = Regular Price – Discount Equation: 2 500 = x – 0.20x 2500 = 0.80x 3125 = x Answer: the regular price of the watch is P3 125.

Practice Solve 1. A markup rate of 255 is used on a cell phone that has a selling price of P15 400. Find the cost of the cell phone.

2. A coffee table costing P1 375 is sold for about P2 500. Find the mark up on the coffee table.

Lesson 4.12 Solving Geometry Problems

Formulas for area and perimeter of quadrilateral and triangles Figure Area Perimeter 2 Square A=s P = 4s Rectangle A = lw P = 2l + 2w Parallelogram A = bh P = 2b + 2a Trapezoid P=a+b+c+d A= Triangle

A=

P=a+b+c

Formulas for the area and circumference of a circle Area : A = circumference: C = 2 Example YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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The perimeter of the isosceles triangle is 27 ft. the length of the third side is 3 ft less than the length of one of the congruent sides. Find the measures of three sides of the angles.

Solution Let x = the length of each of the congruent sides Then, 3 – x = the length of the third side Use the formula for the perimeter of a triangle P=a+b+c 27 = x + x + x – 3 27 = 3x – 3 30 = 3x 10 = x Each of the congruent sides measures 10 ft and the third side measures x –3 or 7 ft.

Practice 1. The perimeter of a rectangular garden is 50 meters. The length of the rectangle is 5 m less than twice the width. Find the length and width of the rectangle.

2. The width of a rectangle is 255 of the length. The perimeter is 300 cm. Find the length and width of the rectangle.

3. The perimeter of a triangle is 160 cm. one side is 10cm less than the second side. The third side is 20 cm more than the second side. Find the length of each side.

Lesson 4.13 Linear Inequalities and open sentences in one variable A mathematical sentence that contains the symbol is called inequality Everyday phrases Translation At most 40 kph The speed is less than or equal to 40 kph Below 5 years old The age is less than 5 years At least 8 glasses The number of glasses is greater than or equal to 8 Between P300 and 500 P300 is less than the amount and the amount is less than P500

inequality S a<5 g 300 < a < 500

Usually, the variable in an equality is read as “each number and name the variable “ for example, the variable x in the inequality x > 8 is read: “ each number x Example b + 8 12 Solution: Each number b plus eight is not equal to twelve An open sentence is a sentence that contains one or more variables. An open sentence may either an equation or inequality. Any value of the variable that makes the sentence true is called solution of open sentence.

Equation: x + 4 = 12 inequality : x > 2 Solution: 8 some solutions: 3, 8 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Every integer greater than 2 is a solution of the inequality x > 2.

Properties of inequality 1. Trichotomy property For all real numbers a and b, only one of the following is true: a < b, a = b or a > b 2. Transitive property of inequality For all real numbers a, b, and c: If a < b and b < c, then a < c If c > b and b > a, then c > a 3. Addition Property of Inequality (API) For all numbers a, b, and c: If a > b then a + c > b + c If a < b, then a + c < b + c 4. Subtraction property of Inequality (SPI) For all numbers a, b, and c If a > b, then a – c > b – c If a < b, then a – c < b – c 5. Multiplication and Division Properties of Inequality by Positive Numbers For all numbers a, b, and c, with c positive If a > b then a x c and

,

Note: the rule is similar for a < b 6. Multiplication and Division Properties of Inequality by Negative Numbers.

For all numbers a, b, and c, with c negative If a > b then a x c < b x c and Note: the rule is similar for a < b Example Solve each inequality 4x X Some inequalities involve more than one operation. To solve the inequality, work backward to undo the operations, just what you did in solving equations with more than one operation.

Example: Solve the inequality 8x + 9 < 2x + 21 8x + 9 – 2x < 2x + 21 – 2x 6x + 9 < 21 6x + 9 – 9 < 21 – 9 6x < 12 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Solution is x < 2 When inequalities contain grouping symbols, you can use the Distributive Property to begin simplifying the inequalities.

Example: 2(6 + x) < 34 12 + 2x < 34 12 + 2x â&#x20AC;&#x201C; 12 < 34 â&#x20AC;&#x201C; 12 2x < 22 X < 11 The solution is x < 11.

We may also express inequalities as intervals. An interval is a set containing all the numbers between its endpoints as well as one endpoint, both endpoints, or either endpoint. Intervals indicate the inclusion or exclusion of endpoints through the use of bracket or parentheses. Brackets [ ], indicate that the endpoints are included in the set. Parentheses, ( ), are used when the endpoints are not included in the set. However brackets and parentheses may be mixed in one interval as shown in the table Inequality interval Words 7 [7,10] The set of numbers between 7 and 10, including 7 and 10 7 [7,10] The set of numbers between 7 and 10, including 7 7< [7,10] The set of numbers between 7 and 10, including 10 7< [7,10] The set of numbers between 7 and 10 Example Robin has scores of 73, 75, and 79 on these exams. What score does he need on the lst exam to get an average of no less than 80?

Solution Let x = the score in the fourth exam.

To find the average score, add the four scores and divide by 4. To earn no less than an average of 80, the average must be greater than or equal to 80.

Average of the four score 80 80 Solve the inequality 80

add 73, 75 and 79 and x

227 + x 320 multiply both sides by 4 X 93 subtract 227 from both sides Practice Write and solve the inequality 1. One less than a number is less than three times the same number plus 5. For what number or number is this true?

2. Twice the sum of a number and one is greater than 16. What is the number?

3. Rose has scores 86, 92, 89, and 92 on four examinations. What is the least score she can get on the fifth examination to have an average of at least 91?

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Lesson 4.14 Absolute values in open sentences Property 1 If |x| = a, then x = a or –x = a that is x = a or x = -a Example |3x – 6| - 2 = 11 |3x – 6| = 13 3x – 6 = 13 or 3x – 6 = -13 3x = 19 3x = -7 X=

x=

The solution set is {

}

Property 2 If |x| < a and a > 0 then –a < x < a Example Solve | x + 2 | < 3 Solution Use the fact that | x + 2 | < 3 is equivalent to -3 < x + 1 < 3 | x + 1| < 3 -3 < x + 1 < 3 -3 – 1 x < 3 – 1 -4 < x < 2 The solution set is ( -4, 2), or the real numbers between – 4 and 2 or {|x| -4 < x < -2} Property 3 If |x| > a and a > 0 then x < -a or x > a Practice Fill the blank to make the statement true 1. |3x + 1 | = 4 is an absolute value _____ 2. |3x + 1| > 4 is an absolute value _____.

3. |x| . 2 is equivalent to _____ or _____

Chapter 5:

Geometry

Lesson 5.1 Logical Reasoning The statement “if a number is even, then it is divisible by 2” is written in conditional form, or in if-then form. A conditional statement has two parts: a hypothesis, denoted by p, and a conclusion, denoted by q. In symbols, the statement, “ if p then q is written as p q.

If it is 9:30 a.m. then it must be daytime P q A definition is a statement of the meaning of a word, or term, or phrase which made use of previous defined terms YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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A postulate is a statement which is accepted o be true without proof A theorem is any statement that can be proved true.

A corollary to a theorem is a theorem that follows easily from a previously proved theorem.

The converse of the conditional statement is formed by interchanging the hypothesis and conclusion.

For instance, the converse of p q is q p If a condition statement and its converse are both true, you can combine them to form one bi conditional statement or a bi conditional. The parts of a bi conditional statement are connected by the phrase if and only if Conditional statement: if p the q p q Converse if q the p q p Bi conditional p if and only q p q Example Write the statement as a bi conditional ”complementary angles are any two angles whose sum of their measures is 90” Solution Conditional statement: if two angles are complementary, then the sum of their measures is 90.

Converse: if the sum of the measure of two angles is 90, then they are complementary Bi conditional: two angles are complementary if and only if the sum of their measure is 90.

Deducting reasoning To deduce means to reason from known facts When you prove theorem you are using deductive reasoning - using existing structures to deduce new parts of the structure. In deductive reasoning, you assume that the hypothesis is true and then write the series of statements that lead to the conclusion.

Each statement is supported by a reason that justifies it. The set of statements and reasons consist the proof.

An argument such as this is known as syllogism. A simple syllogism is an is an argument made up of three statements : a major premise, a minor premise, and a conclusion.

Example X: if two numbers are odd, then the sum is even.

Y: the numbers three and 5 are odd numbers Z: the sum of 3 and 5 is even In all three statements x is called the general statement, y is called the particular statement and z is called the conclusion. In these syllogisms, we reasoned from a statement about a general set to a statement about a particular member or element of that set. This kind of of reasoning is characterized as reasoning from the general to the particular and it is called deductive reasoning.

Practice 1. X: right angles are congruent Y: ______________________ Z: ⦟ 1 and ⦟ 2 are congruent 2. X: if you quit smoking, then you can save your lungs.

Y: Karl quit smoking Z: ___________________________________________ YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Inductive reasoning is a process of observing data, recognizing patterns, and making generalizations from observations.

Most scientific inquiry begins with inductive reasoning. When a mathematician uses inductive reasoning to make a generalization is called a conjecture.

Example Use inductive reasoning to find the next two terms of each sequence. a. 1, 2, 4, 8, 16, …

32, 64 ( each term is 2 times the previous term) b. 1, 4, 9, 16, 25, …

36, 49 ( the sixth and seventh squared numbers) Practice Rewrite each statement in if-then form 1. Perpendicular lines form a right angle 2. A champion is afraid of losing Rewrite each statement as a bi conditional 1. Two angles with the same measure are defined to be congruent.

2. Two segments with the same length are defined to be congruent Lesson 5.2 Building blocks of geometry A definition is a statement that clarifies or explains the meaning of a word or phrase.

A space is the set of all points.

A figure is any set of points.

A set of points that all lie in the same plane is called coplanar Collinear points are points that lie on the same line.

Points that do not lie on the same line are non collinear Points that do not lie in the same plane are non coplanar points A point is between two other points on the same line if its coordinate is between the coordinates of the two points.

The segment with endpoints P and Q denoted by ̅​̅​̅​̅, the set consisting of distinct points P and Q and all points in between P and Q Ray PQ is a set of points consisting of ̅​̅​̅​̅ and all points R such that Q is between P and R. P is called the endpoint of the ray. The symbol ⃗​⃗​⃗​⃗​⃗ is used to name ray PQ An angle is formed by two non collinear rays that have a common endpoint. The endpoint is the vertex of the angle and each ray is the side of the angle A point is in the interior of an angle if it does not lie on the angle itself and it lies on a segment whose endpoints are on the sides of the angle Adjacent angles are two coplanar angles that have a common side between them but have no interior points in common.

Vertical angles are nonadjacent angles formed by two intersecting lines.

Two angles are complementary if the sum of their measures is 90. Each angle is called complement to each other.

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Two angles are supplementary if the sum of their measures is 180. Each angle is called a supplement of the other.

If two angles are adjacent and supplementary, then they are called linear pair.

Practice Find the measure of the angle if its measure is given as: 1. 400 more than its complement 2. Twice that of its complement 3. 500 more than that of its supplement 4. Equal to its supplement 5. The measure of its supplement is thrice of its complement Lesson 5.3 Lines and Planes Relationships Measure of a line The length of a segment AB is the distance between the two endpoints A and B and is denoted by AB. Postulate 1 The distance postulate For every pair of different points there corresponds a unique positive number Postulate 2 The ruler postulate The points on a line can be matched with the set of real numbers in such a way that: 1. To every point on the line there is exactly one real number, known as the coordinate of a point 2. To every real number there is exactly one point on the line and 3. The distance between any two points is the absolute value of the difference of their coordinates Postulate 3 The ruler placement postulate Given two points P and Q on a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and that of Q is a positive number Theorem 1 The point plotting theorem Let ⃗​⃗​⃗​⃗​⃗ be a ray and let r be a positive number. Then, on ray PQ there is exactly one point T such that PT =r Theorem 2 The midpoint theorem Every segment has exactly one midpoint.

Midpoint formula YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates and b is

.

On a coordinate plane, the midpoint of a segment whose endpoints have coordinates ( x 1, y1) and (x1, y1) is the point with coordinates (

)

Example The coordinate of D is -4 and the coordinate of I is 6 on the number line. What is the midpoint of DI Solution Use the midpoint formula to find the coordinate of the midpoint.

Thus,

= or 1

The coordinate of the midpoint of DI is 1.

Practice A. If H is between W and Q. find each indicated measure 1. WQ = 25, WH = 12 HQ = ?

2. WQ = 32.5, HQ = 21.7 WH = ?

B. U is between R and N. find the length of each 1. RU = x + 7 UN = 4x – 9 RN = 33 2. RU = 8x – 4 UN = 2x – 5 RN = 21 Lesson 5.4 Writing a proof A proof is a logical argument in which each statement you make is backed up by a statement that is accepted as true. the reasons for each logical statement is based on established definitions, postulates, theorems, corollaries and properties.

With regard to the sources of reasons mentioned, it is necessary that one must be able to express these into an if-then form using mathematical symbol.

Definitions, postulates and properties: If –then form A. Definitions 1. Betweenness If A-B-C, 2. Midpoint If A is the midpoint of ̅​̅​̅​̅, ̅​̅​̅​̅ 3. Segment bisector If ̅​̅​̅​̅ bisect ̅​̅​̅​̅ at B. 4. Right angle If angle A is a right angle. ⦟ if ⦟BAC is a right ⦟. ̅​̅​̅​̅ ⊥ ̅​̅​̅​̅ 5. Acute angle If ⦟A is an acute ⦟, ⦟A < 90 6. Obtuse angle If ⦟ A is an obtuse ⦟, m ⦟ A > 90 7. Perpendicular line segment If ̅​̅​̅​̅ ⊥ ̅​̅​̅​̅ ⦟ BAC is a right angle 8. Complementary angles If ⦟ A and ⦟ are complementary ⦟s, m⦟ A + m⦟ B = 90 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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If ⦟ A and ⦟ are supplementary ⦟s, m⦟ A + m⦟ B = 180 If ⃗​⃗​⃗​⃗​⃗ and ̅​̅​̅​̅ are opposite rays and ⃗​⃗​⃗​⃗​⃗ is any other ray, ⦟ ⦟ form a linear pair ⃗​⃗​⃗​⃗​⃗ If bisect ⦟ BAC, ⦟ ⦟ If ̅​̅​̅​̅ ̅​̅​̅​̅, If ⦟ A ⦟ , ⦟ ⦟

9. Supplementary angles 10. Linear pair 11. Angle bisector 12. Congruent segments 13. Congruent angles B. Properties of equality 1. Addition Property of equality 2. Multiplication property of equality 3. Subtraction property of equality 4. Reflexive property of equality 5. Symmetric property 6. Transitive property of equality C. Law of substitution

If a = b and c = d, If a = b and c = d, If a = b and c = d, For every real number a, a = a If a = b If a = b and b = c If a + b = c and b = x

Practice A. Supply a valid conclusion for the given hypothesis in (a) and the corresponding reason in (b) 1. If ⦟ Q is a right angle, a. ____________ b. _____________ 2. If ⦟ J and ⦟ P are complementary angles, a. ____________ b. _____________ 3. If m⦟ 1 + m ⦟ 2 = 180 a. ____________ b. _____________ Lesson 5.5 Postulate and theorems on angles Postulate 10 The Angle Addition Postulate If point T is in the interior of angle PQR, then measure of angle PQR = m angle PQT + m angle TQR Postulate 11 The supplement postulate If two angles form a linear pair, then they are supplementary.

Postulate 12 The angle measurement postulate An angle has a measure between 00 and 1800.

Postulate 13 The angle Construction postulate

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Let ⃗​⃗​⃗​⃗​⃗ be a ray on edge of half-plane H1. For every number between 0 and 180, there is exactly one ray AX with point X on H1 such that angle XAB = r Theorem 6 The vertical angle theorem (VAT) Vertical angles are congruent Theorem 7 The supplement theorem Supplements of congruent angles are congruent.

Theorem 8 The complement theorem Complements of congruent angles are congruent.

Theorem 9 If two angles are complementary, then both angles are acute.

Theorem 10 Any two right angles are congruent, or if two angles are right angles, then they are congruent.

Theorem 11 If two angles are supplementary, then each is a right angle.

Perpendicular lines and Oblique lines Two lines that intersect to form a right angle are called perpendicular lines.

Lines that intersect and not perpendicular are called oblique lines

Theorem 12 The four right angles theorem If two perpendicular lines form one angle, they form four right angles.

Practice Fill each blank to make the statement true 1. An angle measures between _____ and _____ because of the angle measurement postulate.

2. If two angles form a _____ then they are supplementary.

3. If the two angles are _____, then the sum of their degree measure is 180.

4. If two angles are complementary., then both are _____.

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Lesson 5.6 Parallel lines and special angles Definition Skew lines – lines that are non coplanar and do not intersect.

Parallel lines – lines that are coplanar and do not intersect.

Oblique lines – lines that intersect Transversal – a line which intersects two coplanar lines at two different points Alternate interior angles – a pair of non adjacent interior angles on opposite sides of a transversal.

Corresponding angles – a pair of non adjacent interior and exterior angles on the same side of a transversal Same side Interior angles – interior angles on the same side of the transversal Postulate 14 PCAC postulate If two parallel lines are cut by a transversal, then corresponding angles are congruent.

Theorem 13 PAIC theorem If two parallel lines are cut by a transversal, then alternate interior angles are congruent.

Theorem 14 PAEC theorem If two parallel lines are cut by a transversal, then alternate exterior angles are congruent.

Theorem 15 PSSIAS theorem If two parallel lines are cut by a transversal, then same-side interior angles are supplementary.

Theorem 16 If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line.

Practice Tell whether each statement is true or false.

1. If two lines do not intersect, they are parallel.

2. Coplanar lines cannot be skew lines.

3. Skew lines do not intersect.

4. Two parallel lines lie in a plane.

5. Two lines that are not skew and do not intersect are parallel.

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Lesson 5.7 Conditions that Guarantee Parallelism Postulate 15 The parallel postulate Given a point and a line not containing it, there is exactly one line through the given point parallel to the given line.

Postulate 16 Given a point and a line not containing it, there is exactly one line through the given point perpendicular to the given line.

Postulate 17 CACP Postulate Given two lines cut by a transversal, if corresponding angles are congruent, then the two lines are perpendicular.

Theorem 17 AICP theorem Given two lines cut by a transversal, if alternate interior angles are congruent, then the lines are parallel.

Theorem 18 SSIAS theorem Given two lines cut by a transversal, if same side interior angles are supplementary, then the lines are parallel.

Theorem 19 AI â&#x20AC;&#x201C; CA theorem Given two lines cut by a transversal, if alternate interior angles are congruent, then angles are congruent.

Theorem 20 The three parallel Lines theorem In a plane, if two lines are both parallel to a third line and then they are parallel.

Theorem 21 If two coplanar lines are perpendicular to a third line, then they are parallel to each other.

Lesson 5.8 Measuring Angles in Triangles Theorem 22 The sum of the measures of the angles of a triangle is 180.

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Measure of angle 1 + 2 + 3 = 180 Theorem 23 The Third Angles Theorem If two angles of one triangle are congruent to two angles of another, then the third angles are congruent.

Theorem 24 If one angle of a triangle is right or obtuse, then the other two angles are acute Theorem 25 The acute angles of a right triangle are complementary Example What is the measure of the smaller angle formed by J. Luna st. and G. del Pilar st?

Solution Apply the angle-sum theorem for triangles to find the angle at which J. Luna st and J. del Pilar st. intersect Measure of angle M + P + L = 180 87 + 65 + measure of L = 180 Measure of L + 152 = 180 Measure of L = 28 The measure of the smaller angle formed by J. Luna and G. del Pilar st is 28 0 M 870 P 650

L

Definition An angle which forms a linear pair with an angle of the triangle is called exterior angle.

The remote interior angles are the two angles in the angles in the triangle that do not have the same vertex as the exterior angle.

Theorem 26 The Exterior Angle theorem (EAT) The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

Corollary The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angle.

Practice Find the measure of all the angles in each triangle ̅​̅​̅, m angle P = 3x – 6 and m angle O = 2x + 4 1. In SOP, ̅​̅​̅​̅ 2. In , m angle A = x, m angle B = 2x, m angle C = x – 20

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Lesson 5.9 Triangle Inequality Theorem 27 The triangle Inequality theorem In a triangle, the sum of the lengths of any two sides is greater than the length of the third side.

Example 1. Can a triangle be constructed with sides of length 4 cm, 8 cm, and 9 cm Solution: yes because 4 + 8 > 9, 8 + 9 > 4, and 4 + 9 > 8 The sum of the length is greater than the third length.

2. Can a triangle be constructed with sides 4 cm, 7 cm, 12 c,?

Solution: No because the sum of 4 and 7 is not greater than 12.

Theorem 28 Unequal side theorem If one side of a triangle is longer than the second, then the angle opposite the longer side is larger than the angle opposite the second side.

Theorem 29 Unequal angles theorem If one angle of a triangle is larger than the second angle, then the side opposite the larger angle is longer the side opposite the second angle Corollary The perpendicular segment from a point to a line is the shortest segment from the point to the line.

Q R S T Theorem 30 The hinge Theorem If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle.

Theorem 31 Converse of the hinge theorem If two sides of one triangle are congruent respectively to the other two sides of another triangle, and the third side of the first triangle is longer than the third side of the second, then the angle opposite the longer side is larger than the angle opposite the third side of the second triangle.

Practice a. Tell whether a triangle can be constructed with segments having these length 1. 3,4,5 2. 7,7,14 3. 3,10, 9 b. Given the sides, identify the largest and smallest angle in each triangle.

1.

BAC : AB = 12, BC = 5, AC = 9 2. DEF: DE = 6, EF = 9, and DF = 14 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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3.

JKL: JK = 3.9, KL = 3.4, and JL = 4.2 Lesson 5.10 Angles in a Polygon

Theorem 32 The sum of the measure of the angles of a convex quadrilateral is 360.

Measure of angle 1 + 2 + 3 + 4 = 360 Theorem 33 Polygon Interior Angles Theorem The sum of the measures of the angles of a convex polygon with n sides is (n – 2) 180 Measure of angle 1 + 2 + … = ( n – 2) 180 Example Find the sum of the measures of the interior angles of a convex hexagon Solution A hexagon has 6 sides. Use a polygon Interior angle theorem and substitute 6 for n. (n – 2) 180 = (6 – 2) 180 = 4 x 180 = 720 The sum of the measures of the interior angles of the interior angles of convex hexagon is 720.

Corollary The measure of each angle of a regular n-gon is Example Find the measure of each angle of a regular decagon.

Solution The sum of the angle measures of a decagon is (10 – 2) 180 = 8 x 180 = 1 440 Thus, the measure of each angle is

= 1440

Theorem 34 Polygon Exterior Angles Theorem The sum of the measures of the exterior, one of each vertex, of any convex polygon is 360.

Practice Find the measures of the angles of each convex polygon 1. A decagon 2. Dodecagon 3. A 50-gon The number of sides of a regular polygon is given. Find the measures of an interior angle and an exterior angle of each polygon.

1. 23 2. 30

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Lesson 5.11 Properties of Quadrilaterals Important types of quadrilateral and its properties Parallelogram – both pair of opposite sides are parallel Rhombus – a parallelogram with four congruent sides Rectangle – a parallelogram with four right angles Square – a parallelogram with four congruent sides and angles Kite – two pairs of adjacent sides congruent and no opposite sides congruent Trapezoid – a quadrilateral with exactly one pair of parallel sides Practice Draw a quadrilateral with the given description 1. No congruent sides 2. No congruent angles 3. Congruent opposite sides 4. Congruent consecutive angles Lesson 5.12 Properties of Parallelogram A parallelogram is a quadrilateral with both pairs of opposite sides parallel. A parallelogram has several properties. The theorems give the characteristics of a parallelogram.

Theorem 35 Each diagonal of a parallelogram divides the parallelogram into two congruent triangles.

If ABCD is a parallelogram with diagonal AC, then ABC CDA ̅​̅​̅​̅ If ABCD is a parallelogram with diagonal , then ABD CDB Theorem 36 Opposite side of a parallelogram are congruent. ̅​̅​̅​̅ If ABCD is a parallelogram, then ̅​̅​̅​̅

̅​̅​̅​̅

̅​̅​̅​̅

Theorem 37 Opposite angles of a parallelogram are congruent.

If ABCD is a parallelogram then A C and B D Theorem 38 Any two consecutive angles of a parallelogram are supplementary If ABCD is a parallelogram, then A and B are supplementary angles Theorem 39 The diagonals of a parallelogram bisect each other. ̅​̅​̅​̅ ̅​̅​̅​̅ ̅​̅​̅​̅ If ABCD is a parallelogram then ̅​̅​̅​̅ To summarize, the properties of a parallelogram 1. Opposite sides are congruent.

2. Opposite angles are congruent Example

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UNIT is a parallelogram with UN = x + 3, NI = 17 and UT = x – 3 what is the perimeter of a parallelogram Solution Draw and mark the parallelogram.

Opposites sides of a parallelogram are congruent ̅​̅​̅​̅ ̅ are opposite sides ̅​̅​̅​̅ ̅​̅​̅ are opposite sides UT = NI, so x – 3 = 17 of x = 20 Un = x + 3, so UN = 20 + 3 = 23 U X+3 N X -3 17 T I Practice Use the parallelogram OPEL to answer the following questions O P A L E 1. If OE = 20, what is OA? 2. If PE = 12, what is OL? 3. If OL = 7x – 10 and PE = 4x – 1, what is OL? Lesson 5.13 Special parallelogram Theorem 40 The diagonal of a rectangle are congruent. ̅​̅​̅​̅ If ABCD is a rectangle then ̅​̅​̅​̅ A B D C Theorem 41 The diagonals of a rhombus are perpendicular.

If ABCD is a rhombus then, ̅​̅​̅​̅ ⊥ ̅​̅​̅​̅.

A C

D C

Theorem 42 The diagonal of a square are congruent and perpendicular. ̅​̅​̅​̅ and ̅​̅​̅​̅ ⊥ ̅​̅​̅​̅ If ABCD is a square, then ̅​̅​̅​̅ YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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A

B

D

C

Theorem 43 Each diagonal of a rhombus bisects two angles of the rhombus.

If ABCD is a rhombus with diagonal ̅​̅​̅​̅ then, ̅​̅​̅​̅ bisects B and D. A B D

C

Theorem 44 If the diagonal of a parallelogram are congruent, then the parallelogram is a rectangle. ̅​̅​̅​̅ then ABCD is a rectangle.

If ABCD is a parallelogram with ̅​̅​̅​̅ A B D

C

Theorem 45 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

If ABCD is a parallelogram with ̅​̅​̅​̅ ⊥ ̅​̅​̅​̅ then ABCD is a rhombus A B D C Theorem 46 If the diagonals of a parallelogram are congruent and perpendicular, then the parallelogram is a square. ̅​̅​̅​̅ and ̅​̅​̅​̅ ⊥ ̅​̅​̅​̅, then ABCD is a square If ABCD is a parallelogram with ̅​̅​̅​̅ Theorem 47 If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus.

If ABCD is a parallelogram with ̅​̅​̅​̅ bisecting B and D then ABCD is a rhombus.

Practice Write yes or no in each cell of the table for the properties of of each type of quadrilateral.

Property Parallelogram Rhombus Rectangle Square The diagonal bisect each other The diagonals are perpendicular The diagonal are congruent Each diagonal bisects a pair of opposite angles

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Lesson 5.14 Trapezoids and Kites A trapezoid is a convex quadrilateral with exactly one pair of parallel sides.

The bases of a trapezoid are the two parallel sides.

The legs of a trapezoid are the two non-parallel sides.

An isosceles trapezoid is a trapezoid with congruent legs.

The median of a trapezoid is the segment that joins the midpoints of the legs An altitude of a trapezoid is any segment from a point on one base perpendicular to the line containing the other base.

The following is the theorem about the midline of a trapezoid.

Theorem 48 The median of the trapezoid is parallel to its bases and its length is half the sum of the lengths of the bases.

A B E F D C If ̅​̅​̅​̅ is a median of trapezoid ABCD, then, ̅​̅​̅​̅ || ̅​̅​̅​̅ || ̅​̅​̅​̅ and EF = ( AB + DC) Theorem 49 The base angles of an isosceles trapezoid are congruent.

A B D C If ABCD is an isosceles trapezoid then ⦟ D ⦟ C Theorem 50 If the base angles of a trapezoid are congruent, then the trapezoid is isosceles.

If ⦟ D ⦟ C in trapezoid ABCD is an isosceles trapezoid.

Theorem 51 The diagonals of an isosceles trapezoid are congruent, then the trapezoid is isosceles.

A B D C Theorem 52 If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles.

Kite Another special quadrilateral that is not a parallelogram is a kite. Like a rhombus, the diagonals of a kite are perpendicular.

Theorem 53 The diagonals of a kite are perpendicular If ABCD is a kite then ̅​̅​̅​̅ ⊥ ̅​̅​̅​̅.

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Practice Using the trapezoid TANK with median OB, find the indicated base/median length.

T A O B K N

1. if AT = 20 and KN = 26, find OB. 2. if AT = 31 and KN = 43, find OB 3. If OB = 17 and AT = 14, find KN Chapter 6 Statistics Lesson 6.1 Statistics and Basic Terms Statistics is a branch of mathematics that deals with the collection, organization, presentation analysis, and interpretation of data.

Population is a complete collection of all elements to be studied.

Census is a collection of data from every element in a population.

Sample is a subcollection of elements drawn from a population.

Types of samples: 1. In a random sample, each member of the population has an equally likely chance of being selected. The members of a sample are chosen independently of each other.

2. A convenience sample is a sample that is chosen so that it will be easy for the researcher.

3. In a stratified random sample, the population is divided into subgroups, so that each population members is in only one subgroup. In here, individuals are chosen randomly from each subgroup.

4. A cluster sample is a sample that consists of items in a group such as a neighborhood or household. The group may be chosen at random.

5. A systematic sample is obtained using an ordered list of the population, thus selecting members systematically from the list.

The nature of data Quantitative data consist of numbers representing counts of measurements.

Qualitative data can be separated into different categories that are distinguished by some nonnumeric characteristics.

The following are examples of qualitative variables: gender, major classification, political party affiliation, religious preference, marital status,â&#x20AC;Ś Quantitative data can either be discrete or continuous.

Discrete data result from either a finite number of possible values or countable number of possible values as 0, or 1, or 2, and so on.

Continuous data results from infinitely many possible values that can be associated with points on a continuous scale in such a way that there are no gaps or interruptions.

Four levels of measurements YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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1. Nominal level of measurement is characterized by data that consist of names, labels, or categories only.

2. Ordinal level of measurement involves data that may be arranged in some order but differences between data values either cannot be determined or are meaningless.

3. Interval level of measurement is like the ordinal level. But meaningful amounts of differences between data can be determined. It has no inherent zero starting point.

4. The ratio level of measurement is the interval level modified to include the inherent zero starting point.

Example Determine which of the four levels of measurements is used.

1. Average annual temperature in Tagaytay. interval 2. Weights of garbage discarded by household. ratio 3. A judge rates some presentations as â&#x20AC;&#x153;goodâ&#x20AC;?. ordinal 4. The political party to which each governor belongs. nominal Practice Determine which of the four levels of measurements 1. Ratings of excellent, above average, below average, or poor for painting exhibits 2. Zip codes 3. Annual income of teachers 4. Cars described as compact, intermediate, or full-sized 5. Final grades for math students Lesson 6.2 Collection of data Collection of data is an important part of statistics. Data should be collected in a manner that they are accurate and convenient to use.

Data is a collection of facts and information.

Methods in gathering data: 1. Conducting surveys 2. Observing the outcomes of events 3. Taking measurements in experiments 4. Reading statistical publication Interview method and the questionnaire method can be used in conducting surveys. The interview method is done when a person solicits information from another person. The person gathering the data is called the interviewer, while the person supplying the information or data is the interviewee.

If a survey involves a series of questions, the interview method will be very tedious one to employ.

Hence, the questionnaire method is advisable. The firm prepare printed questionnaire and distribute these to several respondents who shall in turn answer the questionnaire and give it back to the firm.

Behavioral scientists are primarily concerned with the behavior of either the individual or a group of individuals. The data are gathered either individually or collectively by means of observation. The person who gathers the data is called the investigator while the person being observed is called the subject.

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Example A restaurant manager requests her customer to fill out this questionnaire: Blue Ribbon Restaurant Customer Satisfaction Survey 1. How would you rate our food?

Very good _____ Good _____ Okay _____ Bad _____ 2. How would you rate our service?

Very good _____ Good _____ Okay _____ Bad _____ What type of data collection method does the manager use?

Solution: Conducting Surveys Practice Suggest a way to collect each set of data 1. Population of region 1, region 2, region 3 2. Effectiveness of new secondary curriculum 3. Heights of 20 applicants 4. Opinion of commuters of the color coding scheme.

5. Popularity of Korean telenovelas.

Lesson 6.3 Organization of Data The following data are obtained in the survey on the number of cell phones possessed by each family.

3 4 5 4 3 4 1 6 2 2 4 5 2 3 6 3 3 2 1 3 2 3 4 5 3 2 4 3 4 5 4 2 5 3 4 6 3 5 3 6 1 3 4 2 3 4 2 4 6 3 The data collected above are called raw data. They are the data collected in a survey. We have to organize the data using a table by following the given steps 1. Set up a table using three columns.

2. Read each item in the raw data and mark a stroke (or tally) in the tally column in the same row as its class.

3. The frequency of the class is the number of times each class occurs. Write down the frequency of each class by counting its corresponding tally marks.

A frequency table shows a clear and definite information about the set of data. With the frequency table, we can easily know which class has the lowest frequency.

Example Construct a frequency table for the data in the survey of cell phone possessed by each family.

Number of cell phones tally Frequency 1 ||| 3 2 |||| ||| 8 3 |||| |||| |||| 15 4 |||| |||| || 12 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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5 6

|||| || |||| total

7 5 50

Grouped frequency distribution Steps in constructing: 1. Get the difference between the highest score and the lowest score. Add 1 to the difference to arrive at the total number of scores or potential scores. The difference between the highest score and the lowest score is called the range.

Range = 29 – 10 = 19 Total number of potential scores = 19 + 1 = 20 2. Decide on the number of class intervals which is appropriate to the given set of data. Divide the final number in step 1 by the desired number of class intervals to arrive at the width of class interval ( I ). If 10 is the desired number of class interval, then: 3. Write the lowest score in the set of raw scores as the lower limit in the lowest class interval. Add to this value I – 1 to obtain the upper limit in the lowest class interval. The lowest score is 10.

Thus, the lowest class interval is 10 – 11 since 10 + I – 1 = 10 + 2 – 1 – 11.

4. The next lower limit can be obtained by adding I to the lower limit of the previous class interval.

To get the corresponding upper limit for this class interval, follow step 3 or add I to the preceding upper limit. Thus, the next lower limit is 12 and the corresponding upper limit is 13 since 10 + 2 = 12 and 11 + 2 = 13.

5. Continue step 4 until all scores are included in their corresponding class intervals.

6. Fill out the f column by following what we have done in the frequency distribution.

Class interval f 28 – 29 1 26 – 27 3 24 – 25 3 22 – 23 3 20 – 21 6 18 – 19 6 16 – 17 8 14 – 15 6 12 – 13 10 10 – 11 14 N = 60 True limits of class interval The true limits or class boundaries of a given scores is the score plus or minus one half of the unit of measure or the place value of the given score. Thus, the true limits of the following scores are as follows: 1) 5, 5 0.5. or 4.5 – 5.5 2) 5.2, 5.2 0.05, or 5.15 – 5.25 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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3) 5.23, 5.23 0.005, or 5.225 – 5.235 Example The true limits of the grouped frequency distribution of 60 students in a 30-element Quiz (i=2) Class intervals Apparent limits True limits F 28 – 29 27.5 – 29.5 1 26 – 27 25.5 – 27.5 3 24 – 25 23.5 – 25.5 3 22 – 23 21.5 – 23.5 3 20 – 21 19.5 – 21.5 6 18 – 19 17.5 – 19.5 6 16 – 17 15.5 – 17.5 8 14 – 15 13.5 – 15.5 6 12 – 13 11.5 – 13.5 10 10 – 11 9.5 – 11.5 14 N = 60 Cumulative Frequency Distribution A cumulative frequency distribution can be obtained by adding the frequency starting from the frequency of the lowest class interval up to the frequency of the highest class interval. It is slao possible to do the reverse.

Cumulative frequency distribution of a 30-element Math quiz Apparent limits True limits F <cf >cf 28 – 29 27.5 – 29.5 1 60 1 26 – 27 25.5 – 27.5 3 59 4 24 – 25 23.5 – 25.5 3 56 7 22 – 23 21.5 – 23.5 3 53 10 20 – 21 19.5 – 21.5 6 50 16 18 – 19 17.5 – 19.5 6 44 22 16 – 17 15.5 – 17.5 8 38 30 14 – 15 13.5 – 15.5 6 30 36 12 – 13 11.5 – 13.5 10 24 46 10 – 11 9.5 – 11.5 14 14 60 N = 60 Cumulative percentage distribution To convert a cumulative frequency distribution to a cumulative percentage distribution, simple divide each term in the <cf by N and multiply by 100.. thus, it can be transformed into a cumulative percentage distribution Example Cumulative percentage distribution on a 30-element Quiz(i=2) Apparent limits f <cf >cpf 28 – 29 1 60 100.00 26 – 27 3 59 98.23 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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24 – 25 22 – 23 20 – 21 18 – 19 16 – 17 14 – 15 12 – 13 10 – 11

3 3 6 6 8 6 10 14 N = 60

56 53 50 44 38 30 24 14

93.33 88.33 83.33 73.33 63.33 50.00 40.00 23.33

Cumulative percentage distribution It enables us to determine what percent of the distribution falls below or above a class boundary. In the table shows that 88.33 percent of the students got a score lower than 23.5 Practice The following table shows the marks gained in an arithmetic examination by 100 pupils. The maximum possible mark was 100.

1. Complete the table 2. Identify a. I b. Total frequency c. Lower limit of the interval 31 – 40 d. True upper limit of the interval 71 – 80 3. How many pupils got marks less than 70.5?

4. How many pupils got marks greater than 50.5?

5. What percent of the pupils got score lower than 61.5?

Marks 91 – 100 81 – 90 71 – 80 61 – 70 51 – 60 41 – 50 31 – 40 21 – 30 11 – 20 1 – 10

Frequency 5 11 14 17 23 14 8 4 2 2

True limits <cf

<cpf

Lesson 6.4 Presentation of Data There are several ways in which you can present information in picture forms which represent numbers are called graphs. Different form of graphs: line graph, bar graph, pictograph, and pie graph The line graph YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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A line graph is used to represent changes in data over a period of time. In a line graph, data are represented by points and are joined by line graph may be curved, broken, or straight.

Generally, the horizontal axis is used as the time axis and the vertical axis is used to show the changes on the other quantity.

Example: The table shows the ticket sales of the Cabanas theater during a typical week. Draw a line graph.

Day Sun Mon Tues Wed Thurs Fri Sat Number of tickets sold 255 60 125 150 240 340 310 Solution We use the horizontal axis for the days and the vertical axis for the number of tickets sold.

number of tickets sold

Ticket sales at Cabana Theater 400 350 300 250 200 150 100 50 0 S

M

T

W

T

F

S

days of week

The Bar Graph Constructing Bar graph 1. Decide on what scale to use by choosing appropriate numbers of convenient size.

2. Write the scale at numbers at the left side when using a vertical bar graph and write the scale numbers underneath the horizontal line when using a horizontal bar graph.

3. Write what the scale numbers represents 4. Label what each bar represents.

5. Write the number that each bar represents.

6. Write the title above the bar graph.

7. If the bar graph requires ordering, the most common order is one in which the bar lengths either increase or decrease.

Example Arnold surveyed a sample of people at a basketball game to find out their favorite drink. The results are shown in the table. Represent the data using the bar graph.

Types of drinks frequency Cola 25 Root beer 20 Lemon 10 Fruit 15 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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Iced tea

12

Solution

frequency

favorite drink 30 25 20 15 10 5 0 cola

root beer

lemon

fruit

iced tea

types of drinks

The pie graph A pie graph is used to show how all parts of something are related to the whole. It is represented by a circle divided into slices or sectors of various sizes that show each partâ&#x20AC;&#x2122;s relationship to the whole and the other parts of the circle.

Example Last Friday, Felix spent 12 hours of the day sleeping and playing, 2 hours eating and dressing, 6 hours at school, and 4 hours surfing the internet. What percentage of the day was spent of the day was spent on each?

Solution Sleeping and playing, 12 hours = Eating and dressing, 2 hours

=

In school, 6 hours

=

Surfing the internet, 4 hours

=

Whole day, 24 hours

=

These facts are shown in the pie graph

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surfing the internet 17% sleeping and playing 50%

in school 25%

eating and dressing 8%

The pictograph A pictograph is a graph that uses pictures to illustrate data. To construct a pictograph, the following steps are to be followed: 1. Collect the necessary data.

2. Round off numerical data if necessary.

3. Choose an appropriate symbol to represent the subject.

4. Indicate the quantity each symbol represents.

Example Draw a pictograph for the given facts below The annual budget of School during 2011 â&#x20AC;&#x201C; 2015 Year 2011 2012 2013 2014 2015 Budget P3 000 000 P4 000 000 P5 000 000 P5 000 000 P5 000 000 Solution Choose a symbol to represent the annual budget Let = P1 000 000 Annual budget of school during 2011 â&#x20AC;&#x201C; 2015

2011

2012

2013

2014

2015

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Tables A more effective and more accurate way of presenting data is the use of statistical tables.

A statistical table contains table heading; body;stubs; and boxheads or column captions.

Table heading shows the table number and the table title Body is the main part of the table.

Stubs are the labels or categories which are presented as values of a variable.

Box heads are the captions that appear above the columns.

Number of registered Filipino Emigrants by Occupational Grouping: 1986 and 1987 Occupational grouping 1986 1987 Total 28761 49338 Professional, technical and related workers 2407 4147 Managerial, executive and administrative 137 288 Clerical workers 2415 3394 Sales workers 1471 2109 Service workers 356 1038 Production process workers 31 54 Aircraft and communication workers 41 67 Constructions workers 678 1603 Farmers, loggers and related workers 554 1389 Others 20671 35249 Test 1. What type of graph would you use to represent your weekly allowance? Explain your answer.

2. How would represent the population of your school 3. Draw a pie graph showing the sales of the three leading milk brands L, M, N; given the sectors representing L, M, N are 600, 900, 2100 respectively. If these are 576 kilos sold, calculate how many kilos of each sold.

Lesson 6.5 Measures of Central Tendency The mean (commonly called average) of a set of n number is the sum of all numbers divided by n.

The median is the middle number when the number in a set of data is arranged in descending order.

When there are even numbers of element, the median is the mean of the two middle numbers.

The mode is the number that occurs most often in a set of data. A set of data can have more than one mode. If all the numbers appear the same number of times, there is no mode for that set of data.

The Mean Ě&#x2026; To get the mean, simply add all the scores in a distribution and divide by the total number of vales.

Example mean (ungrouped data) YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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̅=

∑

where ̅ = mean

∑

xi = scores

= sum of the scores

The following is the list of the weekly savings of ten students.

Student Abel Brenda Carlos Donna Edwin Fred Weekly savings Ᵽ 60 50 40 50 70 50 Find the mean Solution: ̅=

∑

=

N = total frequency

Gina Hans 50 80

Izzy Jacob 70 70

= Ᵽ59

Weighted Mean Formula ̅=

∑

where ̅ mean,

= frequency,

x = score

∑ = sum of the product of frequency and score N= total frequency Example Weekly savings Number of students Weekly savings x number of students (x) (f) (fx) 40 1 40 50 4 200 60 1 60 70 3 210 80 1 80 ∑ = 590 N = 10 ̅=

∑

=

= 59

The mean weekly savings per student if P59 The Median ̃ The median is the value in the distribution which divides an arranged distribution into two equal parts.

To find the median, we arrange the measurements in ascending order and take the middle term.

Example The median of the 5 numbers 20, 40, 70, 80 and 90 is the third number, 70 Find the median of the following scores 34, 11, 49, 54, 64, 59, 47 Solution Arrange the numbers in ascending order 11, 34, 47, 49, 54, 49, 64 N=7 ̃

(

)

=(

) =

score = 49 median

The Mode ( ̂ The mode of a distribution is the data with the highest frequency.

Weekly savings Number of students (x) (f) 40 1 YOUNG JI INTERNATIONAL SCHOOL/COLLEGE

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50 60 70 80

4 1 3 1 N = 10

Clearly, the mode is P50 because it has the highest frequency.

One easy way to obtain the mode is to use a stem-and-leaf plot of frequency table Steam-and-leaf plot 4 0 5 0 0 0 0 6 0 7 0 0 0 8 0 Key: 5|0 means 50 Practice Find the mean, median and mode for each data set.

1. 67, 89, 93, 77, 84 2. 32, 43, 31, 52, 28, 39 End

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