-. Plan - Exploring Square Roots and Irrational Numbers Objective To find and estimate square roots and to classify numbers as rational or irrational Examples Finding Square Roots of Perfect Squares Estimating a Square Root Application: Skydiving Classifying Real Numbers Math Understandings: p. 0C Math Background The equation x has two possible solutions: x and x. However, there is only one solution to x, x. By convention, the radical sign,, means the nonnegative square root only. Nonnegative describes all positive numbers and zero. More Math Background p. 0C Lesson Planning and Resources See p. 0E for a list of the resources that support this lesson. 0 Bell Ringer Practice Check Skills You ll Need Use student page, transparency, or PowerPoint. For intervention, direct students to: Powers and Exponents Lesson - Extra Skills and Word Problems Practice, Ch.. Vocabulary Review In a power, the tells how many times a base is used as a factor. exponent Evaluate the expression x for each value of x..... 0 00 Lesson Perfect Squares n n 0 8 0 0 8 00 What You ll Learn Why Learn This? Not every situation can be modeled using the four basic operations. For example, you need square roots to relate the time and distance a skydiver falls. A number that is the square of a whole number is a perfect square. The square root of a number is another number that when multiplied by itself is equal to the given number. 0 Chapter Real Numbers and the Coordinate Plane Special Needs L Students draw a square and a square on grid paper. They count the square units. and Then they try to draw a square with square units. Elicit the fact that some numbers cannot be drawn as perfect whole-number squares. To find and estimate square roots and to classify numbers as rational or irrational New Vocabulary perfect square, square root, irrational numbers, real numbers In the diagram at the right, square tiles form a square with tiles on each side. Since and ( ), has two square roots, and. Since, is a perfect square. Find the two square roots of. Finding Square Roots of Perfect Squares and ( ) The square roots of are and.. Find the square roots of each number. a. b., c. = The symbol means the square root of a number. In this book, means the positive square root, unless stated otherwise. So means the positive square root of, or, and means the opposite of the positive square root of, or. learning style: visual Below Level L Students are asked to notice what the sign should be in multiplications such as the ones below. Then, students identify which can be re-written as a number squared, and rewrite them. ( )( ) positive, ( ) ( )( ) negative learning style: visual
For: Square Roots Activity Use: Interactive Textbook, - For help in using formulas, go to Lesson -, Example. To estimate the square root of a number that is not a perfect square, use the square root of the nearest perfect square. Estimate the value of Estimating a Square Root 8 8 to the nearest integer. Since 8 is closer to than it is to, 8 is closer to than to. You can write 8.. Estimate the value of 8 to the nearest integer. Finding a number s square root is the inverse operation of finding the number s square. So. Application: Skydiving The formula d t represents the approximate distance d in feet a skydiver falls in t seconds before opening the parachute. The formula assumes there is no air resistance. Find the time a skydiver takes to fall 8 feet before opening the parachute. d t 8 t 8 t t t. t d Use the formula for distance and time. d Substitute 8 for d. d Divide each side by to isolate t. d Simplify..8 d Find the positive square root of each side. d Use a calculator. d Round to the nearest tenth. The skydiver takes about. seconds to fall 8 feet.. Find the time a skydiver takes to fall each distance. Round to the nearest tenth of a second. a. 80 ft. s b. ft. s Irrational numbers are numbers that cannot be written in the form where a is any integer and b is any nonzero integer. Rational and irrational numbers form the set of real numbers. a b,. Teach Activity Lab Use before the lesson. Teaching Resources Activity Lab -: Powerful Patterns Guided Instruction Example To help students recognize perfect squares, have them make a table showing the squares of integers from through. Example Remind students that the symbol means approximately equal to. Error Prevention! Students may confuse squaring a number with multiplying a number by. To clarify this, write and on the board. Elicit the fact that the first means which is not the same as. Have students find the values for both expressions, and write them on the board. ; Technology Tip Note that, when presenting Example, on some calculators, taking the square root may be a nd function. This involves first pressing the key and then the key before entering the number. On other calculators, you may first enter the number and then press the key. Have students experiment with finding to see what keystrokes their calculators require. Additional Examples Find the two square roots of 8 and Estimate the value of 0 to the nearest integer. 0 N 8 Advanced Learners L Students find the side of a square with the given area: 8 00 0 learning style: verbal - 0 English Language Learners Students draw a -column table on an index card and label the table Real Numbers. They label the columns Rational Numbers and Irrational Numbers, respectively. Have them provide examples of rational numbers in one column and irrational numbers in the other. learning style: verbal The math class drops a small ball from the top of a stairwell. They measure the distance to the basement as 8 feet. Use the formula d t to find how long it takes the ball to fall. t N. s 0
Guided Instruction Connection to Physics The formula d t in Example is the same for objects of any size and weight. So, in the absence of air resistance, a feather and a hammer fall the same distance in a specified time. This was demonstrated by an astronaut on the moon. Additional Examples Identify each number as rational or irrational. Explain. a.. Rational; the decimal repeats. b. Rational; the ratio is. c. 0 Irrational; 0 is not a perfect square. d..... Irrational; the decimal does not terminate or repeat a group of digits. Teaching Resources Daily Notetaking Guide - Adapted Notetaking - Closure L L Vocabulary Tip The word rational has the word ratio in it. The word irrational means not rational. For help with terminating and repeating decimals, go to Lesson, Example. The diagram below shows the relationships among sets of numbers. Reals Rationals Irrationals Integers Fractions Terminating and repeating decimals The decimal digits of irrational numbers do not terminate or repeat. The decimal digits of p.... do not terminate or repeat, because p is an irrational number. Irrational numbers can also include decimals that have a pattern in their digits, like 0. 0000... For any integer n that is not a perfect square, Classifying Real Numbers Is each number rational or irrational? Explain. n is irrational. a. 0. 888... Irrational; the decimal does not terminate or repeat. b. 0. 8 Rational; the decimal repeats. -, 8, 0, -0 -,, 8,, 0.000000... + p, c. Rational; the number can be written as the ratio d. Irrational; is not a perfect square..,-0., 0.. What is the square root of a given number? A number that when multiplied by itself is equal to the given number. Give several examples of irrational numbers. Sample:,...., Give several examples of rational numbers. Sample:,, 0.. Is 0. rational or irrational? Explain. Rational; the decimal repeats. Check Your Understanding Vocabulary Write all the possible names for each number. Choose from the terms at the right... 0. irrational, real rational, real.. rational, real, rational, real perfect square A. rational number B. irrational number C. real number D. perfect square Find the positive and negative square roots of each number.... 00 00 00 0 0 08 Chapter Real Numbers and the Coordinate Plane 08
Homework Exercises For Exercises See Examples For more exercises, see Extra Skills and Word Problems. Find the square roots of each number.. 0. 00... 00 Estimate the value of each expression to the nearest integer... 0.. 88. 0 0 0. 0. 0 Use s 0 T to estimate the speed of sound s in meters per second for each Celsius temperature T. Round to the nearest integer.. 0 C 0 m/s. 0 C m/s. 0 C. 0 C m/s 0 m/s Is each number rational or irrational? Explain.. See margin.. 0.. 0 0. 000.... 0.. 0. 00000.... Practice Assignment Guide Check Your Understanding Go over Exercises 8 in class before assigning the Homework Exercises. Homework Exercises A Practice by Example B Apply Your Skills C Challenge 0 Test Prep and Mixed Review Homework Quick Check To check students understanding of key skills and concepts, go over Exercises,,,, and. nline Visit: PHSchool.com Web Code: ase-00. Find the closest perfect square to 0, which is. Then take the square root of, which is. a. Yes; the sum of even numbers is an even number. b. Yes; the sum of two irrational numbers is an irrational number. c. No; the sum of two prime numbers can be a composite number.. Guided Problem Solving The area 8 of a square postage stamp is 00 in.. What is the side length of the stamp? What is the formula for the area of a square? How can you use the formula to find the side length of a square?. Boxing The area of a square boxing ring is 8 ft. What is the perimeter of the boxing ring? 88 ft. Geometry A tile is shown at the right. The area of the larger square is in.. Find the area of the smaller square. in.. Open-Ended Give an example of an irrational number that is less than and greater than.. Explain how you know. the number is irrational. 0 in. in. Writing in Math Explain how you can approximate in. in. Answers may vary. Sample: ; is not a perfect square. in. 0. See left.. The Closure Property states that a set of numbers is closed under a given operation if the result of the operation is in the same set of numbers. For example, the set of rational numbers is closed under addition, because the sum of any two rational numbers is a rational number. Is each set of numbers closed under addition? Explain. a. even numbers b. irrational numbers c. prime numbers a c. See left. Adapted Practice - Practice - Find the two square roots of each number.. 8... 8 Find each square root. Round to the nearest tenth if necessary..!0.!8.!.!8 0.!.!0. Identify each number as rational or irrational..!.!. 0... -8 rational rational rational.! 0... 0.000000....!.!0..0 Use s 0! T to estimate the speed of sound s in meters per second for each Celsius temperature T. Round to the nearest integer.. ºC. ºC. ºC 8ºC Find the value of each expression.. "() 0. (!). "(.). "x Estimate the value of each expression to the nearest integer..!.!.!.!.!!0!!0!0 L L..8... rational irrational rational irrational rational irrational rational irrational rational 0. x - Guided Problem Solving GPS Student Page 0, Exercise : Ferris Wheels The formula d."h represents the distance in miles d you can see from h feet above ground. On the London Eye Ferris wheel, you are 0 ft above ground. To the nearest tenth of a mile, how far can you see? Understand. What are you being asked to find? the distance in miles that one can see from 0 ft above ground on the London Eye Ferris Wheel Plan and Carry Out. What is the formula?. What is the height? d."h 0 ft. Substitute known values into the formula. d."0 L lesson quiz, PHSchool.com, Web Code: asa-00. Rational; the decimal terminates.. Irrational; 0 is not a perfect square. Irrational; the decimal does not terminate or repeat. - 0. Rational; is a perfect square. 0. Irrational; is not a perfect square.. Irrational; the decimal does not terminate or repeat.. Simplify using a calculator. Round to the nearest tenth.. mi Check. Use estimation to check your answer. d N. N ( ) ( ) N. N.; It checks. Solve Another Problem. The formula d."h represents the distance in miles d you can see from h feet above ground. At the top of the Ferris wheel at Cedar Point, you are 0 ft above ground. To the nearest tenth of a mile, how far can you see?. mi 0
. Assess & Reteach Find the value of each expression. ( ). ( 0) 0 0. (. ).. ( a) a» Reteaching - The square of is. The square root of is perfect squares because. Á Example: You can use a calculator to find square roots. Find Á and Á to the nearest tenth..8. Á You can estimate square roots like and Á. 0 Á Á Perfect Estimate Á Estimate Á 8 squares Á 8 Á 8 Find each square root. Estimate to the nearest integer if necessary. Use to show that a value is estimated. Á Á8 Á.... Á N N. Á8. Á0. Á00 N 0 Lesson Quiz. Find the two square roots of 00. 0 and 0. Estimate to the nearest integer.. Using d t, find how long it takes a skydiver to fall ft from an airplane.. s. Is rational or irrational? Explain. Rational; it can be 8 written as. Á Á Á8. 0... Á N N 8. Á. Á. Á. N 8 N N. If a whole number is not a perfect square, its square root is an irrational number. List the numbers from exercises that are irrational.!8,!,!8,!0,!8,!,!8,!,!,! Enrichment - Patterns in Numbers Á8 N 0 N Use a calculator or a table of square roots to find the square root of each integer below. Round each answer to the nearest thousandth. The first ten are done for you. N!N N!N N. Use the square roots in the table to find each product. Round the product to the nearest thousandth. a.!!. b.!!.88 c. d.!!. e.!!.8 f.. Look at your answers in Exercise. Compare them to the square roots of other numbers in the table. Describe the pattern you see.. Choose two pairs of two numbers from the table. Multiply to see if your conjecture is true for these numbers. Á!!.!!.0 L!N...0..0..000..8..8.000..000.0... 8.88 8. 8..000..8 0. 0. 0...8.8 Sample answer: The product of the square roots of two integers is equal to the square root of the product of the two integers. Sample answer: " "...0 "!!...000! L For Exercises See Lesson -8 0 Chapter Real Numbers and the Coordinate Plane Test Prep A number that is used as a factor three times is the cube root of the product. Since 8, is the cube root of Find each cube root n.. n. n. n. n 8. The area of a square is What is the length of its side? in.. in.. Ferris Wheels The formula d. h represents the distance in miles d you can see from h feet above ground. On the London Eye Ferris Wheel, you are 0 ft above ground. To the nearest tenth of a mile, how far can you see?. mi Number Sense For what values of n is n a rational number? when n is a perfect square, including 0. Error Analysis A student evaluated the expression and got the answer. What error did the student make? See margin. 0. Challenge Explain how you know that the number,,8,0, cannot be a perfect square. (Hint: What is the units digit?) No integer multiplied by itself ends in. Test Prep and Mixed Review Multiple Choice Resources For additional practice with a variety of test item formats: Test-Taking Strategies, p. Test Prep, p. Test-Taking Strategies with Transparencies Practice. The area of a square is 0 square centimeters. Which best represents the side length of the square? B. cm. cm. cm cm. The diameter of a human hair is about. 0 meters. Which of the following represents this number in standard notation? F 0.0000 0.000,000 0,000. Which problem situation matches the equation x 0? C Jacob travels more than twice as many miles to work as Carrie travels. If Carrie travels 0 miles to work, how many miles x does Jacob travel? Dana s arm is inches longer than Collin s arm. If Dana s arm is 0 inches long, what is twice the length x of Collin s arm? Joel made a $0 phone call to Spain. The call cost $ per minute plus a $ connection fee. How many minutes x did the call last? Alondra invited 0 people to a party. Two people arrived late, and five people could not go. How many people x arrived on time for the party? Write each number in scientific notation.. 8,000 8. 0.,08,000.,00 08. 0. 0 Alternative Assessment Each student in a pair writes an irrational number. Then each partner decides which two whole numbers the other partner s value falls between.. The student took the square root of and added it to the square root of. You must add first and then take the square root.