Real Number System Numbers, Numbers, 1

Real Number SystemNumbers, Numbers, .ACTIVITY1.1SUGGESTED LEARNING STRATEGIES: Activating PriorKnowledge, Graphic Organizer, Quickwrite, Think/Pair/ShareMy NotesYour teacher will use a Venn diagram to help you understand therelationships between different types of numbers including rationalnumbers, integers, whole numbers, and natural numbers. Your teacher will ask for volunteers to name numbers to placeinto the Venn diagram. Listen carefully to the numbers chosen. Try to determine howyour teacher decides where to place each number in the Venndiagram. (See if you can decide where a number will be placedin the Venn diagram before your teacher shows the class.)1. Your goal is to figure out the description for each part of theVenn diagram. Write the numbers in your diagram as yourteacher puts the numbers in the class Venn diagram.WRITING MATHTo show all the numbers in asequence, an ellipsis, whichis three periods in a row (forexample, 1, 2, 3, 4, 5, ), isused. An ellipsis indicates a setof numbers that continues onwithout end. 2010 College Board. All rights reserved.WRITING MATHWhere did these groups of numbers originate? Archaeologicalrecords show that ancient cultures used many different systemsof writing numbers. Most of these cultures had symbols forthe numbers 1, 2, 3, 4, 5, . These numbers are called naturalnumbers, or counting numbers.The set of natural numbers can be written as N {1, 2, 3, 4, 5, }.A set is a collection of numbers or objects. Writing numbers inbrackets and separating them by commas is called set notation.The set of natural numbers isrepresented by the symbol N.ACADEMIC VOCABULARYset notation2. Which name, natural numbers or counting numbers, do youthink is more descriptive? Why?Unit 1 Integers and Rational Numbers3

ACTIVITY 1.1Real Number SystemcontinuedNumbers, Numbers, SUGGESTED LEARNING STRATEGIES: Quickwrite, Think/Pair/ShareMy NotesWRITING MATHThe set of whole numbers isrepresented by the symbol W.Notice that 0 is not in the set of natural numbers. Most ancientcultures did not have a concept of the number 0. The Mayanswere one of the first to use the concept of 0. Adding 0 to the set ofnatural numbers forms the set of whole numbers.3. Consider the set of whole numbers and the set of natural numbers. Describe the relationship between the two sets.4. Use set notation to represent the set of whole numbers.In mathematics, numbers that are opposites are the same distancefrom zero and have different signs. The numbers -3 and 3 areexamples of opposites. On the number line, these two numbers arethe same distance from 0 in opposite directions.3 units–5 –4 –3 –2–13 units012345oppositesMATH TERMS5. Each element in the set of whole numbers has an opposite.Together, all the whole numbers and their opposites form theset of integers.a. What is the opposite of 52?WRITING MATHThe set of integers is represented by the symbol Z. Thissymbol was chosen because inGerman Zahlen is the word fornumbers.b. What is the opposite of –312?c. What is the opposite of 0?6. The set of natural numbers was represented using the notation{1, 2, 3, 4, 5, }. Consider the difference between the set ofnatural numbers and the set of integers.a. How would you describe this difference?b. Write the set of integers using set notation.4SpringBoard Mathematics with MeaningTM Level 2 2010 College Board. All rights reserved.The set of integers is the setcontaining all natural numbers,their opposites, and zero.

Real Number SystemACTIVITY 1.1continuedNumbers, Numbers, SUGGESTED LEARNING STRATEGIES: Graphic Organizer,Think/Pair/Share, Group Presentation, QuickwriteMy NotesVenn diagrams can be used to compare the set of whole numbersand the set of integers.7. Place each number in the appropriate region of the Venn diagram.5, 1, 0, 3, 12, 11, 15, 23, 2, 9Whole numbersWZIntegers8. Write one or more sentences summarizing the results in theVenn diagram in Item 7.CONNECT TO HISTORYThe set of numbers known as integerswas used by ancient Chinesemathematicians more than 2000years ago. They performed computations by manipulating countingrods—short rods approximately10 centimeters long—on a tableor counting board. Red rodsrepresented positive numbers andblack rods represented oppositeor negative numbers.9. Complete this sentence that describes the relationship of thesets in Item 7:Everyis a(n), 2010 College Board. All rights reserved.but not everyis a(n).10. The Venn diagram below also can be used to compare the setof integers and the set of whole numbers.a. Place the names of these sets in the Venn diagram andexplain why you placed them there.Unit 1 Integers and Rational Numbers5

ACTIVITY 1.1Real Number SystemcontinuedNumbers, Numbers, SUGGESTED LEARNING STRATEGIES: Quickwrite,Think/Pair/Share,My Notesb. Compare and contrast the two Venn diagrams. Whichdo you think is a better representation of the relationshipbetween the two sets? Explain your reasoning.WRITING MATHThe set of rational numbersis represented by the symbolQ. The symbol Q was chosenbecause it is the first letter ofquotient.One of the areas in the Venn diagram we used at the beginningof this activity represents the set of rational numbers. The set ofrational numbers is the set of all numbers that can all be written asa ratio of two integers.a,A rational number, x, can be defined using symbols as x bwhere both a and b are integers and b 0. Any rational number maybe represented as a fraction, as shown, or as a decimal.11. Explain why b cannot be 0.12. If rational numbers are defined as fractions, why can arational number be represented by a decimal?A terminating decimal is adecimal that ends.A repeating decimal is adecimal that has one or moredigits following the decimalpoint that repeat endlessly.Some decimals are terminating decimals. To show that thesedecimals are rational numbers, it must be possible to express thesenumbers as the ratio of two integers in fraction form.13. Write each decimal as a ratio of two integers. Express theanswer as a fraction.a. 0.35b. 0.00414. Can all terminating decimals be written as fractions? Justifyyour answer.6SpringBoard Mathematics with MeaningTM Level 2 2010 College Board. All rights reserved.MATH TERMS

Real Number SystemACTIVITY 1.1continuedNumbers, Numbers, SUGGESTED LEARNING STRATEGIES: Quickwrite, Look for aPattern, Activating Prior KnowledgeMy NotesSome decimals are repeating decimals. To show that these decimalsare rational numbers, it must be possible to express these numbersas the ratio of two integers in fraction form.15. Rewrite each rational number as a decimal. Describe anypatterns you observe with the quotients.1a.34b.9CONNECT TO AP5c.6In advanced math courses you willlearn why a repeating decimal canbe expressed as a fraction.16. Now explore some numbers with nines. Study these equalitiesand describe any patterns that you observe.410.41 993520.352 99980.8 9 2010 College Board. All rights reserved.17. Consider the numbers in Items 15 and 16. Would these beconsidered rational numbers? Why or why not?MATH TERMSA subset of a set is anotherset whose elements are all inthe original set. Every set is asubset of itself.18. Explain why natural numbers, whole numbers, and integersare all subsets of the set of rational numbers?TECHNOLOGYIrrational numbers are numbers that cannot be written as the ratioof two integers. Some examples are decimals that do not terminateor repeat. These includepi, represented by the symbol, π, and thesquare root of 2, 2 . An approximate decimal value of pi is3.1415926535897932384626433832795 Pi is often represented by the rounded decimal, 3.14. Pi cannot22 is close, but is not exact.be expressed as a fraction,7 The approximate decimal value of 2 is1.4142135623730950488016887242097 This decimal also does not terminate or repeat and a fractioncannot be written to equal its value.Look at what your calculatorshows as a value for π. Howis it different from the one onthis page?WRITING MATHA symbol that is sometimesused to representthe irrationalnumbers is Q. This symbolmeans “not rational.”Unit 1 Integers and Rational Numbers7

ACTIVITY 1.1Real Number SystemcontinuedNumbers, Numbers, My NotesSUGGESTED LEARNING STRATEGIES: Quickwrite, Think/Pair/Share, Graphic Organizer, Group Presentation, SelfRevision/Peer Revision19. Are all square roots irrational? Explain using at least twoexamples.The set of real numbers isrepresented by the symbol R.The set of real numbers includes all rational and irrational numbers.20. For each number, place a check in the box for any set of whichthe number is a member.Natural WholeRational Irrational RealNumber Numbers Numbers Integers Numbers Numbers Numbers0.253 30 64-33.140.4π25521. Create a Venn diagram to illustrate the relationship betweenreal numbers, rational numbers, irrational numbers, integers,whole numbers, and natural numbers.8SpringBoard Mathematics with MeaningTM Level 2 2010 College Board. All rights reserved.ACADEMIC VOCABULARY

Real Number SystemACTIVITY 1.1continuedNumbers, Numbers, SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge,Think/Pair/Share, Self Revision/Peer Revision, Identify a Subtask,Quickwrite, Create RepresentationsMy NotesSome properties can be helpful when you are solving equations andevaluating expressions with real numbers.22. Write what you recall about the commutative property. Give anexample using only whole numbers in your explanation.23. Write the meaning of the associative property in your own words.Give an example using only integers in your explanation.24. Two students, Nick and Nathaniel, used the distributive propertyto make mental math easier. Look at each student’s work and explainwhat each student did. 2010 College Board. All rights reserved.Nathaniel: 4 (100 21) 4 100 4 21Nick:3 (108) 3 100 3 825. Which property states that when you add a number and its opposite,the sum is zero? Write a problem using rational numbers that are notintegers to illustrate this property.26. What do the identity properties state and does this hold true for allreal numbers? Give examples to justify your answer.Unit 1 Integers and Rational Numbers9

ACTIVITY 1.1Real Number SystemcontinuedNumbers, Numbers, SUGGESTED LEARNING STRATEGIES: Quickwrite, Think/Pair/ShareMy Notes27. List any properties of real numbers that were used to evaluateeach expression.a. 26 1 26b. 2 (a b) 2a 2bc. 14 30 26 30 14 26d. 2 (13 5) (2 5) 13CHECK YOUR UNDERSTANDINGWrite your answers on notebook paper. Showyour work.Write the property illustrated in each example.Name one number for each description.7. 2(3 6) 6 121. A whole number but not a naturalnumber.8. 2 3 ( 2) 2 ( 2) 32. An integer but not a whole number.3. An integer and a natural number.4. An irrational number.5. A rational number but not an integer.10SpringBoard Mathematics with MeaningTM Level 26. 5 0 59. (2 4) 5 2 ( 4 5 )10. MATHEMATICAL Describe the set ofR E F L E C T I O N numbers with which youare most comfortable and explain why.Describe the set of numbers with whichyou are least comfortable and explain why. 2010 College Board. All rights reserved.28. How does using properties of real numbers make it easier foryou to do mental math?

Explain why natural numbers, whole numbers, and integers are all subsets of the set of rational numbers? Irrational numbers are numbers that cannot be written as the ratio of two integers. Some examples are decimals that do not terminate or repeat. Th ese include pi, represe