Report

New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org 8th Grade Math Numerical Roots & Radicals 2012-12-03 www.njctl.org Setting the PowerPoint View Use Normal View for the Interactive Elements To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: • On the View menu, select Normal. • Close the Slides tab on the left. • In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen. • On the View menu, confirm that Ruler is deselected. • On the View tab, click Fit to Window. • On the View tab, click Slide Master | Page Setup. Select On-screen Show (4:3) under Slide sized for and click Close Master View. • On the Slide Show menu, confirm that Resolution is set to 1024x768. Use Slide Show View to Administer Assessment Items To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 8 for an example.) Numerical Roots and Radicals Squares, Square Roots & Perfect Squares Squares of Numbers Greater than 20 Simplifying Perfect Square Radical Expressions Approximating Square Roots Rational & Irrational Numbers Radical Expressions Containing Variables Simplifying Non-Perfect Square Radicands Simplifying Roots of Variables Properties of Exponents Solving Equations with Perfect Square & Cube Roots Common Core Standards: 8.NS.1-2; 8.EE.1-2 Click on topic to go to that section. Squares, Square Roots and Perfect Squares Return to Table of Contents Area of a Square The area of a figure is the number of square units needed to cover the figure. The area of the square below is 16 square units because 16 square units are needed to COVER the figure... Area of a Square The area (A) of a square can be found by squaring its side length, as shown below: 2 A=s Click 2to see if the Aanswer = 4 =found 4 4with formula is =the16Area sq.units correct! The area (A) of a square is labeled as 2 square units, or units , because you cover the figure with squares... 4 units 1 What is the area of a square with sides of 5 inches? A 16 in2 B 20 in2 C 25 in2 D 30 in2 2 What is the area of a square with sides of 6 inches? A 16 in2 B 20 in2 C 24 in2 D 36 in2 3 2 If a square has an area of 9 ft , what is the length of a side? A 2 ft B 2.25 ft C 3 ft D 4.5 ft 4 What is the area of a square with a side length of 16 in? 5 What is the side length of a square with an area of 196 square feet? When you square a number you multiply it by itself. 2 5 = 5 5 = 25 so the square of 5 is 25. You can indicate squaring a number with an exponent of 2, by asking for the square of a number, or by asking for a number squared. What is the square of seven? 49 What is nine squared? 81 Make a list of the numbers 1-15 and then square each of them. Your paper should be set up as follows: Number 1 2 3 (and so on) Square 1 4 Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Square 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 The numbers in the right column are squares of the numbers in the left column. If you want to "undo" squaring a number, you must take the square root of the number. So, the numbers in the left column are the square roots of the numbers in the right column. Square Root 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Square 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 The square root of a number is found by undoing the squaring. The symbol for square root is called a radical sign and it looks like this: Using our list, to find the square root of a number, you find the number in the right hand column and look to the left. So, the 81 = 9 What is 169? Square Root 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Perfect Square 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 When the square root of a number is a whole number, the number is called a perfect square. Since all of the numbers in the right hand column have whole numbers for their square roots, this is a list of the first 15 perfect squares. Find the following. You may refer to your chart if you need to. 6 What is 1 ? 7 What is 81? 8 What is the square of 15 ? 9 What is 256? 10 2 What is 13 ? 11 What is 196? 12 What is the square of 18? 13 What is 11 squared? 14 What is 20 squared? Squares of Numbers Greater than 20 Return to Table of Contents Think about this... What about larger numbers? How do you find ? It helps to know the squares of larger numbers such as the multiples of tens. 2 10 = 100 2 202 = 400 302 = 900 402 = 1600 502 = 2500 602 = 3600 702 = 4900 802 = 6400 90 2= 8100 100 = 10000 What pattern do you notice? For larger numbers, determine between which two multiples of ten the number lies. 2 102 = 100 202 = 400 302 = 900 402 = 1600 50 = 2500 2 602 = 3600 702 = 4900 802 = 6400 90 2= 8100 100 = 10000 2 12 = 1 22 = 4 32 = 9 42 = 16 5 = 25 2 62 = 36 72 = 49 82 = 64 9 2= 81 10 = 100 Next, look at the ones digit to determine the ones digit of your square root. Examples: 2809 Lies between 2500 & 3600 (50 and 60) Ends in nine so square root ends in 3 or 7 Try 53 then 57 2 53 = 2809 7744 Lies between 6400 and 8100 (80 and 90) Ends in 4 so square root ends in 2 or 8 Try 82 then 88 2 82 = 6724 NO! 2 88 = 7744 15 Find. 16 Find. 17 Find. 18 Find. 19 Find. 20 Find. 21 Find. 22 Find. 23 Find. Simplifying Perfect Square Radical Expressions Return to Table of Contents Can you recall the perfect squares from 1 to 400? 2 1 = 2 2 = 2 2 8 = 2 9 = 2 2 15 = 2 16 = 2 3 = 10 = 17 = 2 2 2 4 = 2 5 = 2 6 = 2 7 = 11 = 2 12 = 2 13 = 2= 14 18 = 2 19 = 2 20 = Square Root Of A Number 2 Recall: If b = a, then b is a square root of a. 2 Example: If 4 = 16, then 4 is a square root of 16 What is a square root of 25? 64? 100? 5 8 10 Square Root Of A Number Square roots are written with a radical symbol Positive square root: Negative square root:- =4 = -4 Positive & negative square roots: = 4 Negative numbers have no real square roots no real roots because there is no real number that, when squared, would equal -16. Is there a difference between & Which expression has no real roots? Evaluate the expressions: ? Evaluate the expression is not real 24 25 ? 26 = ? 27 28 = ? 29 A 3 B -3 C No real roots 30 The expression equal to positive integer when b is is equivalent to a A -10 B 64 C 16 D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Square Roots of Fractions a b 16 49 = = b = 4 7 0 Try These 31 A C B D no real solution 32 A C B D no real solution 33 A C B D no real solution 34 A C B D no real solution 35 A C B D no real solution Square Roots of Decimals Recall: To find the square root of a decimal, convert the decimal to a fraction first. Follow your steps for square roots of fractions. = .2 = .05 = .3 36 Evaluate A C B D no real solution 37 Evaluate A .06 B .6 C 6 D No Real Solution 38 Evaluate A .11 B 11 C 1.1 D No Real Solution 39 Evaluate A C .8 B .08 D No Real Solution 40 Evaluate A B C D No Real Solution Approximating Square Roots Return to Table of Contents All 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. A perfect square has a whole number square root. 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? Square Root 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Perfect Square 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 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. Square Root 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Perfect Square 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 When estimating 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: 110 Lies between 100 & 121, closer to 100. So 110 is between 10 & 11, closer to 10. Square Root 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Perfect Square 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 Estimate the following: 30 200 215 Approximating a Square Root Approximate 6 to the nearest integer < < < < Identify perfect squares closest to 38 7 Take square root Answer: Because 38 is closer to 36 than to 49, 7. So, to the nearest integer, =6 is closer to 6 than to Approximate to the nearest integer < < Identify perfect squares closest to 70 < < Take square root Identify nearest integer Another way to think about it is to use a number line. √8 2 2.2 2.1 2.4 2.3 2.6 2.5 2.8 2.7 3.0 2.9 Since 8 is closer to 9 than to 4, √8 is closer to 3 than to 2, so √8 ≈ 2.8 Example: Approximate 10 10.2 10.4 10.6 10.8 11.0 10.1 10.3 10.5 10.7 10.9 41 The square root of 40 falls between which two perfect squares? A 9 and 16 B 25 and 36 C 36 and 49 D 49 and 64 42 Which whole number is 40 closest to? < < Identify perfect squares closest to 40 < < Take square root Identify nearest integer 43 The square root of 110 falls between which two perfect squares? A 36 and 49 B 49 and 64 C 64 and 84 D 100 and 121 44 Estimate to the nearest whole number. 110 45 Estimate to the nearest whole number. 219 46 Estimate to the nearest whole number. 90 47 What is the square root of 400? 48 Approximate to the nearest integer. 49 Approximate 96 to the nearest integer. 50 Approximate 167 to the nearest integer. 51 Approximate 140 to the nearest integer. 52 Approximate 40 to the nearest integer. 53 The expression A 3 and 9 B 8 and 9 C 9 and 10 D 46 and 47 is a number between From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Rational & Irrational Numbers Return to Table of Contents Rational & Irrational Numbers is rational 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: Sort the following numbers. 0 24 25 300 Rational 32 36 625 40 1225 52 64 100 1681 3600 Irrational 200 225 54 Rational or Irrational? A Rational B Irrational 55 Rational or Irrational? A Rational B Irrational 56 Rational or Irrational? A Rational B Irrational 57 Rational or Irrational? A Rational B Irrational 58 Rational or Irrational? A Rational B Irrational 59 Which is a rational number? A B p C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 60 Given the statement: “If x is a rational number, then is irrational.” Which value of x makes the statement false? A B 2 C 3 D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Radical Expressions Containing Variables Return to Table of Contents Square Roots of Variables To take the square root of a variable rewrite its exponent as the square of a power. 12 = (x12)2 = x = (a ) 8 2 8 =a Square Roots of Variables If the square root of a variable raised to an even power has a variable raised to an odd power for an answer, the answer must have absolute value signs. This ensures that the answer will be positive. By Definition... Examples Try These. 5 = |x| 13 = |x| How many of these expressions will need an absolute value sign when simplified? yes yes no no yes yes 61 Simplify A B C D 62 Simplify A B C D 63 Simplify A B C D 64 Simplify A B C D 65 A C B D no real solution Simplifying Non-Perfect Square Radicands Return to Table of Contents What happens when the radicand is not a perfect square? Rewrite the radicand as a product of its largest perfect square factor. Simplify the square root of the perfect square. When simplified form still contains a radical, it is said to be irrational. Try These. Identifying the largest perfect square factor when simplifying radicals will result in the least amount of work. Ex: Not simplified! Keep going! Finding the largest perfect square factor results in less work: Note that the answers are the same for both solution processes 66 Simplify A B C D already in simplified form 67 Simplify A B C D already in simplified form 68 Simplify A B C D already in simplified form 69 Simplify A B C D already in simplified form 70 Simplify A B C D already in simplified form 71 Simplify A B C D already in simplified form 72 Which of the following does not have an irrational simplified form? A B C D Note - If a radical begins with a coefficient before the radicand is simplified, any perfect square that is simplified will be multiplied by the existing coefficient. (multiply the outside) Express in simplest radical form. 73 Simplify A B C D 74 Simplify A B C D 75 Simplify A B C D 76 Simplify A B C D 77 Simplify A B C D 78 When is is written in simplest radical form, the result . What is the value of k? A 20 B 10 C 7 D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 79 When is expressed in simplest form, what is the value of a? A 6 B 2 C 3 D 8 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Simplifying Roots of Variables Return to Table of Contents Simplifying Roots of Variables Remember, when working with square roots, an absolute value sign is needed if: • the power of the given variable is even and • the answer contains a variable raised to an odd power outside the radical Examples of when absolute values are needed: Simplifying Roots of Variables Divide the exponent by 2. The number of times that 2 goes into the exponent becomes the power on the outside of the radical and the remainder is the power of the radicand. Note: Absolute value signs are not needed because the radicand had an odd power to start. Example Simplify Only the y has an odd power on the outside of the radical. The x had an odd power under the radical so no absolute value signs needed. The m's starting power was odd, so it does not require absolute value signs. 80 Simplify A B C D 81 Simplify A B C D 82 Simplify A B C D 83 Simplify A B C D Properties of Exponents Return to Table of Contents Rules of Exponents Materials Exponential Table Questions.pdf Exponential Table.pdf Exponential Test Review.pdf There are handouts that can be used along with this section. They are located under the heading labs on the Exponential page of PMI Algebra. Documents are linked. Click the name above of the document. The Exponential Table x 1 2 3 4 5 6 7 8 1X 2X 3X 4X 5X 6X 7 X 8X 9X 10 X Question 1 x 1X 2X 3X 4X 5X 6X 7X 8X 9X 10X 1 2 16 3 729 4 16 5 6 729 7 8 4 2 Why is 2 equivalent to 4 ? Write the values out in expanded form and see if you can explain why. Question 2 x 1X 2X 3X 4X 5X 6X 7X 8X 9X 10 X 1 2 16 3 64 4 5 1024 6 7 8 16 x 64 = 1024 2 3 5 4 x4 =4 Write the equivalent expressions in expanded form. Attempt to create a rule for multiplying exponents with the same base. Question 3 x 1X 2X 3X 8 27 4X 5X 6X 7X 8X 9X 10 X 1 2 3 216 4 5 6 7 8 8 x 27 = 216 3 3 3 2 x3 =6 Write the equivalent expressions in expanded form. Attempt to create a rule for multiplying exponents with the same power. Question 4 x 1X 2X 3X 4X 5 X 6 X 7 X 8 X 9 X 10 X 1 25 2 3 625 4 5 15625 6 7 8 15625 ÷ 625 = 25 6 4 2 5 ÷5 =5 Write the equivalent expressions in expanded form. Attempt to create a rule for dividing exponents with the same base. A. 1. Explain why each of the following statements is true. 3 2 5 A. 2 x 2 = 2 (2 x 2 x 2) x (2 x 2) = (2 x 2 x 2 x 2 x 2) 4 3 7 3 5 8 B. 3 x 3 = 3 C. 6 x 6 = 6 m n m+n a xa =a B. 1. Explain why each of the following statements is true. 3 3 3 A. 2 x 3 = 6 (2 x 2 x 2) x (3 x 3 x 3) = (2 x 3)(2 x 3)(2 x 3) 3 3 3 B. 5 x 6 = 30 4 4 4 C. 10 x 4 = 40 m m a x b = (ab) m C. 1. Explain why each of the following statements is true. 2 2 2 4 2 2 2 4 A. 4 = (2 ) = 2 B. 9 = (3 ) = 3 2 3 2 6 C. 125 = (5 ) = 5 m n mn (a ) = a D. 1. Explain why each of the following statements is true. A. 5 3 2 3 6 B. 4 5 4 3 = 3 1 = 4 m a C. 10 5 10 5 0 = 5 m-n n a = a Operating with Exponents Examples m n m m m+n m a x b = (ab) mn (a ) = a 2 2 6 2 5 x 3 = 15 3 2 6 (4 ) = 4 5 m a n a 4 3 x3 =3 a xa =a m n 2 m-n =a 3 3 3 =3 2 84 3 5 Simplify: 4 x 4 15 A 4 B 4 C 4 D 47 8 2 85 7 3 Simplify: 5 ÷ 5 2 A 5 B 5 C 5 D 5 4 21 10 86 7 7 Simplify: 4 x 5 A B C D 87 Simplify: A B C D 88 Simplify: A B C D 89 2 3 2 The expression (x z )(xy z) is equivalent to A 2 2 3 xyz B 3 3 4 C xyz D xyz 4 2 5 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 90 Simplify: A B C D 91 Simplify: A B C D simplified 92 The expression is equivalent to 5 A 2w B 2w C 20w D 20w 8 5 8 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 93 2 If x = - 4 and y = 3, what is the value of x - 3y ? A -13 B -23 C -31 D -85 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 94 5 3 When -9 x is divided by -3x , x ≠ 0, the quotient is 2 A –3x B 3x C –27x D 27x 2 15 8 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. By definition: -1 x = , x 0 95 -4 Which expression is equivalent to x ? A 4 B x C -4x D 0 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 96 -3 What is the value of 2 ? A B C -6 D -8 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. 97 -1 2 Which expression is equivalent to x • y ? A B xy2 C D -2 xy From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Solving Equations with Perfect Square and Cube Roots Return to Table of Contents The product of two equal factors is the "square" of the number. The product of three equal factors is the "cube" of the number. When we solve equations, the solution sometimes requires finding a square or cube root of both sides of the equation. When your equation simplifies to: 2 x =# you must find the square root of both sides in order to find the value of x. When your equation simplifies to: 3 x =# you must find the cube root of both sides in order to find the value of x. Example: Solve. Divide each side by the coefficient. Then take the square root of each side. Example: Solve. Multiply each side by nine, then take the cube root of each side. Try These: Solve. ± 10 ±8 ±9 ±7 Try These: Solve. 2 1 4 5 98 Solve. 99 Solve. 100 Solve. 101 Solve. New Jersey Center for Teaching and Learning Progressive Mathematics Initiative (PMI) For additional NJCTL Math content, visit http://njctl.org/courses/math/. Progressive Science Initiative (PSI) For NJCTL Science content, visit http://njctl.org/courses/science/. eInstruction For information about Insight 360™ Classroom Instruction System, visit http://www.einstruction.com. For additional content samples, click here.