Name Period Date. REAL NUMBER SYSTEM Student Pages for Packet 3: Operations with Real Numbers

Transcription

1 Name Period Date REAL NUMBER SYSTEM Student Pages for Packet : Operations with Real Numbers RNS. Rational Numbers Review concepts of experimental and theoretical probability. a 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. RNS. 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. RNS. 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. b RNS STUDENT PAGES RNS.4 Vocabulary, Skill Builder, and Review 9 5 RNS SP

2 WORD BANK (RNS) Word or Phrase Definition or Explanation Picture or Example integers irrational numbers natural numbers radical expression radicand rational numbers real numbers repeating decimal terminating decimal whole numbers RNS SP0

3 . Rational Numbers Ready (Summary) 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. RATIONAL NUMBERS Go (Warmup) Set (Goals) Review concepts of experimental and theoretical probability. Understand why all quotients of integers a (, b 0 b Describe each set of numbers using numbers and words. ) can be written as terminating or repeating decimals. Learn to change a repeating decimal to an equivalent quotient of integers. Number Set Numerical Description Verbal Description. Natural numbers. Whole numbers. Integers 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 RNS SP

4 . Rational Numbers TERMINATING AND REPEATING DECIMALS 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. Examples: = = 0.5; = = 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.) Examples: = 0... = 0. ; = = Write each quotient of integers as an equivalent decimal. Use a calculator or scratch paper if doing long division List the fractions above that can be represented by terminating decimals. 0. List the fractions above that can be represented by repeating decimals.. List the fractions above that are NOT rational numbers. Explain.. Predict the decimal values for 5 and Predict the decimal values for and RNS SP

5 . Rational Numbers ROLL A FRACTION: EXPERIMENTAL Terry and Robin are playing a game called Roll a Fraction. In this game, they roll two six-sided number cubes labeled -6 and a fraction less than or equal to 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.. With a partner, designate one player to be Terry the Terminator and the other player to be Robin the Repeater. Then, roll the cubes 0 times with your partner and record the results in the table. Trial # Numbers Fraction Rolled Formed Winner Trial # Robin (repeating) Terry (terminating) My Pair s Game Data (do this now) Number of Wins Proportion of Wins Percentage of Wins Numbers Rolled Fraction Formed Winner Class Game Data (do this later) Number of Wins Proportion of Wins. Based on My Pair s Game Data results, which represents your experimental probability, do you think this is a fair game? Explain. Percentage of Wins. If you rolled the cubes,000 times instead of 0, how many times would you expect Terry to win? RNS SP

6 . Rational Numbers ROLL A FRACTION THEORETICAL. Make an outcome grid to determine the theoretical probabilities of Terry the Terminator winning and of Robin the Repeater winning. Number Cube Number Cube T R. Determine the theoretical probabilities of wins for Terry and Robin. P (Terry wins) = = = Fraction Decimal Percent P (Robin wins) = = = Fraction Decimal Percent. Based on the theoretical probabilities, do you think that this is a fair game? Explain. 4. Based on the theoretical probabilities, out of,000 rolls, how many times can we expect Terry to win? 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. RNS SP4

7 . Rational Numbers SOME FRACTION-DECIMAL EQUIVALENTS. Write decimal equivalents for these unit fractions. Use any combination of your own previous knowledge, number sense, or the long division algorithm Are there any unit fractions above whose decimal expansions do not terminate or repeat? Explain RNS SP5

8 . Rational Numbers EXPLORING REPEATING DECIMALS. From the previous page you found that 7 =. Your teacher will ask you to use long division to change one of these fractions into an equivalent decimal Compare decimal expansions with your classmates. Record all decimal expansions here. = = = 7 7 =. 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 5. Using the long division work as an example, explain why decimal expansions for 7ths must repeat from some point on. RNS SP6

9 . Rational Numbers EXPLORING REPEATING DECIMALS (continued) 6. From a previous page you found that =. Your teacher will ask you to use long division to change one of these fractions into an equivalent decimal Compare decimal expansions with your classmates. Record all decimal expansions here. = = 4 5 = = 6 8. Do you think will have an equivalent decimal expansion that repeats from some point on? Explain Do you think will have an equivalent decimal expansion that repeats from some point on? Explain. 0. Choose one example of a decimal expansion for ths that repeats. Without performing long division forever, how do you know it must repeat?. 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. RNS SP7

10 . Rational Numbers A CLEVER PROCEDURE The following algebraic process is used to change a repeating decimal to a quotient of integers. Change 0. 6 = to a quotient of integers. Notice that step is above step. The trick is to multiply both sides of the equation in step by a power of 0 that will line up the repeating portion of the decimal. Subtract the expressions in step from step. This results in a terminating decimal (step ). Solve for x and simplify your result into a quotient of integers (step 4). 0x = () Let x = () 9x =. 5 (). 5 5 x = = = Use this clever procedure to change each repeating decimal to a quotient of integers = digit(s) repeat(s), so multiply by 0. 0x = Let x = x = x =.. 9 digit(s) repeat(s), so multiply by = (4) digit(s) repeat(s), so multiply by 00x = Let x = ( ) x = x = = digit(s) repeat(s), so multiply by RNS SP8

11 . Irrational Numbers Ready (Summary) 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. IRRATIONAL NUMBERS Go (Warmup) 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.. Approximate the placement of the following numbers on the number line. Label the locations with the capital letters. (Hint: 0.5 = 0.50 = and 0.5 = 0.50; and you know how to count from to 0) (A) 0.5 (B) 0.5 (C) (D) 0.50 (E) (F) Change each decimal to an equivalent quotient of integers Did all numbers in problem fit on the number line? Explain. RNS SP9

12 . Irrational Numbers RATIONAL NUMBERS ON THE NUMBER LINE Here is the number line from the previous page. Approximate the placement of points A and E on the line Complete the chart, and place more numbers on the line. Any rational number between points Label the point A and E G A and G H A and H J A and J K A and K M Write this rational number as a decimal. Do you think it is always possible to find a rational number between two rational numbers? Explain.. Do you think you could list all the rational numbers between point A and point M? Explain. 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? RNS SP0

13 . Irrational Numbers IRRATIONAL NUMBERS ON THE NUMBER LINE 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. (Hint: 0.50= and 0.50=0.500; and you know how to count from 0 to 0) For the number , 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.. Write another number below that has a pattern but does not repeat (like the number at N in problem). 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:. Why can the numbers in problems - 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.) 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). RNS SP

14 . Irrational Numbers A FAMOUS IRRATIONAL NUMBER 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. 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. Fraction used as approximation for π. Egyptian: Greek: between 7. Hindu: 4. Roman: 5. Chinese: 6. Babylonian:,97, and 7 Use a calculator to find decimal approximations for π (to the nearest ten-thousandth) The decimal approximation of π, correct to seven decimal places, is Round this decimal approximation to the nearest ten-thousandth. 8. Write the number in problem 7 in words 9. Which civilization(s) had the closest approximation(s) for π? RNS SP

15 . Irrational Numbers ANOTHER IRRATIONAL NUMBER You are probably already familiar with other irrational numbers. One example is.. Use your calculator to try to find a decimal such that If n = Then n = Estimate a decimal value for n Is n Find n too high or too low? low 4 high.5. Were you able to find an exact value of n that will satisfy the equation n =? Explain. is an irrational number because it cannot be written as a quotient of two integers. For problems and 4, one unit of length is defined as illustrated on the right triangle as well as the number line.. Use the Pythagorean Theorem to find the hypotenuse of an isosceles right triangle with leg =. 4. How might you use the diagram above to estimate a location for on this number line? 0 RNS SP

16 . Irrational Numbers THE REAL NUMBER SYSTEM 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).,,,. Label the hash marks on the number line.. Complete the table and locate each point on the number line. Find Label the point(s) Write the number(s) a whole number that is not a natural number P a rational number between and a rational number between -5 and -4 an irrational number between and 4 an irrational number between - and , -, -, Q R V W π RNS SP4

17 . Computing with Real Numbers COMPUTING WITH REAL NUMBERS Ready (Summary) 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. Go (Warmup) 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.. What is the set of whole numbers?. What is the set of integers? Each expression below consists of integers. Simplify each expression Is result an integer? Is result an integer? 9. (-6) Is result an integer?. 4 (-) 4. (-)(-) Is result an integer? Is result an integer? Is result an integer? Is result an integer? Is result an integer?. 4 5 Is result an integer? Is result an integer? Is result an integer? Is result an integer? 4. + ( ) 9 ( 5) RNS SP5

18 . Computing with Real Numbers THE CLOSURE PROPERTY 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. 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.. Is the set of whole numbers closed under a. Addition? b. Subtraction? c. Multiplication? d. Division?. Is the set of integers closed under a. Addition? b. Subtraction? c. Multiplication? d. Division?. Under what condition is half of an integer an integer? Support your conjecture with examples. RNS SP6

19 . Computing with Real Numbers OPERATIONS WITH RATIONAL NUMBERS. What is the set of rational numbers? Each expression below consists of rational numbers. Simplify each expression Is result a rational number? 5. Is result a rational number? 8. Is result a rational number? Is result a rational number? - 6. Is result a rational number? 9. 5 Is result a rational number? 4. of (-5) Is result a rational number? Is result a rational number? Is result a rational number? 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.. Is the set of rational numbers closed under a. addition and subtraction? b. multiplication and division?. Is half of a rational number always a rational number? Support your conjecture with examples. RNS SP7

20 . Computing with Real Numbers OPERATIONS WITH IRRATIONAL NUMBERS. 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. Multiplication Property of Square Roots For every number a 0 and b 0, ab = a b Rewrite each expression by removing any perfect squares from under the radical sign = 7 75 = 5 50 ( ) = 7 = Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? = 5 ( ) = = Is result rational or irrational? Is result rational or irrational? 9. ( 6) 4 ( )( 7) Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? Is result rational or irrational?. Do you think that the irrational numbers are closed under multiplication? Explain. RNS SP8

21 . Computing with Real Numbers OPERATIONS WITH IRRATIONAL NUMBERS Here is another property commonly used to rewrite square roots. Division Property of Square Roots a For every number a 0 and b > 0, = b Rewrite each expression by removing any perfect squares from under the radical sign. Note whether the result is a rational or an irrational number = = 5 = = Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? a b 0 Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? 0. Do you think that the irrational numbers are closed under division? Explain. RNS SP9

22 . Computing with Real Numbers GENERALIZING RULES Simplify using properties of square roots 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 In your own words, generalize a rule for dividing expressions that contain rational numbers and irrational numbers in the form of square roots RNS SP0

23 . Computing with Real Numbers EVALUATING EXPRESSIONS WITH ROOTS Evaluate the expression b 4ac if:. a =, b = 6, c = 8. a =, b = 9, c = -5. a =, b = 4, c = 4. a =, b = 6, c = 7 5. a =, b =, c = 6. a =, b = -5, c = -8 Evaluate the expression b 4ac if: a 7. a =, b =, c = 0 8. a =, b = -6, c = a =, b = 0, c = a =, b = -, c = -4. a = 5, b =, c = -. a = 6, b =, c = 0 RNS SP

24 . Computing with Real Numbers RATIONALIZING THE DENOMINATOR Simplest Radical Form An expression containing square roots is in simplest radical form when: The radicand has no perfect square factors other than. The radicand has no fractions. The denominator of the fraction does not contain a radical. Use the process called rationalizing the denominator to write the fraction simplest radical form. 7 in. Is the denominator rational or irrational?. Multiply the fraction by = 7 7 and write in simplest radical form What property of arithmetic is being used here?. Is the new denominator rational or irrational? Write each expression in simplest radical form. If it already is, then circle the expression RNS SP

25 .4 Vocabulary, Skill Builders, and Review FOCUS ON VOCABULARY (RNS) Choose words from this list. Match each number below with all words that could be used to describe it a. Integer (Z) 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. radicand j. whole number (W) in RNS SP

26 .4 Vocabulary, Skill Builders, and Review Compute with rational numbers Solve for x: -x + 5 = -(x 6) + SKILL BUILDER Solve for x in terms of y: x y = 5 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. 6. x is less than - 7. x is greater than or equal to 8. the opposite of x is less than or equal to - 9. is greater than x 0. - is less than the opposite of x RNS SP4

27 .4 Vocabulary, Skill Builders, and Review Evaluate each expression if x = SKILL BUILDER. x. x. x 4. x 0 5. x - 6. x - 7. x - 8. x -4 Examine your answers to problems -8. Explain the following. 9. Why only one has a value equal to. 0. Why three have a value less than.. Why four have a value greater than.. Given two points on a line, (-, -6) and (, -): a. Find the y-intercept. b. Find the x-intercept. c. Find the slope. d. Write the equation of the line in slopeintercept form. RNS SP5

28 .4 Vocabulary, Skill Builders, and Review Compute with rational numbers SKILLBUILDER. Rewrite each number as indicated. Number 7. 9, Approximate the square roots. Number (4-6) Product of a number between and 0, and a multiple of 0 Between square roots of perfect squares: Between two consecutive integers: About (as a fraction): (-5) Scientific notation About (as a decimal): Calculator check (nearest tenth): RNS SP6

29 .4 Vocabulary, Skill Builders, and Review SKILLBUILDER 4 Simplify each expression. Leave results in exponent form (x 0) x 4 ( ) ( 0 ) 0 x x x 0. Find the square roots ( ) 5 x ( 4 ) ( ) 5 7 ( x ) 7 x ( x) 5 5 ( x ). x 5 x x RNS SP7

30 .4 Vocabulary, Skill Builders, and Review. SKILL BUILDER 5 Write using square root or cube root notation. Then compute Write using exponent notation. Then compute Compute ( )( )( ) RNS SP8

31 .4 Vocabulary, Skill Builders, and Review Compute. 4. ( )( ) SKILL BUILDER Write these numbers in order from least to greatest ( )( 7 64 ),,,,, Write each rational number as a quotient of integers Evaluate if x = -.. (x). (x) - 4. x RNS SP9

32 .4 Vocabulary, Skill Builders, and Review SKILLBUILDER 7. Write these numbers in order from least to greatest. A. 0. B. 0. C. 0.0 D. 0. E. F. 0. G. 0. H.. Which of the numbers above are terminating decimals?. Which of the numbers above are repeating decimals? 4. Which of the numbers above are rational numbers? 5. Which of the numbers above are irrational numbers? 6. Two trains start traveling in the same direction at the same time. The Red train starts 00 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? RNS SP0

33 .4 Vocabulary, Skill Builders, and Review Fill in the blanks.. + fraction = decimal = decimal decimal SKILL BUILDER 8. Are the fraction and decimal values at the bottom of problem equivalent? Explain. Use the clever procedure from Lesson. to change each repeating decimal into an equivalent quotient of integers RNS SP

34 .4 Vocabulary, Skill Builders, and Review SKILLBUILDER 9. Use the letter next to each direction when placing the number on the number line. A. Place a natural number. (example above for A is ) B. Place a whole number that is not a natural number. C. Place an integer that is not a whole number. D. Place a rational number that is not an integer 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. Find another fraction between them. H. Place an irrational number between and 4.. Roger changed to a decimal using a calculator and saw Since this number is rational, it must either terminate or repeat. Explain to Roger how to interpret this readout.. In a triangle, the second side is one inch smaller than the first, and the third side is inches larger than the first. The total perimeter of the triangle is 7.5 inches. Find the length of the longest side. A RNS SP

35 .4 Vocabulary, Skill Builders, and Review SKILLBUILDER 0 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). The sum of two integers is an integer.. The difference of two integers is an integer.. The product of two integers is an integer. 4. The quotient of two integers is an integer. 5. The square root of an integer is an integer 6. One-fourth of an integer is an integer. Simplify each expression by removing any perfect squares from under the radical sign. Note whether the result is a rational or irrational number Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? Is result rational or irrational? RNS SP

36 .4 Vocabulary, Skill Builders, and Review Simplify using properties of square roots Evaluate the expression -b + SKILLBUILDER b 4ac if: a =, b = 4, c = 8. a =, b = -5, c = -7 Evaluate the expression -b b 4ac if: 9. a =, b = -, c = - 0. a =, b = 0, c = -6 Write each expression in simplest radical form by rationalizing the denominator RNS SP4

37 .4 Vocabulary, Skill Builders, and Review TEST PREPARATION (RNS Show your work on a separate sheet of paper and choose the best answers.. Choose all phrases that apply to the number 6. A. is a rational number B. is an irrational number C. can be written as a terminating decimal. Choose all statements that are true. D. can be written as a repeating decimal A. All integers are rational numbers. B. All rational numbers are integers. C. All irrational numbers are real numbers. Choose all expressions that are equivalent to 60. A. 6 0 B. 6 0 C Choose all expressions that are equivalent to 0. A. 4 5 B. 40 D. All integers are irrational numbers. C Which one statement disproves the following: irrational numbers are closed under multiplication.? A. 9 = B. = = 6 = 6 C. 4 5 = 5 = 0 D. 7 = 7 = 4 D. D RNS SP5

38 .4 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK (RNS) Show your work on a separate sheet of paper and write your answers on this page.. Rational Numbers. State in your own words how you know whether a number is rational.. Write the decimal equivalent of each rational number below. Note whether it repeats or terminates. a. c Change 0.4. Irrational Numbers b. d. 8 9 to an equivalent quotient of integers. 4. Write two different irrational numbers with patterns in the decimal expansions, using dots at the end ( ). 5. Using a calculator, write as a decimal to as many places as your calculator shows, and explain why this number is irrational.. Computing with Real Numbers Simplify using properties of square roots. Rationalize the denominator as needed RNS SP6

39 HOME-SCHOOL CONNECTION (RNS). Explain what the set of real numbers is, including the difference between rational and irrational numbers.. Change each decimal to an equivalent quotient of integers. a. 0.7 b Simplify the following using properties of square roots. a. 8 0 b Parent (or Guardian) signature RNS SP7

40 COMMON CORE STATE STANDARDS MATHEMATICS 7.SP.6 7.SP.7a 7.SP.8b 8.NS. 8.NS. 8.EE. 8.G.7 N-RN- MP MP MP MP4 MP5 MP6 MP7 MP8 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 or 6 would be rolled roughly 00 times, but probably not exactly00 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. 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 ). For example, by truncating the decimal expansion of, show that is between and, then between.4 and.5, and explain how to continue on to get better approximations. Use square root and cube root symbols to represent solutions to equations of the form x = p 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 is irrational. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in realworld and mathematical problems in two and three dimensions. 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. STANDARDS FOR MATHEMATICAL PRACTICE Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. First Printing 0 DO NOT DUPLICATE RNS SP8