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UNIT 1VocabularyPreviewUse the puzzle to preview key vocabulary from this unit. Unscramblethe circled letters to answer the riddle at the bottom of the CEDI4.LAERSEBMNUR5.NIISICFTCEOITANTON1. Has integers as its square roots. (Lesson 1.1)2. Any number that can be written as a ratio of two integers. (Lesson 1.1) Houghton Mifflin Harcourt Publishing Company3. A decimal in which one or more digits repeat infinitely. (Lesson 1.1)4. The set of rational and irrational numbers. (Lesson 1.2)5. A method of writing very large or very small numbers byusing powers of 10. (Lesson 2.2)Q:A:2Vocabulary PreviewWhat keeps a square from moving?!

23. Critique Reasoning The circumference of a circular region is shown.What type of number best describes the diameter of the circle? Explainπ miyour answer.24. Critical Thinking A number is not an integer. What type of numbercan it be?25. A grocery store has a shelf with half-gallon containers of milk. What typeof number best represents the total number of gallons?FOCUS ON HIGHER ORDER THINKINGWork Area26. Explain the Error Katie said, “Negative numbers are integers.” What washer error?128. Draw ConclusionsThe decimal 0.3 represents 3 . What type of numberbest describes 0.9, which is 3 · 0.3? Explain.29. Communicate Mathematical Ideas Irrational numbers can never beprecisely represented in decimal form. Why is this?20Unit 1 Houghton Mifflin Harcourt Publishing Company27. Justify Reasoning Can you ever use a calculator to determine if anumber is rational or irrational? Explain.

Ordering RealNumbersLESSON1.3?8.NS.1.2Use rational approximationsof irrational numbers tocompare the size of irrationalnumbers, locate themapproximately on a numberline diagram, and estimatethe value of expressions(e.g., π2).ESSENTIAL QUESTIONHow do you order a set of real numbers?Comparing Irrational NumbersBetween any two real numbers is another real number. To compare and orderreal numbers, you can approximate irrational numbers as decimals.Math On the SpotEXAMPL 1EXAMPLE8.NS.1.23 5 . Write , , or .Compare 3 5STEP 1STEP 2my.hrw.comFirst approximate 3 . 3 is between 1 and 2.Next approximate 5 . 5 is between 2 and 3.Use perfect squares to estimatesquare roots.12 1 22 4 32 9My NotesThen use your approximations to simplify the expressions. 3 5 is between 6 and 7.3 5 is between 5 and 6. Houghton Mifflin Harcourt Publishing CompanySo, 3 5 3 5 .Reflect1.If 7 5 is equal to 5 plus a number, what do you know about thenumber? Why?2.What are the closest two integers that 300 is between?YOUR TURNPersonalMath TrainerCompare. Write , , or .3. 2 42 44. 12 612 6Online Assessmentand Interventionmy.hrw.comLesson 1.321

Ordering Real NumbersYou can compare and order real numbers and list them from least to greatest.Math On the Spotmy.hrw.comMy NotesEXAMPLE 28.NS.1.2Order 22 , π 1, and 4 12 from least to greatest.STEP 1First approximate 22 . 22is between 4 and 5. Since you don’t know where it fallsbetween 4 and 5, you need to find a better estimate for 22 soyou can compare it to 4 12 .Since 22 is closer to 25 than 16, use squaresof numbers between4.5 and 5 to find a better estimate of 22 .4.52 20.254.62 21.164.72 22.094.82 23.04Since 4.72 22.09, an approximate value for 22 is 4.7.An approximate value of π is 3.14. So an approximate valueof π 1 is 4.14.STEP 2Plot 22 , π 1, and 4 21 on a number line.142π 144.24.4 224.64.85Read the numbers from left to right to place them in order fromleast to greatest.YOUR TURNOrder the numbers from least to greatest. Then graph them on thenumber line.5. 5 , 2.5, 3Math TalkMathematical Practices00.511.522.533.546. π2, 10, 75PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.com22Unit 188.599.5 10 10.5 11 11.5 12If real numbers a, b, and care in order from least togreatest, what is the orderof their opposites fromleast to greatest?Explain. Houghton Mifflin Harcourt Publishing CompanyFrom least to greatest, the numbers are π 1, 4 12 , and 22 .

Ordering Real Numbers ina Real-World ContextCalculations and estimations in the real world may differ. It can be importantto know not only which are the most accurate but which give the greatest orleast values, depending upon the context.EXAMPL 3EXAMPLEMath On the Spotmy.hrw.com8.NS.1.2Four people have found the distance in kilometers across a canyon usingdifferent methods. Their results are given in the table. Order the distancesfrom greatest to least.Distance Across Quarry Canyon (km)JuanaLee AnnRyneJackson 282345.55 12STEP 1Write each value as a decimal. 28 is between 5.2 and 5.3. Since 5.32 28.09, an approximatevalue for 28 is 5.3.23 5.7545.5 is 5.555 , so 5.5 to the nearest hundredth is 5.56.5 12 5.5STEP 223, 5.5, and 5 21 on a number line.Plot 28 ,4 Houghton Mifflin Harcourt Publishing Company 2855.215 2 5.55.45.62345.86From greatest to least, the distances are:23 km, 5.5 km,5 21 km, 28 km.4YOUR TURN7.Four people have found the distance in miles across a crater usingdifferent methods. Their results are given below.10 103 1Jonathan:3 , Elaine: 3.45, José: 2 , Lashonda:Order the distances from greatest to least.PersonalMath TrainerOnline Assessmentand Interventionmy.hrw.comLesson 1.323

Guided PracticeCompare. Write , , or . (Example 1)1. 3 2 3 32. 11 153. 6 56 54. 9 35. 17 - 37. 7 2-2 8 159 5 10 - 1 36. 10 - 812 - 2 17 33 118.9. Order 3 , 2π, and 1.5 from least to greatest. Then graph them on thenumber line. (Example 2) 3 is betweenπ 3.14, so 2π 00.511.5, so 3 and.22.533.544.55From least to greatest, the numbers are5.566.5,7,.?ESSENTIAL QUESTION CHECK-IN11. Explain how to order a set of real numbers.24Unit 1Forest Perimeter (km)Leon 17-2MikaJasonAshleyπ1 2122.55 Houghton Mifflin Harcourt Publishing Company Image Credits: ElenaElisseeva/Alamy Images10. Four people have found the perimeter of a forestusing different methods. Their results are givenin the table. Order their calculations fromgreatest to least. (Example 3)

NameClassDate1.3 Independent PracticePersonalMath Trainer8.NS.1.2my.hrw.comOnlineAssessment andInterventionOrder the numbers from least to greatest.12.14. 8 7 , 2,2 220 , -10, 100 , 11.513. 10 , π, 3.515.9 8 , -3.75, 3,416. Your sister is considering two different shapes for her garden. One is asquare with side lengths of 3.5 meters, and the other is a circle with adiameter of 4 meters.a. Find the area of the square.b. Find the area of the circle.c. Compare your answers from parts a and b. Which garden would giveyour sister the most space to plant? Houghton Mifflin Harcourt Publishing Company17. Winnie measured the length of her father’s ranchfour times and got four different distances.Her measurements are shown in the table.Distance Across Father’s Ranch (km)1235887.3a. To estimate the actual length, Winnie first 60approximated each distance to the nearesthundredth. Then she averaged the fournumbers. Using a calculator, find Winnie’s estimate.47 35b. Winnie’s father estimated the distance across his ranch to be 56 km.How does this distance compare to Winnie’s estimate?Give an example of each type of number.18. a real number between 13 and 1419. an irrational number between 5 and 7Lesson 1.325

20. A teacher asks his students to write the numbers shownin order from least to greatest. Paul thinks the numbersare already in order. Sandra thinks the order should bereversed. Who is right?115 115 ,11 , and 10.562421. Math History There is a famous irrational number called Euler’s number,symbolized with an e. Like π, its decimal form never ends or repeats. Thefirst few digits of e are 2.7182818284.a. Between which two square roots of integers could you find thisnumber?b. Between which two square roots of integers can you find π?FOCUS ON HIGHER ORDER THINKINGWork Area22. Analyze Relationships There are several approximations used for π,22including 3.14 and7 . π is approximately 3.14159265358979 . . .3.1403.1413.1423.143b. Which of the two approximations is a better estimate for π? Explain.c. Find a whole number x so that the ratio113 is a better estimate for πxthan the two given approximations.23. Communicate Mathematical Ideas If a set of six numbers that includeboth rational and irrational numbers is graphed on a number line, what isthe fewest number of distinct points that need to be graphed? Explain.24. Critique Reasoning Jill says that 12.6 is less than 12.63. Explain her error.26Unit 1 Houghton Mifflin Harcourt Publishing Company Image Credits: 3DStock/iStockPhoto.coma. Label π and the two approximations on the number line.

MODULE QUIZReadyPersonalMath Trainer1.1 Rational and Irrational NumbersOnline Assessmentand InterventionWrite each fraction as a decimal or each decimal as a fraction.71.20my.hrw.com2. 1.273. 1 785. x3 34316. x2 100Solve each equation for x.4. x2 817. A square patio has an area of 200 square feet. How long is each sideof the patio to the nearest 0.05?1.2 Sets of Real NumbersWrite all names that apply to each number.1218.† 121π9.210. Tell whether the statement “All integers are rational numbers” is trueor false. Explain your choice. Houghton Mifflin Harcourt Publishing Company1.3 Ordering Real NumbersCompare. Write , , or .11. † 8 38 †3Order the numbers from least to greatest.13. † 99, π2, 9.812. † 5 1114.†5 † 111, 1, 0.225 4ESSENTIAL QUESTION15. How are real numbers used to describe real-world situations?Module 127

MODULE 1 MIXED REVIEWPersonalMath TrainerAssessmentReadiness6. Which of the following is not true?1. The square root of a number is 9. What isthe other square root?B –3C3D 812. A square acre of land is 4,840 square yards.Between which two integers is the lengthof one side?A between 24 and 25 yardsB between 69 and 70 yardsCbetween 242 and 243 yardsD between 695 and 696 yards3. Which of the following is an integer butnot a whole number?A – 9.6B –4C0D 3.74. Which statement is false?A No integers are irrational numbers.B All whole numbers are integers.CNo real numbers are irrationalnumbers.D All integers greater than 0 are wholenumbers.5. Which set of numbers best describes thedisplayed weights on a digital scale thatshows each weight to the nearest halfpound?A π2 2π 4B 3π 9C17‡ 27 3 2D 5 – ‡ 24 13π7. Which number is between ‡ 21 and2?14A3B 2‡ 6C5D π 18. What number is shown on the graph?66.26.4A π 3B ‡ 4 2.56.66.8C‡ 20 27D 6.149. Which is in order from least to greatest?1011A 3.3, , π,431011B , 3.3, , π43C10π,, 11 , 3.33 41011D , π, 3.3,43Mini-Task10. The volume of a cube is given by V x3,where x is the length of an edge of thecube. The area of a square is given byA x2, where x is the length of a side ofthe square. A given cube has a volume of1728 cubic inches.a. Find the length of an edge.b. Find the area of one side of the cube.A whole numbersB rational numbersCreal numbersD integers28c. Find the surface area of the cube.Unit 1d. What is the surface area in square feet? Houghton Mifflin Harcourt Publishing CompanySelected ResponseA –9my.hrw.comOnlineAssessment andIntervention

RealNumbers?MODULE1LESSON 1.1ESSENTIAL QUESTIONRational andIrrational NumbersHow can you usereal numbers to solvereal-world problems?8.NS.1.1, 8.NS.1.2,8.EE.1.2LESSON 1.2Sets of Real Numbers8.NS.1.1LESSON 1.3Ordering RealNumbers Houghton Mifflin Harcourt Publishing Company Image Credits: DanielHershman/Getty Images8.NS.1.2Real-World VideoLiving creatures can be classified into groups. Thesea otter belongs to the kingdom Animalia andclass Mammalia. Numbers can also be classified intomy.hrw.com groups such as rational numbers and integers.my.hrw.commy.hrw.comMath On the SpotAnimated MathPersonal Math TrainerGo digital with yourwrite-in studentedition, accessible onany device.Scan with your smartphone to jump directlyto the online edition,video tutor, and more.Interactively explorekey concepts to seehow math works.Get immediatefeedback and help asyou work throughpractice sets.3

Are YOU Ready?Complete these exercises to review skills you will needfor this module.PersonalMath TrainerFind the Square of a NumberEXAMPLEmy.hrw.comOnlineAssessment andInterventionFind the square of 23.22 2 23 33 3 49Multiply the number by itself.Simplify.Find the square of each number.1. 72. 213. -34. 455. 2.76. - 147. -5.78. 1 25Exponents53 5 5 5 25 5 125Use the base, 5, as a factor 3 times.Multiply from left to right.Simplify each exponential expression.9. 9213. 43( 13 )10. 2411.14. (-1)515. 4.52212. (-7)216. 105Write a Mixed Number as an Improper FractionEXAMPLE2 25 2 2510 25 512 5Write the mixed number as a sum of a whole number anda fraction.Write the whole number as an equivalent fraction with thesame denominator as the fraction in the mixed number.Add the numerators.Write each mixed number as an improper fraction.17. 3 134Unit 118. 1 5819. 2 3720. 5 56 Houghton Mifflin Harcourt Publishing CompanyEXAMPLE

Reading Start-UpVocabularyReview WordsVisualize Vocabularyintegers (enteros) negative numbers(números negativos) positive numbers(números positivos) whole number (númeroentero)Use the words to complete the graphic. You can put morethan one word in each section of the triangle.Integers1, 45, 192Preview Words0, 83, 308-21, -78, -93Understand VocabularyComplete the sentences using the preview words.1. One of the two equal factors of a number is a2. A Houghton Mifflin Harcourt Publishing Company3. Theof a number.has integers as its square roots.cube root (raiz cúbica)irrational numbers (númeroirracional)perfect cube (cuboperfecto)perfect square (cuadradoperfecto)principal square root (raízcuadrada principal)rational number (númeroracional)real numbers (número real)repeating decimal (decimalperiódico)square root (raíz cuadrada)terminating decimal(decimal finito)is the nonnegative square rootActive ReadingLayered Book Before beginning the lessons in thismodule, create a layered book to help you learn theconcepts in this module. Label the flaps “RationalNumbers,” “Irrational Numbers,” “Square Roots,” and“Real Numbers.” As you study each lesson, writeimportant ideas such as vocabulary, models, andsample problems under the appropriate flap.Module 15

MODULE 1Unpacking the StandardsUnderstanding the standards and the vocabulary terms in thestandards will help you know exactly what you are expected tolearn in this module.Know that numbers that arenot rational are called irrational.Understand informally thatevery number has a decimalexpansion; for rational numbersshow that the decimal expansionrepeats eventually, and convert adecimal expansion which repeatseventually into a rational number.Key Vocabularyrational number (númeroracional)A number that can beexpressed as a ratio of twointegers.irrational number (númeroirracional)A number that cannot beexpressed as a ratio of twointegers or as a repeating orterminating decimal.8.NS.1.2Use rational approximations ofirrational numbers to comparethe size of irrational numbers,locate them approximatelyon a number line diagram,and estimate the value ofexpressions (e.g., π2).Visit my.hrw.comto see all FloridaMath Standardsunpacked.my.hrw.com6Unit 1What It Means to YouYou will recognize a number as rational orirrational by looking at its fraction or decimal form.UNPACKING EXAMPLE 8.NS.1.1Classify each number as rational or irrational.0.3 130.25 41These numbers are rational because they can be written as ratiosof integers or as repeating or terminating decimals. 5 2.236067977 π 3.141592654 These numbers are irrational because they cannot be written asratios of integers or as repeating or terminating decimals.What It Means to YouYou will learn to estimate the values of irrational numbers.UNPACKING EXAMPLE 8.NS.1.2Estimate the value of 8.8 is not a perfect square. Find the two perfect squares closest to 8.8 is betweenthe perfectsquares4 and 9.So 8 is between 4 and 9. 8 is between 2 and 3.8 is close to 9, so 8 is close to 3.22.82.852 8.12252.92 8.41 7.84 8 is between 2.8 and 2.9, but closer to 2.8.A good estimate for 8 is 2.8. Houghton Mifflin Harcourt Publishing Company8.NS.1.1

LESSON1.1Rational and IrrationalNumbers8.NS.1.1Know that numbers thatare not rational are calledirrational. Understandinformally that every numberhas a decimal expansion; .Also 8.NS.1.2, 8.EE.1.2?ESSENTIAL QUESTIONHow do you rewrite rational numbers and decimals, take squareroots and cube roots, and approximate irrational numbers?Expressing Rational Numbersas DecimalsA rational number is any number that can be written as a ratio in the form ba ,where a and b are integers and b is not 0. Examples of rational numbers are6 and 0.5.6 can be written as 61 .Math On the Spotmy.hrw.com0.5 can be written as 12 .Every rational number can be written as a terminating decimal or a repeatingdecimal. A terminating decimal, such as 0.5, has a finite number of digits.A repeating decimal has a block of one or more digits that repeat indefinitely.EXAMPL 1EXAMPLE8.NS.1.1Write each fraction as a decimal.My Notes Houghton Mifflin Harcourt Publishing CompanyA 140.25 4 1.00-820-2001 0.254Remember that the fraction bar means “divided by.”Divide the numerator by the denominator.Divide until the remainder is zero, adding zeros afterthe decimal point in the dividend as needed.1— 0.3333333333333.31B 30.333 3 1.000 910 910 911 0.33Divide until the remainder is zero or until the digits inthe quotient begin to repeat.Add zeros after the decimal point in the dividend asneeded.When a decimal has one or more digits that repeatindefinitely, write the decimal with a bar over therepeating digit(s).Lesson 1.17

YOUR TURNPersonalMath TrainerOnline Assessmentand InterventionWrite each fraction as a decimal.51.112. 183. 2 13my.hrw.comExpressing Decimals asRational NumbersYou can express terminating and repeating decimals as rational numbers.Math On the Spotmy.hrw.comEXAMPLE 28.NS.1.1Write each decimal as a fraction in simplest form.My NotesA 0.825The decimal 0.825 means “825 thousandths.” Write this as a fraction.8251000To write “825 thousandths”, put 825 over 1000.Then simplify the fraction.825 2533 1000 2540Divide both the numerator and the denomi

Math Trainer Online Assessment and Intervention Personal my.hrw.com Math On the Spot my.hrw.com Identifying Sets for Real-World Situations Real numbers can be used to represent real-world quantities. Highways have posted speed limit signs that are represented by natural