Unit 2: Exponents 8 th Grade Math 8A - Mrs. Trinquero 8B - Dr. Taylor 8C - Mrs. Benefield 1
8 th Grade Math Unit 2: Exponents Standards and Elements Targeted in the Unit: NS 1 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. NS 2 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., ). For example, by truncating the decimal expansion of (square root of 2), show that is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. EE 1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, x = = 1/( ) = 1/27. EE 2 Use square root and cube root symbols to represent solutions to equations of the form = p and = 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. EE 3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 x and the population of the world as 7 x, and determine that the world population is more than 20 times larger. EE 4 Perform operations with numbers expressed 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. September and October 2012 Monday Tuesday Wednesday Thursday Friday 18 19 20 How do I estimate the How do you identify How do you identify value of non-perfect rational and irrational rational and irrational square roots? numbers? numbers? 17 How do I find square roots of numbers? CW: Notes (p3) and Activity Where does this number belong? HW: Squares and Square Roots WS (p4) CW: Notes and Investigation (p5) HW: Estimating Non Perfect Square Roots Practice (p6) CW: Notes on Real Number System and examples; SIGN Game; HW: The Real Number System WS (p7) 2 CW: Rational and Irrational Line up; Pizzazz Practice/Number Checklist HW: Quiz Review (p8) 24 25 26 27 28 How do you convert repeating decimals to fractions? How do you simplify expressions with integer exponents? How do you simplify expressions with integer exponents? How do you simplify expressions with integer exponents? CW: Notes and practice (p10) HW: Converting repeating decimals (p11) CW: Alien Attack Activity; Notes and practice (p12) HW: Simplifying Exponent Expressions Practice (p13) CW: Exponent Match-up HW: Simplifying Exponent Expressions Practice (p14 1-15) CW: Exponent MATHO HW: Simplifying Exponent Expressions Practice (p14 16-30) 1 2 3 4 5 How do I express numbers in scientific notation and standard notation? CW: Notes and Practice HW: Scientific Notation Practice (p15) 8 Columbus Day Holiday No School How do I express numbers in scientific notation and standard notation? CW: Pick up a Stick HW: Scientific Notation Practice (p16) 9 How do I use square root and cube root symbols to represent solutions to equations of the form = p and = p? CW: Activity: Building a cube HW: Test Study Guide (p19-20 evens) Giantburger Task Due October 15th How do I express numbers in scientific notation and standard notation? CW: Evaluating Expressions Using Scientific Notation HW: Practice (p17) 10 How do I represent numbers including square/cube roots, exponents, and scientific notation and determine whether they are rational or irrational? CW: Test Review HW: Test Study Guide (p19-20 odds) How do I express numbers in scientific notation and standard notation and express how many times as much one is than the other? CW: Quiz Scientific Notation; How many times more? HW: Practice (p18) 11 How do I represent numbers including square/cube roots, exponents, and scientific notation and determine whether they are rational or irrational? CW: Test Review HW: Study notes and examples for test 21 How do we convert rational numbers to decimals and vice versa? CW: Whiteboard practice Quiz Squares & Square Roots; Rational & Irrational; CCGPS Review Quiz HW: Converting WS (p9) How do you simplify expressions with integer exponents? CW: Quiz Simplifying Exponent Expressions; CCGPS Review Quiz; Square Puzzle HW: Review notes and Examples Survivor Day! 12 How do I represent numbers including square/cube roots, exponents, and scientific notation and determine whether they are rational or irrational? CW: Exponents Unit Test HW: CCGPS Review
Squares and Square Roots Square a number: Perfect Square - Square root of a number - Radical - Radicand - Memorize the first 25 perfect squares: 1 2 = 6 2 = 11 2 = 16 2 = 21 2 = 2 2 = 7 2 = 12 2 = 17 2 = 22 2 = 3 2 = 8 2 = 13 2 = 18 2 = 23 2 = 4 2 = 9 2 = 14 2 = 19 2 = 24 2 = 5 2 = 10 2 = 15 2 = 20 2 = 25 2 = Examples: Find the positive square root of each number. 1. 2. 3. 4. What is the best whole number estimate for each square root below? Hint: Find the two perfect squares from your table above that each square root is located between and choose the one that it is closest to. 5. 6. 7. 8. What are the square ROOTS for each perfect square? Because it asks for square roots, give both the positive and negative square root. 9. 81 10. 121 11. 625 12. 441 Find the point that best represents each square root on the number line below. 13. 14. 15. 16. 3
Squares and Square Roots 1 2 = 6 2 = 11 2 = 16 2 = 21 2 = 2 2 = 7 2 = 12 2 = 17 2 = 22 2 = 3 2 = 8 2 = 13 2 = 18 2 = 23 2 = 4 2 = 9 2 = 14 2 = 19 2 = 24 2 = 5 2 = 10 2 = 15 2 = 20 2 = 25 2 = Find the square root. 1. 2. 3. 4. 5. 6. 7. 8. What is the best whole number estimate of each square root? 9. 10. 11. 12. 13. 14. 15. 16. What are the square roots of each perfect square? 17. 144 18. 100 19. 49 20. 361 21. 25 22. 169 23. 484 24. 289 25. Charlie wants to build a square fence with an area of 289 square feet. How long should he make each side of the fence? 26. Becky framed a square picture with an area of 400 square inches. What is the width of the picture? 27. A square root shows a number under a. 28. The numbers 1, 4, 9, 16, 25 are examples of. 29. The opposite of squaring a number is taking the. 30. The number inside a radical sign is called the. 4
Estimating Non-Perfect Square Roots In order to find the location of a non-perfect square root on a number line, you need to find an approximation. Step 1: Between what two whole numbers does lie? (Use your perfect squares chart.) and Step 2: Determine which of the above numbers is closest to. Step 3: Identify the closest tenths. Investigate the squares of decimals that are closer to. = = Between which 2 numbers does it lie? = and = Which will it be closest to? Step 4: Identify the closest hundredths. Investigate the squares of decimals that are closer to. = = Between which 2 numbers does it lie? = and = Which will it be closest to? Step 5: An approximation of Practice Find an approximation to the nearest hundredth for the following square roots. 1. 2. 3. 5
Estimating Non-Perfect Square Roots Determine the two closest integers for each square root. 1. 2. 3. Plot each square root at its approximate location on the following number line. 0 1 2 3 4 5 6 7 8 9 10 4. 5. 6. Plot each square root at its approximate location on the following number line. 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 7. 8. 9. Complete each sentence. Write decimals to the hundredths place. 10. is between 8.12 and, and it is closer to. 11. is between and, and it is closer to. 12. is between and, and it is closer to. Choose the best answer. 13. What is the best approximation for? a) 1.7 b) 1.73 c) 1.74 d) 1.8 6
The Real Number System State whether each number belongs to the set of whole numbers, integers, rational numbers, or irrational numbers. Remember, a number can belong to more than one set. 1. -25.4 2. 3. 4. 5. - 6. Choose the best answer. 7. Which number is rational? a) -0.26381 b) c) - d) - 8. Which number is irrational? a) b) - c) d) 9. Which numbers are rational? a) only b) and -, -7.25,, - c), -7.25, and - d) -7.25 only Fill in the blanks. 10. A rational number is any number that can be written as a. 11. Integers include the set of whole numbers and their. 12. Another name for the set of natural numbers is the set of. 13. All square roots of non perfect squares are. Tell whether the statement is true or false. If false, tell why? 14. A whole number can be negative. 15. Zero is a natural number. 16. A number can be both rational and irrational. 17. All fractions are rational. 18. All decimals are irrational. 19. All integers are whole numbers. 7
Squares and Square Roots The Real Number System Review Find the positive square root of each number. 1. 2. 3. 4. What is the best whole number estimate for each square root below? 5. 6. 7. 8. What are the square ROOTS for each perfect square? 9. 100 10. 4 11. 169 12. 64 13. Cody wants to build a square fence with an area of 196 square feet. How long should he make each side of the fence? 14. Kylee framed a square picture with an area of 225 square inches. What is the width of the picture? Plot each square root at its approximate location on the following number line. 0 1 2 3 4 5 6 7 8 9 10 15. 16. 17. Plot each square root at its approximate location on the following number line. 8 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9 18. 19. 20. State whether each number belongs to the set of whole numbers, integers, rational numbers, or irrational numbers. Remember, a number can belong to more than one set. 21. 36.5 22. 23. 24. 25. - 26. 8
Express each number as a fraction. Rational Numbers Converting between Decimals and Fractions 1. 21.5 2. -0.9 3. 42% 4. 19 5. -44.045 6. -2.65 Show the decimal expansion of each number, then plot and label a point for each on the number line using the given letter. 7. (A) 8. - (B) 9. 2.5% (C) 10. (D) 11.- (E) 12. 65% (F) -3-2 -1 0 1 2 3 REVIEW Complete each sentence. 13. -11.3 is rational because. 14. is irrational because. 15. is rational because. 16. 2.1371938 is irrational because. Choose the best answer. 17. Which number is NOT equivalent to 13.02? a) 13.002 b) 13.020 c) 13.0200 d) 13.020000 18. Which is an irrational number? a) - b) c) d) 11.2092 Put the following numbers in order from least to greatest. 19. 3.24037,, 20. 9
Converting Repeating Decimals to Fractions Convert to a fraction. Examples: Step 1: n = Use Algebra Step 2: 10n = There is one repeating digit, so multiply n by 10. Step 3: - n = Subtract the number, n, from 10n. 9n = 3 Step 4: 9n = 3 Solve the equation and simplify the result. 9 9 n = Convert to a fraction. Convert to a fraction. Convert 9. to a fraction. Convert 4. to a fraction. Convert 0. to a fraction. Convert 2. to a fraction. 10
Converting Repeating Decimals to Fractions Convert the following decimals to fractions in simplest form. 1) 0. 2) 0.87 3) 0. 4) 6. 5) 1. 6) 2. 7) 0. 8) 4. 9) 0. 10) 0. 11
Simplifying Expressions Containing Integer Exponents Properties of Exponents 1. Product of Powers: If multiplying the same base, add exponents. a is the base m and n are exponents 2. Power of a Power: If a power is raised to an exponent, multiply exponents. the expression is called a power 3. Power of a Product: Distribute the exponent outside to all exponents inside. **Important note: if a base does not have an exponent written, the exponent is 1.** 4. Quotient of Powers: If dividing the same base, subtract the exponents. 5. Zero Exponent: Any number other than 0 to the zero power is 1. 6. Negative Exponent: If a base has a negative exponent in the numerator, put it in the denominator and make the exponent positive. If a base has a negative exponent in the denominator, put it in the numerator and make the exponent positive. Practice: 1. 7. 2. 8. 3. 9. 4. 10. 5. 11. 6. 12. 12
Simplifying Expressions Containing Integer Exponents Use the properties of exponents to simplify each expression. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. Evaluate if x = -4. 18. Evaluate if w = -1. 19. Evaluate if y = 3. 20. Evaluate if b = -2. 13
Simplifying Expressions Containing Integer Exponents Use the properties of exponents to simplify each expression. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. (-4 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 14
Converting between Standard and Scientific Notation A number expressed in scientific notation has 2 factors: o The first factor is a number greater than or equal to 1 and less than 10. o The second factor is a power of 10 (10 with an exponent). A number that is not written in scientific notation is written in standard form. (Ex: 247,000,000) Rules for going from scientific notation to standard notation: o If the exponent is positive, move the decimal right that number of places and the first factor will become greater. o If the exponent is negative, move the decimal left that number of places and the first factor will become smaller. Rules for going from standard notation to scientific notation: o If the decimal moves to the left, the exponent will be positive. o If the decimal moves to the right, the exponent will be negative. Write each number in scientific notation. 1. 3400 2. 56,000,000 3. 0.00923 4. 0.00000004 5. 5000 6. 643,000 7. 0.00041 8. 0.18 9. 32,610,000 Write each number in standard notation. 10. 11. 12. 13. 2.4 14. 15. 16. 17. 18. Decide whether the number is in scientific notation. If it is not, rewrite the number in scientific notation. 19. 20. 21. 22. 23. 24. 15
Converting between Standard and Scientific Notation Circle the best answer for each question. 1. What is in standard notation? a) 87,000 b) 0.000087 c) 870,000 d) 0.00087 2. What is 6,785,000 in scientific notation? a) b) c) d) 3. What is 0.0094 in scientific notation? a) b) c) d) 4. What is in standard notation? a) 1,600 b) 16,000 c) 160,000 d) 1,600,000 5. What is 325 in scientific notation? a) b) c) d) 6. What is in standard notation? a) 1 b) 0 c) 5.8 d) 58 7. Which is larger? or 8. Which is larger? or 9. The monarch butterfly is only one of about species of butterflies in the world. Write this number in standard notation. 10. The smallest butterfly is only about -in. long. Write this length as a decimal, and then express it in scientific notation. Fill in the blanks. 11. When you convert a number from scientific notation to standard notation, move the decimal point to the if the exponent is positive and to the if the exponent is negative. 12. The first factor of a number written in scientific notation is greater than or equal to and less than. 13. The number 0.00082 is in notation. 14. The second factor of a number written in scientific notation is a power of. 15. Scientific notation is a method of writing numbers that are very or very. 16
Evaluating Expressions using Scientific Notation Multiply or divide the expression using your exponent rules then make sure the answer is written in scientific notation o If you multiply, you will need to add the exponents. o If you divide, you will need to subtract the exponents. o Remember that the first factor must be a number greater than or equal to 1 and less than 10. If the first factor is not greater than or equal to 1 and less than 10, you will need to move the decimal to the right or left a certain number of places and change the exponent. o Remember that the second factor must be written as a power of 10. Find each product or quotient. Write your answer in scientific notation. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. The distance between Earth and the sun is about 150,000,000 kilometers. How can you write this number in scientific notation? 14. Some stars in the Milky Way are light-years from Earth. A light-year is miles. Write light-years in miles. 17
How many times more? 1. 2. 3. 4. The population of the world is and the population of Brazil is. How many more times is the population of the world than Brazil? 5. The population of China is and the population of the U.S. is. About how many more times is the population of China than the U.S.? 6. The population of the U.S. is and the population of Italy is. How many times larger is the population of the U.S. compared to Italy? 7. The population of Florida is and the population of the U.S. is. How many times larger is the population of the U.S. compared to Florida? 8. According to the 2011 Census, the population of Georgia is approximately U.S. is. The population of Alaska is approximately U.S. is. How many times greater is the population of Georgia compared to Alaska? 18
Exponents Unit Test Study Guide Find the square root. 1. 2. 3. 4. What is the best whole number estimate of each square root? 5. 6. 7. 8. What are the square roots of each perfect square? 9. 225 10. 81 11. 64 12. 256 Find an approximation to the nearest hundredth for the following square roots. 13. 14. 15. State whether each number belongs to the set of whole numbers, integers, rational numbers, or irrational numbers. Remember, a number can belong to more than one set. 16. -83.7 17. 18. 19. 20. - 21. Express each number as a fraction. 22. 45.5 23. -0.25 24. 56% 25. 186 26. -14.016 27. -5.95 Show the decimal expansion of each number, then plot and label a point for each on the number line using the given letter. 28. (A) 29. (B) 30. 5.5% (C) 31. (D) 32. (E) 33. 15% (F) -3-2 -1 0 1 2 3 19
Convert the following decimals to fractions in simplest form. 34. 0. 35. 0.0 Use the properties of exponents to simplify each expression. 36. 37. 38. 39. 40. 41. Write each number in scientific notation. 42. 8900 43. 126,000,000 44. 0.00753 Write each number in standard notation. 45. 46. 47. Find each product or quotient. Write your answer in scientific notation. 48. 49. 50. 51. 52. Approximately people play the piano in the United States. Approximately 3 people play the drums. About how many more people play the piano than the drums? 20