Unit Essential Questions What are the different representations of exponents? Where do exponents fit into the real number system? How can exponents be used to depict real-world situations?
REAL NUMBERS AND THEIR SUBSETS MGSE8.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. MGSE8.NS.: Use rational approximation of irrational numbers to compare the size or irrational numbers, locate them approximately on a number line, and estimate the value of expressions. TARGET RATING 4 3 1 Warm Up/Vocabulary LEARNING SCALE Provide counterexamples involving the sum and product of real number subsets Classify real numbers into real number subsets Understand the sum and product of real number subsets Classify real numbers into real number subsets with help Understand the sum and product of real number subsets with help Understand that real numbers can be divided into subsets ORDER OF OPERATIONS P E M D PARENTHESIS perform any operations inside grouping symbols, such as parenthesis ( ), brackets [ ], and a fraction bar. EXPONENTS simplify powers MULTIPLY AND DIVIDE from LEFT TO RIGHT (not multiplication before division) A S ADDITION AND SUBTRACTION from LEFT TO RIGHT (not addition before subtraction) Use order of operations to simplify. 1) 3 4 + 6 4 ) 5[( + 5) 3] 3) 40 + 4 8 1 SUBSETS OF REAL NUMBERS NAME DESCRIPTION EXAMPLES Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers
Practice 1: IDENTIFYING SUBSETS OF REAL NUMBERS Your math class is selling pies to raise money to go to a math competition. Which subset of real numbers best describes the number of pies p that your class sells? Practice : CLASSIFYING NUMBERS INTO SUBSETS OF REAL NUMBERS For each number, place a check in the column that the number belongs to. Remember the numbers may belong to more than one set. # Number Real Whole Natural Integer Rational Irrational a) 9 b) 4 c) 81 d) e) 5 10 f) 0 g) 4 h) 3 + 1 RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 4 3 1
Exponents MGSE8.EE.1: Know and apply the properties of integer exponents to generate equivalent numerical expressions. MGSE8.EE.: Use square root and cube root symbols to represent solutions to equations. RATING 4 3 TARGET 1 LEARNING SCALE Produce examples of exponents in real-world situations. Understand the rules for manipulating exponents. Apply the appropriate rule(s) to create and solve numerical expressions. Compute and simplify exponents Recite the rules for manipulating exponents. Name the parts of an exponent and describe its function. KEY CONCEPTS AND VOCABULARY 1. has two parts, base and exponent. a number that shows repeated multiplication 3. a number that is multiplied repeatedly 4. to replace an expression with its simplest form 5. to substitute a given number for each variable, and then simplify Exponent: a quantity representing the power to which a given number or expression is to be raised. Example: 3 4 is the same as writing 3 3 3 3. The base, 3, is being multiplied by itself 4 times. Simplified Form Properties of Exponents: Expanded Form (x 4 ) 3 = (x x x x) (x x x x) (x x x x) = x 4 3 = x 1 1. Multiplying Exponents with the Same Base - Rule: You add the exponents and keep the base. - x 4 x 3 = (x x x x) (x x x) = x 4+3 = x 7 - Example: 3 1 3 8 = 3 0. Dividing Exponents with the Same Base - Rule: You subtract the exponents and keep the base. - x 4 x 3 = (x x x x) = x 4-3 = x 1 or x (anything raised to the power of 1 is (x x x) still that same number). - Example: 3 1 3 8 = 3 4 3. Raising a Power to a Power - Rule: You multiply the exponents and keep the base. - - Example: (3 1 ) 8 = 3 96 4. Negative Exponents - Rule: Take the reciprocal of the exponent and make it positive. - x -4 = 1 x 4 - Example: 3-1 = 1 = 1 3-1 531,441
Practice 1: Expanded vs. Simplified: Write the following exponents in either expanded or simplified form. A. 1 4 B. 3 C. 9 9 9 9 9 9 D. 45,000 Practice : Create the Expanded Form: Write the following exponents in expanded form. A. 17 8 B. 9 3 C. 18 4 D. 10 6 Practice 3: Computation: Evaluate each expression? A. 4 B. 8 5 C. D. (0.3) 3 Practice 4: Simplifying: Simplify each of the following expressions. A. (b 3 b ) 3 B. 5k 4 8k - 3k -5 C. ( 3 d4 3 4 d 10 ) 4 Square Root: a number that produces a specified quantity when multiplied by itself. The number that the square root produces is called a. Taking the square root is the or opposite of squaring. Example: 7, which is read Seven Squared, = 49, so The square root of 49 is 7. - 7 is the square root. - 49 is the perfect square. Computing square roots is denoted by using a. The index tells which root to look for (square, cube, etc.). If there is no index outside the radical, it s automatically assumed to be square root. The radical symbol tells you that a root is about to be computed. The radicand is the number that we re finding the root for. 3 Example: 1. 11 = 11. 64 4 = 4 3. 10,000 = 10
Practice 1: Computing Roots: Evaluate the following root problems. 3 A. 169 B. 7 4 C. 65 Non-Perfect Square Roots: Computing non-perfect square roots is reducing a radical down to its lowest possible form. Example: 16 Step 1: Factor the radicand. --------------------------------- 1,, 3, 6, 7, 9, 14, 18, 1, 4, 63, 16 Step : Find the largest perfect square of the factors. ----------------------------------------- 9 Step 3: Rewrite the problem under the radical. ------------------------------------------------- 9 14 Step 4: Take the square root of the perfect square and put it out side the radical. -- 3 14 Step 5: Record your answer. Reminder: 3 is NOT an index. It is written larger and in front because it is a coefficient, meaning it is being multiplied by the radical. Practice 1: Simplifying: Simplify each of the following expressions. A. 180 B. 175 C. 4 81 RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 4 3 1
Scientific Notation MGSE8.EE.3: Understand and use numbers expressed in scientific notation to estimate very large or small quantities, and to express how many times as much one is than the other. MGSE8.EE.4: Add, subtract, multiply, and divide numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. TARGET RATING 4 3 1 WARM UP/Vocabulary LEARNING SCALE represent a represent and compute scientific notation as it applies to real-world situations represent scientific notation compute scientific notation represent scientific notation with help compute scientific notation with help the difference in regular notation vs. scientific notation. 1. (RN) The regular way that we write numbers. Ex. Two hundred and eighty million is written as. (SN) The Scientific way that we write really large or really small numbers. Ex. 15,000,000 is written as 3. The only number allowed in front of the decimal when written in scientific notation. * There can only be 1, non-zero number in front of the decimal. 4. Scientific Notation is always written to the power of Example. 3.4 x 10 5 = 340,000 Regular notation Scientific Notation If the decimal is moved, then the exponent is and add zeros in the needed spaces. If the decimal is moved, then the exponent is and add zeros in the needed spaces.
Practice 1: Identifying notations: Decide whether the number is written in standard form or scientific notation. A) 7,450 B) 7.5 x 10 3 C) 4 x 10 5 EXAMPLE : Identifying which way to move the decimal (Left or Right): Decide which way to move the decimal, right or left. A. 9.7 x 10 7 B. 7.5 x 10-3 C. 4.6 x 10-8 EXAMPLE 3: Identifying how to convert from standard to scientific notation: Convert each number written in standard notation to scientific notation. A. 7,100,000 B. 800,400 C..00000089 EXAMPLE 4: Identifying how to convert from scientific to standard notation: Convert each number written in scientific notation to standard notation. A. 3 x 10 3 B..07 x 10 6 C. 8.97 x 10-3 Adding and Subtracting: When adding and subtracting in scientific notation, you must ensure that your exponents are the same. Making the exponents the same ensures that you re keeping with the appropriate place value. Example: (7.5 x 10-3 ) + (4.6 x 10-8 ) Step 1: Find the difference in the exponents. -3 (-8) = 5, so you will have to move your decimal 5 times. Step : Move the smaller exponent to the left to make it larger. (7.5 x 10-3 ) + (.000046 x 10-3 ) Step 3: Line the decimals up and add the numbers. 7.5 +.000046 7.50046 Step 4: Keep the exponent the same. 7.50046 x 10-3
Practice 1: Adding and Subtracting in Scientific Notation: A. ( 10 3 ) + (5 10 5 ) B. (4.9 10 ) + (7.8 10 3 ) C. (1 10 6 ) (8 10 4 ) Multiplying and Dividing: Unlike addition and subtraction, you DO NOT have to change the exponents before computing. Multiplication Example: (7.5 x 10-3 ) (4.6 x 10-8 ) Step 1: Multiply the factors. 7.5 4.6 = 33.35 Step : Multiply the exponents. 10-3 10-8 = 10 (-3)+(-8) = 10-11 Step 3: Since you can only have 1 significant digit in front of the decimal, you must move the decimal, thereby changing your exponent.. 33.35 x 10-11 becomes 3.335 x 10-10 Division Example: (.176 x 10-3 ) (3. x 10-8 ) Step 1: Divide the dividend by the divisor..176 3. = 0.68 Step : Divide the exponents. 10-3 10-8 = 10 (-3)-(-8) = 10 5 Step 3: Since you can only have 1 significant digit in front of the decimal, you must move the decimal, thereby changing your exponent.. 0.68 x 10 5 becomes 6.8 x 10 4 Practice 1: Multiplying and Dividing in Scientific Notation: A. ( 10 3 ) (5 10 5 ) B. (4.9 10 ) (7.8 10 3 ) C. (1 10 6 ) (8 10 4 ) RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 4 3 1
TARGET RATING 4 3 1 LEARNING SCALE VARIABLES AND EXPRESSIONS MGSE8.EE.7: Solve linear equations in one variable. write algebraic expressions and apply them to real world situations or more challenging problems that I have never previously attempted write algebraic expressions write algebraic expressions with help understand that algebra uses symbols to represent quantities that are unknown or may vary WARM UP Which of these situations have a value that varies? a) The population of this school b) The number of classrooms in this school c) The time it takes you to get to your next class d) Your high school GPA KEY CONCEPTS AND VOCABULARY Addition Subtraction Multiplication Division sum plus added to more than increased by difference minus subtract less than decreased by less fewer than product times multiply multiplied by of double/ triple quotient divide shared equally divided by divided into anything that can be measured or counted a symbol, usually a letter, that represents the value of a variable quantity a mathematical phrase that includes one or more variables a mathematical phrase involving numbers and operation symbols, but no variables Practice 1: REWRITING A WORD EXPRESSION (ADDITION OR SUBTRACTION) Write an algebraic expression for each word phrase. a) 3 more than f b) 10 less than c c) 5 decreased by p
Practice : REWRITING A WORD EXPRESSION (MULTIPLICATION OR DIVISION) Write an algebraic expression for each word phrase. a) the quotient of 9 and k b) the product of 15 and y c) r divided by 5 d) twice a number s Practice 3: REWRITING A WORD EXPRESSION (WITH VARIOUS OPERATIONS) Write an algebraic expression for each word phrase. a) The sum of 4 and twice y b) 7 less than the product of y and z c) 5 minus the quotient of x and y d) 4 more than twice the number z Practice 4: REWRITING AN ALGEBRAIC EXPRESSION Write the word phrase for each algebraic expression. a) d + 5 b) p 3 c) x d) x/7 e) 100 + 6y f) c - 85 Practice 5: WRITING AN ALGEBRAIC EXPRESSION FOR REAL WORLD SITUATIONS A car salesman gets paid a weekly salary of $300. They are also paid $100 for each car that they sell during the week. Write a rule in words and as an algebraic expression to model the relationship in the table. Cars Sold Total Earned 0 $300 + (0 x $100) 1 $300 + (1 x $100) $300 + ( x $100) n RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 5 4 3 1
TARGET AN INTRODUCTION TO EQUATIONS MGSE8.EE.7: Solve linear equations in one variable. MGSE8.EE.7a: Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. MGSE8.EE.7b: Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. RATING 4 3 1 WARM UP/Vocabulary 1. A mathematical statement that contains algebraic expressions and symbols is an.. An is a mathematical sentence that uses an equal sign (=). 3. A of an equation containing a variable is a value that makes the equation true. 4. - an equation that is true for every value of the variable. 5. is a number, a variable, or the product of a number and one or more variables 6. is a term that has no variable 7. is a numerical factor of a term 8. have the same variable factors (same variables raised to the same power). 9. An expressions is in when it contains no like terms or parenthesis. LEARNING SCALE write and solve equations with one variable and apply them to real world situations or more challenging problems that I have never previously attempted. write and solve equations with one variable. Write and solve equations with infinite solutions. Write and solve equations with no solution. write and solve equations with one variable with help. understand that an equation is a mathematical sentence. THE DISTRIBUTIVE PROPERTY For any numbers a, b and c a (b + c) = ab + ac and (b + c)a = ab + ac and a(b c) = ab ac and (b c)a = ab ac
Obtaining Solutions: When solving equations with one variable, there are two options: 1. Substitution. Isolation Substitution: - Can only be done with the solution for the variable is given. - Example: Is x = 7 a solution of the equation x + 10 = 3? o Step 1: Substitute (plug in) the 7 everywhere you see the variable x. (7) + 10 = 3 o Step : Following order of operations, compute the equations. 14 + 10 = 3 4 = 3 o Step 3: Determine if both sides of the equal sign are the same. If both sides are, it is a solution, but if they aren t, then it IS NOT a solution. Numerical Answer: 4 3, NO 4 does not equal 3, so 7 isn t a solution to the equation. Isolation: - When you must get the variable alone on one side of the equal sign and move all the numbers to the other side. - You must work backwards to undo everything being done to the variable. - Can be done whenever the solution to the variable isn t given. - Example: x + 10 = 3? o Step 1: Simplify the equation by combining any like terms on the same side. x + 10 = 3 ; this problem is already simplified o Step : Following the order of operations, the last thing that would have been done on is adding 10, so we must do the inverse and subtract 10 from both sides of the equal sign. x + 10 = 3-10 - 10 x = 13 o Step 3: The next thing being done to the variable is multiplying it by the coefficient (), so we must divide by the coefficient on both sides of the equal sign. x = 13 x = 6.5 ; 6.5 is the solution to the problem.
Practice 1: Using Mental Math to find Solutions What is the solution of each equation? a) b) c) Practice : Identifying Solutions of an Equation Determine if the given value is a solution to the equation. a) Is x = 8 a solution of the equation 4x + 10 = 4? b) Is x =10 a solution of the equation? Practice 3: APPLYING THE ORDER OF OPERATIONS Solve. a) b) c) d) Practice 4: EVALUATING AN ALGEBRAIC EXPRESSION What is the value of the expression for x = 1 and y = 3? a) b) c) d)
Practice 5: WRITING EQUATIONS Write an equation for each sentence. a) The sum of 3x and 5 is 13. b) The product of x and 4 is 64. Practice 6: REWRITING FRACTION EXPRESSIONS Write each fraction as a sum or difference. a) b) Practice 7: WRITING AND SIMPLIFYING EXPRESSIONS Use the expression twice the sum of 4x and y increased by six times the difference of x and 3y. a) Write an algebraic expression for the verbal expression. b) Simplify the expression. Practice 8: WRITING EQUATIONS FOR REAL WORLD SITUATIONS A grocery store cashier makes $1.50 more per hour than a bagger. Write an equation that relates the amount x that a bagger earns each hour if a cashier makes $10.5 per hour. RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 5 4 3 1