millercfca.weebly.commillercfca.weebly.com/uploads/1/3/2/0/13200028/math_f… · Web viewnot to be ashamed, rightly dividing the word of truth. Romans 12:2 (KJV) = And be not conformed

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 1

Name Grade Class

[MATH]∑ ± = ∆ √ ≥ ∞ ≤ × = ÷ π ≠ ≈ ℮

[FOUNDATIONS]℮ ≈ ≠ π ÷ = × ≤ ∞ ≥ √ ∆ = ± ∑

Through faith we understand that the worlds were framed by the word of God, so that things which are seen were not made of things which do appear.

Hebrews 11:3 (KJV)

Thou art worthy, O Lord, to receive glory and honour and power: for thou hast created all things, and for thy pleasure they are and were created.

Revelation 4:11 (KJV)

The laws of nature are but the mathematical thoughts of God.Euclid

CFCA HS MATH | MR. MILLER | 2013-2014

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 2MATH FOUNDATIONS – TABLE OF CONTENTS

COVER PAGE.............................................................................................................................................. 1

TABLE OF CONTENTS / PREFACE ...............................................................................................................2

MATH FOUNDATIONS # 1 = WHY MATH? .................................................................................................3

MATH FOUNDATIONS # 2 = THE WORLD OF NUMBERS ...........................................................................5

MATH FOUNDATIONS # 3 = OPERATION: ADDITION ...............................................................................8

MATH FOUNDATIONS # 4 = OPERATION: SUBTRACTION .......................................................................10

MATH FOUNDATIONS # 5 = OPERATION: MULTIPLICATION ..................................................................12

MATH FOUNDATIONS # 6 = OPERATION: DIVISION ...............................................................................14

MATH FOUNDATIONS # 7 = DIVISIBILITY RULES .....................................................................................16

MATH FOUNDATIONS # 8 = DERIVED OPERATIONS ................................................................................18

MATH FOUNDATIONS # 9 = ORDER OF OPERATIONS .............................................................................20

MATH FOUNDATIONS # 10 = PROPERTIES OF OPERATIONS ...................................................................22

MATH FOUNDATIONS # 11 = PROPERTIES OF EQUALITY ........................................................................24

MATH FOUNDATIONS # 12 = EXPRESSIONS, EQUATIONS, INEQUALITIES ...............................................25

MATH FOUNDATIONS # 13 = THE FUNDAMENTAL THEOREM OF ARITHMETIC ......................................26

MATH FOUNDATIONS # 14 = LCM & GCF ................................................................................................28

MATH FOUNDATIONS # 15 = WORKING WITH FRACTIONS, DECIMALS, AND PERCENTS ........................31

MATH FOUNDATIONS # 16 = WORKING WITH EXPONENTS/ROOTS/RADICALS .....................................39

MATH FOUNDATIONS # 17 = CALCULATOR BASICS ................................................................................42

MATH FOUNDATIONS # 18 = MEASURES OF CENTRAL TENDENCY..........................................................44

MATH FOUNDATIONS # 19 = GEOMETRY BASICS ...................................................................................46

MATH FOUNDATIONS # 20 = THE COORDINATE PLANE .........................................................................48

MATH FOUNDATIONS # 21 = MATH WRITING SAMPLES ........................................................................50

OTHER # 22________________________________________....................................................................

OTHER # 23________________________________________....................................................................

OTHER # 24________________________________________....................................................................

OTHER # 25________________________________________....................................................................

PREFACEThis packet contains some, but (of course) not all, of the basic mathematical foundations which we will utilize in this HS math course. You are expected to be familiar with these concepts/practices and able to discuss and accurately perform them in a group/class setting and individually. We will elaborate on many, if not all, of these topics (and others) during the course of the year. If additional help is needed with any of these foundations, please see Mr. Miller. Complete this packet and keep it in your math binder for future reference/assistance.

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 3MATH FOUNDATIONS # 1 = WHY MATH?............................................................................................3

We begin our MATH FOUNDATIONS by asking a simple yet profound question – why math? Let’s examine this

using the Bible & deductive reasoning (big picture →little picture). You can also read from the end to the

beginning in using inductive reasoning (little details →big picture). Provide commentary after each section.

~ Why live?

God created us to love us, for us to glorify Him, and for us to tell everyone about Him.

Genesis 2:7 (KJV) = And the LORD God formed man [of] the dust of the ground, and breathed into his nostrils the

breath of life; and man became a living soul.

Genesis 1:27-28 (KJV) = So God created man in his [own] image, in the image of God created he him; male and

female created he them. And God blessed them, and God said unto them, Be fruitful, and multiply, and replenish the

earth, and subdue it: and have dominion over the fish of the sea, and over the fowl of the air, and over every living

thing that moveth upon the earth.

Isaiah 43:7 (KJV) = Even every one that is called by my name: for I have created him for my glory, I have formed him;

yea, I have made him.

Matthew 28:19-20 (KJV) = Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of

the Son, and of the Holy Ghost: Teaching them to observe all things whatsoever I have commanded you: and, lo, I am

with you alway, [even] unto the end of the world. Amen.

What are your thoughts?

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~ Why learn?

God desires for us to serve Him and seek His will in our lives, which includes self-improvement.

1 Thessalonians 4:11 (KJV) = And that ye study to be quiet, and to do your own business, and to work with your own

hands, as we commanded you;

2 Timothy 2:15 (KJV) = Study to shew thyself approved unto God, a workman that needeth not to be ashamed, rightly

dividing the word of truth.

Romans 12:2 (KJV) = And be not conformed to this world: but be ye transformed by the renewing of your mind, that

ye may prove what is that good, and acceptable, and perfect, will of God.

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 4What are your thoughts?

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~ Why math?

Math is a method of describing God’s glorious creation and organizing our lives to His purpose.

Deuteronomy 1:10 (KJV) = The LORD Your God hath multiplied you, and, behold, ye are this day as the stars of

heaven for multitude.

Matthew 10:29-31 (KJV) = Are not two sparrows sold for a farthing? and one of them shall not fall on the ground

without your Father. But the very hairs of your head are all numbered. Fear ye not therefore, ye are of more value

than many sparrows.


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~ Why this year, this unit, this chapter, this assignment, this problem?

Learning math is a comprehensive series of small steps.

Proverbs 16:9 (KJV) = A man's heart deviseth his way: but the Lord directeth his steps.


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~ Can you think of some important numbers mentioned in the Bible?

For example, David killed Goliath with ONE out of FIVE stones. The account is found in 1 Samuel 17.

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CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 5MATH FOUNDATIONS # 2 = THE WORLD OF NUMBERS............................................................................5

Next, it is necessary to examine the “building blocks” of math. We start with the “world” of numbers.

The “world” of numbers is divided into SETS, with each set having certain elements and properties.

Following is an approximate diagram of the relationship between the number sets.

We can examine the “world” of numbers by reading the material below EITHER top to bottom OR bottom to top.

NATURAL NUMBERS (N)

This is the set off numbers used to count objects. There is some debate regarding inclusion of ZERO.

Description/Examples: ___________________________________________________________________________

Complex (C)a + bi

Imaginary (I)i, √(-1), etc.

Real (R)

Rational (Q)ratios of integers

Integers (Z)-2, -1, 0, + 1, +2

Whole (W)0, 1, 2, 835,

etc.

Natural (N)1, 2, 3, 48,

etc.

Zero0

Opposites of Whole Numbers

0, -1, -2, -835, etc.

Fractions1/2, 3/4,

etc.

Decimals (terminating & repeating)0.45, 0.333333, 0.65, etc.

Irrational (not Q)√2, √3, non-terminatingnon-repeating decimals

The Number Sets

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 6WHOLE NUMBERS (W) – These are not represented in the Venn Diagram, only in the Tree Diagram above.

This is a combination of the set of NATURAL NUMBERS (N) and ZERO.

This is according to those who assert that ZERO is not a NATURAL NUMBER (N).


INTEGERS (Z)

This is the set of all positive and negative NATURAL NUMBERS, including ZERO. All NATURAL and WHOLE numbers,

other than ZERO, have an additive inverse (3 and -3).


RATIONAL NUBMERS (Q)

This is the set of all numbers that can be expressed as a fraction a/b – such that a and b are both integers and b ≠ 0.

This set includes all repeating decimals (such as 1/3 = 0.3, 1/9 = 0.1, 1/7 = 0.142857).

This set also includes all terminating decimals (such as 1/8 = 0.125, 2/5 = 0.4).


IRRATIONAL NUMBERS (Q)

This is the set of real numbers which cannot be expressed as a fraction of a/b – such that a and b are both integers

are b ≠ 0. More specifically, this set includes non-terminating, non-repeating decimals. For the time being, we will

not concern ourselves with dividing IRRATIONAL NUMBERS into REAL ALGEBRAIC or TRANSCENDENTAL.

The most “famous” or common examples are:

√2 = 1.4142135 . . . / √3 = 1.7320508 . . . / Π = 3.14159 . . .

(The square root of any prime number is irrational.)


REAL NUMBERS (R)

The set of REAL NUMBERS (R) consists of RATIONAL (Q) and IRRATIONAL (Q) numbers. Every REAL NUMBER is either

a RATIONAL or an IRRATIONAL number. One colloquial way of expressing the set of REAL NUMBERS is “any number

that can be placed on a number line”.


IMAGINARY NUMBERS (I)

This is a set of non-real numbers that include the value i. i is equal to the square root of – 1. We will study these in

time. Their importance lies in higher-order mathematical applications. Imaginary numbers cannot be represented on

a traditional, horizontal, two-dimensional number line. Following are four basic equations using i.

i = √-1 i2 = - 1 i3 = - I i4 = + 1


CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 7COMPLEX NUMBERS (C)

This set includes all REAL (R) and IMAGINARY NUMBERS. Most often, we see combinations of real and imaginary

numbers in the form of: a + bi in which a isa real number, b is a real coefficient, and i = √-1.


However, you will notice that COMPLEX numbers include ALL NUMBERS. For instance, the natural number 4 can be

written in the form a + bi as 4 + 0i. Any number you can think of (for now) belongs in the set of COMPLEX NUMBERS.


~ There are also other descriptors of numbers, such as:

All natural/whole numbers are either even (divisible by two) or odd (not divisible by two).


All natural/whole numbers greater than one are either prime (only having factors of one and itself) or composite

(having more factors than only one and itself)


All natural/whole numbers having a natural/whole number square root are called perfect squares.


At this point, given ANY number, you should be able to identify ALL number sets to which it belongs, as well as

possibly provide other descriptors as they apply.

For example . . .

The number 4

Complex ~ Real ~ Rational (4/1) ~ Integer (+4) ~ Whole ~ Natural ~ It is also even, composite, and a perfect square.

The number .234098 . . .

Complex ~ Real ~ Irrational (non-terminating, non-repeating decimal)

Your turn . . .

The number 576


The number ½


The number √10


CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 8MATH FOUNDATIONS # 3 = OPERATION: ADDITION................................................................................8

There are 4 basic arithmetic operations: ADDITION/subtraction, and multiplication/division (paired as inverses).

2 + 3 = 5

2 is an ADDEND. 3 is an ADDEND. 5 is the SUM. + is a PLUS SIGN or an ADDITION SIGN. = is an EQUALS SIGN.

Adding with integers . . .

(For now, we are only discussing adding integers. Adding with other number sets will be discussed at a later time.)

~ If addends have the same sign, add them and keep the sign for the answer.

6 + 10 = 16....................................both addends are positive, add and keep the positive sign for the sum

~ If addends have different signs, find DIFFERENCE & keep the sign of the addend with the larger absolute value.

6 + (-10) = - 4..........addends different signs, difference is 4, keep sign of larger absolute value (-10) = - 4

Solve for x. Show all work. Circle your final answer. Rewrite the problem as needed. Use back as needed.

1. 3498 + 2340 = x

2. 14 + 29 = x

3. 6,983,442 + 27, 343 = x

4. 999 + 111 = x

5. 132, 230, 343 + 245, 993, 209 = x

6. 6 + (-10) = x

7. 3x + 12x = x

8. -18 + (-27) = x

9. 978 + (-123, 434) = x

10. Design your own problem and solve.

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 9Continue Addition Practice below.

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 10MATH FOUNDATIONS # 4 = OPERATION: SUBTRACTION........................................................................10

There are 4 basic arithmetic operations: addition/SUBTRACTION, and multiplication/division (paired as inverses).

3 - 2 = 1

3 is the MINUEND. 2 is the SUBTRAHEND. 1 is the DIFFERENCE. - is a MINUS SIGN or a SUBTRACTION SIGN. = is

an EQUALS SIGN.

Subtracting with integers . . .

(For now, we are only discussing subtracting integers. Other number sets will be discussed at a later time.)

~ Change operation to addition, change sign of the second number, and follow rules for adding integers.

6 – 12 = ?

6 + 12 = ? ..............................................................................................Change the operation to addition.

6 + (- 12) = ? .................................................................................Change the sign of the second number.

6 + (- 12) = ? ..............................................................................Now, follow the rules for adding integers.

12 - 6 = 6 .................................................................................................... Signs are different, so subtract

Answer is – 6..............................................................................keep the sign of the larger absolute value

Evaluate/Simplify: Show all work. Circle final answer. Rewrite the problem as needed. Use back as needed.

1. 3498 – 2340

2. 14 - 29

3. 6,983,442 - 27, 343

4. 999 - 111

5. 132, 230, 343 - 245, 993, 209

6. 6 - (-10)

7. 3x - 12x

8. -18 - (-27)

9. 978 - (-123, 434)


CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 11Continue Subtraction Practice below.

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 12MATH FOUNDATIONS # 5 = OPERATION: MULTIPLICATION...................................................................12

There are 4 basic arithmetic operations: addition/subtraction, and MULTIPLICATION/division (paired as inverses).

3 * 2 = 6 . . . (3)(2) = 6 . . . 3 x 2 = 6

3 and 2 are FACTORS. * is a TIMES SIGN or a MULTIPLICATION SIGN. X is a TIMES SIGN or a MULTIPLICATION

SIGN. ( ) PARENTHESES may also, at times, be used to notate MULTIPLICATION. = is an EQUALS SIGN.

Multiplying with integers . . .

(For now, we are only discussing integers. Other number sets will be discussed at a later time.)

~ If the factors have the SAME sign, the product will be POSITIVE.

6 x 6 = 36.........................................................both factors have the SAME sign, so the answer is positive

(-6) ∙ (-6) = 36.................................................both factors have the SAME sign, so the answer is positive

~ If the factors have DIFFERENT signs, the product will be NEGATIVE.

8 * (-14) = - 112............................................the factors have DIFFERENT signs, so the answer is negative

(-12)(19) = - 228............................................the factors have DIFFERENT signs, so the answer is negative

Notice in the above examples, there are several different ways to notate the operation of multiplication.

The (x) TIMES SIGN is rarely used to denote multiplication once variables are used (confusion with variable “x”).

Evaluate/Simplify. Show all work. Circle your final answer. Rewrite the problem as needed. Use back as needed.

1. 3498 * 2341

2. (14) (- 29)

3. 6,983,442 x 27, 343

4. (999)(- 111)

5. (-132, 230, 343) x (- 245, 993, 209)

6. 6 ∙ (-10)

7. 3x ∙ 12x

8. (-18) x (-27)

9. (978)(-123, 434)


CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 13Continue Multiplication Practice below.

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 14MATH FOUNDATIONS # 6 = OPERATION: DIVISION................................................................................13

There are 4 basic arithmetic operations: addition/subtraction, and multiplication/DIVISION (paired as inverses).

7/2 = 3 R 1

OR

7 ÷ 2 = 3.5 (or 3 ½)

2 is the DIVISOR. 6 is the DIVIDEND. 3 is the QUOTIENT. 1 is the REMAINDER. / is a DIVISION BAR or a DIVISION

SIGN or a DIVISION SLASH. ÷ is a DIVISION SIGN. - HORIZONTAL BAR or FRACTION BAR may also, at times, be

used to notate DIVISION. = is an EQUALS SIGN.

Dividing with integers . . .

(For now, we are only discussing integers. Other number sets will be discussed at a later time.)

~ If the dividend and divisor have the SAME sign, the quotient will be POSITIVE.

18 ÷ 2 = 9...................both the dividend and the divisor have the SAME sign, so the quotient is POSITIVE

(-16)/(-4) = 4..............both the dividend and the divisor have the SAME sign, so the quotient is POSITIVE

~ If the dividend and divisor have DIFFERENT signs, the quotient will be NEGATIVE.

100 ÷ (-20) = -5...............the dividend and the divisor have OPPOSITE signs, so the quotient is NEGATIVE

(-100)/(2).......................the dividend and the divisor have OPPOSITE signs, so the quotient is NEGATIVE

Notice in the above examples, there are several different ways to notate the operation of multiplication.

Solve for x. Show all work. Circle your final answer. Rewrite the problem as needed. Use back as needed.

1. 4/2 = x

2. (250) ÷ (- 25) = x

3. 10,750 ÷(- 125) = x

4. (999)/(- 111) = x

5. (-13, 068) ÷ (- 18) = x

6. (20,000) ÷ (18) = x Show remainder as a whole number.

7. (20,000) ÷ (18) = x Solve for a quotient in decimal form. What is unique about your quotient?

8. 12y2/ 2y = x

9. 48 ÷ (-48) = x


CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 15Continue Division Practice below.

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 16MATH FOUNDATIONS # 7 = DIVISIBILITY RULES......................................................................................16

There are certain “rules”, “patterns”, or “tricks” (so to speak) that one can use in order to determine if a given

integer is divisible by one or more of the first ten positive integers. Long division is always an option as well.

(Remember, division by zero is undefined. Notice 7 & 8 are a bit more involved, so there are two options for each.)

A given integer is divisible by . . .

1 ....................................................................................................................................... if it is an integer.

2...................................................................................... if it is an even integer (ending in 0, 2, 4, 6, or 8).

3..................................................................................................... if the sum of the digits is divisible by 3

4........................................................................... if the number formed by the last digits is divisible by 4

5...................................................................................................................................... if it ends in 5 or 0

6............................................................................................................. if it is both divisible by 2 and by 3

7........ if you subtract 2 times the last digit from the remaining digits, and the difference is divisible by 7

7..........................if you add five times the last digit to the remaining digits, and the sum is divisible by 7

8................................. if you add the last digit to twice the remaining digits and the sum is divisible by 8

8.................................................................................................... if the last three digits are divisible by 8

9..................................................................................................... if the sum of the digits is divisible by 9

10........................................................................................................................................... if it ends in 0

Example: ...................................................................................................................................23,412

Divisible by:

1 yes...................................................................................................................................... it is an integer

2 yes............................................................................................................................. it is even (ends in 2)

3 yes........................................................................................................sum of digits (12) is divisible by 3

4 yes...........................................................the number formed by the last two digits (12) is divisible by 4

5 no...................................................................................................................... it does NOT end in 5 or 0

6 yes........................................................................................................... it is both divisible by 2 and by 3

7 no........subtracting 2 times last digit (2x2) from remaining digits (2341) yields 2337, NOT divisible by 7

7 no...........................................................five times last digit (5 x 2) yields 10, 2351 is NOT divisible by 7

8 no..........................last digit (2) added to twice remaining digits (2341 x 2) = 4684 is NOT divisible by 8

8 no................................................................................................. last three digits are NOT divisible by 8

9 no.....................................................................................the sum of the digits (12) is NOT divisible by 9

10 no...........................................................................................................the number does NOT end in 0

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 17Divisibility Practice

Is the number 3,628,800 divisible by . . . ? Explain why or why not . . . (Hint: 10!)

1? _________________________________________________________________________________________

2? _________________________________________________________________________________________

3? _________________________________________________________________________________________

4? _________________________________________________________________________________________

5? _________________________________________________________________________________________

6? _________________________________________________________________________________________

7? _________________________________________________________________________________________

8? _________________________________________________________________________________________

9? _________________________________________________________________________________________

10? _________________________________________________________________________________________

Is the number 7,919 divisible by . . . tell why or why not . . .

1? _________________________________________________________________________________________

2? _________________________________________________________________________________________

3? _________________________________________________________________________________________

4? _________________________________________________________________________________________

5? _________________________________________________________________________________________

6? _________________________________________________________________________________________

7? _________________________________________________________________________________________

8? _________________________________________________________________________________________

9? _________________________________________________________________________________________

10? _________________________________________________________________________________________

Should this pattern continue, what special description can we give to the number 7,919?

_______________________________________________________________________________________________

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 18MATH FOUNDATIONS # 8 = DERIVED OPERATIONS.................................................................................18

There are mathematical operations that are “derived” from, or extensions of the four listed above.

Three such operations are described below:

~ Exponents (powers)

Exponents are often used as shorthand notation for repetitive multiplication.

For example . . .

3 * 3 * 3 * 3 = 81 = 34 ..................................................................... (read as “three to the fourth power”)

9*9 = 81 = 92 .....................................(read as “nine squared”, less often as “nine to the second power”)

5*5*5 = 125 = 53 ........................................(read as “five cubed”, less often as “five to the third power”)

6n ................................................................................................................ (read as “six to the nth power”)

a4................................................................................................................... (read as “a to the 4th power”)


~ Roots

Roots, in terms of an operation, are basically the inverse of exponents.

81 = 34, therefore 4√81 = 3 ....................................(read as “the fourth root of eighty-one equals three”)

81 = 92, therefore √81 = 9........................................(read as “the square root of eighty-one equals nine”)

125 = 53, therefore, 3√125 = 5..............(read as “the cubed root of one hundred twenty-five equals five”)n√6 ................................................................................................................. (read as “the nth root of six”)3√x .............................................................................................................. (read as “the cubed root of x”)


~ Absolute Value

The absolute value of a number is, basically, the measurement of its distance from ZERO.

|3| = 3.......................................................................(read as “the absolute value of three equals three”)

|-4| = 4...........................................................(read as “the absolute value of negative four equals four”)

|x| = x .....................................................................................(read as “the absolute value of x equals x”)

|-p| = p.................................................................... (read as “the absolute value of negative p equals p”)

|0| = 0...................................................................................(read as “the absolute value of zero is zero”)


CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 19Evaluate the following derived operation expressions. Show all work. Circle your final answer.

~ Exponents

34 _________________________________________________________________________________________

25 _________________________________________________________________________________________

(-3)2 _________________________________________________________________________________________

(-3)3 _________________________________________________________________________________________

108 _________________________________________________________________________________________

50 _________________________________________________________________________________________

~ Roots

√4 _________________________________________________________________________________________

√676 _________________________________________________________________________________________

√(-1) _________________________________________________________________________________________

√0 _________________________________________________________________________________________

√32 _________________________________________________________________________________________

√(-4) _________________________________________________________________________________________

~ Absolute Value

|- 4| _________________________________________________________________________________________

|27| _________________________________________________________________________________________

|(2)(-6)| ___________________________________________________________________________________

|- x| _________________________________________________________________________________________

|43| _________________________________________________________________________________________

|- π| _________________________________________________________________________________________

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 20MATH FOUNDATIONS # 9 = ORDER OF OPERATIONS..............................................................................20

When more than one type of operation appears in a mathematical expression/equation, performing the

operations in the order that they appear (reading left to right) will NOT consistently yield the accurate

evaluation/solution; rather, multiple operations are to be performed in a specific order – the Order of Operations.

P E M D A S

(Also known by the mnemonic device, Please excuse my dear aunt Sally. I simply pronounce it as a word “pemdas”.)

When solving an equation or evaluating an expression containing multiple operations, here is the order:

Level 1 is PE . . . Parentheses and Exponents are first (from the inside out)

Level 2 is MD . . . Multiplication and Division are next (from left to right)

Level 3 is AS = Addition and Subtraction are last (from left to right)

If a problem contains only one level, perform operations from left to right (or inside out for parentheses).

Each level’s members are equally important, and should be performed from left to right.

........................................................................................................ For example, 6 * 10 ÷ 12 = 60 ÷ 12 = 5

Simplify: 6 – 4 * 2

WRONG: 6 – 4 *2 . . . 2 * 2 . . . -4..................................................................I evaluated from left to right.

RIGHT: 6 – 4*2 . . . 6 – 8 . . . -2...................................................Multiplication comes before subtraction.

Simplify: 18 + (5-3)2 * 7

WRONG: 23 – 32 * 7 . . . 202 * 7 . . . 400*7 . . . 2,800....................................I evaluated from left to right.

RIGHT: 18 + (2)2 * 7....................................................First, operations within parentheses are evaluated.

RIGHT: 18 + 4 * 7.......................................................................................Next, I evaluate the exponents.

RIGHT: 18 + 28.......................................................................................Multiplication precedes addition.

RIGHT: 46............................................................................................................................... Finally, I add.

Simplify: 24 – (6+3)3 * (11-32)

WRONG: 18 + 27 * 11 – 9............................................................................. I evaluated from left to right.

WRONG: 45 * 11 – 9 . . . 495 – 9 . . . 486......................................................I evaluated from left to right.

RIGHT: 24 – (9)3 * (11-9).................................................................First, evaluate PE from the inside out.

RIGHT: 24 – 729 * 2.........................................................................................Continue eliminating all PE.

RIGHT: 24 – 1458..............................................................................Multiplication precedes subtraction.

RIGHT: - 1434..................................................................................................................Finally, I subtract.

**** This is also an important concept to consider when inputting expressions into a calculator. ****

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 21Demonstrate & label the WRONG method of simplification (left to right) and the RIGHT method (using PEMDAS).

Evaluate. Show all work. Circle final answers.

1. 428 – (3 + 7)2 + (6*5) + 13

WRONG (evaluating from left to right)

RIGHT (evaluating using PEMDAS)

2. 172 – (62 + 12) + (3*9)0

WRONG (evaluating from left to right)

RIGHT (evaluating using PEMDAS)

When finding the product of two binomials, we use the process called FOIL: first, outer, inner, last.

Example (a + b)(c + d)

First = ac / Outer = bd / Inner = bc / Last = bd

Answer ac + bd + bc + bd

Example (x + 3)(x – 9)

First = x2 / Outer = - 9x / Inner = 3x / Last = - 27

Answer x2 – 9x + 3x – 27 Simplified x2 – 6x – 27

3. Evaluate (y – 8)(y + 16)

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 22MATH FOUNDATIONS # 10 = PROPERTIES OF OPERATIONS....................................................................22

Certain operations have properties that hold true regardless of the numbers used in the expression/equation.

~ Commutative Property of Addition........................................................................................a + b = b + a

When two numbers are added, the sum is the same regardless of the order of the addends.

Provide TWO Examples: __________________________________________________________________________

~ Commutative Property of Multiplication.............................................................................(a)(b) = (b)(a)

When two numbers are multiplied together, the product is the same regardless of the order of the multiplicands.


~ Associative Property of Addition............................................................................(a + b) + c = a + (b + c)

When three or more numbers are added, the sum is the same regardless of the grouping of the addends.


~ Associative Property of Multiplication............................................................................. (ab)(c) = (a)(bc)

When three or more numbers are multiplied, the product is the same regardless of the grouping of the factors.


~ Identity Property of Addition (Additive Identity).........................................................................a + 0 = a

The sum of any number and zero is the original number.


~ Identity Property of Multiplication (Multiplicative Identity)........................................................a * 1 = a

The product of any number and one is that number.


~ Inverse Property of Addition (Additive Inverse).......................................................................a + (-a) = 0

The sum of a number and its inverse is zero.


CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 23~ Inverse Property of Multiplication (Multiplicative Inverse = reciprocal)................................a * (1/a) = 1

The product of a number and its multiplicative inverse is one.


~ Distributive property (uses both addition and multiplication)......................................a (b + c) = ab + ac

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


Other “rules” . . .

The number ZERO (0) is neither positive nor negative.

Dividing by ZERO is UNDEFINED........................................................................................a/0 is undefined


Any number raised to the 1st power is that number itself..................................................................a1 = a


Any number raised to the 0 power is equal to ONE...........................................................................A0 = 1


Any number raised to a negative power is equal to the reciprocal raised to the same positive power

For example: a (-n) = 1/an 3 (-2) = 1/32 = 1/9


CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 24MATH FOUNDATIONS # 11 = PROPERTIES OF EQUALITY.........................................................................24

Certain operations have properties that hold true regardless of the numbers used in the expression/equation.

~ Addition Property of Equality

If a = b . . . then a + x = b + x .....................................adding the same value to a and b keeps them equal


~ Subtraction Property of Equality

If a = b . . . then a – x = b – x ..........................subtracting the same value from a and b keeps them equal


~ Multiplication Property of Equality

If a = b . . . then a * x = b * x ......................multiplying both a and b by the same value keeps them equal


~ Division Property of Equality

If a = b . . . then a / x = b / x ..............................diving both a and b by the same value keeps them equal


~ Reflexive Property of Equality

a = a.........................................................................................................................a equals a / a is a / etc.


~ Symmetric Property of Equality

If a = b . . . then b = a...........two values remain equal regardless of position in regards to the equals sign


~ Transitive Property of Equality

If a = b . . . and b = c . . . then a = c


~ Substitution Property of Equality

If a = b . . . then a can be substituted for b in an equation


CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 25MATH FOUNDATIONS # 12 = EXPRESSIONS, EQUATIONS, INEQUALITIES................................................25

Combinations of NUMBERS and OPERATIONS are often found in one of three arrangements:

Expressions, Equations, or Inequalities

~ Expressions

When manipulating ONE or more values or numbers, one can write an expression. There is NO equals sign.

3 + 4

5 times a number

62

(x – 27) + 2x

|-12|

Expressions are not SOLVED . . . they are EVALUATED or SIMPLIFIED.


~ Equations

When comparing two or more values or numbers, one can state they are equal.

p = 12

5 + 17 = ___

y – 11 = 22

1/r = 0

x + y = 36

One can SOLVE an equation, or find that there is NO SOLUTION. Which one of the above equations has no solution?


~ Inequalities

When comparing two or more values or numbers, one can state they are unequal.

Unequal quantities can be compared as less than, less than or equal to, greater than, greater than or equal to.

6 > 5...............................................................................................................read “six is greater than five”

6 > y............................................................................................... read “six is greater than OR equal to y”

4 < 12........................................................................................................... read “four is less than twelve”

4 < (12 – p)......................................................................... read “four is less than or equal to 12 minus p”

A ≠ B...............................................................................read “a is not equal to b” or “a does not equal b”

3 < x < 7...............................................read as “three is less than x is less than 7” (compound inequality)


120

2 60

3 20

4 5

2 2

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 26MATH FOUNDATIONS # 13 = THE FUNDAMENTAL THEOREM OF ARITHMETIC......................................26

The FUNDAMENTAL THEOREM OF ARITHMETIC states that every natural number greater than 1 is either a PRIME

number, or can be expressed as a product of PRIME numbers.

For example:

2 is prime..................................................................................................... its only factors are 1 and itself


4 is composite............................................................................................. it is the product of 2 and 2 = 22


6 is composite.................................................................................................... it is the product of 2 and 3


8 is composite....................................................................................it is the product of 2 and 2 and 2 = 23

9 is composite.................................................................................................... it is the product of 3 and 3

10 is composite.................................................................................................. it is the product of 2 and 5

. . . 100 is composite...........................................................it is the product of 2 and 2 and 5 and 5 = 22*52

It is often helpful/necessary to be able to find the prime factors of a composite natural number. There are various

methods to employ to do this. Below is an example of a FACTOR TREE.The underlined numbers are the PRIME FACTORS, at the “ends” of the “branches”.

Thus, the prime factorization of 120 is . . .

120 = 2 x 3 x 5 x 2 x 2GOOD

Written in numerical order . . .

120 = 2 x 2 x 2 x 3 x 5BETTER

Using exponents . . .

120 = 23 x 3 x 5BEST

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 27Give the prime factorization of each number. PF should be in numerical order, using exponents.

Show all work. Circle your final answer. Many numbers can be factored in products of 2, 3, and 5.

1. 4

_________________________________________________________________________________________

2. 10

_________________________________________________________________________________________

3. 20

_________________________________________________________________________________________

4. 44

_________________________________________________________________________________________

5. 85

_________________________________________________________________________________________

6. 110

_________________________________________________________________________________________

7. 1000

_________________________________________________________________________________________

8. 2,700

_________________________________________________________________________________________

9. 144,000

_________________________________________________________________________________________

10. 1,000,000

_________________________________________________________________________________________

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 28MATH FOUNDATIONS # 14 = LCM & GCF................................................................................................28

These two items are often paired, but vastly different from each other – such as peanut butter and jelly.

They have many valuable uses in mathematics, especially when working with fractions.

LCM: Least common multiple – The lowest number that is a multiple of two or more given numbers.

For instance, the LCM of 10 and 15 is 30. 30 is the lowest number that is a multiple of both 10 and 15.

You can find this by listing the multiples of 15 and 30 until there is a common multiple. This is the LCM.

Multiples of 10: 0, 10, 20, 30, 40, 50 . . .

Multiples of 15: 0, 15, 30, 45, 60, etc.

The LCM of 15 and 30 is 30.

Using the LIST METHOD, find the LCM for 12 and 20.

12 = _________________________________________________

20 = _________________________________________________

LCM = _______________________________________________

GCF: Greatest common factor – The highest number that is a factor of two or more given numbers.

For instance, the GCF of 24 and 30 is 6. 6 is the highest number that is a factor of both 24 and 30.

You can find this by listing the factors of 24 and 30 and finding the highest common factor. This is the GCF.

24: 1, 24 . . . 2, 12 . . . 3, 8 . . . 4, 6

30: 1, 30 . . . 2, 15 . . . 3, 10 . . . 5, 6

The GCF of 24 and 30 is 6.

Using the LIST METHOD, find the GCF for 36 and 60.

36 = _________________________________________________

60 = _________________________________________________

GCF = _________________________________________________

Above, I used the LIST METHOD for finding LCM and GCF. It is effective, but not always efficient.

Also, above, I used the FACTOR TREE method for finding the prime factors of a number. Again, it is not always best.

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 29The LADDER METHOD provides the prime factorization (PF) for a single number, the simplified fraction for two

numbers, and the LCM/GCF two or more numbers. As you can see, it is a very valuable tool.

Example: What is the LCM and GCF for 24 and 40? What is 24/40 expressed in simplest form?

Step 1 – Write both numbers (This works for infinitely many numbers, theoretically.)

Step 2 – Draw a “L” shape to the left and underneath both numbers.

Step 3 – The prime factorization of many numbers includes 2, 3, and/or 5. Start by dividing each number by 2.

...................................................If 2 does not evenly divide, try 3. If 3 does not evenly divide, try 5. Etc.

Step 4 – Write 2 (or 3, or 5) on the left and divide, writing quotients underneath the original divisors

Step 5 – Repeat Step 3 until you are left with two numbers containing no common factors under your ladder.

GCF = The product of all numbers on the LEFT side of the ladder.

LCM = The product all of all numbers on the LEFT side and on the BOTTOM of the ladder.

SF = The simplified fraction will be the two numbers at the BOTTOM of the ladder, underneath the original

numerator and denominator, respectively.

To find the PF (prime factorization) place a SINGLE number under the ladder and calculate until the final number under the ladder is prime.

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 30Use the ladder method to find:

1. The PF for 128

2. The LCM and GCF for 48 and 60

3. The simplification of the fraction 36/156

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 31MATH FOUNDATIONS # 15 = WORKING WITH FRACTIONS, DECIMALS, AND PERCENTS........................31

Simplifying Fractions

A proper fraction is considered simplified when there are NO common factors between numerator & denominator.

Examples

5/6 ....................................................................................Simplified – 5 and 6 share NO common factors

4/7.....................................................................................Simplified – 4 and 7 share NO common factors

2/4................................................................................................Not simplified – 2 and 4 have a GCF of 2

...........................................Divide both the numerator & denominator by the GCF to find simplest form.

½............................................................................................................This is the final simplified fraction.

Write the following fractions in simplest form. Show all work.

1/3

2/6

3/8

4/20

Equivalent Fractions

Two or more fractions are considered equivalent when they are identical when in simplest form. To find an

equivalent fraction to a given fraction, use the rule of proportions – the product of the means = the product of the

extremes. This is often used when trying to find a common denominator (adding/subtracting fractions).

Example

1/3 + 1/6 = ?..................................................................We cannot add without a common denominator.

1/3 is equivalent to ?/6............................................................This will give us a COMMON denominator.

1 ? 1 x 6 = 3 x ?

___ = ___ 6 = 3?

3 6 2 = ?

So 1/3 = 2/6.

Find an equivalent fraction.

2/4 = ?/8

3/8 = ?/24

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 32Simplifying Proper Fractions

A fraction is considered proper when the numerator (top number) is < the denominator. It has a value less than 1.

Example

2/3 is a proper fraction. Because 2 and 3 share no common factors (they are, in fact, both prime) it is simplified.

Example

12/24 is a proper fraction. Because 12 and 24 share a common factor, it is not yet in simplified form.

We must find the GCF of 12 and 24. ...............................................................................................It is 12.

We must divide the numerator and denominator by the GCF...............................12/12 = 1 . . . 24/12 = 2

The resulting quotients are the numerator and denominator of the new, simplified fraction.................½

Simplify the following proper fractions.

3/6

14/38

56/140

Simplifying Improper Fractions

A fraction is considered improper when the numerator is > the denominator. It has a value greater than 1.

3/2 is an improper fraction. It has a value greater than one. It is often desirable to simplify Improper fractions –

writing them in the form of a mixed number. A mixed number is a combination of a whole number and a proper

fraction. 2 ½ is a mixed number.

Example

16/5....................................................................divide the numerator by the denominator (16÷5 = 3 R 1)

The quotient is your new WHOLE NUMBER (3) . . . the remainder is the new NUMERATOR (1).

Keep the denominator (5). Make sure the fraction portion of the mixed number is in simplest form.

So, 16/5 = 3 ⅕

Simplify the following improper fractions.

6/3

38/14

140/56

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 33Adding/Subtracting Fractions

To add or subtract fractions, all addends/minuends/subtrahends must have the same/COMMON DENOMINATORS.

The common denominator is the LCM of the denominators given.

1/3 + 1/3 = 2/3.........................................................keep same denominator, add numerators, simplified

2/5 + 1/5 = 3/5.........................................................keep same denominator, add numerators, simplified

1/6 + 1/3.................................................................................need common denominators / 6 is the LCM

1/6 + 2/6......................................................1/3 is equivalent to 2/6 (check via proportion or graphically)

3/6..............................This is the sum, but not yet our final answer. 3 and 6 have a common factor of 3.

½...........................................................................................................This is our final, simplified answer.

Add/subtract the following fractions. Show all work. Answers should be in simplest form.

1/3 + 2/3

2/5 + 2/5

3/8 – 3/16

1/3 – 5/6

Multiplying Fractions

To multiply fractions, find the product of the numerators, find the product of the denominators. Simplify.

Mixed numbers must be converted into improper fractions.

2 1/3..........................................This needs to be converted to an improper fraction in order to multiply.

3*2 = 6.................................................................................Multiply denominator times whole number.

6 + 1 = 7........................................................................................................................Add to numerator.

7/3.......................................................................................................................Keep same denominator.

Whole numbers must be placed over a denominator of ONE.........................................................6 = 6/1.

Evaluate. Write answers in simplest form.

½ x 1/3

2/5 * 2/5

4/9 * 2/7

2 ¼ * 4

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 34Dividing Fractions

To divide fractions, use the SAME first fraction . . . CHANGE the operation to multiplication . . . FLIP the second

fraction . . . follow the rules for multiplying fractions. (This “flipped” fraction is known as the reciprocal.)

½ ÷ 1/3

2/5 ÷ 2/5

4/9 ÷ 2/7

2 ¼ ÷ 4

Adding/Subtracting Decimals

To add/subtract decimals, you must remember to ALIGN the decimal points.

Example

23.45 + 6.789

23.450........................................................We can assume/write a 0 in the thousandths place. Why?

+ 6.789

30.239

Add 34.987 + 4, 231.0038

Multiplying Decimals

These are multiplied in the same manner as you do whole/natural numbers. The final product must contain the

same number of decimal places (numbers to the right of the decimal) as the SUM of decimal places in the factors.

Factor (17.2)....................................................................................................This has ONE decimal place.

Factor (12.56)...............................................................................................This has TWO decimal places.

Answer.................................................................................We know it MUST have three decimal places.

216.032.................................................................................................................. This is our final answer.

Note – multiplying 172 * 1,256 yields an answer with the same digits 216,032 with no decimal places.

Multiply 19.233 * (- 123.7)

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 35Dividing Decimals

The dividend (the number OUTSIDE of the long division bar) cannot contain any decimal places. If it has any decimal

places, you must move the decimal to the RIGHT until all numbers are whole.

For instance, divisor 0.96.....................................Move decimal TWO places to the RIGHT to give you 96.

Then, move the decimal the same number of places in the dividend.

Dividend 88......................................................Move decimal TWO places to the RIGHT to give you 8800.

Then divide as you would two whole numbers.

Example:

100 ÷ 2.5....................................................................................2.5, the divisor MUST be a whole number

25........................................................................Move the decimal ONE place to the RIGHT to give us 25

1,000.......................................Move the decimal ONE place to the right in the dividend, giving us 1,000

1,000 ÷ 25 = 40..............................................................Divide the whole numbers to arrive at a quotient

So, 100 ÷ 2.5 = 40.............................................................................................Here is the final statement.

Divide 180 by 4.5

Fractions vs. Decimals vs. Percents

If I purchased a pizza with 8 slices and ate 4 of them, there are several mathematical statements we could make.

8 – 4 = 4.....................................................................................................................................Subtraction

4 + 4 = 8..........................................................................................................................................Addition

8/2 = 4.....................................................................................Division / Fraction / I ate HALF of the pizza.

8 * ½ = 4........................................................................Multiplication / Fraction / I ate HALF of the pizza.

100% / 2 = 50%...........The WHOLE pizza (100%) was divided in HALF. Now there is ½ or 50% remaining.

½ = 0.5..............................................................................One pizza, divided by two, leaves 0.5 of a pizza.

So . . .

1 / 2 (fraction) = 50% (percentage) = 0.5 (decimal)...............................These are all equivalent values.

We need to be able to convert from one form to another, depending on our mathematical context.

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 36Explanation of Percent

Per cent literally means “per 100”. 0% means nothing, never, or no chance. 100% means all, always, etc.

~ CONVERTING: Decimal to Percent

Simply move the decimal place TWO places to the RIGHT.

0.86 = 86% 0.09 = 9% 1.03 = 103% (Yes, you can have a percentage > 100.)

Convert the following decimals to percentages.

0.7

0.89

12.098

0.007

~ CONVERTING: Percent to Decimal

Simply move the decimal place TWO places to the LEFT.

86% = 0.86 9% = 0.09 12.4% = 0.124

Convert the following percentages to decimals.

82%

12.9%

123.1%

6%

~ CONVERTING: Fraction to Decimal

Simply divide the numerator BY the denominator.

1 / 2 = 1 ÷2 = 0.5

2 / 5 = 2÷ 5 = 0.4

Convert the following fractions to decimals.

1 / 4

5 / 8

2 / 9

7 / 5

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 37~ CONVERTING: Decimal to Fraction

Any number(s) to the left of the decimal will be a whole number in your final fraction answer.

Count the numbers to the right of the decimal . . . divide these digits by that power of 10, ignoring the decimal.

1.23.............................................This has TWO decimal places . . . so we divide “23” by 102 which is 100.

1.23 = 1 23/100.................................................................This is in simplified form. We are done.

13.45 = 13 45/100..........................................This is not yet simplified. 45 and 100 have a GCF of 5.

13.45 = 13 9/20.......................................................................................................................Done.

5.644....................................This has THREE decimal places . . . so we divide “644” by 103 which is 1,000.

5.644 = 5 644/1,000.....................This is not yet simplified. 644/1000 can be reduced by GCF of 4.

5.644 = 5 161/250..................................................................................................................... Done.

Convert the following decimals to fractions in simplest form.

3.25

10.055

2.9

~ CONVERTING: Percent to Fraction

Convert the percent to a decimal (move decimal TWO places to the LEFT), then convert the decimal to a fraction.

45% = 0.45 = 45/100 = 9/20

22.8% = 0.228 = 228/1000 = 57/250

Convert the following percentages to fractions.

40%

112%

8.09%

~ CONVERTING: Fraction to Percent

Convert the fraction to a decimal (by dividing the numerator by the denominator) then move the decimal TWO

places to the RIGHT to find the final percent.

2/5 2 ÷5 = 0.4 40%

1/8 1 ÷ 8 = 0.125 12.5%

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 38Convert the following fractions to percentages.

1/4

3/10

4/5

2/9

Complete the following table.

PERCENT FRACTION DECIMAL50%

1/3

.04

25%

1/5

.89

11%

2/9

1.04

- 19.003%

MATH FOUNDATIONS # 16 = WORKING WITH EXPONENTS/ROOTS/RADICALS......................................39

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 39This sign √ can be called “the root of” or a “radical sign”. Expressions/equations using this sign can also be called

“radicals”. There are rules for working with radicals/roots.

Parts of a radical: kn√akn√a........................................................................................................................n = index

kn√a...........................................................................................√ = radical (or, radical sign)

kn√a...................................................................................................................a = radicand

kn√a..............................................................................................k = constant = coefficient

Parts of an exponential expression: bx bx.............................................................................................................................b = base

bx...................................................................................................x = power (or, exponent)

Simplifying RadicalsBefore performing operations on multiple radicals, individual radicals need to be in simplest form.

~ Step One ......................................................................................Know your perfect squares from 1-20.

12 = 1 22 = 4 32 = 9 42 = 16 52 = 2562 = 36 72 = 49 82 = 64 92 = 81 102 = 100112 = 121 122 = 144 132 = 169 142 = 196 152 = 225162 = 256 172 = 289 182 = 324 192 = 361 202 = 400(1, 4, 9, 16, 25, etc.........................Start to immediately recognize these numbers as perfect squares.)

~ Step Two....................Given a radical, find the LARGEST perfect square that is a factor of the radicand.

√200..................................................the largest perfect square factor is 100~ Step Three....Factor the radicand into the product of the perfect square and its complimentary factor.

√200............................................................................................ (√100)( √2)~ Step Four ..........Evaluate the square root of the perfect square – simplify by changing to a coefficient.

√200.................................................................................(√100)( √2) = 10√2~ Step Five ...................Repeat this process until the radicand has NO factors which are perfect squares.

√200........................................(√100)( √2) = 10√2 (This is in simplified form.)

Adding/Subtracting Radicals k n √a + l n √a = (k+l) n √a

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 40To add or subtract radicals, the indices must be the same (n) and the radicals themselves (a; the number underneath

the radical sign) must be identical. Then, the coefficients can be added or subtracted.

Examples:

√3 + √4 = √3 + √4................................................Cannot be simplified further / radicals are NOT identical

2√3 + 3√3 = 5√3.....................................................Indices and radicals are the same, add the coefficients

Multiplying Radicals n √ab = ( n √a)( n √b) or ( n √a)( n √b) = n √ab To multiply or divide radicals, the indices must be the same. The radicals may be equal or unequal.

Examples:

√20 = (√4)(√5) Twenty is the product of 4 and 5. So, the sqrt. of 20 is the product of sqrt of 4 and sqrt of 5.

2√5 The sqrt of 4 is 2. The sqrt. of 5 cannot be simplified further.

√90 = (√9)( √10) 90 is the product of 9 and 10. So, the sqrt. of 90 is the product of sqrt of 9 and sqrt of 10.

3√10 The sqrt of 9 is 3. The sqrt. of 10 cannot be simplified further.

Dividing Radicals n √(a/b) = ( n √a) / ( n √b) or ( n √a) / ( n √b) = n √(a/b) Examples

√20 / √10 = √2 20/10 = 2. So, the sqrt of 20/10 = the sqrt of 2. Root 2 cannot be simplified further.

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 41Working with radicals and roots. Evaluate. Show all work. Circle your final answer. Use simplest form.

√100

√36

√144

√200

√80

√440

3√6 + 5√6

2√5 + 2√7

4√3 - 6√3

√80 / √4

(√10)( √6)

√ (99/11)

Challenge!

√(-144)

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 42MATH FOUNDATIONS # 17 = CALCULATOR BASICS.................................................................................42

Learning to effectively, efficiently, and accurately use a graphing calculator is an important component of this course

and further mathematical progression. Using the space below, take notes regarding important and common math-

related tasks completed on your calculator, and the specific manner in which the task was completed.

In order to:...............................................................................................Then I need to:

TURN ON ___________________________________________________________________________________

TURN OFF ___________________________________________________________________________________

ADJUST CONTRAST (Darker/Lighter)__________________________________________________________________

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CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 43MATH FOUNDATIONS # 17 = CALCULATOR BASICS..............................................................43 (continued)

Learning to effectively, efficiently, and accurately use a graphing calculator is an important component of this course

and further mathematical progression. Using the space below, take notes regarding important and common math-

related tasks completed on your calculator, and the specific manner in which the task was completed.

In order to:...............................................................................................Then I need to:

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CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 44MATH FOUNDATIONS # 18 = MEASURES OF CENTRAL TENDENCY..........................................................44

When working with a set of data, or a set of numbers, it is possible to make generalizations about the set based on measures of central tendency. There are four common measures used.

Let A be a set of data, for example, a hypothetical set of test scores.

A {85, 100, 70, 35, 44, 98, 15, 12, 86, 82, 70}

~ Step 1: Count the number of elements.

N = 11

~ Step 2: Arrange elements in numerical order.

A {10, 12, 15, 35, 44, 70, 70, 82, 85, 86, 100}

~ Step 3: The MODE is the most common number listed. A set with two modes is BIMODAL. A set with more than two modes is MULTIMODAL.

Mode = 70

~ Step 4: The ARITHMETIC MEAN is the average of the set of numbers, found by dividing the SUM of all numbers by the NUMBER of elements.

Sum = 609 N = 11 Mean = 609/11 = 55.363636…

~ Step 5: The MEDIAN is the middle number in an arranged set containing an odd number of elements. The MEDIAN is the arithmetic average of the two middle numbers in an arranged set containing an even number of elements.

Median = 70

~ It can also be helpful to know the following items:

Maximum (the greatest value in a set) = 100

Minimum (the least value in a set) = 10

Range (the difference between the maximum and minimum values) = 90

What general statements can you make regarding this set of scores? Look at the mean, median, mode, range, etc.

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CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 45Given set B, find/circle/label the number of elements, maximum, minimum, range, mean, median, and mode.

B {12, 89, 92, 100, 25, 4, 78, 79, 52, 44, 27}

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 46MATH FOUNDATIONS # 19 = GEOMETRY BASICS....................................................................................46

PLANE GEOMETRY: 2-dimensional objects

Item Symbol/Picture Name Description/Definition

Point . P (single capital letter) A single specific spot, location

Line l (single, lower case italicized letter) Set of all points on 1 dimension

Plane ABC (named by > 3 points in the plane) Flat surface, no thickness

Ray AB (endpoint first, then another point name) Endpoint, extending forever

Segment AB (named by two endpoints) Set distance between two endpoints

Angle < A (named by vertex) or ABC (named by 3 endpoints) 2 rays with common endpoint

Circle A (named by the center point) Set of all points equidistant from center

PERIMETER: ____________________________________ AREA: _________________________________________

Polygon n/a Named by vertices Closed figure formed by 3+ line segments intersecting

Quadrilateral ABCD (named by 4 vertices) A polygon with four sides

Rectangle ABCD (named by 4 vertices) Quadrilateral with 4 right angles (900)

PERIMETER: ____________________________________ AREA: _________________________________________

Square ABCD (named by 4 vertices) A rectangle with four congruent sides

PERIMETER: ____________________________________ AREA: _________________________________________

Triangle ABC (named by 3 vertices) A three-sided polygon

PERIMETER: ____________________________________ AREA: _________________________________________

Right Triangle AB (named by 3 vertices) A triangle containing one right angle

PERIMETER: ____________________________________ AREA: _________________________________________

AREA = The number of square units that a figure covers (square feet, square miles, square centimeters, etc.)

PERIMETER = Distance completely around a polygon COMPLETE THE FORMULAS ABOVE.

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 47SOLID GEOMETRY: 3-dimensional objects

Sphere

DEFINITION: _______________________________________________________________________________________

VOLUME: ____________________________________ SURFACE AREA: _________________________________________

Cylinder

DEFINITION: _______________________________________________________________________________________


Cone

DEFINITION: _______________________________________________________________________________________


Pyramid

DEFINITION: _______________________________________________________________________________________


Prism

DEFINITION: _______________________________________________________________________________________


Cube

DEFINITION: _______________________________________________________________________________________


SURFACE AREA = The number of square units that all faces of an object cover (square feet, square centimeters, etc.)

VOLUME = The amount of space an objects takes up in three dimensions, or the capacity which it can hold

COMPLETE THE DEFINITIONS and FORMULAS ABOVE. There may be more than one example for a given form.

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 48MATH FOUNDATIONS # 20 = THE COORDINATE PLANE..........................................................................48

The coordinate system is a method for defining the location of a single point on a particular plane.

The coordinate plane, or Cartesian plane (after Rene Descartes) is composed of:

~ an origin (middle point)

~ a horizontal X-axis (which procedes infinitely to the left and right of the origin)

~ a vertical Y-axis (which procedes infinitely above and below the origin)

(There is also a third axis, the Z-axis, allowing for 3-dimensional plotting. We will discuss this at a later time.)

A particular point on the CP (coordinate plane) is identified by an ORDERED PAIR.

This is a pair of numbers, in parentheses, separated by a comma.

The origin is designated (0, 0).

The first number corresponds to the HORIZONTAL distance away from the origin – on the X-axis.

The second number corresponds to the VERTICAL distance away from the origin – on the Y-axis.

Below, there are points drawn at (2, 3) . . . (0, 0) . . . (-3, 1) . . . and (-1.5, -2.5).

(Source: Public Domain, via http://en.wikipedia.org/wiki/File:Cartesian-coordinate-system.svg)

The perpendicular intersection of the X and Y axes at the original divides the CP into four sections or QUADRANTS.

These quadrants are labeled in Roman numerals, beginning with the upper right, rotating counterclockwise.

Points in each quadrant are signed as follows:

Quadrant I .......................................................................................................................................... (+ , +)

Quadrant II ......................................................................................................................................... (+ , -)

Quadrant III.......................................................................................................................................... (- , -)

Quadrant IV......................................................................................................................................... (- , +)

Practice with graphing on the coordinate plane.

Quadrant

IV

QuadrantIII

Quadrant

I

Quadrant

II

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 49Draw a coordinate plane. Draw and label the origin.

Include an X-axis, with units drawn and labeled from (-10) to (+ 10).

Include a Y-axis, with units drawn and labeled from (-10) to (+10).

Label each of the four quadrants. Remember to use Roman numerals.

Graph and label the following points:

A (2, 3) B (-2, 3) C (2, -3) D (-3, -3) E (0, 1) F (0, 3) G (0, 5) H (1, 0) I (3, 0) J (5, 0)

CENTRAL FELLOWSHIP CHRISTIAN ACADEMY – MATH FOUNDATIONS – 2013/2014 SCHOOL YEAR – PAGE 50MATH FOUNDATIONS # 21 = MATH WRITING SAMPLES.........................................................................50

In mathematics we focus on numbers and their various interactions; however, an important component is the ability

to read and write regarding numbers. In addition, your ability to read and write independently in all subject areas is

an invaluable skill which must be constantly practiced in order to be consistently developed. Finally, a crucial

component of your education at CFCA is the understanding, appreciation, and application of Biblical principles to all

walks of life. To that extent, each week you will have a writing assignment to complete and submit for grading,

relating to math and the Bible.

PART 1 – Find a particular passage in the Bible which references a number. Quote and cite/reference the scripture

PART 2 – Describe that number in mathematical terms/concepts.

PART 3 – Explain the significance of the number, in its relating to the context of events in the scripture.

You are expected to work independently, using proper grammar, sentence structure, paragraph formation, and

other composition-related procedures as are taught and expected at CFCA.

Below is an example of a completed writing sample. Length is roughly ½ a page.

PART 1

And he took his staff in his hand, and chose him five smooth stones out of the brook, and put them in a shepherd's

bag which he had, even in a scrip; and his sling was in his hand: and he drew near to the Philistine.

1 Samuel 17:40 (KJV)

PART 2

The number is 5. It is complex, real, rational, an integer, whole, and natural. It is also odd and prime.

PART 3

The context of the passage is the account of David versus Goliath. Goliath was a Philistine soldier who had

challenged for any Israelite to come forward and fight him. The winner would, with his people, rule over the losing

side. For 40 days Goliath’s challenge went unanswered. David challenged Goliath, not using sword and spear, but

simply choosing 5 stones and his slingshot. He defeated the giant with one strike of a single stone. Praise God!

Documents

millercfca.weebly.commillercfca.weebly.com/uploads/1/3/2/0/13200028/math_f… · Web viewnot to be ashamed, rightly dividing the word of truth. Romans 12:2 (KJV) = And be not conformed