Represent and evaluate addition and subtraction applications symbolically that contain more than one operation

Transcription

1 UNIT 3 ORDER OF OPERATIONS AND PROPERTIES INTRODUCTION Thus far, we have only performed one mathematical operation at a time. Many mathematical situations require us to perform multiple operations. The question that arises is, In what order do we perform the operations? In this lesson, we will look at the order of operations and properties of operations that will enable us to perform the operations in the correct order. The table below shows the learning objectives that are the achievement goal for this unit. Read through them carefully now to gain initial exposure to the terms and concept names for the lesson. Refer back to the list at the end of the lesson to see if you can perform each objective. Learning Objective Media Examples You Try Represent and evaluate addition and subtraction applications symbolically that contain more than one operation Represent and evaluate multiplication and division applications symbolically that contain more than one operation Represent and evaluate +,,, applications symbolically that contain more than one operation Represent and evaluate applications symbolically that use parentheses as a grouping symbol 1, 2 3 4, 5 6 7, , Represent applications using the notation of exponents 13 Write the language and symbolism of exponents in multiple ways Use PEMDAS to evaluate expression Use applications to show addition is commutative and subtraction is not commutative 18 Apply the commutative property of addition in context and symbolically 19, Use applications to show multiplication is commutative and division is not commutative 22 Determine what operations have the associative property 23 Use the associative property to evaluate expressions in multiple ways Use the distributive property in context and to evaluate expressions 26, Use additive identities and inverses with addition and subtraction problems Use multiplicative identities, inverses, and the zero property with multiplication and division problems

2 UNIT 3 MEDIA LESSON SECTION 3.1: ADDITION, SUBTRACTION AND THE ORDER OF OPERATIONS Problem 1 MEDIA EXAMPLE Addition, Subtraction and the Order of Operations Solve the problem below. Be sure to indicate every step in the process of your solution. a) Suppose on the first day of the month you start with $150 in your bank account. You make a debit transaction on the second day for $60 and then make a deposit on the third day for $20. What is the balance in your account on the third day? b) What string of operations (written horizontally) can be used to determine the amount in your account? Rule 1: When we need to add or subtract 2 or more times in one problem, we will perform the operations from left to right Problem 2 MEDIA EXAMPLE Addition, Subtraction and the Order of Operations performed in the problem. Then compute the results by using the convention of performing the operations from left to right. a) # of operations b) 12 ( 5) ( 1) # of operations Problem 3 YOU TRY - Addition, Subtraction and the Order of Operations performed in the problem. Then compute the results by using the convention of performing the operations from left to right. a) ( 2) # of operations b) 8 + ( 5) 6 ( 2) + 9 # operations 2

3 SECTION 3.2: MULTIPLICATION, DIVISION AND THE ORDER OF OPERATIONS Problem 4 MEDIA EXAMPLE Multiplication, Division and the Order of Operations Solve the problem below. Be sure to indicate every step in the process of your solution. a) Suppose you and your three siblings inherit $40,000. You divide it amongst yourselves equally. You then invest your portion and make 5 times the amount of your portion. How much money do you have? Be sure to indicate every step in your process. b) What string of operations (written horizontally) can be used to determine the result? Rule 2: When we need to multiply or divide 2 or more times in one problem, we will perform the operations from left to right. Problem 5 MEDIA EXAMPLE Multiplication, Division and the Order of Operations performed in the problem. Then compute the results by using the convention of performing the operations from left to right. a) 6 4 ( 2) 2 # of operations b) ( 3) # operations Problem 6 You Try Multiplication, Division and the Order of Operations performed in the problem. Then compute the results by using the convention of performing the operations from left to right. a) 8( 2) ( 4) ( 2) # of operations b) ( 1)(2) # operations 3

4 SECTION 3.3: THE ORDER OF OPERATIONS FOR +,,, Problem 7 MEDIA EXAMPLE The Order of Operations for +,,, Solve the two problems below. Be sure to indicate every step in your process a) Bill went to the store and bought 3 six-packs of soda and an additional 2 cans. How many cans did he buy in total? What string of operations (written horizontally) can be used to represent this problem? b) Amber went to the store and bought 3 six-packs of cola and an additional 2 six-packs of diet cola. How many cans did she buy in total? What string of operations (written horizontally) can be used to represent this problem? Rule 3: Unless otherwise indicated by parentheses, we perform multiplication and division before addition and subtraction. We continue to perform the operations from left to right. Problem 8 MEDIA EXAMPLE The Order of Operations for +,,, performed in the problem. Perform the operations in the appropriate order. Show all intermediary steps. a) ( 3) # of operations b) ( 3) # operations Problem 9 YOU TRY The Order of Operations for +,,, performed in the problem. Perform the operations in the appropriate order. Show all intermediary steps. a) ( 3) # of operations b) ( 3 )( 4) # operations 4

5 SECTION 3.4: PARENTHESES AS A TOOL FOR CHANGING ORDER There are cases when we want to perform addition and subtraction before multiplication and division in the order of operations. So we need a method of indicating we want to make such a modification. In the next media problem, we will discuss how to show this change. Problem 10 Solve the problems below. MEDIA EXAMPLE Parentheses as a Tool for Changing Order a) Howard bought a $25 comic book and a $35 belt buckle. He paid with a $100 bill. How much change will Howard receive? Be sure to indicate every step in your process. b) What string of operations (written horizontally) can be used to determine the amount in your account? Rule 4: If we want to change the order in which we perform operations in an arithmetic expression, we can use parentheses to indicate that we will perform the operation(s) inside the parentheses first. Problem 11 MEDIA EXAMPLE Parentheses as a Tool for Changing Order performed in the problem. Perform the operations in the appropriate order. Show all intermediary steps. a) 8 ( 4 + 2) # of operations b) 3 [6 (5 + 2)] # operations Problem 12 YOU TRY - Parentheses as a Tool for Changing Order performed in the problem. Then compute the results by using the convention of performing the operations from left to right. a) 6 ( 4 ( 7)) # of operations b) 13 [9 + ( 6 2)] # operations 5

6 Problem 13 SECTION 3.5: EXPONENTS MEDIA EXAMPLE Introduction to Exponents Solve the problem below. Use the rectangle below to represent the problem visually. a) Don makes a rectangular 20 square foot cake for the state fair. After he wins his award, he wants to share it with the crowd. First he cuts the cake into 2 pieces. Then he cuts the 2 pieces into 2 pieces each. Then he cuts all of these pieces into two pieces. He continues to do this a total of 5 times. How many pieces of cake does he have to share? b) Write a mathematical expression that represents the total number of pieces in which Don cut the cake. Terminology We will use exponential expressions to represent problems such as the last one. Exponents represent repeated multiplication just like multiplication represents repeated addition as shown below. Multiplication: 5 2 = = 10 Exponents: 2 5 = = 32 In the exponential expression, is called the base 5 is called the exponent We will say 2 5, as 2 raised to the fifth power or 2 to the fifth Since exponents represent repeated multiplication, and we call the numbers we multiply factors, we will also use this more meaningful language when discussing exponents. 2 5 means 5 factors of 2 We also have special names for bases raised to the second or third power. a) For 3 2, we say 3 squared or 3 to the second power b) For 4 3, we say 4 cubed or 4 to the third power 6

7 Problem 14 MEDIA EXAMPLE Language and Notation of Exponents Represent the given exponential expressions in the four ways indicated. a) 6 2 b) 6 2 Expanded Form Word Name Factor Language Math Equation Expanded Form Word Name Factor Language Math Equation c) ( 6) 2 d) ( 5) 3 Expanded Form Word Name Factor Language Math Equation Expanded Form Word Name Factor Language Math Equation Problem 15 YOU TRY Language and Notation of Exponents Represent the given exponential expressions in the four ways indicated. a) 7 2 b) ( 7) 2 Expanded Form Word Name Factor Language Math Equation Expanded Form Word Name Factor Language Math Equation 7

8 SECTION 3.6: PEMDAS AND THE ORDER OF OPERATIONS Finally, we will consider problems that may contain any combination of parentheses, exponents, multiplication, division, addition and subtraction. Problem 16 MEDIA EXAMPLE PEMDAS and the Order of Operations Rule 5: Exponents are performed before the operations of addition, subtraction, multiplication and division. P E M D A S Simplify items inside Parentheses ( ), brackets [ ] or other grouping symbols first. Simplify items that are raised to powers (Exponents) Perform Multiplication and Division next (as they appear from Left to Right) Perform Addition and Subtraction on what is left. (as they appear from Left to Right) performed in the problem. Then compute the results by using the correct order of operations. Check your results on your calculator. a) (8 3) 2 4 b) c) ( 3) 2 4( 3) + 2 Problem 17 YOU TRY PEMDAS and the Order of Operations performed in the problem. Then compute the results by using the correct order of operations. Check your results on your calculator. a) 7 (2 3) 2 # of operations b) ( 4) 2 + 5( 4) 6 # operations 8

9 SECTION 3.7: THE COMMUTATIVE PROPERTY Problem 18 MEDIA EXAMPLE The Commutative Property of Addition Solve the following problems in Problem Sets A and B and fill in the blanks in the results section. Problem Set A: a) Sheldon had $5 and earned $7 more organizing a closet. How much does he have altogether? b) Leonard had $7 and earned $5 more solving math problems. How much does he have altogether? Problem Set B: a) The temperature in Minnesota was 8. It dropped 5 after sunset. What was the temperature after sunset? b) The temperature in Alaska was 5. It dropped 8 after sunset. What was the temperature after sunset? Results: Fill in the blanks. a) When you add two numbers and reverse the order of the addends, the sums are b) When you subtract two numbers and reverse the order of the minuend and subtrahend the differences are Commutative Property of Addition: Reversing the order of the addends in an addition problem doesn t change the sum. In particular, From Problem Set A, $5 + $7 = $7 + $5 Subtraction is NOT commutative. (Note: means not equal to) From Problem Set B,

10 Problem 19 MEDIA EXAMPLE Applying the Commutative Property of Addition Unit 3 Media Lesson Raj recorded his weekly expenditures and deposits in a notebook. He wrote down the following expression to represent his current balance. a) Find the balance in Raj s account b) Rewrite Raj s expression using only addition below. c) Use your result from problem b to complete the table below. Deposits (+) Withdrawals ( ) Totals (Find the sums) d) Use the chart to find the Raj s Balance. Problem 20 MEDIA EXAMPLE Applying the Commutative Property of Addition Rewrite the following problems changing all subtractions to adding the opposite. Combine the positive terms and negative terms separately. Then add the signed numbers to find the result. a) 8 ( 5) ( 1) b) 10 + ( 8) 2 ( 1) 6 Rewrite as addition: Rewrite as addition: Sum of the positive terms: Sum of the negative terms: Combine the positive and negative results: Sum of the positive terms: Sum of the negative terms: Combine the positive and negative results: 10

11 Problem 21 YOU TRY Applying the Commutative Property of Addition Rewrite the following problems changing all subtractions to adding the opposite. Combine the positive terms and negative terms separately. Then add the signed numbers to find the result. a) 8 + ( 5) 6 ( 2) + 9 b) 7 ( 4) 2 + ( 1) Rewrite as addition: Rewrite as addition: Sum of the positive terms: Sum of the negative terms: Combine the positive and negative results: Sum of the positive terms: Sum of the negative terms: Combine the positive and negative results: Problem 22 MEDIA EXAMPLE The Commutative Property of Multiplication Solve the following problems in Problem Sets A and B and fill in the blanks in the results section. Problem Set A: a) Penny made 3 batches of cookies with 6 cookies per batch. How many cookies did she make in total? b) Amy made 6 batches of cupcakes with 3 cupcakes per batch. How many cupcakes did she make in total? Problem Set B: a) Two people are sharing 4 pizzas. How much pizza does each person get? b) Four people are sharing 2 pizzas. How much does each person get? c) How are these problems similar? How are they different? Results: Fill in the blanks. 1. When you multiply two numbers and reverse the order, the products are 2. When you divide two numbers and reverse the order the quotients are Commutative Property of Multiplication: Reversing the order of the factors in a multiplication problem doesn t change the product. In particular, From Problem Set A, multiplication is commutative 3 6 = 6 3 From Problem Set B, division is NOT commutative

12 SECTION 3.8: THE ASSOCIATIVE PROPERTY Problem 23 MEDIA EXAMPLE The Associative Property Complete the following table by performing the indicated operations by computing the result in the parentheses first as the order of operations necessitates. Operation Problem 1 Problem 2 Are the Results the Same? Addition (5 + 7) (7 + 3) Subtraction (10 5) 4 10 (5 4) Multiplication (2 3) 4 2 (3 4) Division (600 30) (30 5) Results: 1. Addition and Multiplication both enjoy the Associative Property. This means, (a + b) + c = a + (b + c) and (a b) c = a (b c) 2. In general, this is not the case for the operations and subtraction and division. 3. This means that if you have a problem where the operations are either all addition or all multiplication, you can add or multiply in any order you want regardless of the parentheses. You can remove the parentheses altogether to simplify the expression. Problem 24 MEDIA EXAMPLE The Associative Property Use the commutative and associative properties to perform the operations in the order you find most simple. a) ( ) + 7 b) 5 (6 8) 12

13 Problem 25 You Try The Associative Property Determine if the associative property applies in the given problems. If so, rewrite the problem using the associative property to perform the operations. If not, perform the operations as shown. a) ( ) + 12 b) 15 (4 6) c) 4 (5 13) Problem 26 SECTION 3.9: THE DISTRIBUTIVE PROPERTY MEDIA EXAMPLE The Distributive Property Use the diagram and information below to answer the following questions. Linda has a rectangular flower bed that is 10 feet long and 4 feet wide. She is going to plant Morning Glories and Tulips. She wants more Morning Glories than Tulips and decides to divide the garden as shown below. a) Determine the total area of the garden by multiplying its total length times its width. b) Determine the total number of square feet that will be planted with tulips. c) Determine the total number of square feet that will be planted with morning glories. d) Determine the total area of the garden by adding the area planted with tulips and the area planted with morning glories. e) Using the equation below, describe in words how it represents the two ways we found the total area of the garden = 4 (3 + 7) The relationship illustrated in Problem 26 is called the distributive property. Distributive Property of Multiplication over Addition (or Subtraction) Examples A. Multiplication over Addition: 4 (3 + 7) = B. Multiplication over Subtraction: 3 (6 4) =

14 Problem 27 MEDIA EXAMPLE The Distributive Property Evaluate the given expressions in the two ways indicated. a) Problem Perform Parentheses First Rewrite Using Distributive Property and Evaluate Result 5(3 + 4) b) 3(7 5) c) 4(5 + 2) d) (3 7) Problem 28 YOU TRY The Distributive Property Evaluate the given expressions in the two ways indicated. a) Problem 2(3 + 5) Perform Parentheses First Rewrite Using Distributive Property and Evaluate Result b) 3(4 2) 14

15 SECTION 3.10: INVERSES, IDENTITIES, ONES, AND ZEROS Definitions: 1. We call the number zero the additive identity for the operation of addition since adding 0 to any number doesn t change the numbers value. For example, = 3 or = We call the opposite of a number its additive inverse since adding any number and its opposite gives a result of 0. For example, 3 + ( 3) = 0 or ( 3) + 3 = Since subtraction is not commutative, identities do not directly apply to subtraction, but we can use similar ideas in some cases. For example, 3 0 = 3 but 0 3 = Since subtraction is not commutative, inverses do not directly apply to subtraction, but we can use similar ideas in some cases. For example, 3 3 = 0 or ( 3) ( 3) = 0. In particular, any number minus itself is 0. Problem 29 Perform the following operations. 1. Add. MEDIA EXAMPLE Adding and Subtracting with Zeros and Opposites a) = b) = c) 5 + ( 5) = d) ( 5) + 5 = e) ( 5) + 0 = f) 0 + ( 5) = g) = h) ( 5) + ( 5) = 2. Subtract. a) 5 0 = b) ( 5) 0 = c) 5 5 = d) ( 5) ( 5) = e) 0 5 = f) 0 ( 5) = g) 5 ( 5) = h) ( 5) 5 = Problem 30 Perform the following operations. 1. Add. YOU TRY Adding and Subtracting with Zeros and Opposites a) 0 + ( 4) = b) ( 4) + 4 = c) ( 4) + ( 4) = d) = e) = f) 4 + ( 4) = 2. Subtract. a) 0 4 = b) 0 ( 4) = c) ( 4) 4 = d) 4 0 = e) ( 4) 0 = f) ( 4) ( 4) = 15

16 Definitions: 1. We call the number one the multiplicative identity for the operation of multiplication since multiplying any number by 1 doesn t change the numbers value. For example, 3 1 = 3 or 1 3 = Since division is not commutative, identities do not directly apply to division, but we can use similar ideas in some cases. For example, 3 1 = 3 but Multiplicative inverses are fractions, so we will not discuss them here. However, since division is the inverse operation of multiplication, we can divide any nonzero number by itself and the result is the identity 1. For example, 3 3 = 1 or ( 3) ( 3) = 1 4. The zero property of multiplication states that any number multiplied by 0 is 0. For example, 3 0 = 3 and 0 3 = 0 5. Since division is not commutative, the zero property of multiplication does not apply to division, but we can use similar ideas in some cases. For example, 0 3 = 0 but 3 0 does not exist! To see why this is the case, rewrite the division problems as multiplication problems with missing factors. 0 3 =? is equivalent to 3? = 0. Here,? = 0 makes the statement true. However, 3 0 =? is equivalent to 0? = 3. But the zero property of multiplication states that any number multiplied by zero is zero. So there does not exist a number for? that would make 0? = 3 true. So dividing any nonzero number by 0 is undefined. Problem 31 Perform the following operations. 1. Multiply. MEDIA EXAMPLE Multiplying and Dividing with Zeros and Ones a) 1 5 = b) ( 5) 1 = c) 5 ( 1) = d) ( 1) ( 5) = f) 5 0 = f) 0 ( 5) = g) 5 0 = h) ( 5) 0 = 2. Divide. a) 5 1 = b) ( 5) 1 = c) 5 5 = d) ( 5) ( 5) = f) 5 ( 1) = f) ( 5) ( 1) = g) 0 ( 5) = h) ( 5) 0 = Problem 32 Perform the following operations. 1. Multiply. YOU TRY Multiplying and Dividing with Zeros and Ones a) 0 ( 4) = b) 4 ( 1) = c) 4 0 = d) ( 1) ( 4) = 2. Divide. a) 4 ( 1) = b) ( 4) ( 4) = c) ( 4) 0 = d) 0 ( 4) = 16