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Number Theory and Algebraic Reasoning

Learning Objectives. To read exponents properlyTo evaluate positive, negative, and zero powersTo express whole numbers as powers. 2-1 Exponents. Exponent: tells how many times to use the base as a factor (little number)Base: the number you're multiplying (the big number)Power: the exponent de

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Number Theory and Algebraic Reasoning

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    1. Chapter 2 Number Theory and Algebraic Reasoning

    2. Learning Objectives To read exponents properly To evaluate positive, negative, and zero powers To express whole numbers as powers

    3. 2-1 Exponents Exponent: tells how many times to use the base as a factor (little number) Base: the number youre multiplying (the big number) Power: the exponent determines the power 34 = 3 x 3 x 3 x 3 = 81 (three to the fourth power) 57= 5 x 5 x 5 x 5 x 5 x 5 x 5 = 78,125 (five to the seventh power)

    4. Expressing Whole Numbers as Powers Write each number using an exponent and the given base. 49, base 7 7 x 7 =49 72= 49 64, base 2

    5. Practice 26 = 111 = 105 = 73 = 44 = 100= 81, base 3 = 343, base 7 = 625, base 5 = 64, base 2 =

    6. Negative Exponents 4-1 5-3 6-4 2-2 7-3 8-1 9-4

    7. Think and Discuss Describe the relationship between 35 and 36. Tell which power of 8 is equal to 26 . Explain. Explain why any number to the first power is equal to that number. What do you do if the exponent is negative?

    8. Learning Objectives To convert numbers in standard form to scientific notation To convert numbers in scientific notation to standard form To multiply by powers of ten mentally To explain why numbers are written in scientific notation

    9. 2-2 Powers of Ten and Scientific Notation The distance from Venus to the Sun is over 100,000,000 kilometers. You can write this number as a power of ten by using a base of ten and an exponent. 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 108 Scientists use powers of ten to write really big numbers in a shorter way (scientific notation) The powers of 10 represent all of the zeros

    10. Scientific Notation vs. Standard Form Scientific Notation: 3.5 x 109 Decimal times 10 to an exponent Always only one number in front of the decimal point Standard Form: 3,500,000,000 Includes all of the zeros

    11. Standard Form to Scientific Notation The planet Neptune is about 4,500,000,000 km from the sun. Move the decimal point to get one digit in front of the decimal. 4.5 The exponent is equal to the number of places the decimal point is moved. 4.5 x 109

    12. Scientific Notation to Standard Form Pluto is about 3.7 x 109 miles from the Sun. Write this distance in standard form. Since the exponent is 9, move the decimal point 9 places to the right. 3,700,000,000

    13. Practice Write these numbers in scientific notation. 4,340,000 327,000,000 1,262,000,000

    14. Practice Write these numbers in standard form. 212 x 104 31.6 x 103 43 x 106 56 x 107

    15. Think and Discuss Tell whether 15 x 109 is in scientific notation. Explain. Compare 4 x 103 and 3 x 104. Explain how you know which is greater. Why do scientists use scientific notation?

    16. Learning Objectives To memorize the order of operations (PEMDAS) To evaluate expressions using order of operations showing steps To solve story problems using order of operations

    17. 2-3 Order of Operations Please (Parentheses) Excuse (Exponents) My (Multiplication) Dear (Division) Aunt (Addition) Sally (Subtraction) **Multiplication and Division (left to right) **Addition and Subtraction (left to right)

    18. Practice Evaluate 27 18 6 = Evaluate 36 18 2 x 3 + 8 = Evaluate 5 + 62 x 10 =

    19. More Practice Evaluate 36 (2 x 6) 3 = Evaluate {(4 + 12 4) -2}3 Evaluate 44 14 2 x 4 + 6 =

    20. Think and Discuss Apply the order of operations to determine if the expressions 3 + 42 and (3 + 4)2 have the same value. Determine whether grouping symbols should be inserted in the expression 3 + 9 4 x 2 so that its value is 13. Give the correct order of operations for evaluating (5 + 3 x 20) 13 + 32

    21. Anticipation Guide Please indicate True or False for each statement. 1. Prime numbers are those that contain only two factors. 2. Multiples are those that divide evenly into a number. 3.All numbers are either prime or composite. 4. Prime numbers can only be odd numbers. 5. The number one is considered to be a prime number.

    22. Learning Objectives To distinguish between a prime number and composite number To define prime and composite To write a number as its prime factorization using a factor tree To write a number as its prime factorization using the birthday cake method

    23. 2-4 Prime Factorization Prime number: whole number greater than 1 that is divisible by only 1 and itself Examples: Composite number: whole number with more than 2 factors Examples:

    24. Divisibility Rules Divisible by 2 Last digit is even number Divisible by 3 Sum of digits is divisible by 3 Divisible by 5 Last digit is 5 or 0 Divisible by 9 Sum of digits is divisible by 9 Divisible by 10 Last digit is 0

    25. Factor Trees Prime factorization: a composite number can be written as a product of its prime factors. Monkeys are like prime factors! 36 280 252

    26. More Practice Create a factor tree to find the prime factorization. 495 150 476

    27. Think and Discuss Explain how to decide whether 47 is prime. Compare prime numbers and composite numbers. Tell how you know when you have found the prime factorization of a number.

    28. Learning Objectives To define GCF To use a list to find the GCF To use a factor tree to find the GCF To use the birthday cake method to find the GCF

    29. 2-5 Greatest Common Factor Greatest Common Factor (GCF): the greatest whole number that divides evenly into each number Factor: the numbers that divide into a number evenly

    30. Using a List to Find GCF Lets compare two of your favorite movies! Movie A: Movie B:

    31. Listing or Rainbow Method Find the GCF of 24, 36, 48 24: 36: 48: List all the factors Circle the greatest factor that is in all of the lists

    32. Factor Tree Method Make a factor tree for each number Circle the common prime factors Multiply the common prime factors 60 45

    33. Birthday Cake Method Step 1: Write numbers side by side. Step 2: Draw a shelf under the numbers and pick a number to divide them both by. Put the division answers below the numbers. Step 3: Repeat step 2 with another shelf or layer. Step 4: Multiply the numbers on the left side to get GCF! 24 18

    34. Practice Find the GCF of 12 36 54 40, 56 20, 35

    35. Problem Solving Sasha and David are making centerpieces for the Fall Festival. They have 50 small pumpkins and 30 ears of corn. What is the greatest number of matching centerpieces they can make using all of the pumpkins and corn?

    36. Think and Discuss Tell what the letters GCF stand for and explain what the GCF of two numbers means in your own words. Discuss whether the GCF of two numbers could be a prime number.

    37. Learning Objectives To define LCM To find the LCM using a list To find the LCM using a factor tree To find the LCM using the birthday cake method

    38. 2-6 Least Common Multiple Multiple: number that is the product of that number and a whole number (skip-counting) Ex. 5, 10, 15, 20, 25 (multiples of 5) Least Common Multiple (LCM): the common multiple of two or more numbers with the least value

    39. Listing Method (can be time-consuming) Find the LCM of 3 and 5 3: 5: Find the LCM of 4, 6, 12 4: 6: 12:

    40. Birthday Cake Method Do the same as you did for GCF, but you multiply all of the numbers on the left and bottom together (L-shape) Find the LCM of 78 110 Find the LCM of 16 128

    41. Practice Find the LCM of 9 27 45 Find the LCM of 60 130

    42. Problem Solving Charlotte and her brother are running laps on a track. Charlotte runs one lap every 4 minutes, and her brother runs one lap every 6 minutes. They start together. In how many minutes will they be together at the starting line again?

    43. Think and Discuss Tell what the letters LCM stand for and explain what the LCM of two numbers is. Describe a way to remember the difference between GCF and LCM.

    44. 2-7 Variables and Algebraic Expressions Variable: letter that represents a number Ex. x, y, n, etc. Constant: a number because it cannot change Ex. 16, 25, 1954 Algebraic expression: consists of one or more variables and constants and operations Ex. N + 7 Evaluate: to substitute a number in for the variable N = 5 N + 7 5 + 7 = 12

    45. Learning Objectives To distinguish between a constant and a variable To evaluate algebraic expressions containing variables and constants To evaluate algebraic expressions using order of operations To evaluate algebraic expressions with more than one variable

    46. Evaluating Expressions N = 3 N + 7 x = 6 x -3 Y = 12 y 4

    47. Using Order of Operations 3x -2 for x = 5 n 2 + n for n = 4 6y2 + 2y for y = 2

    48. Evaluating with Two Variables 3/n + 2m for n = 3 and m = 4 3x 5y for x = 6 and y = 2

    49. Think and Discuss Write each expression another way. A. 12x B. 4/y C. 3xy/2 Explain the difference between a variable and a constant.

    50. Learning Objectives To translate words into algebraic expressions To translate algebraic expressions into words To translate real world problems into algebraic expressions

    51. 2-8 Translate Words into Math When solving real-world problems, you will need to translate words into algebraic expressions. Example: Although they are closely related, a Great Dane weighs about 40 times as much as a Chihuahua. 40c or 40 x c = Great Danes weight

    52. Addition and Subtraction Verbal Expressions Addition Subtraction Add Plus Sum More than Increased by Subtract Minus Difference Less than Decreased by Take away Less

    53. Multiplication and Division Verbal Expressions Multiplication Division Times Multiplied by Product Divided into Divided by Quotient

    54. Real-World Problems Jed reads p pages each day of a 200-page book. Write an algebraic expression for how many days it will take Jed to read the book. To rent a certain car for a day costs $84 plus $0.29 for every mile the car is driven. Write an algebraic expression to show how much it costs to rent the car for a day.

    55. Practice Write each phrase as an algebraic expression. The quotient of a number and 4 W increased by 5 The difference of 3 times a number and 7 The quotient of 4 and a number, increased by 10

    56. More Practice Mr. Campbell drives at 55 mi/hr. Write an expression for how far he can drive in h hours. On a history test Marissa scored 50 points on the essay.. Besides the essay, each short answer question was worth 2 points. Write an expression for her total points if she answered q short answer questions correctly.

    57. Think and Discuss Write three different verbal expressions that can be represented by 2 y. Explain how you would determine which operation to use to find the number of chairs in 6 rows of 100 chairs each.

    58. Learning Objectives To identify like terms in an algebraic expression or list To combine like terms given an expression To find the perimeter of a shape by combining like terms

    59. 2-9 Combining Like Terms Term: a number, variable, or product of numbers and variables Ex. 4a, 3k5 Coefficient: number that is multiplied by a variable Ex. 4 of 4a Like terms: terms with the same variable raised to the same power Ex. 3x and 2x, 5 and 1.8, 2x2 and 5x2

    60. Identifying Like Terms You cant combine apples and bananas!! You have to group similar objects Identify the like terms in the list 5a t/2 3y2 7t x2 4z k 4.5y2 2t 2/3a

    61. Combining Like Terms 7x + 2x 5x3 + 3y + 7x3 -2y -4x2 3a + 4q2 + 2b 45x -37y + 87

    62. More Practice 6t -4t 3a2 +5b + 11b2 -4b + 2a2 -6 2x + 3 + 3x + 2 + x

    63. Think and Discuss Identify the variable and the coefficient in each term A. 11t B. -3a C. 4/5n Explain whether 5x, 5x2, and 5x3 are like terms. Explain how you know which terms to combine in an expression.

    64. Learning Objectives To determine whether a number is a solution of an equation To determine whether a number is a solution from a story problem

    65. 2-10 Equations and Their Solutions Equation: a mathematical statement that two expressions are equal in value Its like a balanced scale. The left side is equal to the right side. Solution: the value for the variable that makes the equation true Ex. x + 3 = 10 7 is the solution for x

    66. Determine Whether a Number is a Solution 18 = s 7 Is 11 a solution? Is 25 a solution? 9y +2 = 74 Is 8 a solution?

    67. More Practice 13w 2 6w = 103 Does w = 15? 3(50 t) 10t = 104 Does t = 12?

    68. Practice Nicole has 82 CDs. This is 9 more than her friend Jessica has. The equation 82 = j + 9 can be used to represent the number of CDs Jessica has. Does Jessica have 91 CDs, 85 CDs, or 73 CDs?

    69. More Practice Tyler wants to buy a new skateboard. He has $57, which is $38 less than he needs. Does the skateboard cost $90 or $95?

    70. Think and Discuss Compare equations with expressions. Give an example of an equation whose solution is 5.

    71. Learning Objectives To define inverse operations To isolate the variable and solve bye adding or subtracting To identify inverse operations in a story problem

    72. 2-11 Solving Equations by Adding or Subtracting Solve: to find the solution of an equation Isolate the variable: get the variable alone on one side of the equal side Ex. X = 3 + 8 Inverse operations: opposite operations that undo each other Ex. Addition and subtraction Ex. Multiplication and division

    73. Solving an Equation with Addition Solve the equation x 8 = 17 Solve the equation y -11 = 20

    74. Solving an Equation with Subtraction Solve the equation a + 5 = 11 Solve the equation m + 16 = 25

    75. Practice Michael Jordans highest point total for a single game was 70. The entire team scored 117 points in that game. How many points did his teammates score? 70 + p = 117

    76. More Practice B - 7 = 24 T + 14 = 29 C 12 = 35

    77. Think and Discuss Explain how to decide which operation to use in order to isolate the variable in an equation. Describe what would happen if a number were added or subtracted on one side of an equation but not on the other side.

    78. Learning Objectives To solve equations by isolating the variable through multiplication or division To identify inverse operations in a story problem

    79. 2-12 Solving Equations by Multiplication or Division Multiplication and division are inverse operations of each other They undo each other

    80. Solving by Multiplication Solve the equation x/7 = 20 Solve the equation y9 = 2

    81. Solving by Division Solve the equation 240 = 4z Solve the equation 51 = 17x

    82. Real-Life Application If you count your heartbeats for 10 seconds and multiply that number by 6, you can find your heart rate in beats per minute. Lance Armstrong, who won the Tour de France four years in a row, from 1999 to 2002, has a resting heart rate of 30 beats per minute. How many times does his heart beat in 10 seconds? 6b = 30

    83. More Practice h2 = 13 t5 = 20 4x = 48

    84. Think and Discuss Explain how to check your solution to an equation. Describe how to solve 13x = 91. When you solve 5p = 35, will p be greater than 35 or less than 35? When you solve p5 = 35, will p be greater than 35 or less than 35?

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