8th Grade. Slide 1 / 157. Slide 2 / 157. Slide 3 / 157. The Number System and Mathematical Operations Part 2. Table of Contents

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1 Slide 1 / 157 Slide 2 / 157 8th Grade The Number System and Mathematical Operations Part Table of Contents Slide 3 / 157 Squares of Numbers Greater than 20 Simplifying Perfect Square Radical Expressions pproximating Square Roots Rational & Irrational Numbers Real Numbers Properties of Exponents Glossary & Standards Click on topic to go to that section.

2 Table of Contents Slide 3 () / 157 Squares of Numbers Greater than 20 Simplifying Perfect Square Radical Expressions pproximating Square Vocabulary Roots Words are bolded Rational & Irrational in Numbers the presentation. The text Real Numbers box the word is in is then Properties of Exponents linked to the page at the end Glossary & Standards of the presentation with the word defined on it. Teacher Notes Click on topic to go to that section. Slide 4 / 157 Squares of Numbers Greater than 20 Return to Table of Contents Square Root of Large Numbers Slide 5 / 157 Think about this... What about larger numbers? How do you find?

3 Square Root of Large Numbers Slide 6 / 157 It helps to know the squares of larger numbers such as the multiples of tens = = = = = = = = = = What pattern do you notice? Square Root of Large Numbers Slide 6 () / 157 It helps to know the squares of larger numbers such as the multiples of tens. Short cut: Number in tens place squared,then two zeros are added = = = = = = = = = = & Math Practice What pattern do you notice? MP8 Look for and express regularity in repeated reasoning is addressed with the question on this slide. Square Root of Large Numbers Slide 7 / 157 For larger numbers, determine between which two multiples of ten the number lies = = = = = = = = = = = = = = = = = = = = 100 Next, look at the ones digit to determine the ones digit of your square root.

4 Square Root of Large Numbers Slide 8 / 157 Examples: Lies between 2500 & 3600 (50 and 60) Ends in nine so square root ends in 3 or 7 Try 53 then = 2809 List of Squares Lies between 6400 and 8100 (80 and 90) Ends in 4 so square root ends in 2 or 8 Try 82 then = 6724 NO! 88 2 = Find. Slide 9 / 157 List of Squares 1 Find. Slide 9 () / List of Squares

5 2 Find. Slide 10 / 157 List of Squares 2 Find. Slide 10 () / List of Squares 3 Find. Slide 11 / 157 List of Squares

6 3 Find. Slide 11 () / List of Squares 4 Find. Slide 12 / 157 List of Squares 4 Find. Slide 12 () / List of Squares

7 5 Find. Slide 13 / 157 List of Squares 5 Find. Slide 13 () / List of Squares 6 Find. Slide 14 / 157 List of Squares

8 6 Find. Slide 14 () / List of Squares 7 Find. Slide 15 / 157 List of Squares 7 Find. Slide 15 () / List of Squares

9 8 Find. Slide 16 / 157 List of Squares 8 Find. Slide 16 () / List of Squares 9 Find. Slide 17 / 157 List of Squares

10 9 Find. Slide 17 () / List of Squares Slide 18 / 157 Simplifying Perfect Square Radical Expressions Return to Table of Contents Two Roots for Even Powers If we square 4, we get 16. Slide 19 / 157 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) = 16 ND (4) 2 = (4)(4) = 16 This can be written as 16 = ±4, meaning positive or negative 4. This is not an issue with odd powers, just even powers.

11 Slide 20 / 157 Is there a difference between... Slide 21 / 157 &? Which expression has no real roots? Evaluate the expressions: Is there a difference between... Slide 21 () / 157 Math Practice & MP7: Look for and make use of structure is addressed by the question on Which expression this slide has no real roots? Follow up questions: Why does the "Error" message Evaluate the expressions: come up on your calculator? (if they are using them) Why is the square root of negative 25 not possible??

12 Evaluate the Expressions Slide 22 / 157 is not real 10 Slide 23 / B -6 C is not real 10 Slide 23 () / B -6 C is not real

13 11 Slide 24 / B -9 C is not real 11 Slide 24 () / B -9 C is not real C 12 Slide 25 / B -20 C is not real

14 12 Slide 25 () / B -20 C is not real 13 (Problem from ) Slide 26 / 157 Which student's method is not correct? shley's Method B Brandon's Method On your paper, explain why the method you selected is not correct. 13 (Problem from ) Slide 26 () / 157 Brandon's method works for the Which student's square method root of is not 4, but correct? it would not shley's work Method for the square root of 36. B Brandon's Half of Method 36 is 18, but the square of On your paper, 36 is explain 6 times why 6 equals the method 36. shley you selected is not correct. describes the correct way to find the square root of a number. Brandon's method is not correct.

15 14 Slide 27 / Slide 27 () / Slide 28 / 157

16 15 Slide 28 () / Slide 29 / 157 B C D 16 Slide 29 () / 157 B C D B

17 17 Slide 30 / B -3 C No real roots 17 Slide 30 () / B -3 C No real roots C 18 The expression equal to Slide 31 / 157 is equivalent to a positive integer when b is -10 B 64 C 16 D 4 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 2011.

18 18 The expression equal to Slide 31 () / 157 is equivalent to a positive integer when b is -10 B 64 C 16 D 4 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, Square Roots of Fractions Slide 32 / 157 a b = b = = 4 7 Slide 33 / 157

19 19 Slide 34 / 157 B C D no real solution 19 Slide 34 () / 157 B C D no real solution 20 Slide 35 / 157 B C D no real solution

20 20 Slide 35 () / 157 B C D no real solution C Slide 36 / 157 Slide 36 () / 157

21 22 Slide 37 / 157 B C D no real solution 22 Slide 37 () / 157 B C D no real solution D 23 Slide 38 / 157 B C D no real solution

22 23 Slide 38 () / 157 B C D no real solution C Square Roots of Decimals To find the square root of a decimal, convert the decimal to a fraction first. Follow your steps for square roots of fractions. Slide 39 / 157 =.2 =.05 =.3 24 Evaluate Slide 40 / 157 B C D no real solution

23 24 Evaluate Slide 40 () / 157 B C D no real solution C 25 Evaluate Slide 41 / B.6 C 6 D no real solution 25 Evaluate Slide 41 () / B.6 C 6 D no real solution B

24 26 Evaluate Slide 42 / B 11 C 1.1 D no real solution 26 Evaluate Slide 42 () / B 11 C 1.1 D no real solution 27 Evaluate Slide 43 / B 0.08 C D no real solution

25 27 Evaluate Slide 43 () / B 0.08 C D no real solution B Slide 44 / 157 Slide 44 () / 157

26 Slide 45 / 157 pproximating Square Roots Return to Table of Contents Perfect Square Slide 46 / 157 ll of the examples so far have been from perfect squares. What does it mean to be a perfect square? The square of an integer is a perfect square. perfect square has a whole number square root. Perfect Square Slide 46 () / 157 ll of the examples so far have been from perfect squares. MP6 ttend to precision is addressed by the question on What does it mean to be a perfect square? this slide Math Practice The square of an integer is a perfect square. perfect square has a whole number square root.

27 Non-Perfect Squares Slide 47 / 157 You 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 Squares Slide 48 / 157 Square Perfect Root Square Think about the square root of 50. Where would it be on this chart? What can you say about the square root of 50? 50 is between the perfect squares 49 and 64 but closer to 49. So the square root of 50 is between 7 and 8 but closer to 7. Non-Perfect Squares Square Perfect MP1 Make sense of problems and Root Square persevere at solving them Think about the square root of and 2 4 MP5 Use appropriate tools Where would it be on this chart? 3 9 strategically are addressed on this slide 4 16 What can you say about the square 5 25 If students get confused root of or 50? stuck on 6 36 future problems, ask: 7 49 Which tool would be 50 best is between for solving the perfect squares this problem? and 64 but closer to : Square root chart You can also ask the So questions the square given root of 50 is between on this slide to assist and 8 students. but closer to Math Practice Slide 48 () / 157

28 pproximating Non-Perfect Squares Square Perfect Root Square 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. Slide 49 / 157 pproximating Non-Perfect Squares Slide 50 / 157 Square Perfect Root Square pproximate the following: pproximating Non-Perfect Squares Slide 50 () / 157 Square Perfect Root Square pproximate the following: Lies between 25 & 36 closer to 25 but since almost right in the middle, # 5.5 Lies between 196 & 225 closer to 196 so # 14 Lies between 196 & 225. Closer to 225 so # 15

29 pproximating Non-Perfect Squares Slide 51 / 157 pproximate to the nearest integer < < Identify perfect squares closest to 38 6 < < 7 Take square root : Because 38 is closer to 36 than to 49, 7. So, to the nearest integer, = 6 is closer to 6 than to pproximating Non-Perfect Squares Slide 52 / 157 pproximate to the nearest integer < < Identify perfect squares closest to 70 < < Take square root Identify nearest integer pproximating a Square Root Slide 53 / 157 pproximate on a number line Since 8 is closer to 9 than to 4, is closer to 3 than to 2; so # 2.8

30 pproximating a Square Root Slide 54 / 157 nother way to think about it is to use a number line. Example: pproximate lies between and So, it lies between whole numbers and. Imagine the square roots between these numbers, and picture where lies. pproximating a Square Root Slide 54 () / 157 nother way to think about it is to use a number line. Example: pproximate lies between and So, it lies between whole numbers and. Imagine the square roots between these numbers, and picture where lies. 29 The square root of 40 falls between which two perfect squares? 3 and 4 B 5 and 6 C 6 and 7 D 7 and 8 Slide 55 / 157

31 29 The square root of 40 falls between which two perfect squares? 3 and 4 B 5 and 6 C 6 and 7 D 7 and 8 C Slide 55 () / Which integer is closest to? Slide 56 / 157 < < Identify perfect squares closest to 40 < < Take square root Identify nearest integer 30 Which integer is closest to? Slide 56 () / < < Identify perfect squares closest to 40 < < Take square root Identify nearest integer

32 31 The square root of 110 falls between which two perfect squares? 36 and 49 B 49 and 64 C 64 and 84 D 100 and 121 Slide 57 / The square root of 110 falls between which two perfect squares? 36 and 49 B 49 and 64 C 64 and 84 D 100 and 121 D Slide 57 () / Estimate to the nearest integer. Slide 58 / 157

33 32 Estimate to the nearest integer. Slide 58 () / Select the point on the number line that best approximates the location of. Slide 59 / 157 B C E G D F H From PRCC EOY sample test non-calculator #19 33 Select the point on the number line that best approximates the location of. Slide 59 () / 157 B C E G D F H E From PRCC EOY sample test non-calculator #19

34 34 Estimate to the nearest integer. Slide 60 / Estimate to the nearest integer. Slide 60 () / Estimate to the nearest integer. Slide 61 / 157

35 35 Estimate to the nearest integer. Slide 61 () / pproximate to the nearest integer. Slide 62 / pproximate to the nearest integer. Slide 62 () / 157 5

36 37 pproximate to the nearest integer. Slide 63 / pproximate to the nearest integer. Slide 63 () / pproximate to the nearest integer. Slide 64 / 157

37 38 pproximate to the nearest integer. Slide 64 () / pproximate to the nearest integer. Slide 65 / pproximate to the nearest integer. Slide 65 () /

38 40 pproximate to the nearest integer. Slide 66 / pproximate to the nearest integer. Slide 66 () / The expression is a number between: Slide 67 / and 9 B 8 and 9 C 9 and 10 D 46 and 47 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 2011.

39 41 The expression is a number between: Slide 67 () / and 9 B 8 and 9 C 9 and 10 D 46 and 47 C From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, For what integer x is closest to 6.25? Slide 68 / 157 Derived from 42 For what integer x is closest to 6.25? Slide 68 () / Derived from

40 43 For what integer y is closest to 4.5? Slide 69 / 157 Derived from 43 For what integer y is closest to 4.5? Slide 69 () / Derived from 44 Between which two positive integers does lie? Slide 70 / B 2 C 3 D 4 E 5 F 6 G 7 H 8 I 9 J 10 Derived from

41 44 Between which two positive integers does lie? Slide 70 () / B 2 C 3 D 4 E 5 F 6 G 7 H 8 I 9 J 10 G & H Derived from 45 Between which two positive integers does lie? Slide 71 / B 2 C 3 D 4 E 5 F 6 G 7 H 8 I 9 J 10 Derived from 45 Between which two positive integers does lie? Slide 71 () / B 2 C 3 D 4 E 5 F 6 G 7 H 8 I 9 J 10 H & I Derived from

42 46 Between which two labeled points on the number line would be located? Slide 72 / 157 B C D E F G H I J Derived from 46 Between which two labeled points on the number line would be located? Slide 72 () / 157 B C D E F G H I J B & C Derived from Slide 73 / 157 Rational & Irrational Numbers Return to Table of Contents

43 Irrational Numbers Slide 74 / 157 Just as subtraction led us to zero and negative numbers. nd 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. Irrational Numbers Slide 75 / 157 Irrational Numbers are real numbers that cannot be expressed as a ratio of two integers. In decimal form, they extend forever and never repeat. There are an infinite number of irrational numbers between any two integers (think of all the square roots, cube roots, etc. that don't come out evenly). Then realize you can multiply them by an other number or add any number to them and the result will still be irrational. Rational & Irrational Numbers Slide 76 / 157 is rational. This is because the radicand (number under the radical) is a perfect square. If a radicand is not a perfect square, the root is said to be irrational. Ex:

44 Irrational Numbers Slide 77 / 157 Irrational numbers were first discovered by Hippasus about the about 2500 years ago. He was a Pythagorean, and the discovery was considered a problem since Pythagoreans believed that "ll was Number," by which they meant the natural numbers (1, 2, 3...) and their ratios. Hippasus was trying to find the length of the diagonal of a square with sides of length 1. Instead, he proved, that the answer was not a natural or rational number. For a while this discovery was suppressed since it violated the beliefs of the Pythagorean school. Hippasus had proved that 2 is an irrational number. Irrational Numbers Slide 78 / 157 n infinity of irrational numbers emerge from trying to find the root of a rational number. Hippasus proved that 2 is irrational goes on forever without repeating. Some of its digits are shown on the next slide. Square Root of 2 Here are the first 1000 digits, but you can find the first 10 million digits on the Internet. The numbers go on forever, and never repeat in a pattern: Slide 79 /

45 Roots of Numbers are Often Irrational Slide 80 / 157 Soon, thereafter, it was proved that many numbers have irrational roots. We now know that the roots of most numbers to most powers are irrational. These are called algebraic irrational numbers. In fact, there are many more irrational numbers that rational numbers. Principal Roots Slide 81 / 157 Since you can't write out all the digits of 2, or use a bar to indicate a pattern, the simplest way to write that number is 2. 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 would be called the principal root and would be written 2. The negative value would be written as - 2. lgebraic Irrational Numbers There are an infinite number of irrational numbers. Slide 82 / 157 Here are just a few that fall between -10 and

46 Transcendental Numbers Slide 83 / 157 The other set of irrational numbers are the transcendental numbers. These are also irrational. No matter how many decimals you look at, they never repeat. But these are not the result of solving a polynomial equation with rational 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 real transcendental numbers. Pi Slide 84 / 157 We have learned about Pi in Geometry. It is the ratio of a circle's circumference to its diameter. It is represented by the symbol. Discuss why this is an approximation at your table. Is this number rational or irrational? Pi Slide 84 () / 157 We have learned about Pi in Geometry. It is the ratio of a circle's circumference to its diameter. It is represented by the symbol Pi is an. irrational number because it has a decimal representation that never ends and never settles into a permanent repeating pattern. Super computers have found over 12 trillion digits in pi. Discuss why this is an approximation at your table. Is this number rational or irrational?

47 # is a Transcendental Number The most famous of the transcendental numbers is #. Slide 85 / 157 O B # is the ratio of the circumference to the diameter of a circle. People tried to find the value of that ratio for millennia. Only in the mid 1800's was it proven that there was no rational solution /06/pi-day004.jpg Since # is irrational (it's decimals never repeat) and it is not a solution to a equation...it is transcendental. Some of its digits are on the next page. Slide 86 / Other Transcendental Numbers Slide 87 / 157 There are many more transcendental numbers. nother famous one is "e", which you will study in lgebra II as the base of natural logarithms. In the 1800's Georg Cantor proved there are as many transcendental numbers as real numbers. nd, there are more algebraic irrational numbers than there are rational numbers. The integers and rational numbers with which we feel more comfortable are like islands in a vast ocean of irrational numbers.

48 Slide 88 / 157 Sort by the square root being rational or irrational. 47 Rational or Irrational? Slide 89 / 157 Rational B Irrational 47 Rational or Irrational? Slide 89 () / 157 Rational B Irrational

49 48 Rational or Irrational? Slide 90 / 157 Rational B Irrational 48 Rational or Irrational? Slide 90 () / 157 Rational B Irrational B 49 Rational or Irrational? Slide 91 / Rational B Irrational

50 49 Rational or Irrational? Slide 91 () / Rational B Irrational 50 Rational or Irrational? Slide 92 / 157 Rational B Irrational 50 Rational or Irrational? Slide 92 () / 157 Rational B Irrational

51 51 Rational or Irrational? Slide 93 / Rational B Irrational 51 Rational or Irrational? Slide 93 () / Rational B Irrational 52 Rational or Irrational? Slide 94 / 157 Rational B Irrational

52 52 Rational or Irrational? Slide 94 () / 157 Rational B Irrational B 53 Which is a rational number? Slide 95 / 157 B π C D From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, Which is a rational number? Slide 95 () / 157 B π C D C From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 2011.

53 54 Given the statement: If x is a rational number, then is irrational." Which value of x makes the statement false? Slide 96 / 157 B 2 C 3 D 4 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, Given the statement: If x is a rational number, then is irrational." Which value of x makes the statement false? Slide 96 () / 157 B 2 C 3 D 4 D From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, student made this conjecture and found two examples to support the conjecture. If a rational number is not an integer, then the square root of the rational number is irrational. For example, is irrational and is irrational. (Problem from ) Slide 97 / 157 Provide an example of a non-integer rational number that shows that the conjecture is false.

54 (Problem from ) 55 student made this conjecture and found two examples to support the conjecture. If a rational number is not an integer, then the square root of the rational number is irrational. For example, is irrational and is irrational. Slide 97 () / 157 Provide an example of a non-integer rational number that shows that the conjecture is false = 1.5 1/4 = 1/2 Slide 98 / 157 Real Numbers Return to Table of Contents Real Numbers Slide 99 / 157 Real numbers are numbers that exist on a number line.

55 Real Numbers Slide 100 / 157 The Real Numbers are all the numbers that can be found on a number line. That may seem like all numbers, but later in lgebra II we'll be talking about numbers which are not real numbers, and cannot be found on a number line. s you look at the below number line, you'll see certain numbers indicated that are used to label the line. Those are not all the numbers on the line, they just help us find all the other numbers as well...numbers not shown below What type of number is 25? Select all that apply. Slide 101 / 157 Irrational B Rational C Integer D Whole Number E Natural Number F Real Number 56 What type of number is 25? Select all that apply. Slide 101 () / 157 Irrational B Rational C Integer D Whole Number E Natural Number F Real Number B, C, D, E, F

56 57 What type of number is - 5? Select all that apply. 3 Slide 102 / 157 Irrational B Rational C Integer D Whole Number E Natural Number F Real Number 57 What type of number is - 5? Select all that apply. 3 Slide 102 () / 157 Irrational B Rational C Integer D Whole Number B, F E Natural Number F Real Number 58 What type of number is 8? Select all that apply. Slide 103 / 157 Irrational B Rational C Integer D Whole Number E Natural Number F Real Number

57 58 What type of number is 8? Select all that apply. Slide 103 () / 157 Irrational B Rational C Integer, F D Whole Number E Natural Number F Real Number 59 What type of number is - 64? Select all that apply. Slide 104 / 157 Irrational B Rational C Integer D Whole Number E Natural Number F Real Number 59 What type of number is - 64? Select all that apply. Slide 104 () / 157 Irrational B Rational C Integer D Whole Number E Natural Number B, C, F F Real Number

58 60 What type of number is 2.25? Select all that apply. Slide 105 / 157 Irrational B Rational C Integer D Whole Number E Natural Number F Real Number 60 What type of number is 2.25? Select all that apply. Slide 105 () / 157 Irrational B Rational C Integer D Whole Number B, F E Natural Number F Real Number Slide 106 / 157 Properties of Exponents Return to Table of Contents

59 Powers of Integers Slide 107 / 157 Just as multiplication is repeated addition, Exponents is repeated multiplication. For example, 3 5 read as "3 to the fifth power" = In this case "3" is the base and "5" is the exponent. The base, 3, is multiplied by itself 5 times. Powers of Integers Slide 108 / 157 Make sure when you are evaluating exponents of negative numbers, you keep in mind the meaning of the exponent and the rules of multiplication. For example, (-3) 2 = (-3) (-3) = 9, which is the same as 3 2. However, (-3) 2 = (-3)(-3) = 9, and -3 2 = -(3)(3) = -9 which are not the same. Similarly, 3 3 = = 27 and (-3) 3 = (-3) (-3) (-3) = -27, which are not the same. 61 Evaluate: 4 3 Slide 109 / 157

60 61 Evaluate: 4 3 Slide 109 () / Evaluate: (-2) 7 Slide 110 / Evaluate: (-2) 7 Slide 110 () /

61 63 Evaluate: (-3) 4 Slide 111 / Evaluate: (-3) 4 Slide 111 () / Properties of Exponents Slide 112 / 157 The properties of exponents follow directly from expanding them to look at the repeated multiplication they represent. Don't memorize properties, just work to understand the process by which we find these properties and if you can't recall what to do, just repeat these steps to confirm the property. We'll use 3 as the base in our examples, but the properties hold for any base. We show that with base a and powers b and c. We'll use the facts that: (3 2 )= (3 3) (3 3 ) = (3 3 3) (3 5 ) = ( )

62 Properties of Exponents Slide 113 / 157 We need to develop all the properties of exponents so we can discover one of the inverse operations of raising a number to a power...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 others first. Multiplying with Exponents Slide 114 / 157 When multiplying numbers with the same base, add the exponents. (a b )(a c ) = a (b+c) (3 2 )(3 3 ) = 3 5 (3 3 )(3 3 3) = ( ) (3 2 )(3 3 ) = 3 5 (3 2 )(3 3 ) = 3 (2+3) Multiplying with Exponents Slide 114 () / 157 When multiplying numbers with the same base, add the exponents. MP3 Construct viable arguments and critique (a b )(a c ) = the a (b+c) reasoning of others is address on this slide and the (3 2 )(3 next 3 ) = slide 3 5 Math Practice (3 We 3 )(3 are 3 using 3) = examples (3 3 3 to 3 prove 3) that the properties are true. (3 2 )(3 3 ) = 3 5 (3 2 )(3 3 ) = 3 (2+3)

63 Dividing with Exponents Slide 115 / 157 When dividing numbers with the same base, subtract the exponent of the denominator from that of the numerator. (a b ) (a c ) = a (b-c) (3 3 ) (3 2 ) = 31 (3 3 3) (3 3 ) = 3 (3 3 ) (3 2 ) = 3 (3-2) (3 3 ) (3 2 ) = Simplify: Slide 116 / B 5 3 C 5 6 D Simplify: Slide 116 () / B 5 3 C 5 6 D 5 8 C

64 65 Simplify: B 7 2 C 7 10 D 7 24 Slide 117 / Simplify: Slide 117 () / 157 B 7 2 C 7 10 D 7 24 B Slide 118 / 157

65 Slide 118 () / Simplify: Slide 119 / B 8 3 C 8 9 D Simplify: Slide 119 () / B 8 3 C 8 9 D 8 18 C

66 68 Simplify: 5 2 B 5 3 C 5 6 D 5 8 Slide 120 / Simplify: Slide 120 () / 157 B 5 3 C 5 6 D 5 8 Slide 121 / Simplify: 4 3 (4 5 ) 4 15 B 4 8 C D

67 Slide 121 () / Simplify: 4 3 (4 5 ) 4 15 B C B D 4 7 Slide 122 / Simplify: B C D Slide 122 () / Simplify: B C D D 5 4

68 n Exponent of Zero ny base raised to the power of zero is equal to 1. Slide 123 / 157 a (0) = 1 Based on the multiplication rule: (3 (0) )(3 (3) ) = (3 (3+0) ) (3 (0) )(3 (3) ) = (3 (3) ) ny 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) )= 1 This is not just true for base 3, it's true for all bases. 71 Simplify: 5 0 = Slide 124 / Simplify: 5 0 = Slide 124 () / 157 1

69 72 Simplify: = Slide 125 / Simplify: = Slide 125 () / Simplify: (7)(30 0 ) = Slide 126 / 157

70 73 Simplify: (7)(30 0 ) = Slide 126 () / Negative Exponents negative exponent moves the number from the numerator to denominato and vice versa. (a (-b) )= 1 a b Based on the multiplication rule and zero exponent rules: (3 (-1) )(3 (1) ) = (3 (-1+1) ) (3 (-1) )(3 (1) ) = (3 (0) ) (3 (-1) )(3 (1) ) = 1 1 a b = (a (-b) ) But, any number multiplied by its inverse is 1, so Slide 127 / (3 = (1) ) (3 (-1) )(3 (1) ) = 1 Comparing these two equations, we see that 1 (3 (-1) )= 3 1 Negative Exponents Slide 128 / 157 By definition: x -1 =, x 0

71 74 Simplify the expression to make the exponents positive. 4-2 Slide 129 / B Simplify the expression to make the exponents positive. 4-2 Slide 129 () / B B 75 Simplify the expression to make the exponents positive Slide 130 / B 1 4 2

72 75 Simplify the expression to make the exponents positive Slide 130 () / B Simplify the expression to make the exponents positive. x 3 y -4 Slide 131 / 157 x 3 y 4 B y 4 x 3 x 3 C D y 4 1 x 3 y 4 76 Simplify the expression to make the exponents positive. x 3 y -4 Slide 131 () / 157 x 3 y 4 B y 4 x 3 x 3 C D y 4 1 x 3 y 4 C

73 77 Simplify the expression to make the exponents positive. a -5 b -2 Slide 132 / 157 a 5 b 2 B C D b 2 a 5 a 5 b 2 1 a 5 b 2 77 Simplify the expression to make the exponents positive. a -5 b -2 Slide 132 () / 157 a 5 b 2 B b 2 a 5 C D a 5 b 2 1 a 5 b 2 D 78 Which expression is equivalent to x -4? Slide 133 / 157 B x 4 C -4x D 0 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 2011.

74 78 Which expression is equivalent to x -4? Slide 133 () / 157 B x 4 C -4x D 0 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, What is the value of 2-3? Slide 134 / 157 B C -6 D -8 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, What is the value of 2-3? Slide 134 () / 157 B C -6 D -8 B From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 2011.

75 80 Which expression is equivalent to x -1 y 2? Slide 135 / 157 xy 2 C B D xy -2 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, Which expression is equivalent to x -1 y 2? Slide 135 () / 157 xy 2 C B D xy -2 C From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, a) Write an exponential expression for the area of a rectangle with a length of 7-2 meters and a width of 7-5 meters. Slide 136 / 157 b) Evaluate the expression to find the area of the rectangle. When you finish answering both parts, enter your answer to Part b) in your responder.

76 81 a) Write an exponential expression for the area of a rectangle with a length of 7-2 meters and a width of 7-5 meters. Slide 136 () / 157 a) 7-2 x 7-5 = 7-7 m 2 b) Evaluate the expression to find the area of the rectangle. b) When you finish answering both parts, enter your answer to Part b) in your responder. 82 Which expressions are equivalent to? Select all that apply. Slide 137 / B 3-4 C 3 2 D E F From PRCC EOY sample test non-calculator #13 82 Which expressions are equivalent to? Select all that apply. Slide 137 () / B 3-4 C 3 2 D E F B & E From PRCC EOY sample test non-calculator #13

77 83 Which expressions are equivalent to? Select all that apply. Slide 138 / B 3-3 C 3-10 D E F Which expressions are equivalent to? Select all that apply. Slide 138 () / B 3-3 C 3-10 D E B, D & E F Which expressions are equivalent to? Select all that apply. Slide 139 / 157 B C D E F G From PRCC PB sample test non-calculator #1

78 84 Which expressions are equivalent to? Select all that apply. Slide 139 () / 157 B C D E F, C, F G From PRCC PB sample test non-calculator #1 Raising Exponents to Higher Powers Slide 140 / 157 When raising a number with an exponent to a power, multiply the exponents. (a b ) c = a (bc) (3 2 ) 3 = 3 6 (3 2 )(3 2 )(3 2 ) = 3 (2+2+2) (3 3 )(3 3)(3 3) = ( ) (3 2 ) 3 = Simplify: (2 4 ) 7 Slide 141 / B 2 3 C 2 11 D 2 28

79 85 Simplify: (2 4 ) 7 Slide 141 () / B 2 3 C 2 11 D 2 28 D 86 Simplify: (g 3 ) 9 Slide 142 / 157 g 27 B g 12 C g 6 D g 3 86 Simplify: (g 3 ) 9 Slide 142 () / 157 g 27 B g 12 C g 6 D g 3

80 87 Simplify: (x 4 y 2 ) 7 Slide 143 / 157 x 3 y 5 B x 1.75 y 3.5 C x 11 y 9 D x 28 y Simplify: (x 4 y 2 ) 7 Slide 143 () / 157 x 3 y 5 B x 1.75 y 3.5 C x 11 y 9 D x 28 y 14 D Slide 144 / 157

81 Slide 144 () / The expression (x 2 z 3 )(xy 2 z) is equivalent to: Slide 145 / 157 x 2 y 2 z 3 B x 3 y 2 z 4 C x 3 y 3 z 4 D x 4 y 2 z 5 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, The expression (x 2 z 3 )(xy 2 z) is equivalent to: Slide 145 () / 157 x 2 y 2 z 3 B x 3 y 2 z 4 C x 3 y 3 z 4 D x 4 y 2 z 5 B From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 2011.

82 90 The expression is equivalent to: Slide 146 / 157 2w 5 B 2w 8 C 20w 8 D 20w 5 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, The expression is equivalent to: Slide 146 () / 157 2w 5 B 2w 8 C 20w 8 D 20w 5 D From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, When -9 x 5 is divided by -3x 3, x 0, the quotient is Slide 147 / 157 3x 2 B -3x 2 C 27x 15 D 27x 8 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, 2011.

83 91 When -9 x 5 is divided by -3x 3, x 0, the quotient is Slide 147 () / 157 3x 2 B -3x 2 C 27x 15 D 27x 8 From the New York State Education Department. Office of ssessment Policy, Development and dministration. Internet. vailable from accessed 17, June, Lenny the Lizard's tank has the dimensions b 5 by 3c 2 by 2c 3. What is the volume of Larry's tank? Slide 148 / 157 6b 7 c 3 B 6b 5 c 5 C 5b 5 c 5 D 5b 5 c 6 92 Lenny the Lizard's tank has the dimensions b 5 by 3c 2 by 2c 3. What is the volume of Larry's tank? 6b 7 c 3 B 6b 5 c 5 & Math Practice C 5b 5 c 5 B b 5 3c 2 2c 3 = 6b 5 c 5 MP.2: Reasoning quantitatively and abstractly. MP.4: Model with mathematics. D 5b 5 c 6 sk: How can you represent the problems with symbols and numbers? What connections do you see? Write a number sentence to describe this situation. Slide 148 () / 157

84 93 food company which sells beverages likes to use exponents to show the sales of the beverage in a 2 days. If the daily sales of the beverage is 5a 4, what is the total sales in a 2 days? Slide 149 / 157 a 6 B 5a 8 C 5a 6 D 5a 3 93 food company which sells beverages likes to use exponents to show the sales of the beverage in a 2 days. If the daily sales of the beverage C is 5a 4, what is the total sales in a 2 days? 5a 4 a 2 = 5a 6 a 6 & Math Practice B 5a 8 C 5a 6 D 5a 3 MP.2: Reasoning quantitatively and abstractly. MP.4: Model with mathematics. sk: How can you represent the problems with symbols and numbers? What connections do you see? Write a number sentence to describe this situation. Slide 149 () / rectangular backyard is 5 5 inches long and 5 3 inches wide. Write an expression for the area of the backyard as a power of 5. Slide 150 / in 2 B 8 5 in 2 C 25 8 in 2 D 5 8 in 2

85 94 rectangular backyard is 5 5 inches long and 5 3 inches wide. Write an expression for the area of the backyard as a power of 5. D Slide 150 () / 157 MP.2: Reasoning quantitatively and 5 15 in 2 abstractly. & Math Practice MP.4: Model with mathematics. B 8 5 in 2 sk: How can you represent the C 25 8 problems in 2 with symbols and numbers? What connections do you see? Write a D 5 8 in 2 number sentence to describe this situation. 95 Express the volume of a cube with a length of 4 3 units as a power of 4. Slide 151 / units 3 B 4 6 units 3 C 12 6 units 3 D 12 9 units 3 95 Express the volume of a cube with a length of 4 3 units as a power of units 3 MP.2: Reasoning quantitatively and B 4 6 units 3 abstractly. MP.4: Model with mathematics. & Math Practice C 12 6 units sk: 3 How can you represent the problems with symbols and numbers? D 12 9 units What 3 connections do you see? Write a number sentence to describe this situation. Slide 151 () / 157

86 Slide 152 / 157 Glossary & Standards Return to Table of Contents Slide 152 () / 157 Teacher Notes Vocabulary Words are bolded in Glossary the presentation. & The text box the word is in is then Standards linked to the page at the end of the presentation with the word defined on it. Return to Table of Contents Irrational Numbers number that cannot be expressed as a ratio of integers. radical of a non perfect square. Slide 153 / π e Back to Instruction

87 Radicand Slide 154 / 157 The value inside the radical sign. The value you want to take the root of. 9 = 3 29 = = 12 Back to Instruction Rational Numbers number that can be expressed as a fraction. Slide 155 / symbol for rational numbers Back to Instruction Real Numbers ll the numbers that can be found on a number line. Slide 156 / Back to Instruction

88 Standards for Mathematical Practice Slide 157 / 157 MP1 Making sense of problems & persevere in solving them. MP2 Reason abstractly & quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 ttend to precision. MP7 Look for & make use of structure. MP8 Look for & express regularity in repeated reasoning. Click on each standard to bring you to an example of how to meet this standard within the unit.