8th Grade - Woodstown

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Slide 1 / 156Slide 2 / 1568th GradeThe Number System and MathematicalOperations Part 22015-08-31www.njctl.orgSlide 3 / 156Slide 4 / 156Table of ContentsSquares of Numbers Greater than 20Click on topicto go to thatsection.Simplifying Perfect Square Radical ExpressionsApproximating Square RootsRational & Irrational NumbersReal NumbersProperties of ExponentsSquares of NumbersGreater than 20Glossary & StandardsReturn toTable ofContentsSlide 5 / 156Square Root of Large NumbersSlide 6 / 156Square Root of Large NumbersIt helps to know the squares of larger numbers such as the multiples oftens.Think about this.What about larger numbers?How do you find?102 100202 400302 900402 1600502 2500602 3600702 4900802 6400902 81001002 10000What pattern do you notice?

Slide 7 / 156Slide 8 / 156Square Root of Large NumbersFor larger numbers, determine between which twomultiples of ten the number lies.Examples:102 100202 400302 900402 1600502 2500602 3600702 4900802 6400902 81001002 10000Lies between 2500 & 3600 (50 and 60)Ends in nine so square root ends in 3 or 7Try 53 then 57532 280912 122 432 942 1652 2562 3672 4982 6492 81102 100List ofSquaresSquare Root of Large NumbersLies between 6400 and 8100 (80 and 90)Ends in 4 so square root ends in 2 or 8Try 82 then 88822 6724 NO!882 7744Next, look at the ones digit to determine the ones digitof your square root.Slide 9 / 156Slide 10 / 1562 Find.List ofSquaresList ofSquares1 Find.Slide 11 / 156Slide 12 / 156List ofSquares4 Find.List ofSquares3 Find.

Slide 13 / 156Slide 14 / 156List ofSquares6 Find.List ofSquares5 Find.Slide 15 / 156Slide 16 / 1568 Find.Slide 17 / 156List ofSquaresList ofSquares7 Find.Slide 18 / 156List ofSquares9 Find.Simplifying Perfect SquareRadical ExpressionsReturn toTable ofContents

Slide 19 / 156Slide 20 / 156Two Roots for Even PowersIf we square 4, we get 16.If we take the square root of 16, we get two answers: -4 and 4.That's because, any number raised to an even power, such as 2, 4,6, etc., becomes positive, even if it started out being negative.So, (-4)2 (-4)(-4) 16AND(4)2 (4)(4) 16This can be written as 16 4, meaning positive or negative 4.This is not an issue with odd powers, just even powers.Slide 21 / 156Slide 22 / 156Is there a difference between.&Evaluate the Expressions?Which expression has no real roots?Evaluate the expressions:is not realSlide 23 / 15610Slide 24 / 15611A 6A 9B -6B -9C is not realC is not real

Slide 25 / 15612Slide 26 / 156(Problem from13A 20B -20C is not realWhich student's method is not correct?A Ashley's MethodB Brandon's MethodOn your paper, explain why the method you selected isnot correct.Slide 27 / 15614Slide 28 / 15615Slide 29 / 15616Slide 30 / 15617AA3BB-3CCNo real rootsD)

Slide 31 / 156Slide 32 / 156Square Roots of Fractions18 The expression equal toa bis equivalent to a positive integer when b isA -10b 0B 64C 16D 416 49 47From the Ne w York S ta te Educa tion De pa rtme nt. Office of As s e s s me nt P olicy, De ve lopme nt a ndAdminis tra tion. Inte rne t. Ava ila ble from www.nys e dre ge nts .org/Inte gra te dAlge bra ; a cce s s e d 17, J une , 2011.Slide 33 / 156Slide 34 / 15619Slide 35 / 15620ACBDno real solutionACBDno real solutionSlide 36 / 156

Slide 37 / 15622Slide 38 / 15623ACACBD no real solutionBDSlide 39 / 156Square Roots of DecimalsTo find the square root of a decimal, convert the decimal to afraction first. Follow your steps for square roots of fractions.no real solutionSlide 40 / 15624 EvaluateABCDno real solution .2 .05 .3Slide 41 / 15625 EvaluateSlide 42 / 15626 EvaluateA.06B.6A0.11B11C6Dno real solutionC1.1Dno real solution

Slide 43 / 156Slide 44 / 15627 EvaluateAC0.8B0.08Dno real solutionSlide 45 / 156Slide 46 / 156Perfect SquareAll of the examples so far havebeen from perfect squares.ApproximatingSquare RootsWhat does it mean to be a perfect square?The square of an integer is a perfect square.A perfect square has a whole number square root.Return toTable ofContentsSlide 47 / 156Slide 48 / 156Non-Perfect SquaresYou know how to find the square root of a perfect square.What happens if the number is not a perfect square?Does it have a square root?What would the square root look like?Non-Perfect e149162536496481100121144169196225Think about the square root of 50.Where would it be on this chart?What can you say about the squareroot of 50?50 is between the perfect squares 49and 64 but closer to 49.So the square root of 50 is between 7and 8 but closer to 7.

Slide 49 / 156Slide 50 / 156Approximating Non-Perfect e149162536496481100121144169196225When approximating square roots ofnumbers, you need to determine:· Between which two perfect squares it lies(and therefore which 2 square roots).· Which perfect square it is closer to (andtherefore which square root).Example:Lies between 100 & 121, closer to 100.Sois between 10 & 11, closer to 10.Approximating Non-Perfect e149162536496481100121144169196225Slide 51 / 156Slide 52 / 156Approximating Non-Perfect SquaresApproximating Non-Perfect SquaresApproximateApproximate6to the nearest integer Identify perfect squares closest to 387Take square rootAnswer: Because 38 is closer to 36 than to 49,7. So, to the nearest integer, 6to the nearest integer Identify perfect squares closest to 70 Take square rootIdentify nearest integeris closer to 6 than toSlide 53 / 156Approximating a Square RootApproximateApproximate the following:Slide 54 / 156Approximating a Square RootAnother way to think about it is to use a number line.on a number line.Example: Approximate2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.010.0 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11.0Since 8 is closer to 9 than to 4,so# 2.8is closer to 3 than to 2;115 lies between andSo, it lies between whole numbers and .Imagine the square roots between these numbers, andpicture wherelies.

Slide 55 / 15629 The square root of 40 falls between which two perfectsquares?Slide 56 / 15630 Which integer isclosest to?A 3 and 4B 5 and 6C 6 and 7D 7 and 8 Identify perfect squares closest to 40 Take square rootIdentify nearest integerSlide 57 / 15631 The square root of 110 falls between which two perfectsquares?Slide 58 / 15632 Estimate to the nearest integer.A 36 and 49B 49 and 64C 64 and 84D 100 and 121Slide 59 / 15633 Select the point on the number line that bestapproximates the location of.B C EADGF HFrom PARCC EOY sample test non-calculator #19Slide 60 / 15634 Estimate to the nearest integer.

Slide 61 / 15635 Estimate to the nearest integer.Slide 62 / 15636 ApproximateSlide 63 / 15637 Approximateto the nearest integer.Slide 64 / 15638 ApproximateSlide 65 / 15639 Approximateto the nearest integer.to the nearest integer.to the nearest integer.Slide 66 / 15640 Approximateto the nearest integer.

Slide 67 / 15641 The expressionSlide 68 / 156is a number between:42 For what integer x isclosest to 6.25?A 3 and 9B 8 and 9C 9 and 10D 46 and 47Derived fromFrom the Ne w York S ta te Educa tion De pa rtme nt. Office of As s e s s me nt P olicy, De ve lopme nt a ndAdminis tra tion. Inte rne t. Ava ila ble from www.nys e dre ge nts .org/Inte gra te dAlge bra ; a cce s s e d 17, J une , 2011.Slide 69 / 15643 For what integer y isSlide 70 / 15644 Between which two positive integers doesclosest to 4.5?A 1F 6B 2G 7C 3H 8D 4I 9E 5J 10Derived fromDerived fromSlide 71 / 15645 Between which two positive integers doesA 1F 6B 2G 7C 3H 8D 4I 9E 5J 10Derived fromlie?Slide 72 / 156lie?46 Between which two labeled points on thenumber line wouldABDerived fromCDbe located?EFGHIJ

Slide 73 / 156Slide 74 / 156Irrational NumbersJust as subtraction led us to zero and negative numbers.Rational & IrrationalNumbersReturn toTable ofContentsAnd division led us to fractions.Finding the root leads us to irrational numbers .Irrational numbers complete the set of Real Numbers.Real numbers are the numbers that exist on the number line.Slide 75 / 156Slide 76 / 156Rational & Irrational NumbersIrrational NumbersIrrational Numbers are real numbers that cannot be expressedas a ratio of two integers.In decimal form, they extend forever and never repeat.There are an infinite number of irrational numbers between anytwo integers (think of all the square roots, cube roots, etc. thatdon't come out evenly).Then realize you can multiply them by an other number or addany number to them and the result will still be irrational.Slide 77 / 156Irrational Numbersis rational.This is because the radicand (number under theradical) is a perfect square.If a radicand is not a perfect square, the root is said to be irrational.Ex:Slide 78 / 156Irrational NumbersIrrational numbers were first discovered by Hippasus about theabout 2500 years ago.An infinity of irrational numbers emerge from trying to find theroot of a rational number.He was a Pythagorean, and the discovery was considered aproblem since Pythagoreans believed that "All was Number," bywhich they meant the natural numbers (1, 2, 3.) and their ratios.Hippasus proved that 2 is irrational goes on forever withoutrepeating.Hippasus was trying to find the length of the diagonal of a squarewith sides of length 1. Instead, he proved, that the answer wasnot a natural or rational number.For a while this discovery was suppressed since it violated thebeliefs of the Pythagorean school.Hippasus had proved that 2 is an irrational number.Some of its digits are shown on the next slide.

Slide 79 / 156Square Root of 2Here are the first 1000 digits, but you can find the first 10 million digitson the Internet. The numbers go on forever, and never repeat in 2951848847208969.Slide 80 / 156Roots of Numbers are Often IrrationalSoon, thereafter, it was proved that many numbers haveirrational roots.We now know that the roots of most numbers to most powers areirrational.These are called algebraic irrational numbers.In fact, there are many more irrational numbers that rationalnumbers.Slide 81 / 156Principal RootsSince you can't write out all the digits of 2, or use a bar toindicate a pattern, the simplest way to write that number is 2.Slide 82 / 156Algebraic Irrational NumbersThere are an infinite number of irrational numbers.Here are just a few that fall between -10 and 10.But when solving for the square root of 2, there are two answers: 2 or - 2.These are in two different places on the number line.To avoid confusion, it was agreed that the positive value wouldbe called the principal root and would be written 2.-0.6 852- 2 2 952 95The negative value would be written as - 2.-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10- 2 8 50- 80- 810Slide 83 / 156Transcendental NumbersThe other set of irrational numbers are the transcendentalnumbers.Slide 84 / 156PiWe have learned about Pi in Geometry. It is the ratio of acircle's circumference to its diameter. It is represented bythe symbol.These are also irrational. No matter how many decimals you lookat, they never repeat.But these are not the result of solving a polynomial equation withrational coefficients, so they are not due to an inverse operation.Some of these numbers are real, and some are complex.But, this year, we will only be working with the realtranscendental numbers.Discuss why this is an approximation at your table.Is this number rational or irrational?

Slide 85 / 156Slide 86 / 156# is a Transcendental NumberThe most famous of the transcendentalnumbers is # .OB # is the ratio of the circumference to thediameter of a circle.People tried to find the value of that ratiofor millennia.AOnly in the mid 1800's was it proven thatthere was no rational 06/pi-day004.jpgSince # is irrational (it's decimals neverrepeat) and it is not a solution to aequation.it is transcendental.Some of its digits are on the next i-day004.jpgSlide 87 / 156Slide 88 / 156Other Transcendental NumbersThere are many more transcendental numbers.Another famous one is "e", which you will study in Algebra II asthe base of natural logarithms.In the 1800's Georg Cantor proved there are as manytranscendental numbers as real numbers.And, there are more algebraic irrational numbers than there arerational numbers.The integers and rational numbers with which we feel morecomfortable are like islands in a vast ocean of irrational numbers.Sort by the square root being rational or irrational.Slide 89 / 15647 Rational or Irrational?A RationalSlide 90 / 15648 Rational or Irrational?B IrrationalA RationalB Irrational

Slide 91 / 15649 Rational or Irrational?Slide 92 / 15650 Rational or Irrational?0.141414.B IrrationalA RationalSlide 93 / 15651 Rational or Irrational?B IrrationalA RationalSlide 94 / 15652 Rational or Irrational?1.222.A RationalB IrrationalA RationalSlide 95 / 15653 Which is a rational number?AB πCDB IrrationalSlide 96 / 15654 Given the statement: “If x is a rational number,thenis irrational." Which value of x makesthe statement false?AB 2C 3D 4From the Ne w York S ta te Educa tion De pa rtme nt. Office of As s e s s me nt P olicy, De ve lopme nt a ndAdminis tra tion. Inte rne t. Ava ila ble from www.nys e dre ge nts .org/Inte gra te dAlge bra ; a cce s s e d 17, J une , 2011.From the Ne w York S ta te Educa tion De pa rtme nt. Office of As s e s s me nt P olicy, De ve lopme nt a ndAdminis tra tion. Inte rne t. Ava ila ble from www.nys e dre ge nts .org/Inte gra te dAlge bra ; a cce s s e d 17, J une , 2011.

Slide 97 / 156Slide 98 / 156(Problem from)55 A student made this conjecture and found two examplesto support the conjecture.If a rational number is not an integer, then the squareroot of the rational number is irrational.For example,is irrational andis irrational.Real NumbersProvide an example of a non-integer rational number thatshows that the conjecture is false.Return toTable ofContentsSlide 99 / 156Slide 100 / 156Real NumbersReal numbers are numbers that exist on a number line.Real NumbersThe Real Numbers are all the numbers that can be found on anumber line.That may seem like all numbers, but later in Algebra II we'll betalking about numbers which are not real numbers, and cannotbe found on a number line.As you look at the below number line, you'll see certainnumbers indicated that are used to label the line.Those are not all the numbers on the line, they just help us findall the other numbers as well.numbers not shown below.-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10Slide 101 / 15656 What type of number is 25? Select all that apply.Slide 102 / 15657 What type of number is - 5 ? Select all that apply.3A IrrationalA IrrationalB RationalB RationalC IntegerC IntegerD Whole NumberD Whole NumberE Natural NumberE Natural NumberF Real NumberF Real Number

Slide 103 / 15658 What type of number is 8? Select all that apply.Slide 104 / 15659 What type of number is - 64? Select all that apply.A IrrationalA IrrationalB RationalB RationalC IntegerC IntegerD Whole NumberD Whole NumberE Natural NumberE Natural NumberF Real NumberF Real NumberSlide 105 / 156Slide 106 / 15660 What type of number is 2.25? Select all that apply.A IrrationalB RationalC IntegerD Whole NumberProperties ofExponentsE Natural NumberF Real NumberReturn toTable ofContentsSlide 107 / 156Powers of IntegersSlide 108 / 156Powers of IntegersExponents is repeated multiplication.Make sure when you are evaluating exponents of negativenumbers, you keep in mind the meaning of the exponent andthe rules of multiplication.For example, 35 read as "3 to the fifth power" 3 3 3 3 3For example, (-3)2 (-3) (-3) 9, which is the same as 32.In this case "3" is the base and "5" is the exponent.However, (-3)2 (-3)(-3) 9, and -32 -(3)(3) -9The base, 3, is multiplied by itself 5 times.which are not the same.Just as multiplication is repeated addition,Similarly, 33 3 3 3 27and (-3)3 (-3) (-3) (-3) -27,which are not the same.

Slide 109 / 15661 Evaluate: 43Slide 110 / 15662 Evaluate: (-2)7Slide 111 / 156Slide 112 / 156Properties of Exponents63 Evaluate: (-3)4The properties of exponents follow directly from expandingthem to look at the repeated multiplication they represent.Don't memorize properties, just work to understand the processby which we find these properties and if you can't recall what todo, just repeat these steps to confirm the property.We'll use 3 as the base in our examples, but the properties holdfor any base. We show that with base a and powers b and c.We'll use the facts that:(32) (3 3)(33) (3 3 3)(35) (3 3 3 3 3)Slide 113 / 156Properties of ExponentsWe need to develop all the properties of exponents so we candiscover one of the inverse operations of raising a number to apower.finding the root of a number to that power.That will emerge from the final property we'll explore.But, getting to that property requires understanding the othersfirst.Slide 114 / 156Multiplying with ExponentsWhen multiplying numbers with the same base,add the exponents.(ab)(ac) a(b c)(32)(33) 35(3 3 )(3 3 3) (3 3 3 3 3)(32)(33) 35(32)(33) 3(2 3)

Slide 115 / 156Dividing with ExponentsWhen dividing numbers with the same base, subtract theexponent of the denominator from that of the numerator.(a ) (a ) abc(b-c)(33)1(32) 3(3 3 3)(3 3 ) 3Slide 116 / 15664 Simplify: 54 52A 52B 53C 56D 58(33) (32) 3(3-2)(33) (32) 31Slide 117 / 156Slide 118 / 156Slide 119 / 156Slide 120 / 156665 Simplify: 774A 71.5B 72C 710D 72467 Simplify: 86 83468 Simplify: 525A 82A 52B 83B 53C 89C 56D 818D 58

Slide 121 / 156Slide 122 / 15670 Simplify: 57 5369 Simplify: 43 (45)A415A52B48B510C42C521D47D54Slide 123 / 156An Exponent of ZeroAny base raised to the power of zero is equal to 1.Slide 124 / 15671 Simplify: 50 a(0) 1Based on the multiplication rule:(3(0))(3(3)) (3(3 0))(3(0))(3(3)) (3(3))Any number times 1 is equal to itself(1)(3(3)) (3(3))(3(0))(3(3)) (3(3))Comparing these two equations, we see that(3(0)) 1This is not just true for base 3, it's true for all bases.Slide 125 / 15672 Simplify: 80 1 Slide 126 / 15673 Simplify: (7)(300)

Slide 127 / 156Slide 128 / 156Negative ExponentsNegative ExponentsA negative exponent moves the number from the numerator todenominator, and vice versa.(a(-b)) 1abab By definition:1(a(-b))Based on the multiplication rule and zero exponent rules:x-1 (3(-1))(3(1)) (3(-1 1)), x 0(3(-1))(3(1)) (3(0))(3(-1))(3(1)) 1But, any number multiplied by its inverse is 1, so1(3(1)) 131(3(-1))(3(1)) 1Comparing these two equations, we see that1(3(-1)) 13Slide 129 / 15674 Simplify the expression to make the exponentspositive. 4-2A 42Slide 130 / 15675 Simplify the expression to make the exponentspositive.14-2A 42B 124B 124Slide 131 / 15676 Simplify the expression to make the exponentspositive. x3 y-4Slide 132 / 15677 Simplify the expression to make the exponentspositive. a-5 b-2A x3 y4A a5 b24B y3xx3Cy41D 3 4xy2B b5aa5Cb21D 5 2ab

Slide 133 / 156Slide 134 / 15678 Which expression is equivalent to x-4?79 What is the value of 2-3?AAB x4BC -4xC -6D 0D -8From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available fromwww.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.From the Ne w York S tate Education De partme nt. Office of As s e s s me nt Policy, De ve lopme nt andAdminis tration. Inte rne t. Available from www.nys e dre ge nts .org/Inte grate dAlge bra; acce s s e d 17,June , 2011.Slide 135 / 156Slide 136 / 15680 Which expression is equivalent to x-1 y2?A xy2CBD xy-281 a) Write an exponential expression for the area of arectangle with a length of 7-2 meters and a width of7-5 meters.b) Evaluate the expression to find thearea of the rectangle.When you finish answering both parts, enter youranswer to Part b) in your responder.From the Ne w York S tate Education De partme nt. Office of As s e s s me nt Policy, De ve lopme nt andAdminis tration. Inte rne t. Available from www.nys e dre ge nts .org/Inte grate dAlge bra; acce s s e d 17,June , 2011.Slide 137 / 15682 Which expressions are equivalent toSelect all that apply.Slide 138 / 156?83 Which expressions are equivalent toSelect all that 2F13-10From PARCC EOY sample test non-calculator #13?

Slide 139 / 156Raising Exponents to Higher PowersWhen raising a number with an exponent to a power,multiply the exponents.(a ) ab cSlide 140 / 15684 Simplify: (24)7A 21.75(bc)(32)3 36(32)(32)(32) 3(2 2 2)(3 3 )(3 3)(3 3) (3 3 3 3 3 3)B 23C 211D 228(32)3 36Slide 141 / 15685 Simplify: (g3)9Slide 142 / 15686 Simplify: (x4y2)7A g27A x3y5B g12B x1.75y3.5C g6C x11y9D g3D x28y14Slide 143 / 156Slide 144 / 15688 The expression (x2z3)(xy 2z) is equivalent to:A x2y2z3B x3y2z4C x3y3z4D x4y2z5From the Ne w York S ta te Educa tion De pa rtme nt. Office of As s e s s me nt P olicy, De ve lopme nt a ndAdminis tra tion. Inte rne t. Ava ila ble from www.nys e dre ge nts .org/Inte gra te dAlge bra ; a cce s s e d 17, J une , 2011.

Slide 145 / 15689 The expressionis equivalent to:Slide 146 / 15690 When -9 x5 is divided by -3x3, x 0, the quotient isA 3x2B -3x2A 2w5B 2w8C –27x15D 27x8C 20w8D 20w5From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available fromwww.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.From the New York State Education Department. Office of Assessment Policy, Development andAdministration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17,June, 2011.Slide 147 / 15691 Lenny the Lizard's tank has the dimensions b5 by 3c2 by2c3. What is the volume of Larry's tank?A 6b7c3Slide 148 / 15692 A food company which sells beverages likes to useexponents to show the sales of the beverage in a2 days.If the daily sales of the beverage is 5a4, what is the totalsales in a2 days?B 6b5c5A a6C 5b5c5B 5a8D 5b5c6C 5a6D 5a3Slide 149 / 15693 A rectangular backyard is 55 inches long and 53inches wide. Write an expression for the area ofthe backyard as a power of 5.A 515 in2B 85 in2C 258 in2D 5 in82Slide 150 / 15694 Express the volume of a cube with a length of 43units as a power of 4.A 49 units3B 46 units3C 126 units3D 129 units3

Slide 151 / 156Slide 152 / 156Irrational NumbersA number that cannot be expressedas a ratio of integers.A radical of a non perfect square.Glossary &Standardsπ3.14159. 21.41421.e2.71828.Return toTable ofContentsBack toInstructionSlide 153 / 156Slide 154 / 156RadicandRational NumbersA number that can be expressed as afraction.The value inside the radical sign.The value you want to take the root of.9 329 5.3851.14144 1225.3symbol forrationalnumbersBack toBack toInstructionInstructionSlide 155 / 156Slide 156 / 156Real NumbersStandards for Mathematical PracticeAll the numbers that can be found ona number line.MP1 Making sense of problems & persevere in solving them.MP2 Reason abstractly & quantitatively.-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10MP3 Construct viable arguments and critique the reasoning ofothers.MP4 Model with mathematics.MP5 Use appropriate tools strategically.MP6 Attend to precision.MP7 Look for & make use of structure.MP8 Look for & express regularity in repeated reasoning.Back toInstructionClick on each standard to bringyou to an example of how to meetthis standard within the unit.

When approximating square roots of numbers, you need to determine: · Between which two perfect squares it lies (and therefore which 2 square roots). · Which perfect square it is closer to (and therefore which square root). Example: Lies between 100 & 121, closer to 100. So is between 10 & 11, closer to 10. Square Perfect Root Square

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