Name Period Date. RNS1.3 Scientific Notation Read and write large and small numbers. Use scientific notation to write numbers and solve problems.

Transcription

1 Name Period Date REAL NUMBER SYSTEM Student Pages for Packet : RNS. Conjectures About Make conjectures about multiplication with eponents. Use eponent definitions and rules to simplify epressions. RNS. Zero and Negative Understand the meaning of zero and negative eponents. Use patterns to make conjectures about eponent division rules. Use eponent definitions and rules to simplify epressions. RNS. Scientific Notation Read and write large and small numbers. Use scientific notation to write numbers and solve problems. RNS STUDENT PAGES RNS. Vocabulary, Skill Builder, and Review 7 RNS SP

2 WORD BANK (RNS) Word or Phrase Definition or Eplanation Picture or Eample conjecture cube of a number eponential notation factor multiplicative identity property product scientific notation square of a number RNS SP0

3 . Conjectures about CONJECTURES ABOUT EXPONENTS Ready (Summary) We will use patterns to make conjectures about rules for multiplying epressions involving eponents. Go (Warmup) Eponential Notation Set (Goals) Make conjectures about multiplication with eponents. Use eponent definitions and rules to simplify epressions. b n (read as b to the power n ) is used to epress n factors of b. The number b is the base, and the number n is the eponent. We will refer to an epression in the form b n as in eponent form.. The epression is in eponent form. The base is and the eponent is.. The epression is in eponent form. The base is and the eponent is.. Five to the second power is written as, which is equal to =. to the third power is written as, which is equal to = Write an epression in eponent form to represent each of the following.. The area of a square with side length equal to. 6. The volume of a cube with side length equal to. RNS SP

4 . Conjectures about USING THE EXPONENT DEFINITION TO SIMPLIFY Write each eponential epression as a product of factors. Then write it in eponent form.. = ( ) ( ) =. = ( ) ( ) = 6. =. 6 =. For problems -, what do you notice about the bases in the original epressions compared the bases in the result? 6. For problems -, what do you notice about the eponents in the original epressions compared the eponents in the result? 8 = ( )( )( ) = ( )( )( ) = 7. ( ) 8. ( ) = ( )( )( ) = ( )( )( ) = ( ) = ( ) =. For problems 7-0, what do you notice about the eponents in the original epressions compared the eponents in the result? RNS SP

5 . Conjectures about THE EXPONENT PRODUCT RULE Complete problems - and write observations in the space provided about eponent relationships. Then use these observations to make a conjecture in words and symbols.. ( )( ) 6 ( )( ) = = Are the bases the same? How do the eponents in the original epression ( and ) relate to the eponent in the product (6)?. ( )( ) ( )( ) = =. ( )( ). ()( ) = = (Remember that: = ) ( )( ) ( )( ) = =. Write a conjecture in words related to the multiplication in problems - above: 6. Write a conjecture in symbols: ( )( ) = a b We will call this conjecture the Eponent Product Rule. RNS SP

6 . Conjectures about THE EXPONENT POWER RULE Complete problems - and write observations in the space provided about eponent relationships. Then use these observations to make a conjecture in words and symbols.. ( ) = ( ) ( ) ( ) ( ) ( )( )( )( ) = = How is this epression different from the epression in problem on the previous page? How do the eponents in the original epression ( and ) relate to the eponent in the product (8)?.. ( ) = ( ) ( ) ( ) ( ) ( )( )( )( ) = = ( ) = ( ) ( ) ( ) ( ) ( ) ( )( )( )( )( ) = =. Write a conjecture in words related to the multiplication in problems - above:. Write a conjecture in symbols: a b ( ) = We will call this conjecture the Eponent Power Rule. 8 RNS SP

7 . Conjectures about PRACTICE USING EXPONENT RULES Write each epression in eponent form. Circle the rule used (Eponent Product Rule or Eponent Power Rule.). 6 = = Product Rule. (7 )(7 ) Product Rule Product Rule 7. ( )( 6 ) Product Rule 9. y 0 y 0 Product Rule. 0 0 Product Rule. 0 0 Product Rule. 8 Product Rule Power Rule Power Rule Power Rule Power Rule Power Rule Power Rule Power Rule Power Rule. ( 6 ) = =. (7 ) Product Rule Product Rule 6. (9 8 ) 6 8. ( ) 6 Product Rule Product Rule 0. (y 0 ) 0 Product Rule. (0 ) Product Rule. (0 ) 6. ( ) 8 Product Rule Power Rule Power Rule Power Rule Power Rule Power Rule Power Rule Power Rule Product Rule Power Rule RNS SP

8 . Conjectures about EXPONENTS CHALLENGE Compute. Results should contain no eponents ( )( ) ( + ) 7. Why can both the eponent product rule and eponent power rule NOT be used to compute problems and? 8. Write three more epressions equivalent to 8 that include eponents. Write each epression in the form 9. n Write each epression in the form ab n, where a is a constant and b n is eponent form.. ( )( ). () a = a = b = n = b = n =. ( ) a =. ( ) b = n = a = b = n = RNS SP6

9 . Zero and Negative ZERO AND NEGATIVE EXPONENTS Ready (Summary) We will use patterns to eplore the meaning of zero eponents and negative eponents. Write each epression in eponent form. Go (Warmup) Set (Goals) Understand the meaning of zero and negative eponents. Use patterns to make conjectures about eponent division rules. Use eponent definitions and rules to simplify epressions... ( ).. ( ) Compute Why is it not possible to write + in eponent form? 7. RNS SP7

10 . Zero and Negative PAPER FOLDING EXPERIMENT Record the results from folding a piece of paper. Number of Folds Number of Sections Write the Number of Sections as to a Power 6 The paper folding eperiment suggests that 0 = Rule: 0 =, 0, RNS SP8

11 . Zero and Negative EXPONENT QUOTIENT PATTERNS Complete the table. Following patterns down each column may be helpful. I. Epression 0 II. Epanded Form III. Power of 0 (fractions okay) IV. Power of 0 (no fractions) V. Value (fractions okay) The epressions in column I are quotients with the same bases. The epressions in column IV are in eponent form. What about the eponents in column I determine when the eponents in column IV will be a. Positive? b. Negative? c. Zero? 8. How is 0 - related to 0? 9. Do any of the epressions above with negative eponents result in a negative value? RNS SP9

12 . Zero and Negative 0. EXPONENT QUOTIENT PATTERNS (continued) Complete the table. Following patterns down each column may be helpful. I. Epression 0 II. Epanded Form III. Power of (fractions okay) IV. Power of (no fractions) V. Value (fractions okay) 6. Patti thinks that a base number to a negative power must result in a negative value. Is Patti correct? Eplain. 7. How is - related to? 8. Make conjectures: If 0 and a > 0, then 0 = - = -a 7 = RNS SP0

13 . Zero and Negative THE EXPONENT QUOTIENT RULE Complete problems - and write observations in the space provided about eponent relationships. Then use these observations to make a conjecture in words and symbols.. = = Are the bases the same? How do the eponents in the original epression ( and ) relate to the eponent in the quotient ()?... = = = = = = = =. Write a conjecture in words related to the division in problems - above: If 0, then a b = We will call this conjecture the Eponent Quotient Rule. RNS SP

14 . Zero and Negative EXPONENT PRACTICE. A number to a negative power is the same as the of the number opposite / reciprocal to a positive power. In other words: Compute. n b = = b b.... ( ) ( ) ( ) 8 Write each epression in eponent form ( 0) ( ) 0 Write each epression in eponent form (b n where b > ). ( ) ( ) ( ) (hint: negative eponent reciprocal) RNS SP

15 . Zero and Negative. Using the eponent quotient rule, write EXPONENT EXPLORATIONS 8 as to a single power. Now write this epression with a positive eponent. Write each epression in eponent form (b n ) where b is given b = b = b = b = b = b = 9 Compute. 0. (-). (-). (-). (-). (-) -. (-) - 6. (-) - 7. (-) b = 6 9 b = 8. Karin thinks that a negative number to a negative power must be negative. Is she correct? Eplain. RNS SP

16 . Scientific Notation Ready (Summary) We will write large and small numbers using scientific notation. We will solve problems involving large numbers. trillions SCIENTIFIC NOTATION Go (Warmup). Fill in the place value chart above and locate the decimal point. Write in words: hundred thousands Set (Goals) Read and write large and small numbers. Use scientific notation to write numbers and solve problems..,00,00,00, Round to two significant digits.. 9, ,, hundreds ones hundredths RNS SP

17 . Scientific Notation LARGE NUMBERS Scientific notation is a system that can be used for writing large numbers. In scientific notation, a number is written as a decimal that is greater than or equal to and less than 0, multiplied by a power of 0. Standard notation Product of a number between and 0, and a multiple of 0 Scientific notation E.,00., ,000, ,000, , ,000, ,000..6,000, Eplain why is. 0 is NOT in scientific notation. Then rewrite it in scientific notation. RNS SP

18 . Scientific Notation THE NATIONAL DEBT The national debt refers to the amount of money that our nation s government owes. Find websites that give the national debt as of today and the current population of the United States. These websites may also give the amount of the debt per person. Make a note of that number, and then check it against the class calculations.. As of, the national debt was $ (Date). Approimate the national debt by rounding it to two significant digits.. Write an approimation of the national debt in words.. Write an approimation of the national debt in scientific notation.. There are currently about citizens in the United States. 6. Approimate the number of citizens by rounding to two significant digits. 7. Write an approimation of the number of citizens in words. 8. Write an approimation of the number of citizens in scientific notation. 9. Approimate each citizen s portion of the national debt. 0. If the internet stated the national debt per person, does it agree with your answer? Eplain. RNS SP6

19 . Scientific Notation HOW MUCH IS A BILLION?. Make a guess about how many years do you think there are in a billion seconds? To find the number of seconds in year: 6 days hours 60 minutes 60 seconds year =,6,000 seconds year day hour minute. Approimate the number of seconds in one year to three significant digits.. Write an approimation of the number of seconds in one year in words.. Write this approimation in scientific notation.. Write one billion in scientific notation. 6. Calculate the number of years in one billion seconds. 7. How close was your estimate? What historical events were taking place one billion seconds ago? This may require research on the internet. The National Debt is over 0 trillion dollars. 8. How many billions are in 0 trillion? 9. About how many years ago was 0 trillion seconds? RNS SP7

20 . Scientific Notation SMALL NUMBERS Scientific Notation is also useful for writing very small numbers. In scientific notation, a number is written as a decimal greater than or equal to and less than 0, multiplied by a power of 0. Standard notation Product of a number between and 0, and a multiple of 0 E Scientific notation Eplain why is notation is NOT in scientific notation. Then rewrite it in scientific RNS SP8

21 . Scientific Notation DIGITAL MEMORY In the world of computers, cameras, and MPs, digital memory is measured in bytes.. Write fractions that behave as for each digital storage measurement. kilobyte (KB) 000 bytes. 000 bytes KB or KB 000 bytes megabyte (MB) 000 kilobytes gigabyte (GB) 000 megabytes A terabyte (TB) 000 gigabytes. How many bytes are in a kilobyte? Write this number in scientific notation. Write this number in words.. How many bytes are in a megabyte? Write this number in scientific notation. Write this number in words.. How many gigabytes are in a byte? Write this number in scientific notation. Write this number in words.. How many terabytes are in a byte? Write this number in scientific notation. Write this number in words. RNS SP9

22 . Scientific Notation DIGITAL MEMORY (continued) 6. An MP music file is typically about megabytes. How many music files can be stored on an MP player that has 8 gigabytes of digital memory? Think: How many megabytes are in a gigabyte? 7. Ellie s workstation computer had a 00 gigabyte hard drive. Her new workstation has a terabyte hard drive. Which workstation holds more data? By how much? Think: How many gigabytes are in a terabyte? 8. Rudy has a gigabyte memory stick for his camera. According to the manufacturer, the memory stick will hold 9 pictures. About how many megabytes are required for each picture? Think: How many are in a? RNS SP0

23 . Scientific Notation PRACTICE WITH SCIENTIFIC NOTATION Determine whether each number is in scientific notation. If NOT, write it in scientific notation Circle the greater number and state how many times greater it is than the other number Use symbols <, =, or > to compare each pair of numbers Jorge multiplied 60 million by billion on his calculator. The calculator display showed:. E7 Eplain what you think this answer means. RNS SP

24 . Vocabulary, Skill Builders, and Review Across FOCUS ON VOCABULARY (RNS) Down the result of multiplication. (0 6 ) is eample of this notation. 7 in the epression 7 another name for eponent 6 a measure of digital memory the value of 6 o 8 6 is a of. condensed way to show multiplication by the same number: eponential 9 property that applies the big one 6 in the epression 7 7 = applies the eponent rule. 7 RNS SP

25 . Vocabulary, Skill Builders, and Review ten thousands thousands hundreds SKILL BUILDER tens ones. What does the 7 represent in 8.7?. What does the represent in 8.7?. Write 0.9 in words. tenths. Circle the digit in the tenths place Circle the digit in the tens place Round 7.68 to the nearest tenth. 7. Round 79.0 to the nearest hundredth. Write epanded forms of each number in three different ways. Standard Form Epanded Form # Epanded Form # Epanded Form # E.. () () + (0.) + (0.0) decimal point hundredths thousandths ten thousandths RNS SP

26 . Vocabulary, Skill Builders, and Review SKILL BUILDER Use an area model to multiply. Check by division Solve for. Check by substitution.. ( + 6) = 6. = 6 Identify the variable, write an equation, solve the equation, answer the question, and check the solution in the contet of the problem.. The length of the second side of a triangle is less than times the length of the first side. The third side is times longer than the first side. The perimeter of the triangle is 0 units. How long is each side? Does this problem seem realistic, or do you think it was created to help you learn algebra? RNS SP

27 . Vocabulary, Skill Builders, and Review Draw line segments with the following slopes.. Line AB with a slope of.. Line CD with a slope of -.. Line LM with a slope of.. Line PQ with a slope of -. Given each set of ordered pairs, find the slope of the line that goes through them.. E(, -) and F(7, ) 6. G(-, 0) and H(-, 8) 7. J(-, -) and K(-, ) 8. N(6, ) and R(6, ) SKILL BUILDER A C L P RNS SP

28 . Vocabulary, Skill Builders, and Review. Compute. SKILL BUILDER a. = b. = c. = d. = e. = f. 6 = g. 7 = h. 8 =. Compute: +. Can this value be written as raised to a whole number power? Use some eamples to justify your answer. Write each epression in eponent form ( 0). E. = (6 ) ( ).. 9. ( ) 7 ( ) 7. ( ) 0. 9 RNS SP6

29 . Vocabulary, Skill Builders, and Review Write in eponent form... ( ) Compute SKILLBUILDER (8 ) (-8) ERROR ALERT! Be careful with eponential epressions that include a minus sign. The minus sign is part of the base: (-) = (-)(-) = 6. The minus sign is not part of the base: - = -()() = -6.. Here are four epressions that contain eponents: (-) - 0 The epressions that are equal to 9 are (show work to justify): The epressions that are equal to -9 are (show work to justify):. RNS SP7

30 . Vocabulary, Skill Builders, and Review Complete the table. Standard form. 8,000.,70,000 SKILL BUILDER 6 Product of a number between and 0, and a multiple of ,000,000 Scientific notation ,000 Write each epression in eponent form Compute ( ). ( ) ( ) ( ) 6. 6 ( ) RNS SP8

31 . Vocabulary, Skill Builders, and Review Write each epression in eponent form ( ) ( ) 6 SKILL BUILDER 7. ( ). ( ) - ( ) Write each epression in the form a n, where a is a constant ( ) 6. ( ) 8 9. ( 0 ) 7. Write - in three different but equivalent ways that include eponents. 8. Consider the numbers and a. Write them both in scientific notation b. Circle the greater number. c. How many times greater is the number you circled compared to the other one? RNS SP9

32 . Vocabulary, Skill Builders, and Review SKILL BUILDER 8 Write each epression in eponent form ( 0) Write each epression in eponent form (b n where b > ) ( ) ( ) () 6() ( 0 ) ( 0 ) Compute ( ) 7 (0 ) 7 ( 0 ) ( 0 ) (0 ) The population of the United States is about three hundred million. The population of the world is about seven billion. 7. Write each of these numbers in scientific notation. 8. About how many times larger is the world population than the United States population? 7 RNS SP0

33 . Vocabulary, Skill Builders, and Review TEST PREPARATION (RNS) Show your work on a separate sheet of paper and choose the best answer(s).. Find all epressions equivalent to? A. ( ) B. 6 C. - D. 8. Find all epressions equivalent to ( ). A. 7 B. 0 C. D.. Find all epressions equivalent to A. 0 7 B. 0 6 C.. Find all epressions equivalent to y 7 A. y B.. Find all symbols that make y? (0 ) D. 0 - y C. y 9 D. (7 ) true? A. < B. > C. = D. 6. Epress in standard notation. A. 0,000 a.,00,000 b c Epress 7,000,000 in scientific notation. A B C..7 0 D y y RNS SP

34 . Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK (RNS) Show your work on a separate sheet of paper and write your answers on this page.. Conjectures About Compare. Use >, <, or = to complete each statement.. (7 )(7 ) (7 ) Write in eponent form.. 7. ( ) 7. Zero and Negative. Complete the equation = Write in eponent form without a fraction bar. Write in eponent form ( ) 7 ( ) 7. Compute. Scientific Notation Write the number in scientific notation. (9 ) Write the number in standard notation. RNS SP

35 HOME-SCHOOL CONNECTION (RNS) Here are some questions from this week s lessons to review with your young mathematician.. Consider the following epression, and write it in eponent form in the given ways: (8 ). Complete the equation: a. As a power of 8 (in the form 8 n ): b. As a power of (in the form n ): 0 = 0.. Give an eample a numerical epression in eponent form, b n, where n is a negative number but the value of the epression is positive.. Give an eample of a base number with a negative eponent where the result is negative.. Eplain why is not in scientific notation, and then write it in scientific notation. Parent (or Guardian) signature RNS SP

36 COMMON CORE STATE STANDARDS MATHEMATICS 6.EE. 8.EE. 8.EE. 8.EE. N-Q- N-Q- MP MP MP MP MP MP6 MP7 MP8 STANDARDS FOR MATHEMATICAL CONTENT Write and evaluate numerical epressions involving whole-number eponents. Know and apply the properties of integer eponents to generate equivalent numerical epressions. For eample, = = / = /7. Use numbers epressed in the form of a single digit times an integer power of 0 to estimate very large or very small quantities, and to epress how many times as much one is than the other. For eample, estimate the population of the United States as 0 8 and the population of the world as 7 0 9, and determine that the world population is more than 0 times larger. Perform operations with numbers epressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 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 epress regularity in repeated reasoning. First Printing DO NOT DUPLICATE 0 RNS SP