MILLER CFCA



NameGradeClass[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.EuclidCFCA HS MATH | MR. MILLER | 2013-2014MATH FOUNDATIONS – TABLE OF CONTENTSCOVER PAGE1TABLE OF CONTENTS / PREFACE 2MATH FOUNDATIONS # 1 = WHY MATH? 3MATH FOUNDATIONS # 2 = THE WORLD OF NUMBERS 5MATH FOUNDATIONS # 3 = OPERATION: ADDITION 8MATH FOUNDATIONS # 4 = OPERATION: SUBTRACTION 10MATH FOUNDATIONS # 5 = OPERATION: MULTIPLICATION 12MATH FOUNDATIONS # 6 = OPERATION: DIVISION 14MATH FOUNDATIONS # 7 = DIVISIBILITY RULES 16MATH FOUNDATIONS # 8 = DERIVED OPERATIONS 18MATH FOUNDATIONS # 9 = ORDER OF OPERATIONS 20MATH FOUNDATIONS # 10 = PROPERTIES OF OPERATIONS 22MATH FOUNDATIONS # 11 = PROPERTIES OF EQUALITY 24MATH FOUNDATIONS # 12 = EXPRESSIONS, EQUATIONS, INEQUALITIES 25MATH FOUNDATIONS # 13 = THE FUNDAMENTAL THEOREM OF ARITHMETIC 26MATH FOUNDATIONS # 14 = LCM & GCF 28MATH FOUNDATIONS # 15 = WORKING WITH FRACTIONS, DECIMALS, AND PERCENTS 31MATH FOUNDATIONS # 16 = WORKING WITH EXPONENTS/ROOTS/RADICALS 39MATH FOUNDATIONS # 17 = CALCULATOR BASICS 42MATH FOUNDATIONS # 18 = MEASURES OF CENTRAL TENDENCY44MATH FOUNDATIONS # 19 = GEOMETRY BASICS 46MATH FOUNDATIONS # 20 = THE COORDINATE PLANE 48MATH FOUNDATIONS # 21 = MATH WRITING SAMPLES 50OTHER # 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.MATH FOUNDATIONS # 1 = WHY MATH?3We 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? _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________~ 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.What are your thoughts? _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________~ 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.What are your thoughts? _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________~ 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.What are your thoughts? _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________~ 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.____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________MATH FOUNDATIONS # 2 = THE WORLD OF NUMBERS5Next, 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.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: ___________________________________________________________________________WHOLE 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).Description/Examples: ___________________________________________________________________________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). Description/Examples: ___________________________________________________________________________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).Description/Examples: ___________________________________________________________________________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.)Description/Examples: ___________________________________________________________________________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”.Description/Examples: ___________________________________________________________________________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 Description/Examples: ___________________________________________________________________________COMPLEX 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. Description/Examples: ___________________________________________________________________________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.Description/Examples: ___________________________________________________________________________~ 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).Description/Examples: ___________________________________________________________________________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)Description/Examples: ___________________________________________________________________________All natural/whole numbers having a natural/whole number square root are called perfect squares.Description/Examples: ___________________________________________________________________________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 4Complex ~ 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 576Description/Examples: ___________________________________________________________________________The number ?Description/Examples: ___________________________________________________________________________The number √10Description/Examples: ___________________________________________________________________________MATH FOUNDATIONS # 3 = OPERATION: ADDITION8There are 4 basic arithmetic operations: ADDITION/subtraction, and multiplication/division (paired as inverses).2 + 3 = 52 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 = 16both 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) = - 4addends different signs, difference is 4, keep sign of larger absolute value (-10) = - 4Solve for x. Show all work. Circle your final answer. Rewrite the problem as needed. Use back as needed. 3498 + 2340 = x14 + 29 = x6,983,442 + 27, 343 = x999 + 111 = x132, 230, 343 + 245, 993, 209 = x6 + (-10) = x3x + 12x = x-18 + (-27) = x978 + (-123, 434) = xDesign your own problem and solve.Continue Addition Practice below.MATH FOUNDATIONS # 4 = OPERATION: SUBTRACTION10There are 4 basic arithmetic operations: addition/SUBTRACTION, and multiplication/division (paired as inverses).3 - 2 = 13 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 subtractAnswer is – 6keep the sign of the larger absolute valueEvaluate/Simplify: Show all work. Circle final answer. Rewrite the problem as needed. Use back as needed.3498 – 234014 - 296,983,442 - 27, 343999 - 111132, 230, 343 - 245, 993, 2096 - (-10)3x - 12x-18 - (-27)978 - (-123, 434)Design your own problem and solve.Continue Subtraction Practice below.MATH FOUNDATIONS # 5 = OPERATION: MULTIPLICATION12There are 4 basic arithmetic operations: addition/subtraction, and MULTIPLICATION/division (paired as inverses).3 * 2 = 6 . . . (3)(2) = 6 . . . 3 x 2 = 63 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 = 36both factors have the SAME sign, so the answer is positive(-6) ? (-6) = 36both factors have the SAME sign, so the answer is positive~ If the factors have DIFFERENT signs, the product will be NEGATIVE.8 * (-14) = - 112the factors have DIFFERENT signs, so the answer is negative(-12)(19) = - 228the factors have DIFFERENT signs, so the answer is negativeNotice 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.3498 * 2341(14) (- 29)6,983,442 x 27, 343(999)(- 111)(-132, 230, 343) x (- 245, 993, 209)6 ? (-10)3x ? 12x(-18) x (-27)(978)(-123, 434)Design your own problem and solve.Continue Multiplication Practice below.MATH FOUNDATIONS # 6 = OPERATION: DIVISION13There are 4 basic arithmetic operations: addition/subtraction, and multiplication/DIVISION (paired as inverses).7/2 = 3 R 1OR7 ÷ 2 = 3.5 (or 3 ?)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 = 9both the dividend and the divisor have the SAME sign, so the quotient is POSITIVE(-16)/(-4) = 4both 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) = -5the 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.4/2 = x(250) ÷ (- 25) = x10,750 ÷(- 125) = x(999)/(- 111) = x(-13, 068) ÷ (- 18) = x(20,000) ÷ (18) = x Show remainder as a whole number.(20,000) ÷ (18) = x Solve for a quotient in decimal form. What is unique about your quotient?12y2/ 2y = x48 ÷ (-48) = xDesign your own problem and solve.Continue Division Practice below.MATH FOUNDATIONS # 7 = DIVISIBILITY RULES16There 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 34 if the number formed by the last digits is divisible by 45 if it ends in 5 or 06 if it is both divisible by 2 and by 37 if you subtract 2 times the last digit from the remaining digits, and the difference is divisible by 77if you add five times the last digit to the remaining digits, and the sum is divisible by 78 if you add the last digit to twice the remaining digits and the sum is divisible by 88if the last three digits are divisible by 8 9 if the sum of the digits is divisible by 910 if it ends in 0Example: 23,412Divisible by:1 yesit is an integer2 yesit is even (ends in 2)3 yessum of digits (12) is divisible by 34 yesthe number formed by the last two digits (12) is divisible by 45 noit does NOT end in 5 or 06 yesit is both divisible by 2 and by 37 nosubtracting 2 times last digit (2x2) from remaining digits (2341) yields 2337, NOT divisible by 77 nofive times last digit (5 x 2) yields 10, 2351 is NOT divisible by 78 nolast digit (2) added to twice remaining digits (2341 x 2) = 4684 is NOT divisible by 88 nolast three digits are NOT divisible by 89 nothe sum of the digits (12) is NOT divisible by 910 nothe number does NOT end in 0Divisibility PracticeIs 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?_______________________________________________________________________________________________MATH FOUNDATIONS # 8 = DERIVED OPERATIONS18There 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”)Description/Examples: ___________________________________________________________________________~ RootsRoots, 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”)Description/Examples: ___________________________________________________________________________~ Absolute ValueThe 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”)Description/Examples: ___________________________________________________________________________Evaluate the following derived operation expressions. Show all work. Circle your final answer.~ Exponents34 _________________________________________________________________________________________25_________________________________________________________________________________________(-3)2_________________________________________________________________________________________(-3)3_________________________________________________________________________________________108_________________________________________________________________________________________50_________________________________________________________________________________________~ Roots√4_________________________________________________________________________________________√676_________________________________________________________________________________________√(-1)_________________________________________________________________________________________√0_________________________________________________________________________________________√32_________________________________________________________________________________________√(-4)_________________________________________________________________________________________~ Absolute Value|- 4|_________________________________________________________________________________________|27|_________________________________________________________________________________________|(2)(-6)|___________________________________________________________________________________|- x|_________________________________________________________________________________________|43|_________________________________________________________________________________________|- π|_________________________________________________________________________________________MATH FOUNDATIONS # 9 = ORDER OF OPERATIONS20When 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 = 5Simplify: 6 – 4 * 2WRONG: 6 – 4 *2 . . . 2 * 2 . . . -4I evaluated from left to right.RIGHT: 6 – 4*2 . . . 6 – 8 . . . -2Multiplication comes before subtraction.Simplify: 18 + (5-3)2 * 7WRONG: 23 – 32 * 7 . . . 202 * 7 . . . 400*7 . . . 2,800I evaluated from left to right.RIGHT: 18 + (2)2 * 7First, operations within parentheses are evaluated.RIGHT: 18 + 4 * 7Next, I evaluate the exponents.RIGHT: 18 + 28Multiplication precedes addition.RIGHT: 46Finally, I add.Simplify: 24 – (6+3)3 * (11-32)WRONG: 18 + 27 * 11 – 9I evaluated from left to right.WRONG: 45 * 11 – 9 . . . 495 – 9 . . . 486I evaluated from left to right.RIGHT: 24 – (9)3 * (11-9)First, evaluate PE from the inside out.RIGHT: 24 – 729 * 2Continue eliminating all PE.RIGHT: 24 – 1458Multiplication precedes subtraction.RIGHT: - 1434Finally, I subtract.**** This is also an important concept to consider when inputting expressions into a calculator. ****Demonstrate & label the WRONG method of simplification (left to right) and the RIGHT method (using PEMDAS).Evaluate. Show all work. Circle final answers. 428 – (3 + 7)2 + (6*5) + 13WRONG (evaluating from left to right)RIGHT (evaluating using PEMDAS)172 – (62 + 12) + (3*9)0WRONG (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 = bdAnswer ac + bd + bc + bdExample (x + 3)(x – 9)First = x2/Outer = - 9x/Inner = 3x/Last = - 27Answerx2 – 9x + 3x – 27Simplifiedx2 – 6x – 27 Evaluate(y – 8)(y + 16)MATH FOUNDATIONS # 10 = PROPERTIES OF OPERATIONS22Certain operations have properties that hold true regardless of the numbers used in the expression/equation.~ Commutative Property of Additiona + b = b + aWhen 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. Provide TWO Examples: __________________________________________________________________________~ 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. Provide TWO Examples: __________________________________________________________________________~ 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. Provide TWO Examples: __________________________________________________________________________~ Identity Property of Addition (Additive Identity)a + 0 = aThe sum of any number and zero is the original number. Provide TWO Examples: __________________________________________________________________________~ Identity Property of Multiplication (Multiplicative Identity)a * 1 = aThe product of any number and one is that number. Provide TWO Examples: __________________________________________________________________________~ Inverse Property of Addition (Additive Inverse)a + (-a) = 0The sum of a number and its inverse is zero.Provide TWO Examples: __________________________________________________________________________~ Inverse Property of Multiplication (Multiplicative Inverse = reciprocal)a * (1/a) = 1The product of a number and its multiplicative inverse is one.Provide TWO Examples: __________________________________________________________________________~ Distributive property (uses both addition and multiplication)a (b + c) = ab + acThe sum of two numbers times a third number is equal to the sum of each addend times the third number. Provide TWO Examples: __________________________________________________________________________Other “rules” . . . The number ZERO (0) is neither positive nor negative.Dividing by ZERO is UNDEFINED.a/0 is undefinedProvide TWO Examples: __________________________________________________________________________Any number raised to the 1st power is that number itself.a1 = aProvide TWO Examples: __________________________________________________________________________Any number raised to the 0 power is equal to ONE.A0 = 1Provide TWO Examples: __________________________________________________________________________Any number raised to a negative power is equal to the reciprocal raised to the same positive powerFor example: a (-n) = 1/an3 (-2) = 1/32 = 1/9Provide TWO Examples: __________________________________________________________________________MATH FOUNDATIONS # 11 = PROPERTIES OF EQUALITY24Certain operations have properties that hold true regardless of the numbers used in the expression/equation.~ Addition Property of EqualityIf a = b . . . then a + x = b + x adding the same value to a and b keeps them equalProvide TWO Examples: __________________________________________________________________________~ Subtraction Property of EqualityIf a = b . . . then a – x = b – x subtracting the same value from a and b keeps them equalProvide TWO Examples: __________________________________________________________________________~ Multiplication Property of EqualityIf a = b . . . then a * x = b * x multiplying both a and b by the same value keeps them equalProvide TWO Examples: __________________________________________________________________________~ Division Property of EqualityIf a = b . . . then a / x = b / x diving both a and b by the same value keeps them equalProvide TWO Examples: __________________________________________________________________________~ Reflexive Property of Equalitya = aa equals a / a is a / etc.Provide TWO Examples: __________________________________________________________________________~ Symmetric Property of EqualityIf a = b . . . then b = atwo values remain equal regardless of position in regards to the equals signProvide TWO Examples: __________________________________________________________________________~ Transitive Property of EqualityIf a = b . . . and b = c . . . then a = cProvide TWO Examples: __________________________________________________________________________~ Substitution Property of EqualityIf a = b . . . then a can be substituted for b in an equationProvide TWO Examples: __________________________________________________________________________MATH FOUNDATIONS # 12 = EXPRESSIONS, EQUATIONS, INEQUALITIES25Combinations of NUMBERS and OPERATIONS are often found in one of three arrangements:Expressions, Equations, or Inequalities~ ExpressionsWhen manipulating ONE or more values or numbers, one can write an expression. There is NO equals sign.3 + 45 times a number62(x – 27) + 2x|-12|Expressions are not SOLVED . . . they are EVALUATED or SIMPLIFIED.Provide TWO Examples: __________________________________________________________________________~ EquationsWhen comparing two or more values or numbers, one can state they are equal.p = 125 + 17 = ___y – 11 = 221/r = 0x + y = 36One can SOLVE an equation, or find that there is NO SOLUTION. Which one of the above equations has no solution?Provide TWO Examples: __________________________________________________________________________~ InequalitiesWhen 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 > 5read “six is greater than five”6 > yread “six is greater than OR equal to y”4 < 12read “four is less than twelve”4 < (12 – p)read “four is less than or equal to 12 minus p”A ≠ Bread “a is not equal to b” or “a does not equal b”3 < x < 7read as “three is less than x is less than 7” (compound inequality)Provide TWO Examples: __________________________________________________________________________MATH FOUNDATIONS # 13 = THE FUNDAMENTAL THEOREM OF ARITHMETIC26The 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 primeits only factors are 1 and itself3 is primeits only factors are 1 and itself4 is compositeit is the product of 2 and 2 = 225 is primeits only factors are 1 and itself6 is compositeit is the product of 2 and 37 is primeits only factors are 1 and itself8 is compositeit is the product of 2 and 2 and 2 = 239 is compositeit is the product of 3 and 310 is compositeit is the product of 2 and 5. . . 100 is compositeit is the product of 2 and 2 and 5 and 5 = 22*52It 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 2GOODWritten in numerical order . . . 120 = 2 x 2 x 2 x 3 x 5BETTERUsing exponents . . . 120 = 23 x 3 x 5BEST1202603204522Give 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. 4_________________________________________________________________________________________ 10_________________________________________________________________________________________ 20_________________________________________________________________________________________ 44_________________________________________________________________________________________ 85_________________________________________________________________________________________ 110_________________________________________________________________________________________ 1000_________________________________________________________________________________________ 2,700_________________________________________________________________________________________ 144,000_________________________________________________________________________________________1,000,000_________________________________________________________________________________________MATH FOUNDATIONS # 14 = LCM & GCF28These 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, 630: 1, 30 . . . 2, 15 . . . 3, 10 . . . 5, 6The 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.The 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 divisorsStep 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.Use the ladder method to find: The PF for 128 The LCM and GCF for 48 and 60 The simplification of the fraction 36/156MATH FOUNDATIONS # 15 = WORKING WITH FRACTIONS, DECIMALS, AND PERCENTS31Simplifying FractionsA proper fraction is considered simplified when there are NO common factors between numerator & denominator.Examples5/6 Simplified – 5 and 6 share NO common factors4/7Simplified – 4 and 7 share NO common factors2/4Not simplified – 2 and 4 have a GCF of 2Divide both the numerator & denominator by the GCF to find simplest form.?This is the final simplified fraction.Write the following fractions in simplest form. Show all work.1/32/63/84/20Equivalent FractionsTwo 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).Example1/3 + 1/6 = ?We cannot add without a common denominator.1/3 is equivalent to ?/6This 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 = ?/83/8 = ?/24Simplifying Proper FractionsA fraction is considered proper when the numerator (top number) is < the denominator. It has a value less than 1.Example2/3 is a proper fraction. Because 2 and 3 share no common factors (they are, in fact, both prime) it is simplified.Example12/24 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 = 2The resulting quotients are the numerator and denominator of the new, simplified fraction.?Simplify the following proper fractions.3/614/3856/140Simplifying Improper FractionsA 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 ? is a mixed number.Example16/5divide 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 ?Simplify the following improper fractions.6/338/14140/56Adding/Subtracting FractionsTo 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 = 2/3keep same denominator, add numerators, simplified2/5 + 1/5 = 3/5keep same denominator, add numerators, simplified1/6 + 1/3need common denominators / 6 is the LCM1/6 + 2/61/3 is equivalent to 2/6 (check via proportion or graphically)3/6This is the sum, but not yet our final answer. 3 and 6 have a common factor of 3.?This is our final, simplified answer.Add/subtract the following fractions. Show all work. Answers should be in simplest form.1/3 + 2/32/5 + 2/53/8 – 3/161/3 – 5/6Multiplying FractionsTo multiply fractions, find the product of the numerators, find the product of the denominators. Simplify.Mixed numbers must be converted into improper fractions. 2 1/3This needs to be converted to an improper fraction in order to multiply.3*2 = 6Multiply denominator times whole number. 6 + 1 = 7Add to numerator. 7/3Keep same denominator.Whole numbers must be placed over a denominator of ONE.6 = 6/1.Evaluate. Write answers in simplest form.? x 1/32/5 * 2/54/9 * 2/72 ? * 4Dividing FractionsTo 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.)? ÷ 1/32/5 ÷ 2/54/9 ÷ 2/72 ? ÷ 4Adding/Subtracting DecimalsTo add/subtract decimals, you must remember to ALIGN the decimal points.Example23.45 + 6.789 23.450We can assume/write a 0 in the thousandths place. Why?+ 6.789 30.239Add 34.987 + 4, 231.0038Multiplying DecimalsThese 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.AnswerWe know it MUST have three decimal places.216.032This 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)Dividing DecimalsThe 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 0.96Move decimal TWO places to the RIGHT to give you 96.Then, move the decimal the same number of places in the dividend.Dividend 88Move decimal TWO places to the RIGHT to give you 8800.Then divide as you would two whole numbers.Example:100 ÷ 2.52.5, the divisor MUST be a whole number25Move the decimal ONE place to the RIGHT to give us 251,000Move the decimal ONE place to the right in the dividend, giving us 1,0001,000 ÷ 25 = 40Divide the whole numbers to arrive at a quotientSo, 100 ÷ 2.5 = 40Here is the final statement.Divide 180 by 4.5Fractions vs. Decimals vs. PercentsIf I purchased a pizza with 8 slices and ate 4 of them, there are several mathematical statements we could make.8 – 4 = 4Subtraction4 + 4 = 8Addition8/2 = 4Division / Fraction / I ate HALF of the pizza.8 * ? = 4Multiplication / Fraction / I ate HALF of the pizza.100% / 2 = 50%The WHOLE pizza (100%) was divided in HALF. Now there is ? or 50% remaining.? = 0.5One 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.Explanation of PercentPer cent literally means “per 100”. 0% means nothing, never, or no chance. 100% means all, always, etc.~ CONVERTING: Decimal to PercentSimply 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.0.70.8912.0980.007~ CONVERTING: Percent to DecimalSimply move the decimal place TWO places to the LEFT.86% = 0.86 9% = 0.09 12.4% = 0.124Convert the following percentages to decimals.82%12.9%123.1%6%~ CONVERTING: Fraction to DecimalSimply divide the numerator BY the denominator.1 / 2 = 1 ÷2 = 0.52 / 5 = 2÷ 5 = 0.4Convert the following fractions to decimals.1 / 45 / 82 / 97 / 5~ CONVERTING: Decimal to FractionAny 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.1.23This has TWO decimal places . . . so we divide “23” by 102 which is 100.1.23 = 1 23/100This is in simplified form. We are done.13.45 = 13 45/100This is not yet simplified. 45 and 100 have a GCF of 5.13.45 = 13 9/20Done.5.644This has THREE decimal places . . . so we divide “644” by 103 which is 1,000.5.644 = 5 644/1,000This is not yet simplified. 644/1000 can be reduced by GCF of 4.5.644 = 5 161/250Done.Convert the following decimals to fractions in simplest form.3.2510.0552.9~ CONVERTING: Percent to FractionConvert the percent to a decimal (move decimal TWO places to the LEFT), then convert the decimal to a fraction.45% = 0.45 = 45/100 = 9/2022.8% = 0.228 = 228/1000 = 57/250Convert the following percentages to fractions.40%112%8.09%~ CONVERTING: Fraction to PercentConvert 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%Convert the following fractions to percentages.1/43/104/52/9Complete the following table.PERCENTFRACTIONDECIMAL50%1/3.0425%1/5.8911%2/91.04- 19.003%MATH FOUNDATIONS # 16 = WORKING WITH EXPONENTS/ROOTS/RADICALS39This 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√an = indexkn√a√ = radical (or, radical sign)kn√aa = radicandkn√ak = constant = coefficientParts of an exponential expression: bx bxb = basebxx = 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 = 122 = 432 = 942 = 1652 = 2562 = 3672 = 4982 = 6492 = 81102 = 100112 = 121122 = 144132 = 169142 = 196152 = 225162 = 256172 = 289182 = 324192 = 361202 = 400(1, 4, 9, 16, 25, etc.Start to immediately recognize these numbers as perfect squares.)~ Step TwoGiven a radical, find the LARGEST perfect square that is a factor of the radicand.√200the largest perfect square factor is 100~ Step ThreeFactor 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 kn√a + ln√a = (k+l)n√aTo 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 + √4Cannot be simplified further / radicals are NOT identical2√3 + 3√3 = 5√3Indices and radicals are the same, add the coefficientsMultiplying Radicals n√ab = (n√a)(n√b) or (n√a)(n√b) = n√abTo 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√5The 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√10The sqrt of 9 is 3. The sqrt. of 10 cannot be simplified further.Dividing Radicalsn√(a/b) = (n√a) / (n√b) or (n√a) / (n√b) = n√(a/b)Examples√20 / √10 = √220/10 = 2. So, the sqrt of 20/10 = the sqrt of 2. Root 2 cannot be simplified further.Working with radicals and roots. Evaluate. Show all work. Circle your final answer. Use simplest form.√100√36√144√200√80√4403√6 + 5√62√5 + 2√74√3 - 6√3√80 / √4(√10)( √6)√ (99/11)Challenge!√(-144)MATH FOUNDATIONS # 17 = CALCULATOR BASICS42Learning 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)___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________MATH FOUNDATIONS # 17 = CALCULATOR BASICS43 (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: _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________MATH FOUNDATIONS # 18 = MEASURES OF CENTRAL TENDENCY44When 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 = 609N = 11Mean = 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) = 100Minimum (the least value in a set) = 10Range (the difference between the maximum and minimum values) = 90What general statements can you make regarding this set of scores? Look at the mean, median, mode, range, etc._____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________Given 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}MATH FOUNDATIONS # 19 = GEOMETRY BASICS46PLANE GEOMETRY: 2-dimensional objectsItem Symbol/Picture NameDescription/DefinitionPoint .P (single capital letter)A single specific spot, locationLinel (single, lower case italicized letter)Set of all points on 1 dimensionPlaneABC (named by > 3 points in the plane)Flat surface, no thicknessRayAB (endpoint first, then another point name)Endpoint, extending foreverSegmentAB (named by two endpoints)Set distance between two endpointsAngle< A (named by vertex) or ABC (named by 3 endpoints)2 rays with common endpointCircleA (named by the center point)Set of all points equidistant from centerPERIMETER: ____________________________________AREA: _________________________________________Polygonn/aNamed by verticesClosed figure formed by 3+ line segments intersectingQuadrilateralABCD (named by 4 vertices)A polygon with four sidesRectangleABCD (named by 4 vertices)Quadrilateral with 4 right angles (900)PERIMETER: ____________________________________AREA: _________________________________________SquareABCD (named by 4 vertices)A rectangle with four congruent sidesPERIMETER: ____________________________________AREA: _________________________________________TriangleABC (named by 3 vertices)A three-sided polygonPERIMETER: ____________________________________AREA: _________________________________________Right TriangleAB (named by 3 vertices)A triangle containing one right anglePERIMETER: ____________________________________AREA: _________________________________________AREA = The number of square units that a figure covers (square feet, square miles, square centimeters, etc.)PERIMETER = Distance completely around a polygonCOMPLETE THE FORMULAS ABOVE.SOLID GEOMETRY: 3-dimensional objectsSphereDEFINITION:_______________________________________________________________________________________VOLUME: ____________________________________SURFACE AREA: _________________________________________CylinderDEFINITION:_______________________________________________________________________________________VOLUME: ____________________________________SURFACE AREA: _________________________________________Cone DEFINITION:_______________________________________________________________________________________VOLUME: ____________________________________SURFACE AREA: _________________________________________PyramidDEFINITION:_______________________________________________________________________________________VOLUME: ____________________________________SURFACE AREA: _________________________________________PrismDEFINITION:_______________________________________________________________________________________VOLUME: ____________________________________SURFACE AREA: _________________________________________CubeDEFINITION:_______________________________________________________________________________________VOLUME: ____________________________________SURFACE AREA: _________________________________________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 holdCOMPLETE THE DEFINITIONS and FORMULAS ABOVE. There may be more than one example for a given form.MATH FOUNDATIONS # 20 = THE COORDINATE PLANE48The 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).QuadrantIIQuadrantIQuadrantIIIQuadrantIV(Source: Public Domain, via )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.Draw 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)MATH FOUNDATIONS # 21 = MATH WRITING SAMPLES50In 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 scripturePART 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 ? a page.PART 1And 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 2The number is 5. It is complex, real, rational, an integer, whole, and natural. It is also odd and prime.PART 3The 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! ................
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