Name ___________________________ Period __________ Date ___________ RNS3 uc e STUDENT PAGES od REAL NUMBER SYSTEM Student Pages for Packet 3: Operations with Real Numbers ep r RNS3.1 Rational Numbers • Review concepts of experimental and theoretical probability. 1 a b Understand why all quotients of integers ( , b ≠ 0 ) can be written • as terminating or repeating decimals. Learn to change a repeating decimal to an equivalent quotient of integers. ot R • 9 o N RNS3.2 Irrational Numbers • Recognize that there are an infinite number of rational numbers represented on the real number line. • Understand that there are real numbers that are not rational, which are called irrational numbers. • Explore some properties of irrational numbers. 15 RNS3.4 Vocabulary, Skill Builder, and Review 23 Sa m pl e: D RNS3.3 Computing with Real Numbers • Simplify expressions that involve rational numbers. • Simplify expressions that involve square roots. • Make conjectures about the sums or products that result from computing with rational and irrational numbers. Real Number System (Student Packet) RNS3 – SP Operations with Real Numbers WORD BANK (RNS3) Definition or Explanation Picture or Example uc e Word or Phrase integers ep r od irrational numbers natural numbers ot R radical expression pl e: D rational numbers o N radicand real numbers m repeating decimal Sa terminating decimal whole numbers Real Number System (Student Packet) RNS3 – SP0 Operations with Real Numbers 3.1 Rational Numbers RATIONAL NUMBERS Set (Goals) We will deepen our understanding of rational numbers. We will learn that all rational numbers have decimal expansions that terminate or repeat. We will learn to change repeating decimals into equivalent quotients of integers. • Review concepts of experimental and theoretical probability. • Understand why all quotients of integers uc e Ready (Summary) a b od ( , b ≠ 0 ) can be written as terminating ot R Go (Warmup) ep r or repeating decimals. • Learn to change a repeating decimal to an equivalent quotient of integers. Describe each set of numbers using numbers and words. Number Set Numerical Description N 1. Natural numbers Verbal Description pl e: D o 2. Whole numbers 3. Integers m a A rational number is number that can be written as a quotient of integers , b ≠ 0 . b Write the following as quotients of integers to justify that they are rational. Sa 4. 0.7 5. 0.53 Real Number System (Student Packet) 6. -2 7. 1 3 4 RNS3 – SP1 Operations with Real Numbers 3.1 Rational Numbers TERMINATING AND REPEATING DECIMALS Examples: 1 = 0.5000… = 0.5; 2 uc e A terminating decimal is a decimal whose digits are 0 from some point on. The final 0’s in the expression for a terminating decimal are usually omitted. 3 = 0.75000… = 0.75 4 ep r 2 2 = 0.181818... = 0.18 Examples: ; = 0.222222... = 0.2 11 9 od A repeating decimal is a decimal that ends with repetitions of the same pattern of digits. (A “repeat bar” can be placed above the digits that repeat.) ot R Write each quotient of integers as an equivalent decimal. Use a calculator or scratch paper if doing long division. 3 5 2. 5 8 3. 17 40 4. 2 3 5. 5 6 6. 4 9 7. 3 11 8. 4 11 9. List the fractions above that can be represented by terminating decimals. pl e: D o N 1. 10. List the fractions above that can be represented by repeating decimals. m 11. List the fractions above that are NOT rational numbers. Explain. Sa 12. Predict the decimal values for 13. Predict the decimal values for Real Number System (Student Packet) 5 6 and . 9 9 5 11 and 6 . 11 RNS3 – SP2 Operations with Real Numbers 3.1 Rational Numbers ROLL A FRACTION: EXPERIMENTAL uc e Terry and Robin are playing a game called “Roll a Fraction.” In this game, they roll two six-sided number cubes labeled 1-6 and a fraction less than or equal to 1 is formed from the values on the two number cubes. If the fraction results in a terminating decimal, Terry gets a point If the fraction results in a repeating decimal, Robin gets a point. Numbers Rolled Fraction Formed Winner Trial # Fraction Formed Winner 11 2 12 3 13 ot R 1 4 14 5 15 6 16 17 N 7 8 pl e: D o 9 10 Numbers Rolled ep r Trial # od 1. With a partner, designate one player to be Terry the Terminator and the other player to be Robin the Repeater. Then, roll the cubes 20 times with your partner and record the results in the table. 18 19 20 My Pair’s Game Data (do this now) Number of Wins Proportion Percentage of Wins of Wins Class’ Game Data (do this later) Number of Wins Proportion Percentage of Wins of Wins m Robin (repeating) Terry (terminating) Sa 2. Based on “My Pair’s Game Data” results, which represents your experimental probability, do you think this is a fair game? Explain. 3. If you rolled the cubes 1,000 times instead of 20, how many times would you expect Terry to win? Real Number System (Student Packet) RNS3 – SP3 Operations with Real Numbers 3.1 Rational Numbers ROLL A FRACTION – THEORETICAL _______ Number Cube 1 3 4 1 →R 3 1 →T 1 2 6 ep r 3 5 od _______ Number Cube 1 2 uc e 1. Make an outcome grid to determine the theoretical probabilities of Terry the Terminator winning and of Robin the Repeater winning. 4 ot R 5 6 N 2. Determine the theoretical probabilities of wins for Terry and Robin. P (Terry wins) = __________ = __________ = __________ Decimal o Fraction Percent P (Robin wins) = __________ = __________ = __________ Decimal Percent pl e: D Fraction 3. Based on the theoretical probabilities, do you think that this is a fair game? Explain. 4. Based on the theoretical probabilities, out of 1,000 rolls, how many times can we expect Terry to win? Sa m 5. Go back to “My Pair’s Game Data” on the previous page. How does this experimental probability compare to the theoretical probability calculated on this page? Explain. 6. Combine your individual data with others in the class and record on the previous page to arrive at “Class’ Game Data”. How does this experimental probability compare to the theoretical probability calculated on this page? Explain. Real Number System (Student Packet) RNS3 – SP4 Operations with Real Numbers 3.1 Rational Numbers SOME FRACTION-DECIMAL EQUIVALENTS 1 3 1 7 1 4 1 5 1 9 1 8 1 6 od 1 2 o 1 11 1 12 m pl e: D 1 10 N ot R ep r 1 1 uc e 1. Write decimal equivalents for these unit fractions. Use any combination of your own previous knowledge, number sense, or the long division algorithm. Sa 2. Are there any unit fractions above whose decimal expansions do not terminate or repeat? Explain. Real Number System (Student Packet) RNS3 – SP5 Operations with Real Numbers 3.1 Rational Numbers EXPLORING REPEATING DECIMALS that uc e 1. From the previous page you found 1 7 = __________________. 3 7 4 7 5 7 ep r 2 7 od Your teacher will ask you to use long division to change one of these fractions into an equivalent decimal. 3 = 7 4 = 7 5 = 7 N 2 = 7 ot R 2. Compare decimal expansions with your classmates. Record all decimal expansions here. pl e: D o 3. What do you notice about the sequence of digits for the decimal expansions for 7ths? 4. Predict the decimal expansion for 6 . Check your prediction with a calculator. 7 6 = 7 Sa m 5. Using the long division work as an example, explain why decimal expansions for 7ths must repeat from some point on. Real Number System (Student Packet) RNS3 – SP6 Operations with Real Numbers 3.1 Rational Numbers EXPLORING REPEATING DECIMALS (continued) 6. From a previous page you found that uc e 1 = ________________________. 12 3 12 4 12 5 12 2 = 12 3 = 12 4 = 12 5 = 12 6 will have an equivalent decimal expansion that repeats from some 12 N 8. Do you think o point on? Explain. 7 will have an equivalent decimal expansion that repeats from some 12 pl e: D 9. Do you think ot R 7. Compare decimal expansions with your classmates. Record all decimal expansions here. ep r 2 12 od Your teacher will ask you to use long division to change one of these fractions into an equivalent decimal. point on? Explain. m 10. Choose one example of a decimal expansion for 12ths that repeats. Without performing long division forever, how do you know it must repeat? Sa 11. Do you think any quotient of integers whose decimal expansions does not terminate must repeat from some point on? Use the long division process to help you explain. Real Number System (Student Packet) RNS3 – SP7 Operations with Real Numbers 3.1 Rational Numbers A CLEVER PROCEDURE uc e The following algebraic process is used to change a repeating decimal to a quotient of integers. Change 0.16 = 0.166666... . to a quotient of integers. 10 x = 1.66666... Let x = 0.16666... od (2) (1) 9 x = 1.5 = x (3) 1.5 15 1 = = 9 90 6 ep r Notice that step 2 is above step 1. • The “trick” is to multiply both sides of the equation in step 1 by a power of 10 that will “line up” the repeating portion of the decimal. • Subtract the expressions in step 1 from step 2. This results in a terminating decimal (step 3). • Solve for x and simplify your result into a quotient of integers (step 4). (4) 1. ot R Use this clever procedure to change each repeating decimal to a quotient of integers. 0.4 = 0.44444... 2. ____ digit(s) repeat(s), so multiply by 10. Let x = 0.44444… pl e: D 3. 100x = Let x = 0.272727… (_____) • x = o 9x = x = ____ digit(s) repeat(s), so multiply by ____ N 10x = 0.27 = 0.272727... = x = 9 1.232323… 0.345 ____ digit(s) repeat(s), so multiply by ____ Sa m ____ digit(s) repeat(s), so multiply by ____ 4. Real Number System (Student Packet) RNS3 – SP8 Operations with Real Numbers 3.2 Irrational Numbers IRRATIONAL NUMBERS Set (Goals) • • Recognize that there are an infinite number of rational numbers represented on the real number line. Understand that there are real numbers that are not rational, which are called irrational numbers. Explore some properties of irrational numbers. Go (Warmup) od • ep r We will learn that there are real numbers that are not rational, and that these are the irrational numbers. We will identify some irrational numbers, and learn more about some famous irrational numbers. uc e Ready (Summary) ot R 1. Approximate the placement of the following numbers on the number line. Label the locations with the capital letters. (Hint: 0.5 = 0.50 = 0.500 and 0.51 = 0.510; and you know how to count from 1 to 10) (B) 0.51 (C) 0.505 (D) 0.502 (E) 0.5025 (F) 0.5050505… o pl e: D 0.5 N (A) 0.5 0.51 Change each decimal to an equivalent quotient of integers. 0.51 3. 0.505 4. 0.50505050… Sa m 2. 5. Did all numbers in problem 1 fit on the number line? Explain. Real Number System (Student Packet) RNS3 – SP9 Operations with Real Numbers 3.2 Irrational Numbers RATIONAL NUMBERS ON THE NUMBER LINE uc e Here is the number line from the previous page. Approximate the placement of points A and E on the line. 0.5 1. Complete the chart, and place more numbers on the line. Label the point E G A and G H A and H A and J A and K ot R A and Write this rational number as a decimal ep r Any rational number between points od 0.51 J N K o M pl e: D 2. Do you think it is always possible to find a rational number between two rational numbers? Explain. m 3. Do you think you could list all the rational numbers between point A and point M? Explain. Sa 4. Do you think that the rational numbers will fill up the number line without any spaces or “holes?” In other words, do you think that every point on the line corresponds to a quotient of integers? Real Number System (Student Packet) RNS3 – SP10 Operations with Real Numbers 3.2 Irrational Numbers IRRATIONAL NUMBERS ON THE NUMBER LINE uc e Every rational number has a decimal expansion or “address” and can be represented on the number line. Here is a magnified version of a portion of the line on the previous page. Circle this portion on the previous page. Label the rational numbers on the hash marks. ep r od (Hint: 0.501= 0.5010 and 0.502=0.5020; and you know how to count from 10 to 20) 0.501 0.502 ot R 1. For the number 0.5010010001…., the decimal expansion follows a pattern, but this pattern does not repeat. Estimate the location of this number on the number line as precisely as possible, and label this location N. N 2. Write another number below that has a pattern but does not repeat (like the number at N in problem1). Then estimate its location on the number line and label it P. Be sure to write a number that will fit on this number line. P: __________________________ pl e: D o 3. Why can the numbers in problems 1-2 NOT be written as quotients of integers? In other words, why are they NOT rational? (Hint: show that the clever procedure from the following lesson will not work.) Sa m The above problems indicate that there are also “addresses” on the number line for decimal numbers like those at N and P that are not rational. Every point on the number line that does not represent a rational number represents an irrational number. The Real Number System Together the rational numbers and irrational numbers make up the real numbers. Each real number has a location on the real number line. Every point on the line has a decimal name (address). Real Number System (Student Packet) RNS3 – SP11 Operations with Real Numbers 3.2 Irrational Numbers A FAMOUS IRRATIONAL NUMBER uc e Many civilizations over the centuries have observed that the ratio of the circumference to the diameter of a circle is constant. For example, the Romans observed that the number of paces around the outer portion of their circular temples was about three times the number of paces through the center. In mathematics, the Greek letter π (pronounced “pi”) is used to represent this ratio. 2. Greek: between 3. Hindu: 3,927 1,250 4. Roman: 377 120 5. Chinese: 355 113 22 223 and 7 71 o pl e: D 25 8 ep r 256 81 Use a calculator to find decimal approximations for π (to the nearest ten-thousandth) N 1. Egyptian: π ot R Fraction used as approximation for od There is no quotient of integers that represents the exact ratio of a circle’s circumference to its diameter, or π . Therefore, it is an irrational number. Here are some rational approximations used by different civilizations over the ages. 6. Babylonian: The decimal approximation of π , correct to seven decimal places, is 3.1415926. 7. Round this decimal approximation to the nearest ten-thousandth. Sa m 8. Write the number in problem 7 in words 9. Which civilization(s) had the closest approximation(s) for Real Number System (Student Packet) π? RNS3 – SP12 Operations with Real Numbers 3.2 Irrational Numbers ANOTHER IRRATIONAL NUMBER 1. Use your calculator to try to find a decimal such that If n = 2 Then n2 = 2 2. Were you able to find an exact value of n that will satisfy the equation n2 = 2? Explain. 1 1 low 2 4 high ep r Find n od Is n2 too high or too low? Estimate a decimal value for n 2 uc e You are probably already familiar with other irrational numbers. One example is 2 . ot R 2 is an irrational number because it cannot be written as a quotient of two integers. 1.5 Sa m pl e: D o N For problems 3 and 4, one unit of length is defined as illustrated on the right triangle as well as the number line. Real Number System (Student Packet) 3. Use the Pythagorean Theorem to find the hypotenuse of an isosceles right triangle with leg = 1. 1 1 4. How might you use the diagram above to estimate a location for 2 on this number line? 0 1 2 RNS3 – SP13 Operations with Real Numbers 3.2 Irrational Numbers THE REAL NUMBER SYSTEM ep r od uc e Label the sorting diagram below to show the relationship between the various subsets of the real number system: the natural numbers (N), whole numbers (W), integers (Z), rational numbers (Q), irrational numbers (IR), and real numbers (R). 0.12122122212222… 1, 2, 3, … 1.37562987189673… π ot R 0 -1, -2, -3, … N 1. Label the hash marks on the number line. 5 o pl e: D -5 2. Complete the table and locate each point on the number line. Find a whole number that is not a natural number m a rational number between 1 and 2 Sa a rational number between -5 and -4 Label the point(s) P Q R an irrational number between 3 and 4 V an irrational number between -3 and -2 W Real Number System (Student Packet) Write the number(s) RNS3 – SP14 Operations with Real Numbers 3.3 Computing with Real Numbers COMPUTING WITH REAL NUMBERS Set (Goals) • • Simplify expressions that involve rational numbers. Simplify expressions that contain square roots. Make conjectures about the sums or products that result from computing with rational numbers and irrational numbers. ot R Go (Warmup) od • ep r We will review techniques for computing with integers and rational numbers. We will learn some techniques for simplifying expressions that involve square roots. We will explore the closure property for subsets of real numbers under addition, subtraction, multiplication, or division. uc e Ready (Summary) 1. What is the set of whole numbers?_____________________________________ 2. What is the set of integers? ____________________________________________ -12 + 14 – 26 Is result an integer? pl e: D Is result an integer? 4. (-31)(-3) o 3. N Each expression below consists of integers. Simplify each expression. 45 6 6. Is result an integer? (-6)2 m 9. Is result an integer? Sa 12. 4 – (-23) Is result an integer? Real Number System (Student Packet) 7. -3 + 3 Is result an integer? 10. -62 Is result an integer? 13. −8 + 24 2 Is result an integer? 5. 45 −3 Is result an integer? 8. −4 20 Is result an integer? 11. 14 – 25 Is result an integer? 14. 1+ 3( 2) 9 − ( −5 ) Is result an integer? RNS3 – SP15 Operations with Real Numbers 3.3 Computing with Real Numbers THE CLOSURE PROPERTY uc e A set of numbers is said to be closed, or to have the closure property, under a given operation (such as addition, subtraction, multiplication, or division), if the result of this operation on any numbers in the set is also a number in that set. od Answer each question. If the answer is “yes”, then explain what the statement means in your own words and give an example. If the answer is “no”, then, give a counterexample. The problems in the warmup may help you. ep r 1. Is the set of whole numbers closed under a. Addition? ot R b. Subtraction? c. Multiplication? N d. Division? pl e: D a. Addition? o 2. Is the set of integers closed under b. Subtraction? c. Multiplication? m d. Division? Sa 3. Under what condition is half of an integer an integer? Support your conjecture with examples. Real Number System (Student Packet) RNS3 – SP16 Operations with Real Numbers 3.3 Computing with Real Numbers OPERATIONS WITH RATIONAL NUMBERS uc e 1. What is the set of rational numbers?_____________________________________ Each expression below consists of rational numbers. Simplify each expression. 3. Is result a rational number? 6. Is result a rational number? Is result a rational number? 2 9. 2 − 5 1 18 2 -5 2 Is result a rational number? 10. −5 + 14 2 Is result a rational number? Is result a rational number? o Is result a rational number? Is result a rational number? 7. N 1 2 8. 2 -1 ÷ 3 3 1 of (-25) 2 ep r 2 ÷3 3 5. 4. ot R Is result a rational number? -1 5 − 3 6 od -2 5 + 3 3 2. pl e: D Answer each question. If the answer is “yes”, then explain what the statement means in your own words and give an example. If the answer is “no”, explain the statement and give a counterexample. 11. Is the set of rational numbers closed under a. addition and subtraction? Sa m b. multiplication and division? 12. Is half of a rational number always a rational number? Support your conjecture with examples. Real Number System (Student Packet) RNS3 – SP17 Operations with Real Numbers 3.3 Computing with Real Numbers OPERATIONS WITH IRRATIONAL NUMBERS 1 uc e 1. What is the set of irrational numbers? _____________________________ Irrational numbers may occur as expressions containing square roots. Often we rewrite the expression to preserve its exact value using the multiplication property of square roots. For every number a ≥ 0 and b ≥ 0 , od Multiplication Property of Square Roots ab = a b ( 7) = 25 = ( ) 2 = ____ 3 2 3 ot R = ep r Rewrite each expression by removing any perfect squares from under the radical sign. 2. 3. 4. 75 = 25 3 50 49 = 7 = _________ Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? 5. 6. 7. − 80 16 + 9 pl e: D o N 2 • 200 Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? 8. 9. 10. m 5 • 20 Sa Is result rational or irrational? ( −6) 2 − 4(1)(7) Is result rational or irrational? − 4+9 Is result rational or irrational? 11. Do you think that the irrational numbers are closed under multiplication? Explain. Real Number System (Student Packet) RNS3 – SP18 Operations with Real Numbers 3.3 Computing with Real Numbers OPERATIONS WITH IRRATIONAL NUMBERS 2 Division Property of Square Roots a a = For every number a ≥ 0 and b > 0 , b b uc e Here is another property commonly used to rewrite square roots. ot R = = • 2 ep r od Rewrite each expression by removing any perfect squares from under the radical sign. Note whether the result is a rational or an irrational number. 1. 2. 3. 88 − 6 • • 2 18 = = 11 3 25 • Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? 4. 5. 6. 44 48 75 N − 121 pl e: D o 9 120 Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? 7. 8. 9. 3 12 m 9 Is result rational or irrational? 13 49 Is result rational or irrational? 16 − 421 8 Is result rational or irrational? Sa 10. Do you think that the irrational numbers are closed under division? Explain. Real Number System (Student Packet) RNS3 – SP19 Operations with Real Numbers 3.3 Computing with Real Numbers GENERALIZING RULES Simplify using properties of square roots. 1. 2. 7 8 6 7 3 6 uc e 3. ep r od 73 6 ot R 4. In your own words, generalize a rule for multiplying expressions that contain rational numbers and irrational numbers in the form of square roots. Rewrite using properties of square roots 5. 6. 6 N 4 6 2 4 6 2 8. 6 2 pl e: D o 2 7. Sa m 9. In your own words, generalize a rule for dividing expressions that contain rational numbers and irrational numbers in the form of square roots. Real Number System (Student Packet) RNS3 – SP20 Operations with Real Numbers 3.3 Computing with Real Numbers EVALUATING EXPRESSIONS WITH ROOTS a = 1, b = 6, c = 8 2. a = 2, b = 9, c = -5 3. 4. a = 1, b = 6, c = 7 5. a = 1, b = 2, c = 1 6. a = 3, b = 4, c = 3 a = 2, b = -5, c = -8 ot R ep r 1. uc e b 2 − 4ac if: od Evaluate the expression a = 2, b = 12, c = 10 8. a = 1, b = -6, c = -7 9. a = 1, b = 0, c = -9 12. a = 6, b = 12, c = 0 pl e: D o 7. N − b 2 − 4ac if: Evaluate the expression 2a a = 2, b = -1, c = -4 11. a = 5, b = 2, c = -2 Sa m 10. Real Number System (Student Packet) RNS3 – SP21 Operations with Real Numbers 3.3 Computing with Real Numbers RATIONALIZING THE DENOMINATOR An expression containing square roots is in “simplest radical form” when: • The radicand has no perfect square factors other than 1. • The radicand has no fractions. • The denominator of the fraction does not contain a radical. uc e Simplest Radical Form 2 7 in od Use the process called “rationalizing the denominator” to write the fraction simplest radical form. 2 7 • 7 7 7 7 and write in simplest radical form → ot R 2. Multiply the fraction by ep r 1. Is the denominator rational or irrational? ______________________________ = N What property of arithmetic is being used here? _____________________________ 3. Is the new denominator rational or irrational? _______________________________ pl e: D o Write each expression in simplest radical form. If it already is, then circle the expression. 4. 5. 6. 2 5 11 5 5 Sa m 7. 13 4 Real Number System (Student Packet) 18 8. 20 9. 5 30 6 18 RNS3 – SP22 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review FOCUS ON VOCABULARY (RNS3) b. rational number (Q) c. irrational number d. real number (R) e. natural number (N) f. repeating decimal g. radical expression h. terminating decimal i. j. radicand 2. 25 whole number (W) 3. ot R 1. ep r od a. Integer (Z) 0 5. 0.101010101… 6. 25 0.25 pl e: D o 0.101001000100001… N 4. uc e Choose words from this list. Match each number below with all words that could be used to describe it. 7. 8. “25” in 25 11. m 10. 3 11 11 2 12. 0.33 0.3 Sa - 25 9. Real Number System (Student Packet) RNS3 – SP23 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review 1 - ÷ 6 2 - 3 3. 5 6 1 3 Solve for x: -x + 5 = -2(x – 6) + 5. Solve for x in terms of y: 3x – 2y = 15 N ot R 4. ep r od Compute with rational numbers. 3 1 1 1. + • -1 2. 4 2 4 uc e SKILL BUILDER 1 6. 7. x is less than -3 x is greater than or equal to 2 the opposite of x is less than or equal to -3 m 8. pl e: D o Write each statement using symbols. If the variable is on the right side, change it to the left side using appropriate properties. Then graph each. 1 is greater than x Sa 9. 10. -2 is less than the opposite of x Real Number System (Student Packet) RNS3 – SP24 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review 1 2 x3 2. x2 3. x1 5. x-1 6. x-2 7. x-3 4. x0 8. x-4 ep r 1. od Evaluate each expression if x = uc e SKILL BUILDER 2 Examine your answers to problems 1-8. Explain the following. Why only one has a value equal to 1. ot R 9. 10. Why three have a value less than 1. 12. o N 11. Why four have a value greater than 1. Given two points on a line, (-3, -6) and (3, -2): b. Find the x-intercept. c. Find the slope. d. Write the equation of the line in slopeintercept form. Sa m pl e: D a. Find the y-intercept. Real Number System (Student Packet) RNS3 – SP25 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review SKILLBUILDER 3 Compute with rational numbers. 24 – 52 + 2. 36 -3 5. Rewrite each number as indicated. 192,800 8. 6. N Product of a number between 1 and 10, and a multiple of 10 pl e: D 7. -8 - 2(4 - 6) 4 - 23 3. 5 6 1 3 5 -8 • 1 4 -101 – 99 + 4(-25) Scientific notation o Number 1 1 1 - ÷ + 6 8 4 uc e 3 • 1 4 od 1 2 3 5 ep r 4. − ot R 1. 0.0007 Approximate the square roots. m Number 34 10. 58 Sa 9. Between square roots of perfect squares: Real Number System (Student Packet) Between two consecutive integers: About (as a fraction): About (as a decimal): Calculator check (nearest tenth): RNS3 – SP26 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review SKILLBUILDER 4 Simplify each expression. Leave results in exponent form (x ≠ 0). 35 105 3. 1. 2. 36 10 2 5. (10 ) (10 ) 1 7. 2 3 2 8. uc e od (5 ) 3 x4 x4 6. (4 ) 3 2 2 9. 48 57 (x ) 7 2 x3 pl e: D x3 • x5 x10 o N 10 2 5 ep r 4. (3) ot R 5 6 • 53 5 2 • 511 34 m 10. 11. (2x ) 5 (2x ) 3 5 12. x5 x3 • x 6 15. 18 50 Find the square roots. Sa 13. 1 121 14. Real Number System (Student Packet) 64 9 RNS3 – SP27 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review Write using square root or cube root notation. Then compute. 92 2. −27 1 1 3 3. Write using exponent notation. Then compute. 169 3 125 6. 3 −216 ep r 5. Compute. 7. 9 + 16 10. 20 + 5 25 + 144 1 8 9. 3 12. 2 16 + 3 4 15. 3 2 11. pl e: D 4 81 o N 8. ot R 4. 343 3 od 1 1. uc e SKILL BUILDER 5 3 -1 14. 54 6 ( 3 )( 3 )( 3 ) 3 3 3 8 27 Sa m 13. 4 Real Number System (Student Packet) RNS3 – SP28 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 6 Compute. 13 ) 9+ 9 5. 9− 9 3 8. 64 9. )( 64 7 − 64 ) 3 −64 ep r 7. 3 49 (7 + 6. uc e )( od 13 ot R ( 4. 10. Write these numbers in order from least to greatest. 0.6 2 3 0.67 0.6666 . 0.6 N 0.7 _______, o _______, pl e: D _______, _______, _______, _______ Write each rational number as a quotient of integers. -7 12. 0.23 13. 1 1 3 14. -4 3 4 m 11. Sa Evaluate if x = -3 12. (x)2 Real Number System (Student Packet) 13. (x)-2 14. x3 RNS3 – SP29 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review 1. Write these numbers in order from least to greatest. 0.3 B. 0.3131131113… C. 0.30 D. E. 1 3 F. 0.3333 G. 0.3 H. ______ ______ ______ ______ 1 ______ 3 ______ ______ ep r ______ 0.33 od A. uc e SKILLBUILDER 7 ot R 2. Which of the numbers above are terminating decimals? 3. Which of the numbers above are repeating decimals? N 4. Which of the numbers above are rational numbers? pl e: D o 5. Which of the numbers above are irrational numbers? Sa m 6. Two trains start traveling in the same direction at the same time. The Red train starts 100 miles west of the Yellow train. The Yellow train travels at a rate of 65 miles per hour. The Red train travels at a rate of 50 miles per hour. How long does it take for the Yellow train to catch the Red train? Real Number System (Student Packet) RNS3 – SP30 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review SKILL BUILDER 8 Fill in the blanks. = decimal 2. Are the fraction and decimal values at the bottom of problem 1 equivalent? Explain. uc e 2 3 ______________ ______________ fraction od = decimal ep r + 1 3 decimal ot R 1. Use the clever procedure from Lesson 3.2 to change each repeating decimal into an equivalent quotient of integers. 0.666… 4. 0.4666… pl e: D o N 3. 1.30303030… 6. 0.99999… Sa m 5. Real Number System (Student Packet) RNS3 – SP31 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review SKILLBUILDER 9 2 A. Place a natural number. (example above for A is 2) C. Place an integer that is not a whole number. ep r B. Place a whole number that is not a natural number. od A uc e 1. Use the letter next to each direction when placing the number on the number line. D. Place a rational number that is not an integer ot R E. Place a rational number that can be renamed as a terminating decimal. F. Place a rational number that can be renamed as a repeating decimal. G. Place two fractions between 0 and 1. Find another fraction between them. N H. Place an irrational number between 3 and 4. o 3 to a decimal using a calculator and saw 13 0.2307692308 pl e: D 2. Roger changed Since this number is rational, it must either terminate or repeat. Explain to Roger how to interpret this readout. Sa m 3. In a triangle, the second side is one inch smaller than the first, and the third side is 2 inches larger than the first. The total perimeter of the triangle is 17.5 inches. Find the length of the longest side. Real Number System (Student Packet) RNS3 – SP32 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review SKILLBUILDER 10 uc e Write one example and one counterexample for each statement below. If no example or counterexample is possible for a statement, write “not possible.” One One example Statement counterexample (if possible) (if possible) od 1. The sum of two integers is an integer. ot R 3. The product of two integers is an integer. ep r 2. The difference of two integers is an integer. 4. The quotient of two integers is an integer. N 5. The square root of an integer is an integer o 6. One-fourth of an integer is an integer. pl e: D Simplify each expression by removing any perfect squares from under the radical sign. Note whether the result is a rational or irrational number. 7. 32 m Is result rational or irrational? 3 8 4 Sa 10. Is result rational or irrational? Real Number System (Student Packet) 8. 24 -2 Is result rational or irrational? 11. 4+9 Is result rational or irrational? 9. 3 • 12 Is result rational or irrational? 12. 81+19 Is result rational or irrational? RNS3 – SP33 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review SKILLBUILDER 11 Simplify using properties of square roots. 3 24 -6 3 24 -6 5. 3 27 36 3. 6. 6 • 18 uc e 4. 2. od 5 28 56 7 ot R ep r 1. b 2 − 4ac if: Evaluate the expression -b + a = 1, b = 4, c = 3 8. a = 2, b = -5, c = -7 o N 7. pl e: D Evaluate the expression -b – a = 1, b = -2, c = -3 10. a = 2 , b = 0, c = -6 m 9. b 2 − 4ac if: Sa Write each expression in simplest radical form by rationalizing the denominator. 11. 3 3 Real Number System (Student Packet) 12. 2 3 20 RNS3 – SP34 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review TEST PREPARATION (RNS3 uc e Show your work on a separate sheet of paper and choose the best answers. 1 . 6 1. Choose all phrases that apply to the number is a rational number B. is an irrational number C. can be written as a terminating decimal D. can be written as a repeating decimal B. All rational numbers are integers. All integers are rational numbers. C. All irrational numbers are real numbers ot R A. ep r 2. Choose all statements that are true. od A. D. All integers are irrational numbers. 6•10 B. 6 • 10 C. 2 15 D. 120 2 D. 60 3 pl e: D o A. N 3. Choose all expressions that are equivalent to 60 . 4. Choose all expressions that are equivalent to A. 4 5 3 B. 40 3 2 20 . 3 4 15 3 C. Sa m 5. Which one statement disproves the following: “irrational numbers are closed under multiplication.”? A. C. 9• 2 = 3 2 4 • 25 = 2• 5 = 10 Real Number System (Student Packet) B. 3 • 12 = 3•12 = D. 2• 7 = 36 = 6 2• 7 = 14 RNS3 – SP35 Operations with Real Numbers 3.4 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK (RNS3) 3.1 Rational Numbers 1. State in your own words how you know whether a number is rational. uc e Show your work on a separate sheet of paper and write your answers on this page. 2 5 b. 3 8 c. 5 6 d. 9 11 ep r a. od 2. Write the decimal equivalent of each rational number below. Note whether it repeats or terminates. ot R 3. Change 0.34 to an equivalent quotient of integers. 3.2 Irrational Numbers N 4. Write two different irrational numbers with patterns in the decimal expansions, using dots at the end (…). pl e: D o 5. Using a calculator, write 3 as a decimal to as many places as your calculator shows, and explain why this number is irrational. 3.3 Computing with Real Numbers Simplify using properties of square roots. Rationalize the denominator as needed. 5 • 10 7. 8 44 8. 54 6 9. 18 5 Sa m 6. Real Number System (Student Packet) RNS3 – SP36 Operations with Real Numbers HOME-SCHOOL CONNECTION (RNS1) ep r od uc e 1. Explain what the set of real numbers is, including the difference between rational and irrational numbers. 2. Change each decimal to an equivalent quotient of integers. 0.7 b. 0.7 N ot R a. 8 • 20 b. 56 2 7 Sa m pl e: D a. o 3. Simplify the following using properties of square roots. Parent (or Guardian) signature __________________________________________ Real Number System (Student Packet) RNS3 – SP37 Operations with Real Numbers COMMON CORE STATE STANDARDS – MATHEMATICS STANDARDS FOR MATHEMATICAL CONTENT Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly200 times. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy: Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation: Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event. uc e 7.SP.6 od 7.SP.7a ep r 7.SP.8b Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π 2 ). For example, by truncating the decimal expansion of 2 , show that 2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. 2 Use square root and cube root symbols to represent solutions to equations of the form x = p 3 and x = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions. ot R 8.NS.1 8.NS.2 N 8.EE.2 8.G.7 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. pl e: D o N-RN-3 STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. MP7 Look for and make use of structure. Sa m MP1 MP8 Look for and express regularity in repeated reasoning. First Printing © 2012 Real Number System (Student Packet) DO NOT DUPLICATE RNS3 – SP38

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