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Chapter Focus 1 1 Using the Interactive Student Guide The Interactive Student Guide (ISG) can be used in conjunction with Algebra 1. ISG Lesson Expressions, Equations, and Functions CHAPTER FOCUS Learn about some of the Common Core State Standards that you will explore in this chapter. Answer the preview questions. As you complete each lesson, return to these pages to check your work. What You Will Learn N.RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Extend 10–2 1.2 Lesson 1–1 1.3 Lesson 1–2 1.4 Lessons 1–3, 1–4 1.5 Lesson 1–5 1.6 Lesson 1–6 1.7 Lesson 1–7 SMP 3 When you multiply two rational numbers, is the result always another rational number? Why or why not? The result is always another rational number because the result can be expressed as a quotient of two integers. Algebra 1 1.1 Preview Question Lesson 1.1: Rational and Irrational Numbers SMP 7 Give an example of a product of two irrational numbers that is also irrational and a product of two irrational numbers that is rational. _ _ _ Sample answer: √2 · √3 = √6 is irrational. _ _ √2 · √2 = 2 is rational. Lesson 1.2: Variables and Expressions A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. SMP 2 The expression 14x + 5 represents the cost of ordering x copies of a book online, including a flat fee for shipping. Which term in the expression represents the shipping fee? How do you know? The 5 in the expression is the shipping fee because this term of the expression does not depend on the number of books ordered. Lesson 1.3: Order of Operations A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. SMP 6 How do you evaluate the expression 52 - 18 ÷ 6 + 3? Do you get a different result if the expression is written as 52 - 18 ÷ (6 + 3)? Why or why not? Per order of operations, exponents are simplified first followed by multiplication and division from left to right and then addition and subtraction Copyright © McGraw-Hill Education from left to right. With parentheses, the value is 23 rather than 25 because (6 + 3) is found first. 2 CHAPTER 1 Expressions, Equations, and Functions 002_003_ALG1_C1FP_ISG_672788.indd 2 2 CHAPTER 1 Expressions, Equations, and Functions 2/20/15 5:21 PM Copyright © McGraw-Hill Education SMP 2 Teaching Tip The preview question for Lesson 1.2 gives students a taste of SMP 2 (Reason abstractly and quantitatively). Point out to students that the expression 14x + 5, by itself, is an “abstract” mathematical expression. In other words, it can be manipulated and understood using mathematical rules, but it does not necessarily carry a real-world meaning. Explain that the scenario of ordering books online gives the expression a “context.” Students will see many additional examples of this throughout the course and have many opportunities to move back and forth between abstract mathematics and contextualized mathematics. What You Will Learn Preview Question Lesson 1.4: Properties of Numbers A.SSE.2 Use the structure of an expression to identify ways to rewrite it. A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. SMP 7 How can you rewrite the expression 8 · 89 so it can be simplified using mental math? 8 · 89 = 8(90 - 1) = 8 · 90 − 8 · 1 = 720 - 8 = 712 Lesson 1.5: Equations A.CED.1 Create equations and inequalities in one variable and use them to solve problems. A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. SMP 4 Jonas spent $7.75, excluding tax, at the store when he bought a notebook that cost $2.50 and 3 pens. Write and solve an equation to find the price of each pen. Let x = the cost of each pen, 3x + 2.50 = 7.75; x = 1.75; each pen costs $1.75 Lesson 1.6: Relations A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). SMP 1 Use set notation to write the domain and range of the relation {(-2, 3), (4, 4), (4, 5), (5,-8)}. Domain: {-2, 4, 5}; Range: {3, 4, 5, -8} Also addresses: F.IF.1 Copyright © McGraw-Hill Education Lesson 1.7: Functions F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. SMP 3 A student said that the relation {(-2, 3), (4, 4), (4, 5), (5,-8)} is a function. Do you agree? Why or why not? No; the domain value 4 is paired with two SMP 4 Teaching Tip The preview question for Lesson 1.5 can serve as a jumping-off point for a brief discussion of SMP 4 (Model with mathematics). Explain to students that mathematical modeling is the process of using mathematics to describe a real-world situation and/or solve a real-world problem. Ask students to state the real-world situation in this question in their own words. Then ask them how they think mathematics could be useful in solving the problem. Tell students that equations are one way to model a real-world situation and explain that they will have many opportunities to develop and use this type of model throughout this course. different range values, so the relation cannot be a function. SMP 2 The amount of money in Trina’s bank account, in dollars, n months after she opens the account is given by the function b(n) = 50n + 100. What are the meanings of the values 50 and 100? What is the balance in Trina’s account after 10 months? The 50 means that Trina adds $50 to her account each month. The 100 means that Trina opened her account with a deposit of $100. Trina will have $600 in her account after 10 months. CHAPTER 1 Chapter Focus 3 2/24/15 11:53 PM Copyright © McGraw-Hill Education 002_003_ALG1_C1FP_ISG_672788.indd 3 CHAPTER 1 Chapter Focus 3 1.1 Rational and Irrational Numbers 1.1 Rational and Irrational Numbers STANDARDS Objectives STANDARDS Content: N.RN.3 Practices: 2, 3, 4, 5, 6, 7 Use with Extend 10–2 • Explain why the sum or product of two rational numbers is rational. N.RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. • Explain why the sum of a rational number and an irrational number is irrational. • Explain why the product of a nonzero rational number and an irrational number is irrational. a A rational number is a number that can be written in the form __b , where a and b are integers and b ≠ 0. The decimal form of a rational number is a repeating decimal or a terminating decimal. a An irrational number is a number that cannot be expressed in the form __b , where a and b are integers and b ≠ 0. The decimal form of an irrational number is neither repeating nor terminating. EXAMPLE 1 Standards for Mathematical Practice: 2, 3, 4, 5, 6, 7 Sums of Rational and Irrational Numbers N.RN.3 EXPLORE The table shows examples of rational and irrational numbers. a. CALCULATE ACCURATELY Choose two of the rational numbers from the table and find their sum. Is the sum rational or irrational? Explain how you know. SMP 6 1 1 2 11 1 __ __ __ __ Sample answer: __ 3 + 0.4 = 3 + 5 = 15 . The sum of 3 and 0.4 is Rational Numbers __1 __3 3 0.4- 7- 5 Irrational Number __ __ √2 π-√5 a rational because the sum can be written as __ b , where a and b PREREQUISITES are integers b ≠ 0. • Identify rational and irrational numbers • Use basic properties of square roots Teaching Tip The sum of two rational numbers is another rational number. c. USE TOOLS Choose a rational number and an irrational number from the table and find their sum using a calculator. Does the sum appear to be rational or SMP 5 irrational? Why? SMP 5 Part c offers an opportunity to address SMP 5 (Use appropriate tools strategically). In particular, you may want to discuss the limitations of using a calculator to decide whether a number is rational or irrational. Remind students that calculators can only show a limited number of digits, but that they can look for terminating or repeating decimals to make an educated guess about whether a number is rational. neither repeating nor terminating. d. MAKE A CONJECTURE Compare your result for part c with those of other students. Then make a conjecture about the sum of a rational number and an irrational number. SMP 3 The sum of a rational number and an irrational number is an irrational number. 4 CHAPTER 1 Expressions, Equations, and Functions Math Background 004_007_ALG1_C1L1_ISG_672788.indd 4 22/08/14 9:18 PM Every point on the number line represents a real number. Every real number is either rational or irrational. In this lesson, students state and justify generalizations about certain combinations of rational and irrational numbers, such as the sum of two rational numbers. Students may ask about other combinations, such as the sum of two irrational numbers. In this case, the results are more complicated. In fact, the sum of_two irrational numbers may be rational or irrational. _ For example, √2 + √_3 is irrational. _ However, the sum of the irrational numbers √2 and (1 - √ 2 ) is 1, which is rational. Students will explore these combinations, and other examples of irrational numbers (such as e), in later mathematics courses. • In part c, how can a calculator help you decide if the sum is rational or irrational? Look at the decimal form of the sum to see if it repeats or terminates. 4 CHAPTER 1 Expressions, Equations, and Functions Copyright © McGraw-Hill Education Scaffolding Questions • What should you do to decide if the sum of the two rational numbers you chose is rational or irrational? Check to see if the sum can be written as a ratio of two integers. 1 Sample answer: __ 3 + π ≈ 3.474925987. The sum appears to be irrational because the decimal is Copyright © McGraw-Hill Education EXAMPLE 1 b. MAKE A CONJECTURE Compare your result for part a with those of other students. SMP 3 Then make a conjecture about the sum of two rational numbers. EXAMPLE 2 You can use reasoning and the definitions of rational and irrational numbers to construct logical arguments to show that the conjectures you made are true. EXAMPLE 2 Justify a Conjecture Teaching Tip N.RN.3 Explain to students that they based their conjecture on several specific examples, but a general proof of a conjecture must rely on definitions and properties. Therefore, variables are used to represent the numbers in this argument. This ensures that the argument will hold for all rational numbers. Complete the following argument to explain why the sum of two rational numbers is rational. a. USE STRUCTURE Suppose x and y are both rational numbers. You must show that x + y is also a rational number. Since x and y are rational numbers, you can write a c x = __b and y = __d . What must be true about a, b, c, and d? SMP 7 a, b, c, and d are integers, with b ≠ 0 and d ≠ 0. b. USE STRUCTURE Write the sum of x and y as a single fraction in terms of a, b, c, and d. ad + bc a c __ ______ x + y = __ b+d= bd c. CONSTRUCT ARGUMENTS Explain why this shows that x + y is a rational number. SMP 7 SMP 3 ad + bc is a sum of products of integers, so it is an integer; bd is a product of nonzero integers, so it is a nonzero integer; therefore, x + y is a rational number because it can be written as a quotient of integers. EXAMPLE 3 Justify a Conjecture Using a Contradiction Scaffolding Questions • What do you think we will do in order to show that x + y is a rational number? Show that it can be written as a quotient of integers. N.RN.3 Complete the following argument to explain why the sum of a rational number and an irrational number is irrational. a. USE STRUCTURE Suppose x is a rational number and y is an irrational number. You must show that x + y is irrational. In this argument, you will assume that the sum is a rational number and show that this leads to a contradiction. Suppose x + y = z and z is a c a rational number. Then you can write x = __b and z = __d . What must be true about SMP 7 a, b, c, and d? a b. USE STRUCTURE Solving for y shows that y = z - x. Use this to write y as a single fraction in terms of a, b, c, and d. SMP 7 a cb - da Copyright © McGraw-Hill Education __ ______ y = z - x = __ d-b= db c. USE STRUCTURE What does this say about y? Explain why this is a contradiction. c + ___ as a single • How do you write ___ b d fraction? Find a common denominator (bd) and add. a, b, c, and d are integers, with b ≠ 0 and d ≠ 0. c SMP 3 • How do you know that bd ≠ 0? Since b ≠ 0 and d ≠ 0, the product cannot be 0. SMP 7 This says that y is a rational number because it is written as a quotient of two integers. This is a contradiction because y is an irrational number. d. CONSTRUCT ARGUMENTS What does the contradiction allow you to conclude? Why? SMP 3 The contradiction shows that the assumption that x + y is a rational number must be false. EXAMPLE 3 Therefore, x + y is an irrational number, which is what was to be proved. 1.1 Rational and Irrational Numbers 5 Emphasizing the Standards for Mathematical Practice Example 3 is an essential connection to SMP 3 (Construct viable arguments and critique the reasoning of others) because it introduces students to a form of reasoning that they will use in future courses. Proof by contradiction, also known as indirect proof, can be _ used to show that √2 is irrational and that there are infinitely many prime numbers. In geometry, students will use indirect proof to explain why a triangle cannot have two right angles. Copyright © McGraw-Hill Education 004_007_ALG1_C1L1_ISG_672788.indd 5 22/08/14 9:18 PM Be sure to highlight the main steps of the process for students: (1) Assume the opposite of the conjecture that you are trying to prove; (2) Show that this leads to a contradiction of a known fact; (3) Conclude that since the assumption is false, the original conjecture must be true. Teaching Tip SMP 3 Discuss why the different methods of proof are used for Examples 2 and 3. Focus the discussion on the idea that there is a form in which all rational numbers can be expressed, but there is no form in which to easily express irrational numbers. This is why the indirect proof is used for Example 3. Scaffolding Questions • What are the known facts at the beginning of this argument? x is rational and y is irrational. • What are you trying to show? x + y is irrational. • How do you solve x + y = z for y? Subtract x from both sides of the equation. 1.1 Rational and Irrational Numbers 5 PRACTICE PRACTICE Connecting Exercises to Standards Exercises 1–10 address the requirements of N.RN.3. Exercises 1–4, 9, and 10 require students to prove statements about rational and irrational numbers or to provide a counterexample to a given statement. 1. CONSTRUCT ARGUMENTS Explain why the product of two rational numbers N.RN.3, SMP 3 is rational. a c __ Suppose x and y are both rational numbers. Then x = __ b and y = d , where a, b, c, and d are a __ c ac __ integers, with b ≠ 0 and d ≠ 0. So, xy = __ b · d = bd . Both ac and bd are integers since they are the product of integers. Therefore, xy is a rational number because it can be written as the quotient of two integers. 2. The product of a nonzero rational number and an irrational number is irrational. a. CONSTRUCT ARGUMENTS Explain why this statement is true. Use an argument that leads to a contradiction. N.RN.3, SMP 3 Suppose x is a nonzero rational number and y is an irrational number. Assume that xy = z and a c __ that z is a rational number. Then x = __ b and z = d , where a, b, c, and d are integers, with a ≠ 0, c Exercises 5–7 require students to determine whether the means of given sets of numbers can be an irrational number. a c b cb __ __ __ __ b ≠ 0, and d ≠ 0. Solving for y shows that y = z ÷ x = __ d ÷ b = d · a = da . This shows that y is a rational number since it is written as the quotient of two integers. This is a contradiction, so z must be an irrational number. b. COMMUNICATE PRECISELY Why is the word nonzero important in the statement? How do you use this fact in your argument? N.RN.3, SMP 6 The statement is not true when the rational number is 0. For example, the product of 0 and Exercise 8 requires students to find two irrational numbers that satisfy given conditions. π is 0, which is rational. You use the fact that the rational number x is nonzero in the argument when you divide both sides of xy = z by x. 3. CRITIQUE REASONING Alyssa said the square of an irrational number must also be irrational. Do you agree or disagree? Justify your answer. N.RN.3, SMP 3 __ Addressing the Standards Dual Coding CCSS SMP 1 N.RN.3 3 2 N.RN.3 3, 6 3–4 N.RN.3 3 5–7 N.RN.3 6 8 N.RN.3 4 9 N.RN.3 2 10 N.RN.3 3 4. CONSTRUCT ARGUMENTS A set is closed under an operation if for any numbers in the set, the result of the operation is also in the set. For example, the set of integers is closed under addition because the sum of two integers is another integer. Is the set of irrational numbers closed under multiplication? If so, explain why. If not, give a N.RN.3, SMP 3 counterexample. __ __ No; the product of two irrational numbers is not necessarily rational. For example, √2 and √8 are __ __ Copyright © McGraw-Hill Education Exercise __ Disagree; √2 is irrational, but (√2 )2 = 2, which is rational. __ irrational, but √2 · √8 = √16 = 4, which is rational. 6 CHAPTER 1 Expressions, Equations, and Functions Common Errors Students may not understand that a conjecture must be proven to be true for all cases, but that it can be shown to be false with a single well-chosen counterexample. In Exercise 3, some students might use their calculator to evaluate π2, notice that this value appears to be irrational, and conclude that Alyssa’s statement is true. Point out to students that they would need to give a general argument, similar to those in Exercises 1 and 2, in order to show that Alyssa’s statement is true. On the other hand, a single example of an irrational number whose square is rational is enough to conclude that Alyssa’s statement is false. 004_007_ALG1_C1L1_ISG_672788.indd 6 Copyright © McGraw-Hill Education 6 CHAPTER 1 Expressions, Equations, and Functions 22/08/14 9:18 PM COMMUNICATE PRECISELY The table shows how Ms. Rodriguez assigns scores in her biology class. Determine whether each statement is always, sometimes, or never true. Explain. N.RN.3, SMP 6 5. The average of a student’s midterm exam score and ﬁnal exam score is a whole number. Sometimes; the average of two whole numbers is either a whole number or halfway between two whole numbers. Category Common Errors In Exercise 4, students may recognize that the set of irrational numbers is not closed under multiplication, but they may have trouble giving a valid counter example. For instance, _ 4 students may offer the numbers √ _ and √9 as a counter example, since _ the product of these numbers, √36 , is rational. Help students understand that the first step in finding a counter example consists of choosing two _ irrational _ numbers. The numbers √ 4 and √9 do not work as a counterexample since these values are rational. A brief class discussion about what works as a valid counterexample is an excellent way to address SMP 3 (Construct viable arguments and critique the reasoning of others). Type of Score Midterm Exam Whole number from 0 to 100 Final Exam Whole number from 0 to 100 Quizzes Rational number from 0 to 10 Homework 0, 4 , 2 , 4 , 1 3 __ 1 __ 1 __ 6. The average of a student’s quiz scores is a rational number. Always; the sum of rational numbers is rational, and dividing this sum by a whole number results in a rational number. 7. The average of a student’s homework scores is an irrational number. Never; the scores are rational numbers and the sum of rational numbers divided by a whole number cannot be an irrational number. y ft 8. USE A MODEL Determine irrational values for x and y so that the area in square feet of the rectangular carpet is a rational number greater than 100 but less than 200. Justify your answer. N.RN.3, SMP 4 ____ ____ Sample answer: x = √125 , y = √180 ; the area of the carpet ____ is √125 · ____ x ft _______ √180 = √22,500 = 150 ft2. 9. REASON QUANTITATIVELY Without performing any calculations, determine if the 2 represents a rational number. Justify your answer. expression 5.323232 . . . + 6 ___ 3 N.RN.3, SMP 2 Yes; a repeating decimal is a rational number, and so is a mixed fraction with integers. The sum of Copyright © McGraw-Hill Education two rational numbers is a rational number. 10. CRITIQUE ___ ___ REASONING Amanda claims that the product √3 · (√3 - 7) is irrational. Her argument is shown at the right. Do you agree with Amanda’s argument? What about her conclusion? Explain. N.RN.3, SMP 3 Amanda’s reasoning is flawed. In her final statement, she should not have concluded that the product of The difference √3 – 7 can be written as sum of an irrational and (nonzero) rational number, as √3 + (–7). This represents an irrational number, because the sum of an irrational and rational number is irrational. The number √3 is also irrational, so the product of two irrational numbers is also irrational. two irrational numbers is irrational. Her conclusion is __ __ correct, because the product can be written as 3 + (-7 √3 ). -7 √3 is the product of a rational number and an irrational number, and is therefore irrational, and the sum of a rational number __ and an irrational number is irrational, so 3 + (-7 √3 ) is irrational. 1.1 Rational and Irrational Numbers 7 Emphasizing the Standards for Mathematical Practice Exercises 5–7 provide an opportunity for students to work with SMP 6 (Attend to precision), because they will need to “make explicit use of definitions” of different types of numbers. For example, students may decide to explore Exercise 5 by finding the average of some pairs of whole numbers that represent test scores, such as 80 and 82. In this case, the average is 81, which is another whole number. Students who think carefully about their calculations might search for a pair of whole numbers that results in a different outcome. The whole numbers 90 and 91 have an average of 90.5, which is not a whole number. Stepping through this thought process requires an understanding of the definition of a whole number. Remind students to consult the glossary in their text if they are unsure about any definitions. Copyright © McGraw-Hill Education 004_007_ALG1_C1L1_ISG_672788.indd 7 8/26/14 11:37 AM 1.1 Rational and Irrational Numbers 7 1.2 Variables and Expressions 1.2 Variables and Expressions STANDARDS Objectives STANDARDS A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. Content: A.SSE.1a, A.SSE.2 Practices: 2, 3, 4, 7, 8 Use with Lesson 1–1 • Write algebraic expressions. • Use the structure of an expression to identify ways to rewrite it. • Interpret parts of an expression. An algebraic expression consists of sums and/or products of numbers and variables. A term of an expression may be a number, a variable, or a product or quotient of numbers and variables. For example, in the expression 24m + 5n + 0.1, there are three terms: 24m, 5n, and 0.1. EXAMPLE 1 Write an Algebraic Expression A.SSE.2 EXPLORE The Watkins family is designing a new path for their garden. They use black and white square tiles to make a pattern that they will use to build the path. Standards for Mathematical Practice: 2, 3, 4, 7, 8 Stage 1 PREREQUISITES • Perform operations with rational numbers EXAMPLE 1 Scaffolding Questions • How many white tiles would you need for Stage 10? How many black tiles? 20; 13 • How can you check that the expressions you wrote are correct? Check to see if the expressions give the correct number of tiles when n = 1, 2, and 3. Stage 1 2 3 4 5 Number of White Tiles 2 4 6 8 10 Number of Black Tiles 4 5 6 7 8 Total Tiles 6 9 12 15 18 b. FIND A PATTERN Suppose the Watkins family wants to make stage n of the pattern. Write an expression for the number of white tiles they will need. Explain how you wrote SMP 8 the expression. 2n; the number of white tiles is 2 times the stage number; this is 2 × n or 2n. c. FIND A PATTERN Write an expression for the number of black tiles they will need to make stage n of the pattern. Explain how you wrote the expression. SMP 8 n + 3; the number of black tiles is 3 more than the stage number; this is n + 3 or 3 + n. d. USE STRUCTURE Write an expression for the total number of tiles they will need to make stage n of the pattern. Is there more than one way to write the expression? Explain. SMP 7 Sample answer: 2n + n + 3; you can also write this as n + 3 + 2n if you start with the number of black tiles and then add the number of white tiles. 8 CHAPTER 1 Expressions, Equations, and Functions Math Background 008_011_ALG1_C1L2_ISG_672788.indd 8 An algebraic expression is a way to record a series of one or more computations with numbers and/or variables. In this lesson, the emphasis is on interpreting the parts of an expression. Students who are fluent mathematical thinkers are able to see an expression such as 3y + 7 as “3 times a number plus 7,” but they are also able to think of it as two distinct part: 3y, which might represent the cost of y sandwiches that cost $3 each, and 7, which might represent the cost of a $7 vegetable platter. This lesson develops this idea using simple expressions, but students will gain more experience, with more complicated expressions, as they progress through the course. Note that students sometimes confuse expressions and equations. An equation is a mathematical statement that two expressions are equal. 8 CHAPTER 1 Expressions, Equations, and Functions 22/08/14 9:18 PM Copyright © McGraw-Hill Education • How does each stage of the pattern compare to the one before? A vertical row consisting of one black tile and two white tiles is added to the right side of the previous stage Stage 3 SMP 8 Copyright © McGraw-Hill Education SMP 5 Teaching Tip You may wish to connect this to SMP 5 (Use appropriate tools strategically). Kinesthetic learners in particular may benefit from using concrete manipulatives to build the stages of the pattern themselves. Using tiles or other objects to make Stages 4 and 5 of the pattern can also help students check that they have filled in the table correctly. Stage 2 a. FIND A PATTERN Complete the table. EXAMPLE 2 e. USE STRUCTURE Evaluate your expression from part d for n = 1, 2, 3, 4, 5 to verify that it produces the same values as in the bottom row of the table in part a. SMP 7 2(1) + 1 + 3 = 6; 2(2) + 2 + 3 = 9; 2(3) + 3 + 3 = 12; 2(4) + 4 + 3 = 15; 2(5) + 5 + 3 = 18 Teaching Tip Some students may have trouble interpreting the coefficients of the expression as dollar amounts. It may be helpful to rewrite the coefficient 12.5 as 12.50 to make it clear that this coefficient represents a cost of $12.50 per cap. A verbal expression like “3 more than the number of white tiles” may be written as the algebraic expression n + 3 because the words “more than” correspond to addition. KEY CONCEPT Complete the table by writing the operation that corresponds to each set of verbal phrases. Verbal Phrases more than, sum, plus, increased by, added to less than, subtracted from, difference, decreased by, minus product of, multiplied by, times, of quotient of, divided by EXAMPLE 2 Interpret Parts of an Expression Operation Addition Subtraction Multiplication Division Scaffolding Questions • How many terms are in the given expression? 3 • What makes some of the terms different from the others? The first two terms, 8x and 12.5y, contain a variable. The last term, 6, is a constant. • How would you use the expression to find the total cost of ordering 4 T-shirts and 6 caps? Substitute x = 4 and y = 6, then simplify the expression. A.SSE.1a, SMP 7 The expression 8x + 12.5y + 6 gives the total cost in dollars of ordering x T-shirts and y caps from a Web site. The cost includes a fee for shipping that is the same no matter how many shirts or caps you order. a. USE STRUCTURE What does the coefficient 8 represent in the expression? What does the term 8x represent? Explain. The 8 represents the cost of each shirt; 8x represents the total cost of the shirts; you multiply the number of shirts x by the cost per shirt to get the total cost. b. USE STRUCTURE What does the coefficient 12.5 represent in the expression? What does the term 12.5y represent? Explain. The 12.5 represents the cost of each cap; 12.5y represents the total cost of the caps; you multiply the number of caps y by the cost per cap to get the total cost. c. USE STRUCTURE What is the fee for shipping? Explain how you know. Copyright © McGraw-Hill Education SMP 7 $6; this part of the expression does not depend on the number of shirts or caps you order. d. USE STRUCTURE How would the expression for the total cost be different if the company decides to increase the price of each shirt and cap by $1.50? Explain. 9.5x + 14y + 6; the price of each shirt is $9.50, the price of each cap is $14, and the shipping fee does not change. 1.2 Variables and Expressions 9 Emphasizing the Standards for Mathematical Practice Example 2 provides an opportunity to address SMP 7 (Look for and make use of structure) . The standard states that students should be able to “see complicated things, such as some algebraic expressions, as single objects or as composed of several objects.” Help students see the expression in the example as composed of two parts: a variable that depends on the number of shirts and caps ordered, and a fixed part that does not depend on the quantities ordered. In addition, students should be able to go one level deeper to analyze the individual terms, 8x and 12.5y. In each case, the coefficient represents the unit cost of a shirt or a cap. Copyright © McGraw-Hill Education 008_011_ALG1_C1L2_ISG_672788.indd 9 2/20/15 5:21 PM 1.2 Variables and Expressions 9 PRACTICE PRACTICE Connecting Exercises to Standards In Exercises 1, 4, 5, and 7 students must write an expression for a given situation in two different ways, satisfying A.SSE.2. In Exercises 2, 3, 6, and 8 students interpret the terms of given expressions in terms of their context or write expressions to match a given situation, satisfying A.SSE.1a. 1. Diego used gray and white counters to make the pattern shown here. Stage 1 Stage 2 A.SSE.2 Stage 3 a. FIND A PATTERN Write an expression for the number of gray counters at stage n and an expression for the number of white counters at stage n. What do the SMP 7 expressions tell you about the number of counters of each color at stage n? gray: n2; white: 2n + 1; the number of gray counters is the stage number times itself; the number of white counters is 1 more than 2 times the stage number. b. USE STRUCTURE Write two different expressions for the total number of counters at stage n. Explain how you know the two expressions are both correct. SMP 7 n2 + 2n + 1 or 2n + 1 + n2; you can start with the number of gray counters and add the number of white counters, or you can start with the number of white counters and add the number of gray counters. Addressing the Standards Exercise Dual Coding CCSS SMP 1 A.SSE.2 7, 8 2 A.SSE.1a 7 A.SSE.1a 2 4 A.SSE.2 7 5 A.SSE.2 3 6 A.SSE.1a 7 7 A.SSE.2 4 8 A.SSE.1a 7 a. What is the coefficient of d? What does it represent? 13.25; it represents the cost per day of renting the bike without the helmet ($13.25). b. How would the expression be different if the cost of the helmet were doubled? The expression would be 13.25d + 13 3. REASON ABSTRACTLY Gabrielle makes a pattern using pennies. The pattern grows in a predictable way at each stage of the pattern. The expression 3n + 1 gives the total number of pennies at each stage of the pattern. Make a sketch to show what Gabrielle’s pattern might look like. Draw stages 1, 2, and 3. A.SSE.1a, SMP2 Sample answer: Stage 1 Stage 2 Copyright © McGraw-Hill Education 3 2. USE STRUCTURE The expression 13.25d + 6.5 gives the total cost in dollars of renting a bicycle and helmet for d days. The fee for the helmet does not depend upon A.SSE.1a, SMP 7 the number of days. Stage 3 10 CHAPTER 1 Expressions, Equations, and Functions Common Errors In Exercise 1, students may write an expression that works for some stages of the pattern, but not all stages of the pattern. For instance, a student might notice that there are 3 white counters in stage 1, and write 3n or n + 2 for the number of white counters. Both of these expressions are incorrect, since they do not work for the other stages of the pattern. 008_011_ALG1_C1L2_ISG_672788.indd 10 10 CHAPTER 1 Expressions, Equations, and Functions Copyright © McGraw-Hill Education If students have difficulty writing correct expressions, suggest that they perform the intermediate step of making a table and looking for a pattern, as they did in Example 1. Then have them check that their expressions work for all stages in the table. 22/08/14 9:18 PM The table shows the prices of several items at an office supply store. Use the table for Exercises 4–6. Item 4. USE STRUCTURE Jemma buys s staplers and g gel pens. She has a coupon for $2 off the total cost of her purchase. Write two different expressions that can be used to find her final cost before tax. A.SSE.2, SMP 7 Common Errors In Exercise 7, students may write the expression nb + c for the area of the apartment. This is incorrect because the expression is missing parentheses. The expression should be n(b + c). To convince students that these two ways of writing the expression are different, have them find the area of the apartment using specific numerical values for n, b, and c. Note that students will gain additional experience using parentheses in the next two lessons. Price Stapler $5.99 Tape dispenser $3.50 Sticky notes $3.25 Gel pen $1.75 Sample answer: 5.99s + 1.75g - 2; 1.75g + 5.99s - 2 5. CRITIQUE REASONING DeMarco buys x tape dispensers and x packs of sticky notes. He says he can use the expression 6.75x to find the total cost of the items before tax. A.SSE.2, SMP 3 Do you agree? Why or why not? Agree; a tape dispenser and a pack of sticky notes costs $3.50 + $3.25 = $6.75 and he buys x pairs of tape dispensers and packs of sticky notes, which costs 6.75x. 6. USE STRUCTURE Tyler buys s packs of sticky notes and one gel pen. He uses the expression 1.08(3.25s + 1.75) to find the final cost. What do you think the 1.08 in the expression represents? Explain. A.SSE.1a, SMP 7 There is a sales tax of 8%; multiplying by 1.08 gives the total cost including sales tax. b ft 7. USE A MODEL The figure shows a floor plan for a two-room apartment. Write an expression for the area of the apartment, in square feet, by first finding the area of each room and then adding. Then describe how you can write the expression in a different way. A.SSE.2, SMP 4 bn + cn; you can also find the length of each side of the c ft n ft apartment, n and b + c, and then multiply: n(b + c) 8. USE STRUCTURE Describe a situation that could be represented by A.SSE.1a, SMP 7 each expression. a. 9.95 + 0.75b Sample answer: The total cost of a pizza, if a cheese pizza costs $9.95 plus $0.75 for each Copyright © McGraw-Hill Education additional topping, b. b. 15x - 5x Sample answer: The cost of purchasing x hats, if the regular price of each hat is $15 and they are on sale for $5 off. c. 59c - 25c - 30 Sample answer: a store’s profit on tablet cases, if the store pays a $30 shipping charge plus $25 for each tablet case and sells them for $59 each 1.2 Variables and Expressions 11 Emphasizing the Standards for Mathematical Practice Use Exercise 3 to address SMP 2 (Reason abstractly and quantitatively). This exercise begins in the abstract, by providing an algebraic expression that represents the number of pennies needed to build each stage of a pattern. Help students move to the concrete by discussing the fact that the expression 3n + 1 has a “growing part” (3n) and a “constant part” (1). This provides an important clue to the pattern. Students may wish to begin developing a suitable pattern by placing one penny at the edge of the pattern; this represent the part that does not change. Then they can add rows of pennies that grow from stage to stage according to the rule 3n. Copyright © McGraw-Hill Education 008_011_ALG1_C1L2_ISG_672788.indd 11 2/20/15 5:21 PM 1.2 Variables and Expressions 11 1.3 Order of Operations 1.3 Order of Operations STANDARDS Objectives STANDARDS A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. • Evaluate expressions by using the order of operations. • Use the structure of an expression to identify ways to rewrite it. • Interpret parts of an expression. To evaluate an expression means to find its value. When you evaluate a complicated expression containing multiple operations, you must perform the operations in the correct order to get the correct value for the expression. EXAMPLE 1 • Perform operations with rational numbers • Find areas and perimeters of simple figures • Calculate percentages Evaluate Expressions A.SSE.2 EXPLORE Rima created a game. She wrote expressions on slips of paper that represent different sums of money. The player has 10 seconds to match each expression to its sum. Standards for Mathematical Practice: 1, 2, 3, 4, 5, 6, 7 PREREQUISITES Content: A.SSE.1b, A.SSE.2 Practices: 1, 2, 3, 4, 5, 6, 7 Use with Lesson 1–2 26 - 15 + 4 - 1 + 14 48 ÷ 2 + 4 × 2 5[10 - (2 + 5)] (1 + 6)2 - (3 + 7) $15 $32 $39 $28 a. PLAN A SOLUTION Evaluate the expressions. Draw a line from each expression to the SMP 1 correct value. b. CRITIQUE REASONING Ben said that to evaluate 26 - 15 + 4 - 1 + 14 you perform SMP 3 the two additions and then the two subtractions. Do you agree? Explain. No; this gives 26 - 19 - 15 = -8, which is incorrect. To get the correct value, add and subtract from left to right. c. USE TOOLS Use your calculator to evaluate 48 ÷ 2 + 4 × 2. Does it give the correct value? If so, in what order does it perform the operations? If not, in what order do you SMP 5 think it performs the operations? Answer depends upon calculator; calculators that give the correct value multiply and divide EXAMPLE 1 SMP 1 Teaching Tip This Explore offers a connection to SMP 1 (Make sense of problems and persevere in solving them). If students are not sure how to proceed, encourage them to try evaluating the expressions in different ways. When they evaluate an expression correctly, they will find the value among the given choices. Letting students grapple with the expressions in this way gives them a chance to find their own path for navigating the problem. • For 26 − 15 + 4 − 1 + 14, do you get the same value if you add 26 + 4 + 14 and then subtract the remaining quantities versus performing the operations left to right? Yes; both values are 28. • How do you know which of those methods is correct? The second method is correct because $28 is one of the answer choices. Evaluate the expression inside the innermost set of grouping symbols first; then evaluate exponents. 12 CHAPTER 1 Expressions, Equations, and Functions Math Background 012_015_ALG1_C1L3_ISG_672788.indd 12 3/5/15 7:17 AM The order of operations is a standard procedure for evaluating numerical expressions. By using these agreed-upon steps, everyone who evaluates an expression will get the same result. Some students may have seen the mnemonic PEMDAS (or Please Excuse My Dear Aunt Sally) for remembering the order of operations: parentheses, exponents, multiply/divide, add/subtract. This may cause students to think that multiplication is performed before division and that addition is performed before subtraction. If students use the mnemonic, be sure they understand that all multiplications and divisions should be performed, in the order they occur, from left to right. The same is true for all additions and subtractions. 12 CHAPTER 1 Expressions, Equations, and Functions Copyright © McGraw-Hill Education Scaffolding Questions d. COMMUNICATE PRECISELY What rules can you state about grouping symbols, such as parentheses, and exponents in order to get the correct values for 5[10 - (2 + 5)] SMP 6 and (1 + 6)2 - (3 + 7)? Copyright © McGraw-Hill Education before adding; calculators that give incorrect values perform operations from left to right. EXAMPLE 2 The rule that describes the sequence in which you should perform operations is called the order of operations. Teaching Tip KEY CONCEPT Complete the table by writing the order of operations. Operations Order of Operations Multiply and/or divide from left to right. Step 1: Evaluate expressions inside grouping symbols. Add and/or subtract from left to right. Step 2: Evaluate all powers. Evaluate expressions inside grouping symbols. Step 3: Multiply and/or divide from left to right. Evaluate all powers. Step 4: Add and/or subtract from left to right. EXAMPLE 2 Write and Evaluate an Expression Jared is buying carpet for a square room with sides that are s feet long. The table shows the price of the carpet and the price of the metal strip that holds down the edge of the carpet. Item Price Carpet $2.65 per square foot Metal Strip $0.20 per foot a. USE A MODEL The metal strip holds down the carpet around the entire perimeter of the room, except at the doorway, which is 3 feet wide. Write an expression for the total A.SSE.1b, SMP 4 length of the strip that Jared will need. Explain. 4s - 3; the perimeter of the room is 4 times the length of a side; then subtract 3 feet for the doorway. b. USE A MODEL Write an expression that Jared can use to calculate the total cost of the carpet and the metal strip for a room with sides s feet long. Explain what each term A.SSE.1b, SMP 4 of your expression represents. 2.65s2 + 0.2(4s - 3); the total cost is the cost of the carpet (2.65 times the area of the carpet, s2) plus the cost of the metal strip (0.2 times the length of the metal strip). c. USE STRUCTURE Explain how you can write the expression in a different way. Copyright © McGraw-Hill Education 0.2(4s - 3) + 2.65s2. A.SSE.2, SMP 6 Use the order of operations to evaluate the expression for s = 16; 2.65 · 162 + 0.2(4 · 16 - 3) = $690.60. e. USE STRUCTURE How would the expression from part b be different if Jared had a coupon for 10% off the total cost? (Hint: 10% off the total cost means Jared pays 90% of the original cost). How much would Jared pay in this case? A.SSE.1b, SMP 7 0.9[2.65s2 + 0.2(4s - 3)]; $621.54 1.3 Order of Operations 13 Emphasizing the Standards for Mathematical Practice Use Example 2 to address SMP 4 (Model with mathematics). A key element of the standard states that students “are able to identify important quantities in a practical situation.” You may want to have students read the entire problem, and then have them reread each part to identify important quantities. Students can use a highlighter to mark any numerical information that they think will be important in the solution of the problem. For example, the fact that the doorway is 3 feet wide is an essential piece of information that is embedded within the statement of part a. Highlighting this will help students remember to take it into account when they write an expression for the length of the metal strip. Copyright © McGraw-Hill Education 012_015_ALG1_C1L3_ISG_672788.indd 13 Part e is an excellent opportunity to address structure, as well as standards A.SSE.1b and A.SSE.2. Students may want to apply the 10% discount by writing the original expression for the cost minus an expression for 10% of the cost. This results in the expression 2.65s2 + 0.2(4s − 3) − 0.1[2.65s2 + 0.2(4s - 3)], which is complicated but correct. Help students see that an equivalent expression, based on taking 90% of the original cost, results in an expression that is much easier to work with. Scaffolding Questions • What expression represents the perimeter of the room including the doorway? 4s A.SSE.2, SMP 7 Sample answer: Find the cost of the metal strip first; then add the cost of the carpet: d. COMMUNICATE PRECISELY Explain how you can use the expression from part b to find the total cost of the carpet and metal strip for a room with sides 16 feet long. SMP 7 2/20/15 5:21 PM • What should you do to the expression to account for the doorway? Why? Subtract 3 from the expression 4s, since the metal edge does not cross the doorway. • What expression gives the total cost of the metal strip? Do you need to use parentheses? Explain. 0.2(4s − 3); you need parentheses because you multiply the length of the strip, 4s − 3, by 0.2, and 0.2(4s − 3) is different from 0.2 · 4s − 3. 1.3 Order of Operations 13 PRACTICE PRACTICE Connecting Exercises to Standards CALCULATE ACCURATELY Evaluate each expression. Exercises 1−6, 8, 14, and 15 give students additional practice in evaluating a numerical expression and evaluating an algebraic expression after substituting values for the variables, satisfying A.SSE.2. CALCULATE ACCURATELY Evaluate each expression if x = 2, y = 7, and z = −1. Exercises 7, 9, and 13 address A.SSE.1b, requiring students to evaluate numerical expressions for real-world situations. 1. 102 − 20 + 43 2. 32 − (7 − 1)2 -4 144 4. z(2x − y) A.SSE.2, SMP 6 3. 64 ÷ (2 + 6) − 14 5. y2 − 4y ÷ 2 3 -6 A.SSE.2, SMP 6 6. x(z + 5y) − 2x2 35 60 7. The table shows how scores are calculated at diving competitions. Each of the five judges scores each dive from 1 to 10 in 0.5-point increments. a. CALCULATE ACCURATELY Roberto performs a dive with a degree of difficulty of 2.5. His scores from the judges are 8.0, 7.5, 6.5, 7.5, and 7.0. Write and evaluate an expression to find his score for the dive. A.SSE.2, SMP 6 Calculating a Diving Score Step 1 Drop the highest and lowest of the five judges’ scores. Step 2 Add the remaining scores to find the raw score. Step 3 Multiply the raw score by the degree of difficulty. 2.5(7.5 + 7.5 + 7.0); 55 b. CONSTRUCT ARGUMENTS Jennifer performs a dive and uses the expression (7.5 + 8.5 + 8.0) × 3.2 to find her score. What is her score for the dive? What can you conclude about the highest score she received from the five judges? Explain. A.SSE.1b, SMP 3 76.8; the highest score was 8.5 or greater since the highest score is dropped and does not appear in the expression. Exercises 10–12 require students to evaluate an algebraic expression and determine whether given statements about the expression are true, satisfying A.SSE.1b. Exercise Dual Coding CCSS SMP 1−6 A.SSE.2 6 7 A.SSE.1b, A.SSE.2 2, 3, 6 8 A.SSE.2 3 9 A.SSE.1b 5 10−12 A.SSE.1b 2 13 A.SSE.1b 4 14−15 A.SSE.2 2 Sample answer: 7, 7, 6, 8, 5; her score is 3.5(7 + 7 + 6 ) = 70 d. REASON QUANTITATIVELY Eli also performs a dive with a degree of difficulty of 3.5. His score is 75.25. What scores could Eli have received from the five judges? Explain. A.SSE.1b, SMP 2 Sample answer: 7, 7, 7, 7.5, 8; his score is 3.5(7.5 + 7 + 7) = 75.25 e. REASON QUANTITATIVELY Skylar does a dive with a degree of difficulty of 3.3. Four of his scores are 7.5, 7.0, 6.5, and 6.5. Toby does a dive with a degree of difficulty of 3.1 and receives scores of 7.0, 7.5, 7.5, 6.0, and 8.0. If Skylar’s final score was greater than Toby’s, what can you say about Skylar’s fifth score? Explain. A.SSE.1b, SMP 2 Skylar’s fifth score must be 7.5 or greater; Toby’s score is 3.1(7.5 + 7.5 + 7.0) = 68.2; If Skylar’s Copyright © McGraw-Hill Education Addressing the Standards c. REASON QUANTITATIVELY Mai performs a dive with a degree of difficulty of 3.5. Her score for the dive is 70. What scores could Mai have received from the five A.SSE.1b, SMP 3 judges? Explain. fifth score is 7.0 or lower, then the 7.5 score would be dropped, and the highest his score could be is 3.3 (7.0 + 7.0 + 6.5) = 67.65. If his fifth score is 7.5 or higher, then his score is 3.3(7.5 + 7.0 + 6.5) = 69.3. 14 CHAPTER 1 Expressions, Equations, and Functions Emphasizing the Standards for Mathematical Practice Exercise 7c addresses SMP 3 (Construct viable arguments and critique the reasoning of other). Be sure students understand that this is an open-ended problem with many possible answers. A key part of the problem is having students justify their answers to show that the judges’ scores they choose result in a score of 70 for the dive. Remind students that they must provide five scores, even though only three are used in calculating the score for the dive. You may want to ask some follow-up “what if” questions. For example, “What if one of your judges’ scores were lower? Would that change the score for the dive? Why or why not?” 012_015_ALG1_C1L3_ISG_672788.indd 14 Copyright © McGraw-Hill Education 14 CHAPTER 1 Expressions, Equations, and Functions 2/20/15 5:21 PM 8. CRITIQUE REASONING A student was asked to evaluate an expression. The student’s work is shown at right. Critique the student’s work. If there are any errors, describe them and ﬁnd the A.SSE.2, SMP 3 correct value of the expression. Common Errors In Exercise 10, students may correctly rewrite the expression as 100 − 5(b + 2) and then conclude that the value of the expression is always less than 100. Students who make this error may be thinking only of positive values of b, causing them to see (b + 2) as a positive quantity. Suggest that students try evaluating the expression with a variety of values for b. Once students try a negative value, such as b = −10, they should recognize that −5(b + 2) is positive and that the value of the expression is greater than 100. 62 - 5 x 2 + 2(9 - 7) = 62 - 5 x 2 + 2(2) = 36 - 5 x 2 + 2(2) The next to last line is incorrect. The student should have = 36 - 10 + 4 added/subtracted from left to right. The correct value is 30. = 36 - 14 = 22 9. USE TOOLS Kelly buys 3 video games that cost $18.95 each. She also buys 2 pairs of earbuds that cost $11.50 each. She has a coupon for $2 oﬀ the price of each video game. Kelly uses a calculator, as shown, to ﬁnd that the total cost of the items is $77.85. The cashier tells her that the total cost is $73.85. Who is A.SSE.1b, SMP 5 correct? Explain. The cashier is correct. Kelly should have entered the expression into her calculator as 3(18.95 − 2) + 2(11.50). REASON QUANTITATIVELY Determine whether each statement about the expression A.SSE.1b, SMP 2 a2 - 5(b +2) is always, sometimes, or never true. Explain. 10. If a = 10, then the value of the expression is less than 100. Sometimes; when b + 2 is negative, the value of the expression is greater than 100. 11. If b = 1, then the expression is equivalent to a2 - 15. Always; a2 - 5(1 + 2) = a2 - 5(3) = a2 - 15. 12. The value of the expression is 0. Sometimes; if a = 0 and b = -2, then the value of the expression is 02 - 5(-2 + 2) = 0. b1 13. USE A MODEL The side panel of a skateboard ramp is a trapezoid, as h (b + b2) can be used to ﬁnd the area shown in the ﬁgure. The expression ___ 2 1 of a trapezoid. Write and evaluate an expression to ﬁnd the amount of wood needed to build the two side panels of a skateboard ramp where A.SSE.1b, SMP 1 h = 24 inches, b1 = 30 inches, and b2 = 50 inches. 24 2 2 • __ 2 (30 + 50 ); 1920 in h b2 14. REASON QUANTITATIVELY Write an expression that includes the numbers 2, 4, and 5, and has a value of 50. Your expression should include one set of parentheses. A.SSE.2, SMP 2 Copyright © McGraw-Hill Education Sample answer: 52(4 - 2) 15. REASON QUANTITATIVELY Isabel wrote the expression 6 + 3 × 5 - 6 + 8 ÷ 2 and asked Tamara to evaluate it. When Tamara evaluated it, she got a value of 19. Isabel told Tamara that her value was incorrect and said that the value should have been 38. With whom do you agree? Explain. A.SSE.2, SMP 2 Tamara is correct. When evaluating this expression, first perform the multiplication and division, and then the addition and subtraction. 1.3 Order of Operations 15 Emphasizing the Standards for Mathematical Practice Exercise 13 is an opportunity for students to work with SMP 1 (Make sense of problems and persevere in solving them). Some students may miss the fact that two side panels are being built, and this may cause them to arrive at the wrong answer. Encourage students to read the problem carefully, mark the important given information, and persevere in the process of writing and evaluating an appropriate expression. Copyright © McGraw-Hill Education 012_015_ALG1_C1L3_ISG_672788.indd 15 2/20/15 5:21 PM 1.3 Order of Operations 15 1.4 Properties of Numbers 1.4 Properties of Numbers STANDARDS Objectives STANDARDS A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. Standards for Mathematical Practice: 2, 3, 4, 6, 7, 8 PREREQUISITES • Perform operations with rational numbers EXAMPLE 1 • Based on the results of parts c and d, what two numerical expressions must be equal? 3.4(3.6) + 3.4(2.1) = 3.4(3.6 + 2.1) • Use the structure of an expression to identify ways to rewrite it. • Interpret parts of an expression. EXAMPLE 1 Explore Properties of Numbers SMP 7 EXPLORE Rectangle ABCD represents Arletta’s garden. She plants part of the garden with vegetables and part of the garden with flowers, as shown. a. USE STRUCTURE Arletta wants to put a straight path along the garden from D to C. Write two different expressions she can use to find the length of the path. Explain why it makes sense A.SSE.2 that the two expressions give the same length. A 3.6 m 2.1 m vegetables ﬂowers D B 3.4 m C 3.6 + 2.1 or 2.1 + 3.6; you get the same length regardless of whether you add the length of the flower plot to the length of the vegetable plot, or vice versa. b. USE STRUCTURE Arletta also wants to put a fence along the border of the garden from A to B to C. Write an expression she can use to find the length of the fence. Then show two different ways she can group a pair of numbers in the expression. Does she A.SSE.2 get a different result depending on the grouping? Explain. 3.6 + 2.1 + 3.4; (3.6 + 2.1) + 3.4 or 3.6 + (2.1 + 3.4); with either grouping the length of the fence is the same, 9.1 m. c. USE STRUCTURE Arletta wants to find the area of the garden. She finds the area of the vegetable plot and the area of the flower plot then the sum of these areas. Write A.SSE.1b and evaluate an expression to show how she finds the area. 3.4(3.6) + 3.4(2.1) = 12.24 + 7.14 = 19.38 m2 d. USE STRUCTURE Arletta’s friend, Troy, finds the area of the garden by adding to find the distance from A to B and then multiplying by the distance from B to C. Write and evaluate an expression to show how Troy finds the area. Does he get the same result as A.SSE.1b Arletta? Explain why this makes sense. 3.4(3.6 + 2.1) = 3.4(5.7) = 19.38 m2; he gets the same result as Arletta; it makes sense that the results are the same because both expressions represent the same total area. 16 CHAPTER 1 Expressions, Equations, and Functions Math Background 016_019_ALG1_C1L4_ISG_672788.indd 16 22/08/14 9:19 PM The properties of numbers in this lesson play an important role in students’ growth as mathematical thinkers. Students sometimes see these properties as little more than complicated names for “obvious” facts. However, these properties serve as the logical underpinnings of numerical and algebraic computations. As students progress through high school mathematics courses, they will be called upon to give increasingly sophisticated logical arguments. Students will be expected to justify every statement with a valid reason. Having students justify algebraic steps by citing a property of real numbers can help develop a habit of mind that will serve them well when they are asked to justify more complicated statements down the road. 16 CHAPTER 1 Expressions, Equations, and Functions Copyright © McGraw-Hill Education Scaffolding Questions • In part c, how do you find the area of a rectangle? Find the product of the length and width of the rectangle. • Evaluate expressions by using properties of numbers. Copyright © McGraw-Hill Education SMP 7 Teaching Tip One aspect of SMP 7 (Look for and make use of structure) is recognizing equivalent numerical expressions by considering properties of equality and the Distributive Property. A geometric model, such as the garden shown in the example, is a good way to illustrate why the expressions are equivalent. Have students that are having difficulties finding different ways to write the expressions label the lengths of each side on the diagram. Content: A.SSE.1b, A.SSE.2 Practices: 2, 3, 4, 6, 7, 8 Use with Lessons 1–3, 1–4 EXAMPLE 2 In the previous exploration, you may have discovered the following properties of numbers. KEY CONCEPT Teaching Tip Complete the table by using numbers to write examples for each property. Property Commutative Property For any numbers a and b, a + b = b + c and a b = b a. Associative Property For any numbers a, b, and c, (a + b) + c = a + (b + c) and (ab)c = a(bc). Distributive Property For any numbers a, b, and c, a(b + c ) = ab + ac and a(b - c) = ab - ac. EXAMPLE 2 Examples Sample answers: 5+9=9+5 59=95 3 + (4 + 10) = (3 + 4 ) + 10 (3 4)10 = 3(4 10) 8(5 + 2) = 8 5 + 8 2 8(5 - 2) = 8 5 - 8 2 Use Mental Math Giovanni is buying equipment for his soccer team. The table shows the price of some of the items he is buying. Item Price Soccer Balls $22 each Portable Goal $93 per pair a. USE STRUCTURE Giovanni is buying 7 soccer balls. Write an expression for the total cost of the soccer balls. Then explain how he can use the Distributive Property to rewrite the expression to find the cost using mental math. A.SSE.1b, SMP 7 Sample answer: 7(22); rewrite the expression as 7(20 + 2); by the Distributive Property, this equals 7(20) + 7(2) = 140 + 14 = 154. b. USE STRUCTURE Giovanni is buying 6 pairs of portable goals. Show two different ways he can find the cost of the goals using the Distributive Property. A.SSE.2, SMP 7 Sample answer: 6(93) = 6 (90 + 3) = 6(90) + 6(3) = 540 + 18 = 558; 6(93) = 6(100 - 7) = 6(100) - 6(7) = 600 - 42 = 558. Copyright © McGraw-Hill Education c. USE STRUCTURE Fred’s business donated 3 soccer balls and 3 portable goals to Giovanni’s team. Write an expression using the Distributive Property for the cost of the donated items. SMP 8 Sample answer: 3(22 + 93) = 3(20 + 2 + 90 + 3) = 3(110 + 5) = 330 + 15 = 345. d. DESCRIBE A METHOD Describe a general method for using the Distributive Property A.SSE.1b, SMP 8 to find a product by mental math. Write the greater of the two factors as a sum or difference where one term is a multiple of 10 and the other is a single digit. Then apply the Distributive Property. 1.4 Properties of Numbers 17 Emphasizing the Standards for Mathematical Practice Example 2 part c connects to SMP 8 (Look for and express regularity in repeated reasoning). Once students have completed parts a and b, they should begin to recognize a general procedure. If students have trouble recognizing this general method, suggest that they try a few more specific examples, such as 5(72) and 4(89). Then help students break the process into steps. Ask: “What is the first thing you do?” If necessary, help students see that a key step is writing the larger factor as a sum in which one of the numbers is a multiple of 10. If students have trouble seeing this, ask whether it would be helpful to rewrite 5(72) as 5(59 + 13), and why or why not. Copyright © McGraw-Hill Education 016_019_ALG1_C1L4_ISG_672788.indd 17 2/20/15 5:21 PM SMP 7 This example requires students to work fluently with whole numbers by breaking them into parts that are convenient for calculations. You may want to spend a moment working with students to express a number, such as 74, as the sum of a multiple of 10 and a single digit. Be sure students understand that they can use addition (70 + 4) or subtraction (80 − 6). Scaffolding Questions • In general, how do you find the total cost when you buy several of the same item? Multiply the price of one item by the number of items you buy. • In part a, why is it convenient to write 7(22) as 7(20 + 2)? It is difficult to calculate 7(22) directly using only mental math, but it is easier to calculate 7(20 + 2) because you can use the Distributive Property to write it as 7(20) + 7(2). Each of the products in this sum is easier to calculate than the original one. • How can you check each of the mental math calculations? Sample answer: work out each problem in a different way or use a calculator. 1.4 Properties of Numbers 17 EXAMPLE 3 Teaching Tip SMP 6 In this example, students use and identify properties of numbers. If students have trouble remembering the names of the properties, work with them to develop mnemonics. For example, when you commute, you travel back and forth to work or school; in the Commutative Property, numbers or variables travel back and forth around the operation symbol. Like terms are terms that contain the same variables, with corresponding variables having the same exponent. In the expression 5x2 + 4x + 7x2 + 2y2, the terms 5x2 and 7x2 are like terms. You can use properties of numbers to simplify expressions by combining like terms. EXAMPLE 3 Combine Like Terms Follow these steps to simplify the expression -4m3 + 4m + 6n3 + 2m. a. USE STRUCTURE What are the like terms in the expression? Explain how A.SSE.2, SMP 7 you know. 4m and 2m; they have the same variable raised to the same power. b. COMMUNICATE PRECISELY Which property allows you to rewrite the expression as A.SSE.2, SMP 6 -4m3 + 4m + 2m + 6n3? Why? Commutative Property; the property says that you can add in any order. c. USE STRUCTURE Show how to use the Distributive Property to rewrite the middle A.SSE.1b, SMP 7 two terms (4m and 2m) as a single term. What is the simplified expression? 4m + 2m = (4 + 2)m = 6m; -4m3 + 6m + 6n3 d. DESCRIBE A METHOD Describe how you can simplify an expression by combining like A.SSE.2, SMP 8 terms. For each group of like terms, add the coefficients to get the coefficient of the corresponding term in the simplified expression. Scaffolding Questions • Why is it useful to rewrite the original expression as −4m3 + 4m + 2m + 6n3? Once you put the terms 4m and 2m next to each other, you can combine them using the Distributive Property. USE STRUCTURE Which property is illustrated by each equation? 1. 9y + 3x + 6y = 3x + 9y + 6y Commutative Property 2. c(5 + d) = 5c + cd A.SSE.2, SMP 7 3. 4(5m) = (4 5)m Associative Property Distributive Property USE STRUCTURE The table shows the prices of tickets to a theme park. Use the table for Exercises 4–6. For each situation, explain how to use the Distributive Property and mental math to A.SSE.1b, SMP 7 find the total cost of the tickets. Type of Ticket 4. 7 tickets for adults Price Adults $29 Students $21 Seniors $18 Sample answer: 7(29) = 7(30 - 1) = 7(30) - 7(1) = 210 - 7 = $203 5. 9 tickets for students Sample answer: 9(21) = 9(20 + 1) = 9(20) + 9(1) = 180 + 9 = $189 6. 12 tickets for seniors Sample answer: 12(18) = 12(20 - 2) = 12(20) - 12(2) = 240 - 24 = $216 USE STRUCTURE Simplify each expression. 7. -2k + 2k2 - 3k + k3 -5k + 2k2 + k3 Copyright © McGraw-Hill Education • How could you check that the final expression is equivalent to the original one? Sample answer: Substitute the same values for the variables in both expressions and check that you get the same result. PRACTICE A.SSE.2, SMP 7 8. 2n2 + 5n4 - 6n2 - 3n4 -4n2 + 2n4 9. -b + 3a + b2 - 2b + 6a - 9a -3b + b2 18 CHAPTER 1 Expressions, Equations, and Functions 18 CHAPTER 1 Expressions, Equations, and Functions 2/20/15 5:21 PM Copyright © McGraw-Hill Education Common Errors In Exercise 1, some students may see that the expression on each side of the equal sign has three terms and quickly decide that the relevant property must be the Associative Property. Encourage students to look closely at the expressions and ask them to describe how they are different. Once students see that two terms have “traded places,” they should be able to connect the equation more easily to the Commutative Property. 016_019_ALG1_C1L4_ISG_672788.indd 18 10. CRITIQUE REASONING Angela is shipping 8 bags of granola to a customer. Each bag weighs 22 ounces and the maximum weight she can ship in one box is 10 pounds 5 ounces. She makes the calculation at right and decides that she can ship the bags in one box. Do you agree? Explain. A.SSE.1b, SMP 3 PRACTICE = 8(20 + 2) 8(22) = 8(20) + 2 Connecting Exercises to Standards = 160 + 2 = 162 ounces No; 10 pounds 5 ounces is 10(16) + 5 = 165 ounces, but Angela used the Distributive Property incorrectly. She should have written Exercises 1–3, 7–9, 12, and 13 require students to recognize how properties of numbers allow expressions to be written in different ways, per A.SSE.2. 8(20 + 2) = 8(20) + 8(2) = 160 + 16 = 176 ounces. m seats 11. A theater has m seats per row on the left side of the aisle and n seats per row on the right side of the aisle. There are A.SSE.1b r rows of seats. a. USE A MODEL Explain how you can use the Distributive Property to write two diﬀerent expressions that represent the total number of seats in the theater. SMP 4 Aisle n seats r rows The total number of seats is the number of seats per row, m + n, times the number of rows, r. This is r(m + n). By the Distributive Property, this is also rm + rn. b. CONSTRUCT ARGUMENTS Suppose you double the number of seats in each row on the left side of the aisle. Does this double the number of seats in the theater? Use one of the expressions you wrote in part a to justify your answer. SMP 3 No; the original number of seats in the theater is rm + rn. If you double m, you double only the Exercises 4–6, 10, and 11 require students to rewrite an expression using the Distributive Property before evaluating, satisfying A.SSE.1b. first term of the expression, but not the second term, so you do not double the total number of seats in the theater. Addressing the Standards 12. CONSTRUCT ARGUMENTS Is there a Commutative Property or Associative Property for subtraction? Explain why or why not for each property. A.SSE.2, SMP 3 Exercise Neither property exists; a counterexample shows that there is no Commutative Property for CCSS SMP 1−3 A.SSE.2 7 4−6 A.SSE.1b 7 7−9 A.SSE.2 7 10 A.SSE.1b 3 11 A.SSE.1b 3, 4 12 A.SSE.2 3 13 A.SSE.2 2 subtraction: 7 - 4 ≠ 4 - 7; a counterexample shows that there is no Associative Property for Copyright © McGraw-Hill Education subtraction: 10 - (2 - 1) ≠ (10 - 2) - 1. 13. REASON QUANTITATIVELY Provide a counterexample to show that there is no Commutative Property or Associative Property for division. What is the relationship A.SSE.2, SMP 2 between the results when the order of division of two numbers is switched? 4 ÷ 8 ≠ 8 ÷ 4, so there is no Commutative Property for division 16 ÷ (8 ÷ 4) ≠ (16 ÷ 8) ÷ 4, so there is no Associative Property for division. As long as neither number is 0, when the order of division of two numbers is switched, the results are multiplicative inverses of each other. Dual Coding 1.4 Properties of Numbers 19 Emphasizing the Standards for Mathematical Practice Exercise 12 offers a chance to address SMP 3 (Construct viable arguments and critique the reasoning of others). In particular, you may want to use the exercise as the jumping-off point for a brief discussion about counterexamples. Be sure students understand that it takes only one counterexample to show that a conjecture is false. Therefore, a single example in which the Commutative Property does not work for subtraction is enough to conclude that this property does not exist. Similarly, students only need to find a single set of numbers where the Associative Property does not work with subtraction. Copyright © McGraw-Hill Education 016_019_ALG1_C1L4_ISG_672788.indd 19 22/08/14 9:19 PM Common Errors Exercise 10 highlights one of the most common errors in algebra classes. In the calculation shown, the 8 is not completely “distributed” to all of the terms in the second factor. If students make this error, remind them that the Distributive Property states that the factor outside the parentheses must be fully distributed to each of the terms inside the parentheses. It may help to show students an example with several terms in the parentheses. For instance, 5(4 + 2 + 8) may be rewritten as 5(4) + 5(2) + 5(8). 1.4 Properties of Numbers 19 1.5 Equations 1.5 Equations STANDARDS Objectives STANDARDS Content: A.CED.1, A.REI.1, A.REI.3 Practices: 1, 2, 3, 4, 7, 8 Use with Lesson 1–5 • Write equations in one variable and use them to solve problems. A.CED.1 Create equations and inequalities in one variable and use them to solve problems. A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. • Solve linear equations in one variable and explain the steps of the solution. A mathematical sentence that contains an equals sign (=) is an equation. An equation states that two expressions are equal. EXAMPLE 1 Investigate Equations EXPLORE A group of friends rented bicycles from different bike shops. The table shows the expression that each shop uses to calculate the cost of renting one of their bikes for h hours. Shop Cost for h Hours ($) Real Wheels Easy Bike a. USE A MODEL Kaden rented his bike from Easy Bike and Pedal Power he paid $26 for the rental. Write an equation that relates the expression the bike shop uses to calculate the cost and the amount Kaden paid. Do you think Kaden rented his bike for 6 hours? Justify your A.CED.1, SMP 4 answer using the equation you wrote. 5(h + 1) 3.5h + 8.5 2(3h - 1) 3.5h + 8.5 = 26; no; if he rented the bike for 6 hours, then the expression on the left side of the equation would be 3.5(6) + 8.5 = 29.5, which does not equal 26. b. CONSTRUCT ARGUMENTS Do you think Kaden rented his bike for 5 hours? Justify A.REI.3, SMP 3 your answer using the equation you wrote. Also addresses: A.REI.3 Standards for Mathematical Practice: 1, 2, 3, 4, 7, 8 Yes; when h = 5, the left side of the equation is 3.5(5) + 8.5 = 26, so the two sides of the equation are the same. c. CRITIQUE REASONING Megan rented her bike from Real Wheels and she paid $25. She claims that she rented the bike for 5 hours. Do you agree? Use an equation to A.CED.1, SMP 3 explain why or why not. which does not equal 25; so the number of hours cannot be 5. • Perform operations with rational numbers • Use order of operations • Use properties of numbers d. USE REASONING Kim-Ly rented her bike from Pedal Power and she paid $22. Did she rent the bike for 3, 4, or 5 hours? Use an equation to explain your answer. A.CED.1, SMP 2 Copyright © McGraw-Hill Education No; the equation is 5(h + 1) = 25; when h = 5, the left side of the equation is 5(5 + 1) = 5(6) = 30, PREREQUISITES 4 hours; the equation is 2(3h - 1) = 22; only h = 4 makes the left side of the equation equal to the right side. 20 CHAPTER 1 Expressions, Equations, and Functions EXAMPLE 1 Teaching Tip SMP 4 An equation is one type of model that can be used to describe a real-world situation. Help students write an appropriate model by giving them an “equation frame.” For example, [expression Easy Bike uses] = [amount Kaden paid]. • In part a, what should you do to decide if Kaden rented his bike for 6 hours? Substitute h = 6 in the equation and check to see if both sides are equal. 22/08/14 9:19 PM A solution of an equation is a value of the variable that makes the equation true. An equation may have no solution, one solution, several solutions, or infinitely many solutions. In this lesson, students will see equations with one solution, no solution, or infinitely many solutions (i.e., all real numbers). When students study quadratic equations, they will work with many equations that have two solutions. For example, the equation x2 − 1 = 8 has the solutions x = 3 and −3. Students can recognize equations with no solutions by simplifying both sides of the equation until they recognize a contradiction. For example, the equation m + 7 = m + 9 has no solution. This makes sense since adding 7 to a number cannot give the same result as adding 9 to the number. The solution set in this case is the empty set, which may be written as { } or Ø. 20 CHAPTER 1 Expressions, Equations, and Functions Copyright © McGraw-Hill Education Scaffolding Questions • Is there another correct way to write the equation in part a? Yes; for example, you could write it as 26 = 3.5h + 8.5. Math Background 020_025_ALG1_C1L5_ISG_672788.indd 20 EXAMPLE 2 A solution of an equation is a value of the variable that makes the equation true. A set of numbers from which replacements for a variable may be chosen is called a replacement set. A solution set is the set of all solutions in the replacement set. EXAMPLE 2 Teaching Tip Solve an Equation Students will simplify the given equation to get the equivalent equation 3p = 12. Encourage students to solve this equation by inspection. Later, students will be expected to solve the equation by dividing both sides by 3 (Division Property of Equality), but for now they can simply ask themselves, “3 times what number equals 12?” Complete these steps to solve the equation (82 ÷ 4 - 11)p - 2p = 12. a. USE STRUCTURE Use the order of operations to simplify the expression in parentheses and write the resulting equivalent equation. Explain your steps. A.REI.1, SMP 7 First evaluate the exponent: 82 ÷ 4 - 11 = 64 ÷ 4 - 11 Next, perform the division: 64 ÷ 4 - 11 = 16 - 11. Finally, subtract: 16 - 11 = 5. The resulting equation is 5p - 2p = 12. b. USE STRUCTURE Explain how to simplify the resulting equation. What property A.REI.1, SMP 7 justifies this step of the process? 5p - 2p = 12 can be simplified by combining like terms: 3p = 12. This is justified by the Substitution Property. c. USE STRUCTURE What is the solution of the equation? How do you know? A.REI.3, SMP 7 p = 4; this is the only value for p that can be multiplied by 3 to get a product of 12. d. INTERPRET PROBLEMS Explain how you can check that your solution is correct. A.REI.3, SMP 1 Check that substituting p = 4 in the original equation results in a true statement. Scaffolding Questions • What is the first step in simplifying the expression in parentheses? Why? Evaluate the exponent; this is the first step in the order of operations. (82 ÷ 4 - 11)4 - 2(4) = (5)4 - 2(4) = 20 - 8 = 12, so the solution checks. e. CRITIQUE REASONING The set {3 < p < 5} is given as the replacement set for the equation (82 ÷ 4 - 11)p - 2p = 15. This set is shown on the number line. Martina says that this changes the solution of the equation. Do you agree with Martina? Explain your answer. A.REI.3, SMP 3 0 1 2 3 4 5 6 7 8 9 10 Sample answer: Yes; the equation simplifies to 3p = 15, so the solution is p = 5. Since 5 is not a Copyright © McGraw-Hill Education value in the replacement set, the equation has no solution. f. DESCRIBE A METHOD If no replacement set is given for an equation, describe how you could go about solving the equation. How does having a replacement set given change A.REI.1, SMP 8 your approach? Sample answer: If there is no replacement set given and if the equation is simple enough, I can try values and check them in the equation, refining my guess each time. If the equation is not simple, I can simplify the equation using properties of real numbers until the equation is simple enough that I can tell what the solution is. If a replacement set is given, I may be able to check the values from the set in the equation instead of simplifying the equation first. 1.5 Equations 21 Emphasizing the Standards for Mathematical Practice An important element of SMP 1 (Make sense of problems and persevere in solving them) is checking that answers make sense. Explain to students that a solution of an equation makes the equation “balance.” That is, a value of the variable is a solution if it makes the two sides of the equation equal. In order to check a solution, substitute the value of the variable in the original equation. Ask students why they think it is important that they substitute the value in the original equation, and help students understand that this ensures that no errors were made in the intermediate steps of the solution. Copyright © McGraw-Hill Education 020_025_ALG1_C1L5_ISG_672788.indd 21 SMP 7 22/08/14 9:19 PM • When you have the equation 5p − 2p = 12, are there any like terms? Explain. Yes; 5p and −2p are like terms since they have the same variable raised to the same power. • In part e, what are some examples of numbers that are included in the shaded portion of the number line? 1 Sample answers: 3.25, 3__ 2, 4.11111, etc. 1.5 Equations 21 EXAMPLE 3 Teaching Tip SMP 7 It may be beneficial for visual learners to use a highlighter to mark each step of the solution to show which part of the equation has changed from one line to the next. Scaffolding Questions • Why does 9x + 3 = 9x + 2 have no solution? Adding 3 to a quantity cannot give the same result as adding 2 to a quantity. • If the constant terms in the simplified equation were different, would the equation have a solution? As long as the constants are different from each other, the equation will have no solution. Teaching Tip SMP 7 Students may find it strange that an equation can have any real number as a solution. In order to help students become familiar with this idea, you may want to ask them to write their own examples of such equations. Students might start by writing identical expressions on either side of the equation, such as x + 4 = x + 4, and then modify one or both sides to make them look different from each other. For example, x + 4 = 2 + x + 2. Solve an Equation a. USE STRUCTURE Use one or more properties to justify each step of the solution process shown below. A.REI.1, SMP 7 5x + 4x + 3 = 2x + 2 + 7x Original equation 9x + 3 = 2x + 2 + 7x Substitution Property 9x + 3 = 2x + 7x + 2 Commutative Property 9x + 3 = 9x + 2 Substitution Property b. CONSTRUCT ARGUMENTS What is the solution to the original equation? A.REI.3, SMP 3 Justify your answer. There is no solution; there is no value of x for which 9x + 3 can have the same value as 9x + 2. c. REASON QUANTITATIVELY Change one term in the original equation so that the equation has exactly one solution, when x = 1. How do you know that you are correct? A.REI.1, SMP 2 Sample answer: 5x + 4x + 3 = 3x + 2 + 7x; the equation simplifies to 9x + 3 = 10x + 2; substitute 1 for x: 9 + 3 = 10 + 2. EXAMPLE 4 Solve an Equation Complete these steps to solve the equation -3b + 9b + 17 = 5b + 15 + b + 2. a. USE STRUCTURE Use one or more properties to justify each step of the solution process shown below. A.REI.1, SMP 7 -3b + 9b + 17 = 5b + 15 + b + 2 Original equation -3b + 9b + 17 = 5b + b + 15 + 2 Commutative Property -3b + 9b + 17 = 6b + 17 Substitution Property 6b + 17 = 6b + 17 Substitution Property b. CONSTRUCT ARGUMENTS What is the solution to the original equation? Justify your answer. A.REI.3, SMP 3 The solution is any real number; for any value of b, the left side of the equation is equal to the right side. c. REASON QUANTITATIVELY Suppose the 17 in the original equation was an 18. Without solving another equation, what would the solution to this equation be? Explain how you know. A.REI.1, SMP 2 There would be no solution; the final equation would be 6b + 18 = 6b + 17, which is never true for any value of b. 22 CHAPTER 1 Expressions, Equations, and Functions Differentiating Instruction 020_025_ALG1_C1L5_ISG_672788.indd 22 22/08/14 9:19 PM If students have difficulty identifying the properties that justify each step in these solutions, ask them to first explain how each line of the solution is different from the previous line. Once students have verbalized what has changed, they may have an easier time connecting this to one of the properties of numbers they have seen. You may want to post a list of the properties of numbers on a wall of the classroom. This will give students a “menu” of properties to choose from. • Why is it helpful to use the Commutative Property first when you simplify the expression 5b + 15 + b + 2? This allows you to put like terms next to each other; then you can combine like terms. 22 CHAPTER 1 Expressions, Equations, and Functions Copyright © McGraw-Hill Education Scaffolding Questions • When you solve an equation, does it matter which side of the equation you simplify first? No EXAMPLE 3 Complete these steps to solve the equation 5x + 4x + 3 = 2x + 2 + 7x. Copyright © McGraw-Hill Education EXAMPLE 4 Some equations have no solution. Other equations have more than one solution. An equation that is true for every value of the variable is called an identity. For example, x + 3 = 3 + x is an identity. EXAMPLE 5 Write and Solve an Equation EXAMPLE 5 The table shows the fees charged for overdue items at the Cedarville Library. a. USE A MODEL Let d be the number of days that a book is overdue. Let F be the total late fee that is charged for the book. Write an equation of the form F = [expression] that shows how to calculate the late fee. Explain how you wrote the equation. A.CED.1, SMP 4 Teaching Tip Item Late Fees Book $1.10 plus $0.25 per day CD $1.50 plus $0.75 per day DVD $3.00 plus $1.25 per day F = 0.25d + 1.10; the late fee is 0.25 times the number of days, d, plus the fee of $1.10, which does not depend on the number of days. b. USE A MODEL Jamar has a book that is 5 days overdue. Show how to solve an equation to find the late fee for the book. Use the spaces provided to show the steps of A.REI.3, SMP 4 your solution and write an explanation of each step in the spaces at the right. F = 0.25d + 1.10 Original equation from part a F = 0.25(5) + 1.10 Substitute 5 for d. F = 1.25 + 1.10 Multiply. F = 2.35 Add. The total late fee for Jamar’s book is $2.35. d. USE A MODEL Write an equation that shows how to calculate the total late fee F for a CD that is d days overdue. Then show how to use the equation to find the total fee for a A.CED.1, SMP 4 CD that is 7 days overdue. F = 0.75d + 1.50; F = 0.75(7) + 1.50 = 5.25 + 1.50 = 6.75; the late fee is $6.75. Copyright © McGraw-Hill Education Encourage students to express their equation, F = 0.25d + 1.10, in words. (”The fee is $0.25 times the number of days, plus $1.10.”) Working back and forth between different representations—in this case, an equation and a verbal description—is an essential skill for success in algebra. Scaffolding Questions • In part b, do you substitute 5 for F or for d in the equation? Why? d; d represents the number of days, and Jamar’s book is 5 days overdue. c. REASON QUANTITATIVELY Write a sentence explaining what the last line of your A.REI.3, SMP 2 solution tells you. e. USE A MODEL Write an equation that shows how to calculate the total late fee F for a DVD that is d days overdue. Hailey has a DVD with a late fee of $10.50. Use your equation to determine whether Hailey’s DVD is 9 days overdue. Explain your answer. SMP 4 • How do you simplify 0.25(5) + 1.10? Why? First multiply, then add. This is required by the order of operations. A.CED.1, SMP 4 F = 1.25d + 3.00; d = 9 is not a solution of the equation 10.50 = 1.25d + 3.00, so the DVD is not 9 days overdue. f. REASON QUANTITATIVELY Liza returns a CD to the library and his charged a late fee of $6.00. How many days overdue was Liza’s CD? Explain your reasoning. A.REI.1, SMP 2 Sample answer: 6 days; An equation for the situation is 6 = 1.5 + 0.75d. I can substitute whole number values for d until I find one that makes the equation true. 1.5 Equations 23 ELL Strategies 020_025_ALG1_C1L5_ISG_672788.indd 23 22/08/14 9:19 PM Copyright © McGraw-Hill Education Example 5 includes some vocabulary that may be unfamiliar to ELL students. You may want to begin with a brief discussion in which you ask the class if anyone can provide synonyms for these words. This will help students understand that a fee is an amount that is charged. Also, be sure students realize that overdue is a synonym for late that is often used for items that have been borrowed. 1.5 Equations 23 PRACTICE PRACTICE Connecting Exercises to Standards Three Web sites sell food for dogs that require a special diet. The table shows expressions that give the total cost of ordering the dog food from each Web site. Use the table for Exercises 1–3. In Exercises 1–3 and 8 students must write an equation to represent a given real world situation and solve a problem, satisfying A.CED.1. 1. USE A MODEL Natasha bought dog food from Super Chow and Super Chow paid a total of $16.50. Write an equation that relates the expression the Web site uses to calculate the cost and the amount Natasha paid. Then use the equation to explain whether you think Natasha A.CED.1, SMP 4 bought 7 pounds of dog food. In Exercises 4, 6, and 7 students must justify each step in the solution of a linear equation, satisfying A.REI.1. Web Site Cost for p pounds ($) Canine Kitchen 1.75(p- 1) Pet Zone 2.50(p + 2) 1.50p + 6 1.50p + 6 = 16.50; Yes, Natasha bought 7 pounds of dog food because when p = 7, the left side of the equation is 1.50(7) + 6 = 16.50, so the two sides of the equation are the same. 2. REASON QUANTITATIVELY Isaac bought dog food from Pet Zone and paid a total of $32.50. Did he buy 10, 11, or 12 pounds of dog food? Use an equation to justify your A.CED.1, SMP 2 answer. 11 pounds; the equation is 2.50(p + 2) = 32.50; only p = 11 makes the left side of the equation equal to the right side. In Exercise 5 students solve an equation using values from a provided replacement set, satisfying A.REI.3. 3. CONSTRUCT ARGUMENTS The equation 1.75(p - 1) = 42 can be used to find the amount of dog food the Cheng family ordered from Canine Kitchen. Did the Chengs order more or less than 20 pounds of dog food? Justify your response. A.CED.1, SMP 3 More; when p = 20, the left side of the equation is 1.75(20 - 1) = 33.25, since this is less than $42, the Chengs must have ordered more than 20 pounds of dog food. Addressing the Standards Dual Coding CCSS SMP 1 A.CED.1 4 2 A.CED.1 2 3 A.CED.1 3 4 A.REI.1 7 5 A.REI.3 7 6–7 A.REI.1 7 8 A.CED.1 2, 3, 4 Evaluate the exponent: 16 = x + 4 + (16 - 6); evaluate the expression in parentheses: 16 = x + 4 + 10; add: 16 = x + 14; the solution is x = 2; the solution is correct because substituting x = 2 in the original equation results in a true statement. 5. USE STRUCTURE The values on the number line are the replacement set for the equation m2 + 1 = 5. Which of the values, if any, are solutions of the equation? Explain. A.REI.3, SMP 7 -5 -4 -3 -2 -1 0 1 2 3 4 5 2 and -2; substituting m = 2 or m = -2 in the equation makes the left side of the equation equal Copyright © McGraw-Hill Education Exercise 4. USE STRUCTURE Describe the steps you use to solve the equation 16 = x + 4 + (24 - 6). Then explain how you know your solution is correct. A.REI.1, SMP 7 to 5, so 2 and -2 are both solutions. 24 CHAPTER 1 Expressions, Equations, and Functions Common Errors In Exercise 5, students might check the given values on the number line, working from left to right. Once they determine that x = -2 is a solution of the equation, they may stop, assuming that they have found “the solution.” Remind students that an equation may have more than one solution and that they should continue to check the other values. 020_025_ALG1_C1L5_ISG_672788.indd 24 22/08/14 9:19 PM Copyright © McGraw-Hill Education 24 CHAPTER 1 Expressions, Equations, and Functions USE STRUCTURE Solve each equation. First simplify the equation and use a property to A.REI.1, SMP 7 justify each step. Then write the solution and explain how you found it. 6. c + 12 + 6c = 10c - 3c + 11 + 1 Original equation c + 12 + 6c = 7c + 12 Substitution Property c + 6c + 12 = 7c + 12 Commutative Property 7c + 12 = 7c + 12 Substitution Property Solution: The solution is any real number; for any value of c, the left side of the equation is equal to the right side. 7. 2n + 2 + 2n = 6n - 2n - 2 Original equation 2n + 2 + 2n = 4n - 2 Substitution Property 2n + 2n + 2 = 4n - 2 Commutative Property 4n + 2 = 4n - 2 Substitution Property Solution: There is no solution; there is no value of n for which 4n + 2 can have the same value as 4n - 2. 8. A Web site offers its members a special rate for online movies, as shown in the advertisement. Let m be the number of movies you watch and let C be the total cost to watch the movies. A.CED.1 a. USE A MODEL Write an equation that relates the total cost to the number of movies you watch. SMP 4 Movie Mania One-time sign-up fee: $6.85 Then watch as many movies as you like for just $2.99 per movie! C = 2.99m + 6.85 b. USE A MODEL Jeffrey watches 16 movies this month. Explain how to use the SMP 4 equation to find his total cost. Common Errors Students who are familiar with solving an equation by performing the same operation on each side may make an error in Exercise 6. Once they simplify the equation to 7c + 12 = 7c + 12, they may subtract 7c from both sides and subtract 12 from both sides, leaving them with 0 = 0. Or they may subtract 12 from each side and then divide each side by 7c to get 1 = 1. Students might conclude that the original equation therefore has no solution. Encourage students to stop at an earlier stage of the process when they can recognize that any value of the variable makes the two sides equal. Copyright © McGraw-Hill Education Substitute m = 16 in the equation; C = 2.99(16) + 6.85 = $54.69 c. CRITIQUE REASONING Madison said she joined the site and paid exactly $19 to watch some movies. Her sister said this is impossible. Who is correct? Explain. SMP 3 Her sister is correct. The cost of 4 movies is 2.99(4) + 6.85 = $18.81 and the cost of 5 movies is 2.99(5) + 6.85 = $21.80, so no number of movies costs exactly $19. d. REASON QUANTITATIVELY SuperFlix has no sign-up fee, just a flat rate per movie. If renting 13 movies at MovieMania costs the same as renting 9 movies at SuperFlix, what does SuperFlix charge per movie? SMP 2 2.99(13) + 6.85 = 45.72; 45.72 = 9p; p = 5.08; SuperFlix charges $5.08 per movie. 1.5 Equations 25 Emphasizing the Standards for Mathematical Practice You may want to use Exercise 8 to discuss aspects of SMP 4 (Model with mathematics). One part of the standard is reflecting on whether results make sense. Once students have used their model (equation) to determine the cost of watching 16 movies, ask them whether their result is reasonable, and why. To answer this, students might decide to determine the cost without using the model. For instance, they could use the text of the advertisement to reason that watching 16 movies costs $2.99 per movie, which is $47.84, plus the sign-up fee of $6.95, which is a total of $54.69. This should match the result students got by solving an equation, which shows that the answer makes sense. Copyright © McGraw-Hill Education 020_025_ALG1_C1L5_ISG_672788.indd 25 22/08/14 9:19 PM 1.5 Equations 25 1.6 Relations 1.6 Relations STANDARDS Objectives A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Also addresses: F.IF.1 Standards for Mathematical Practice: 1, 2, 3, 4, 6 • Represent a relation in multiple ways. • Identify the domain and range of a relation. • Interpret the graph of a relation. A relation is a set of ordered pairs. The set of the first numbers in the ordered pairs is the domain. The set of the second numbers in the ordered pairs is the range. EXAMPLE 1 Represent a Relation EXPLORE A newspaper reporter asked five teenagers about the last time they babysat. Each teenager gave the number of hours he or she babysat and the amount he or she earned. The graph shows the results. a. USE A MODEL Write the set of ordered pairs shown in the graph. What do the numbers in each ordered pair represent? A.REI.10, SMP 4 {(1, 10), (2, 20), (3, 30), (3, 35), (6, 40)}; the first number is the Amount Earned Babysitting 50 y Amount Earned ($) STANDARDS Content: A.REI.10, F.IF.1 Practices: 1, 2, 3, 4, 6 Use with Lesson 1–6 40 Lee Chantel Reynaldo 30 20 Eliza 10 Jordan x number of hours, the second number is the amount earned. 0 2 4 6 8 10 Time (hours) PREREQUISITES • Plot points on the coordinate plane EXAMPLE 1 • What would the ordered pair (5.5, 52) represent? A teenager babysat for 5.5 hours and was paid $52. y 1 10 2 20 3 30 3 35 6 40 Domain Range 1 2 3 6 10 20 30 35 40 c. INTERPRET PROBLEMS Write the domain and range for the relation. Write each as a F.IF.1, SMP 1 set within brackets, { }. Domain: {1, 2, 3, 6}; range: {10, 20, 30, 35, 40} d. USE A MODEL How can you tell from the graph which of the teenagers babysat for the A.REI.10, SMP 4 same number of hours? Chantel and Reynaldo babysat for the same number of hours because the points for their ordered pairs lie on the same vertical line. 26 CHAPTER 1 Expressions, Equations, and Functions Math Background 026_029_ALG1_C1L6_ISG_672788.indd 26 2/25/15 12:02 AM Lessons 1.6 and 1.7 introduce students to relations and functions. A relation is simply a set of ordered pairs. The ordered pairs can be specified by a list, in which case the list is usually written using set notation, such as {(-2, 4), (3, -5), (0, 4)}. The ordered pairs may also be given in a table, a graph, or an equation. For example, the equation y = x + 3 describes the relation with an infinite set of ordered pairs that includes (-4, -1), (8, -1), and (π, π + 3). The set of the x-values of the ordered pairs is the domain; the set of the y-values in the ordered pairs is the range. Note that elements of the domain and range are usually written in ascending order and repeated elements are only listed once. For the first relation given above, the domain is {-2, 0, 3} and the range is {-5, 4}. • Why are there only 4 domain values? Two of the teenagers babysat for 3 hours each. 26 CHAPTER 1 Expressions, Equations, and Functions Copyright © McGraw-Hill Education Scaffolding Questions • How do you find the ordered pair for Jordan? Start at the origin and count units to the right to find the x-value; then count units moving upward to find the y-value. x Copyright © McGraw-Hill Education Teaching Tip SMP 4 As students work through this exploration, you may want to have a brief discussion about the pros and cons of each representation of the given relation. For example, the graph may make it easy to see any patterns in the ordered pairs, while the mapping makes explicit the connection between domain values and their corresponding range values. b. USE A MODEL You can use a table or a mapping to represent a relation. A mapping is a diagram that shows how each element in the domain is paired with an element in the F.IF.1, SMP 4 range. Complete the table and mapping shown below. EXAMPLE 2 e. REASON QUANTITATIVELY Which teenager was paid the highest hourly rate? Justify your answer. F.IF.1, SMP 2 Teaching Tip Chantel; she made $35 for 3 hours, which is $11.67 per hour. Jordan, Eliza, and Reynaldo each made $10 per hour, and Lee made $6.67 per hour. In a relation, the value that determines the output is the independent variable. The variable with a value that is dependent on the value of the independent variable is the dependent variable. Interpret a Graph Michelle’s Walk a. USE A MODEL What are the independent variable and dependent variable? Explain. F.IF.1, SMP 4 Independent variable: time; dependent variable: distance; the distance depends on the amount of time Michelle has been walking Distance from Home EXAMPLE 2 Michelle started at her house and went for a long walk. The graph represents her distance from home since the walk began. Time b. COMMUNICATE PRECISELY Describe what happens in the graph. A.REI.10, SMP 6 As time increases, distance increases, this shows Michelle walking further from home. The graph becomes a horizontal line, so the distance from home is not changing but time increases. This may mean she stops for a rest. Then she continues to walk further from home until the peak of the graph. At this point, she begins to walk back to her house until she arrives at home. house. Then she reaches her house and starts to walk away from her house, stops, walks further away from her house, and stops again. In the first graph, Michelle started and ended at her house. In the second graph, Michelle started and ended somewhere Michelle’s Walk Distance from Home Copyright © McGraw-Hill Education There would be a horizontal segment on the downward-slanting part of the graph. Sample answer: As time increases, Michelle walks toward her • What is happening to Michelle’s distance from home during the horizontal part of the graph? Her distance from home is not changing (it remains constant). Time besides her house. 1.6 Relations 27 Emphasizing the Standards for Mathematical Practice Example 2 can foster an interesting discussion related to SMP 4 (Model with mathematics). In particular, the example requires students to think carefully about one representation of a relation (a graph) so that they can translate it into another representation (words). You might challenge students by asking them if the horizontal portion of the graph must mean that Michelle stopped walking. Although this is one possibility, the horizontal portion of the graph only shows that her distance from home is not changing. It is possible that she is walking in a circle, at a fixed distance around her home. While this second option may not seem likely within the real-world context, recognizing this possibility requires students to have a deep understanding of what the graph represents. Copyright © McGraw-Hill Education 026_029_ALG1_C1L6_ISG_672788.indd 27 Some students may look at the graph of Michelle’s walk, notice its shape, and assume that she was walking on a steep mountain. Be sure students understand that the graph is not a picture or map of Michelle’s route. It only tells her relative distance from home at various times. To help students understand this, have them slowly trace the graph from left to right with their finger. As they do so, ask them to describe how Michelle’s distance from home is changing. Scaffolding Questions • Why does it make sense that the graph starts at the origin? At time 0, the distance from home is 0 because Michelle starts at her house. c. USE A MODEL How would the graph be different if Michelle decided to stop at her aunt’s house on the way home and spend the night there? A.REI.10, SMP 4 d. COMMUNICATE PRECISELY The graph shows another walk that Michelle took. What does this graph show? Compare the starting and ending points to those in the first graph. A.REI.10, SMP 6 SMP 4 22/08/14 9:20 PM • What point along the graph represents the moment when Michelle is farthest from home? How do you know? At the peak of the graph she is farthest from home because that is when the distance from home is the greatest. 1.6 Relations 27 PRACTICE PRACTICE Exercises 1 and 3–6 provide students with an opportunity to practice interpreting a graph that represents a relation between two quantities, satisfying A.REI.10. Addressing the Standards Exercise Dual Coding SMP 1 A.REI.10, F.IF.1 1, 4 2 F.IF.1 1, 3, 4 3–4 A.REI.10 6 5 A.REI.10 3 6 A.REI.10 4 7 F.IF.1 3 8 F.IF.1 4 F.IF.1, SMP 1 Domain: {2, 3, 5.5, 8}; range: {0.5, 2, 4.5} b. USE A MODEL What are the independent and dependent variables? F.IF.1, SMP 4 Independent variable: length; dependent variable: weight c. USE A MODEL Complete the table, the mapping, and the graph below. x y 2 0.5 3 0.5 3 2 5.5 4.5 8 4.5 Domain Range 2 3 5.5 8 0.5 2 4.5 F.IF.1, SMP 4 Snake Sizes 5 y 4 3 2 1 x 0 2 4 6 8 10 Length (ft) d. USE A MODEL How can you tell from the graph which snakes have the same weight? A.REI.10, SMP 4 If the points lie on the same horizontal line, they have the same weight. e. USE A MODEL What does it mean if there are two points on the graph that lie on a vertical line? A.REI.10, SMP 4 If there are two points on any vertical line on the graph, then two snakes of the same length have different weights. 2. The graph shows the number of items that eight customers bought at a supermarket and the total cost of the items. F.IF.1 a. INTERPRET PROBLEMS Write the domain and range for the relation. SMP 1 Domain: {2, 3, 5, 6, 8, 9}; range: {2, 6, 8, 10, 12, 14, 16, 18} b. USE A MODEL Explain how you can tell from the graph how many of the customers spent more than $12. SMP 4 3 customers; find the number of points that lie above the horizontal line y = 12. Supermarket Costs 20 y 16 12 8 4 x 0 2 4 6 8 Number of Items 10 Copyright © McGraw-Hill Education CCSS a. INTERPRET PROBLEMS Write the domain and range for the relation. Total Cost ($) Exercises 1, 2, 7, and 8 provide students with an opportunity to practice recognizing the domain and range of a relation, satisfying F.IF.1. 1. The following ordered pairs give the length in feet and the weight in pounds of five snakes at the reptile house of a zoo: {(5.5, 4.5), (3, 0.5), (3, 2), (8, 4.5), (2, 0.5)} Weight (lb) Connecting Exercises to Standards c. CRITIQUE REASONING A student said you can add the values in the domain to find the total number of items these customers bought. Do you agree? Explain. SMP 3 No; two customers bought 5 items and 8 items, so you cannot add domain values. 28 CHAPTER 1 Expressions, Equations, and Functions Common Errors In Exercise 1a, some students may write the domain as {2, 3, 3, 5.5, 8}. Remind students that in a set, repeated elements are only listed once. Thus, the set {2, 3, 3, 5.5, 8} is actually the same as the set {2, 3, 5.5, 8}. In addition, students should be aware that changing the order of the elements does not change a set, so the domain could be written as {8, 2, 5.5, 3}. However, it is customary to list set elements in ascending order. 026_029_ALG1_C1L6_ISG_672788.indd 28 Copyright © McGraw-Hill Education 28 CHAPTER 1 Expressions, Equations, and Functions 2/25/15 12:10 AM 3. COMMUNICATE PRECISELY Describe what happens in Tim’s graph. SMP 6 Tim drives away from the pizzeria, stops to make a delivery, continues to drive away from the pizzeria, stops to make another Pizza Deliveries Distance from Pizzeria Tim and Lauren use their cars to deliver pizzas. The graph represents their distance from the pizzeria starting at 6 pm. Use the graph for Exercises 3–6. A.REI.10 Lauren Tim Time delivery, and then returns to the pizzeria. 4. COMMUNICATE PRECISELY Describe what happens in Lauren’s graph. SMP 6 At 6 PM, Lauren is not at the pizzeria; she drives directly there, without stopping. 5. CRITIQUE REASONING A student said that Tim’s and Lauren’s graphs intersect, so their cars must have crashed at some time after 6 pm. Do you agree or disagree? SMP 3 Explain. Disagree; the intersection point represents a time when Tim and Lauren were both at the same distance from the pizzeria. 6. USE A MODEL After 6 pm, which delivery person was the first to return to the SMP 4 pizzeria? How do you know? Lauren; Her graph intersects the x-axis before Tim’s graph. This means her distance from the Common Errors In Exercise 5, some students may see the graph as a ”road map” that shows the paths of the cars. In this case, students may conclude that the point of intersection represents a point of collision. Help students understand that the vertical axis represents the drivers’ distances from the pizzeria. The point of intersection shows that there is a moment when both drivers are at the same distance from the pizzeria. However, they could be on separate roads or even in different towns. pizzeria was 0 before Tim’s distance from the pizzeria was 0. 7. CRITIQUE REASONING Cameron said that for any relation, the number of elements in the domain must be greater than or equal to the number of elements in the range. Do you agree? If so, explain why. If not, give a counterexample. F.IF.1, SMP 3 Disagree; sample counterexample: in the relation {(1, 2), (1, 3)}, the domain is {1}, so it has one 8. USE A MODEL The graph shows the height of an elevator above the ground. Describe the domain and range for this relation in words and by using inequalities. Then give three ordered pairs in the relation. F.IF.1, SMP 2 Domain: all real numbers from 0 to 4, {0 ≤ x ≤ 4}; Range: all real numbers from 0 to 80, {0 ≤ y ≤ 80}; Sample answer: (0, 0), (1, 20), (2, 40) Height of an Elevator 100 y 80 Height (ft) Copyright © McGraw-Hill Education element, while the range is {2, 3}, which has two elements. 60 40 20 x 0 2 4 6 8 10 Time (s) 1.6 Relations 29 Emphasizing the Standards for Mathematical Practice You can use Exercise 8 to connect to SMP 2 (Reason abstractly and quantitatively). Students may be able to look at the graph and describe the domain as ”all real numbers from 0 to 4,” but not be able to translate this into a meaningful statement about the real-world situation. In this case, the domain shows that the elevator traveled for 4 seconds. Similarly, the range shows that the elevator ascended from a height of 0 feet (ground level) to a height of 80 feet. The fact that all real numbers from 0 to 80 are included in the range shows that the elevator passed through all the possible heights from 0 feet to 80 feet. Copyright © McGraw-Hill Education 026_029_ALG1_C1L6_ISG_672788.indd 29 2/25/15 12:13 AM 1.6 Relations 29 1.7 Functions 1.7 Functions F.IF.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Also addresses: F.IF.2, F.IF.5 Standards for Mathematical Practice: 1, 2, 3, 4, 6 Content: F.IF.1, F.IF.2, F.IF.5 Practices: 1, 2, 3, 4, 6 Use with Lesson 1–7 • Understand the definition of a function and identify relations that are functions. • Use and interpret function notation. • Relate the domain of a function to its graph. A function is a relation in which there is exactly one output for each input. In other words, each element of the domain is assigned to exactly one element of the range. EXAMPLE 1 Identify Functions F.IF.1 EXPLORE Tristan surveyed students at some local high schools. At each school, he asked six students how long they studied for their last exam and the score they received on the exam. His data is shown in the table, mapping, and graph below. Central High Score Time (h) Score 0.5 81 1 81 3 92 1.5 75 2 90 1.5 94 Westlake High School Time (h) Score 1 1.5 2 2.5 3 4 77 Miller High School 100 y 78 80 85 60 93 Score STANDARDS STANDARDS Objectives 40 20 x 0 1 2 3 4 5 Time (h) a. USE A MODEL For each school, is the relation a function? Why or why not? SMP 4 Central: No; the input value 1.5 is assigned to two output values, 75 and 94. Westlake: Yes; each input value is assigned to exactly one output value. • Plot points on the coordinate plane EXAMPLE 1 SMP 4 Teaching Tip Help students make sense of each model by first asking them to identify the input (the amount of time a student studies) and the output (the student’s score). • In the graph, how can you tell that an input is assigned to two outputs? Two points lie on the same vertical line. Yes; the relation is not a function because the input value 3 is assigned to two output values, 81 and 87. To make the relation a function, Tristan could change the ordered pair (3, 81) to (3.5, 81). 30 CHAPTER 1 Expressions, Equations, and Functions Math Background Functions are a key tool for describing real-world phenomena. In many real-world situations, it makes sense that each “input” value has exactly one “output” value. For example, a function might provide the temperature in downtown Houston at any given time. In this case, the input is a specific time of day, and it makes sense that there can only be one temperature (output) corresponding to that time of day. 030_035_ALG1_C1L7_ISG_672788.indd 30 22/08/14 9:20 PM Students may wonder why they need to use function notation. Point out that it provides a shorthand for naming functions and distinguishing among multiple functions. Students will also see the utility of function notation in later courses when they work with f(x2) - f(x1) complex expressions such as _______________ x -x . 30 CHAPTER 1 Expressions, Equations, and Functions 2 1 Copyright © McGraw-Hill Education Scaffolding Questions • In the mapping, how can the arrows help you decide whether the mapping is a function? If no input value has more than one arrow, the mapping is a function. Miller: No; the input value 3.5 is assigned to two output values, 80 and 90. b. CRITIQUE REASONING Tristan surveyed six students at Chavez High School, and wrote the data as this set of ordered pairs: {(3, 87), (4, 98), (2.5, 70), (1.5, 70), (0.5, 67), (3, 81)}. He claimed that the relation is not a function, but he said that he could change just one input value or one output value and make it a function. Do you agree with Tristan? Explain. SMP 3 Copyright © McGraw-Hill Education PREREQUISITES EXAMPLE 2 A graph that consists of points that are not connected is a discrete function. A function with a graph that is a line or a smooth curve is a continuous function. EXAMPLE 2 Teaching Tip Graph a Function Tickets to the county fair cost $5 each. F.IF.1, SMP 4 Yes; each input value is assigned to exactly one output value. 25 y 20 Total Cost ($) b. USE A MODEL Is the relation a function? Explain. If students have trouble going directly from the verbal description to the graph, suggest that they first make a table of values and then plot the ordered pairs from their table. County Fair Tickets a. USE A MODEL Make a graph that shows the relationship between the number of tickets you buy and the total cost F.IF.1, SMP 4 of the tickets. 15 10 5 c. COMMUNICATE PRECISELY Is the function a discrete or continuous function? Why? Use the real-world context to explain why your F.IF.1, SMP 6 answer makes sense. x 0 2 4 6 8 10 Number of Tickets Discrete; the graph consists of points that are not connected. This makes sense because you can only buy a whole number of tickets to the fair. d. REASON QUANTITATIVELY What is the domain of the function? How is the domain related to the real-world context? F.IF.5, SMP 2 The domain is the set of whole numbers, {0, 1, 2, ...}. You can only buy a whole number of tickets. e. REASON QUANTITATIVELY What is the range of the function? How is the range related to the real-world context? F.IF.1, SMP 2 The range is the multiples of 5, {0, 5, 10, ...}. Since the total cost is $5 times the number of tickets, the total cost must be a multiple of $5. f. CRITIQUE REASONING A student said that if you write the function as a set of ordered pairs, the ordered pair (25, 100) will be an element of the set. Do you agree or F.IF.1, SMP 3 disagree? Explain. Scaffolding Questions • For each point on the graph, how is the y-value related to the x-value? The y-value is 5 times the x-value. • Would it make sense to plot the point (1.5, 7.5)? Why or why not? No; you cannot buy 1.5 tickets to the fair. • Does it make sense to include the point (0, 0) in the graph? Why or why not? Yes; if you buy 0 tickets, you pay 0 dollars. Disagree; the first number in the ordered pair is the number of tickets you buy; if you buy 25 tickets, then the cost is 5(25) = $125, not $100, so (25, 100) is not in the set. Copyright © McGraw-Hill Education SMP 4 g. COMMUNICATE PRECISELY Describe a situation that could be modeled by a function that includes the same ordered pairs as the function for the cost of county fair tickets, but is of the type that you did not select as your answer to part c. Explain why this situation leads to a function of the other type. F.IF.1, SMP 6 Sample answer: Malia rides her bike at a speed of 5 miles per hour away from her home. The distance she has traveled is 5 times the number of hours she has been biking. This situation would be represented by a continuous function. It includes ordered pairs such as (1, 5) and (2, 10), like the ticket cost situation, but the function also makes sense for all values of time between whole numbers of hours. • What do you notice about all the points on your graph? They all lie along a straight line that passes through the origin. 1.7 Functions 31 ELL Strategies ELL students may be unfamiliar with the word context. Explain that a context is a real-world setting for a problem. In Example 2, the context is a county fair where tickets cost $5 each. You may want to check for understanding by asking students to describe the context of Example 1 in their own words. If students have difficulty, suggest that they reread the given information and underline any words that provide information about the real-world setting of the problem. Copyright © McGraw-Hill Education 030_035_ALG1_C1L7_ISG_672788.indd 31 22/08/14 9:20 PM 1.7 Functions 31 EXAMPLE 3 Teaching Tip SMP 1 If students need help getting started with their table, suggest that they first fill in the row for the values of x and use equally-spaced values, such as x-values that increase by 1. Then have students find the corresponding y-values. By choosing x-values that are equally-spaced, students may be able to identify patterns that can help them fill in the y-values. The vertical line test is a way to check whether a graph represents a function. If there is a vertical line that intersects the graph in more than one point, then the graph is not a function. Otherwise, the relation is a function. EXAMPLE 3 Graph a Function Follow these steps to graph the equation y - 2x = 4 and determine whether the equation represents a function. a. INTERPRET PROBLEMS Complete the table below by finding five ordered pairs that satisfy the equation. F.IF.1, SMP 1 Sample ordered pairs shown. x -2 -1 0 1 2 y 0 2 4 6 8 b. INTERPRET PROBLEMS Use the table to help you graph the equation on the F.IF.1, SMP 1 coordinate plane provided. c. CONSTRUCT ARGUMENTS Is the relation a function? Use your graph to justify your answer. F.IF.1, SMP 3 Yes; there is no vertical line that intersects the graph in more than one point, so the relation is a function by the vertical line test. Scaffolding Questions • For this relation, what happens to the y-value as the x-value increases by 1? The y-value increases by 2. • Why does it make sense to draw the graph as a continuous function? The domain is all real numbers. −8 −6 −4 O y 2 4 6 8x −4 −6 −8 d. REASON ABSTRACTLY What are the domain and range of the function? Explain how F.IF.5, SMP 2 these are related to the graph of the function. The domain and range are all real numbers. The line continues infinitely to the left and to the right, showing that the domain is all real numbers. The line continues infinitely up and down, showing that the range is all real numbers. e. COMMUNICATE PRECISELY Describe how you can use the graph of the function to find the output value that corresponds to the input value -6. F.IF.1, SMP 6 Find the value -6 on the x-axis. Then move down vertically to the graph of the function. Then move horizontally to the y-axis and read the corresponding value. The output value is -8. f. COMMUNICATE PRECISELY How did you know to draw the graph as a continuous function? Use the given equation in your justification. How can you tell if a relation is discrete? F.IF.5, SMP 6 Sample answer: The graph of y − 2x = 4 is continuous because the domain and range are all real numbers. A relation would be discrete if the equation or the context of the situation resulted in Copyright © McGraw-Hill Education • How can you check that you drew the graph correctly? Choose a point on the graph and check that the x- and y-coordinates of the point satisfy the given equation. 8 6 4 2 outputs that are not connected when graphed. 32 CHAPTER 1 Expressions, Equations, and Functions Differentiating Instruction Kinesthetic learners may benefit from using a physical object to help them perform the vertical line test. For example, students might use a pen or pencil or an uncooked strand of spaghetti to represent the vertical line. Have them hold it vertically and slowly pass it across the graph from left to right. This may make it easier for students to determine if there is ever an x-value for which the line intersects the graph in more than one point. 030_035_ALG1_C1L7_ISG_672788.indd 32 Copyright © McGraw-Hill Education 32 CHAPTER 1 Expressions, Equations, and Functions 22/08/14 9:20 PM EXAMPLE 4 Function notation is a way to use an equation to write a rule for a function. For example, in function notation, the equation y = x + 7 is written as f(x) = x + 7. The graph of the function f(x) is the graph of the equation y = f(x). Teaching Tip In function notation, f(x) denotes the element of the range corresponding to the element x of the domain. For example, for the function f(x) = x + 7, f(9) represents the output value that corresponds to the input value x = 9. Therefore, f(9) = 9 + 7 = 16. EXAMPLE 4 SMP 2 It might be helpful for students to think of a function as a “machine” that takes an input, performs one or more operations on it, and produces an “output.” A simple sketch based on the function in this example is shown below. 17.5 Use and Interpret Function Notation Candace runs a company that installs fences. She calculates the total cost C of installing a fence using the function rule C(x) = 5x + 25, where x is the length of the fence in feet. a. REASON QUANTITATIVELY What is the value of C(17.5)? What does C(17.5) F.IF.2, SMP 2 represent? C(17.5) = 112.5; C(17.5) represents the cost of installing a fence that is 17.5 feet long. So it costs $112.50 to install a fence 17.5 feet long. b. USE A MODEL Explain how Candace can use the function rule to find the cost of the F.IF.2, SMP 4 installing a fence that is 11 yards long. 11 yards is 33 feet, so calculate C(33) by substituting x = 33 in the equation; 5x + 25 C(33) = 5(33) + 25 = 190; the cost of installing the fence is $190. c. USE A MODEL Graph C(x) on the coordinate plane at the right. F.IF.1, SMP 4 d. REASON ABSTRACTLY What are the domain and range of the function? Explain how these are related to the graph of F.IF.5, SMP 2 the function. 40 The range is all real numbers greater than or equal to 25; the line starts at y = 25 and continues upward infinitely. e. COMMUNICATE PRECISELY Did you draw the graph as a discrete function or as a continuous function? Justify F.IF.1, SMP 6 your choice. Total Cost ($) The domain is all real numbers greater than or equal to 0; the line starts at the y-axis and continues infinitely to the right. 112.5 Cost of Installing a Fence 50 y Scaffolding Questions 30 20 10 x 0 2 4 6 8 10 Length of Fence (ft) • How do you use the order of operations to evaluate C(17.5)? First multiply 17.5 by 5; then, add 25 to the result. Copyright © McGraw-Hill Education Continuous; you can order a fence with a length that is any • In this problem, what units are associated with the inputs? What units are associated with the outputs? feet; dollars real number greater than or equal to 0, so it makes sense to draw a continuous graph. f. COMMUNICATE PRECISELY Would your answer to part e change if the fencing material were only sold in one-foot increments? If someone needed to fence a length that was not a whole number, what should they do? Explain. F.IF.5, SMP 6 Yes, I would draw a discrete function if the material could only be purchased in one-foot increments. The domain would only consist of whole numbers. If someone needed to fence a length that was not a whole number, they should buy a length of fence that is the next whole number greater than the length they need. 1.7 Functions 33 Emphasizing the Standards for Mathematical Practice Use what students have learned about functions to address SMP 3 (Construct viable arguments and critique the reasoning of others). For example, you might ask students to discuss the connection between relations and functions: Is every relation a function? Is every function a relation? Why? Students might draw a simple Venn diagram, as shown below, to help them respond to these questions and justify their answers. Copyright © McGraw-Hill Education 030_035_ALG1_C1L7_ISG_672788.indd 33 22/08/14 9:20 PM • In the graph, why does it make sense that the line slopes upward as you move from left to right? This shows that the cost of the fence increases as the length of the fence increases. relations functions 1.7 Functions 33 PRACTICE PRACTICE Exercise 1 requires students to work with multiple representations to identify relations that are functions, per F.IF.1. In Exercise 2, students graph a function based on a verbal description of a real-world situation, satisfying F.IF.5. 1. USE A MODEL Mario collected data about some of the players on a women’s basketball team. The data is shown in the table, mapping, and graph. Is each relation a function? Why or why not? F.IF.1, SMP 4 Age and Height Team History Years on Team Games Played 1 24 2 45 3 82 3 88 5 120 Age Height (in.) 22 70 71 72 73 74 23 25 Stats from Last Game 20 y Total Points Scored Connecting Exercises to Standards 16 12 8 4 x 0 2 4 6 8 10 Number of Free Throws Made Team history: No; the input value 3 is assigned to two output values, 82 and 88. Age and height: No; the input values 22 and 25 are both assigned to two output values. Stats from last game: Yes; each input value is assigned to exactly one output value. a. USE A MODEL Make a graph that shows the relationship between the number of servings and the number of eggs. F.IF.1, SMP 4 b. USE A MODEL Is the relation a function? Explain. F.IF.1, SMP 4 Homemade Pasta Dough 10 y 8 6 4 2 Yes; each input value is assigned to exactly one output value. x 0 2 4 6 8 10 Number of Servings c. REASON QUANTITATIVELY What is the domain of the function? How is the domain related to the real-world context? F.IF.5, SMP 2 The domain is all whole numbers greater than or equal to 1. You can only make 1 or more servings, and the number of servings must be a whole number since you can only use a whole Exercise 6 is a reasoning exercise that requires students to think carefully about the definition of a function, addressing F.IF.1. Addressing the Standards Exercise Dual Coding SMP 1 F.IF.1 4 2 F.IF.1, F.IF.5 2, 4 3–4 F.IF.1 1 5 F.IF.2, F.IF.5 2 6 F.IF.1 3 number of eggs. 34 CHAPTER 1 Expressions, Equations, and Functions Common Errors In Exercise 2 part c, students may state that the domain is all real numbers greater than 0. Students might argue that is possible to use proportional reasoning to multiply or divide quantities in a recipe and make any number of servings, such as 0.7 servings or 8.5 servings. While this is theoretically true, the recipe depends on using a whole number of eggs, which restricts the number of servings that can be made to whole numbers greater than or equal to 1. 030_035_ALG1_C1L7_ISG_672788.indd 34 34 CHAPTER 1 Expressions, Equations, and Functions 22/08/14 9:20 PM Copyright © McGraw-Hill Education CCSS Copyright © McGraw-Hill Education In Exercise 5, students interpret function notation, perform calculations using function notation, and interpret the domain of the function in the context of the situation, satisfying both F.IF.2 and F.IF.5. 2. A recipe for homemade pasta dough says that the number of eggs you need is always one more than the number of servings you are making. Number of Eggs Exercises 3 and 4 give students practice in graphing a function from an equation, addressing F.IF.1. INTERPRET PROBLEMS Graph each equation. Then explain whether or not the equation represents a function. F.IF.1, SMP 1 3. 2x + y = 6 8 6 4 2 −8 −6 −4 O 4. y = x2 y 8 6 4 2 2 4 6 8x −8 −6 −4 −4 −6 −8 O y 2 4 6 8x −4 −6 −8 Function; there is no vertical line that Function; there is no vertical line that intersects the graph in more than one point. intersects the graph in more than one point. 5. REASON QUANTITATIVELY The height h of a balloon, in feet, t seconds after it is released is given by the function h(t) = 2t + 6. SMP 5 a. What is the value of h(20), and what does it tell you? F.IF.2 h(20) = 46; the height of the balloon 20 seconds after it is released is 46 feet. b. Explain how to use the function to find the height of the balloon 2 minutes after it F.IF.2 is released. 2 minutes is 2(60) = 120 seconds, so calculate h(120) by substituting t = 120 in the equation; h(120) = 2(120) + 6 = 246; the height of the balloon is 246 feet. c. What is the height of the balloon just before it is released? How do you know? F.IF.2 6 feet; t = 0 before the balloon is released, and h(0) = 6. d. Are there any restrictions on the values of t that can be used as inputs for the function? If so, how would this affect the graph of the function? Explain. F.IF.5 Common Errors In Exercise 6, students may state that the missing value cannot be 3 or 5. Point out that if the missing value were 3 or 5, one of the ordered pairs would be repeated within the set. Although elements of a set are usually listed only once, these repeated elements would not mean that the relation is not a function. Suggest that students reread the definition of a function and ask them to think about whether there is a way to replace the question mark with a value so that a single input is assigned to more than one output. This may help students realize that the missing value cannot be −4 or −3. Sample answer: The values of t must be greater than or equal to zero because a negative value for the time does not make sense for the given situation. The graph would start at the vertical Copyright © McGraw-Hill Education axis and go only to the right. 6. CONSTRUCT ARGUMENTS The following set of ordered pairs represents a function, but one of the values is missing and has been replaced by a question mark: {(-4, -1), (-3, -1), (3, 2), (5, 2), (?, 2)}. What conclusions can you make about the missing F.IF.1, SMP 3 value? Explain. The missing value cannot be -4 or -3. If the missing value were either of these, then the set of ordered pairs would no longer be a function since there would be an input value assigned to two output values. 1.7 Functions 35 Emphasizing the Standards for Mathematical Practice You may want to use Exercise 5 to address SMP 5 (Use appropriate tools strategically). For example, students might enter the function h(t) = 2t + 6 as function Y1 in their graphing calculator. Students can use the calculator to display a table for the function and look for patterns in the y-values. Then students can display a graph and use it (or the table) to check their answers to the questions in this exercise. 2/20/15 5:22 PM Copyright © McGraw-Hill Education 030_035_ALG1_C1L7_ISG_672788.indd 35 1.7 Functions 35 1 Performance Task Performance Task Finding a Sale Price Finding a Sale Price Students will use a graph to write functions, find domain and range, and calculate sale prices. Provide a clear solution to the problem. Be sure to show all of your work, include all relevant drawings, and justify your answers. Sander’s Market is having a special on cherries. For every pound of cherries purchased beyond 3 pounds and up to 6 pounds, the price per pound is discounted by $1. Sander’s Market limits customers to 6 pounds of cherries. The graph shows the cost y in dollars for purchasing x pounds of cherries. y STANDARDS A.CED.1, A.REI.10, F.IF.1, F.IF.2, F.IF.5 Cost ($) 10 8 6 4 2 O Standards for Mathematical Practice: The Chapter 1 Performance Task reinforces Mathematical Practices SMP 1, SMP 2, SMP 3, SMP 4, and SMP 6. 2 4 6 x Weight of Cherries (lb) Part A Find the domain and range. Describe their meaning in the context of this situation. • Name some ordered pairs that lie on the graph. Sample answer: (0, 0), (2, 5), (4, 9) Copyright © McGraw-Hill Education Jump Start If students have trouble getting started with the task, suggest that they first focus on the given graph. Use some or all of the following questions to help students understand the graph. 36 CHAPTER 1 Expressions, Equations, and Functions Emphasizing the Standards for Mathematical Practice This Performance Task is closely aligned to SMP 2 (Reason abstractly and quantitatively). Throughout the task, students will need to move back and forth between the abstract mathematics and the contextual, real-world situation. For example, to highlight the abstract nature of the task, ask students to identify the point on the graph with an x-value of 2. Students should find that the point has coordinates (2, 5). They might check that this point is a solution of the equation y = 2.5x, which describes the portion of the graph for 0 ≤ x ≤ 3. To highlight the contextual side of the problem, ask students what the point (2, 5) represents. Be sure students understand that this ordered pair carries information about the real-world situation; namely, 2 pounds of cherries cost $5. 036_037_ALG1_C1PT1_ISG_672788.indd 36 • Why does it make sense that (0, 0) lies on the graph? 0 pounds of cherries cost 0 dollars. Next, have students read Part A. If they difficulty understanding the question, have them use their glossary, as needed, to look up any unfamiliar terms, such as domain, range, rational, or irrational. 36 CHAPTER 1 Expressions, Equations, and Functions Copyright © McGraw-Hill Education • How can you tell that the price per pound decreases for more than 3 pounds? The graph becomes less steep for more than 3 pounds. 22/08/14 9:10 PM SMP 1 Teaching Tip Part C provides an opportunity to connect to SMP 1 (Make sense of problems and persevere in solving them). Ask students why it makes sense that this real-world relation is a function. If necessary, help students see that for any weight of cherries, there is only one corresponding cost. In other words, each input (weight) is paired with exactly one output (cost). It would not make sense if a single weight had more than one cost associated with it. Part B Use the graph to determine the cost per pound for buying 3 or fewer pounds of cherries. Write and solve an equation to find the total cost for purchasing 2.5 pounds of cherries. Describe how the graph can be used to check your answer. Part C Explain whether the graph represents a relation and/or a function. Part D Copyright © McGraw-Hill Education Write a function f(x) that gives the cost for purchasing x pounds of cherries, where 3 < x ≤ 6. Explain how you arrived at your answer. CHAPTER 1 Performance Task 37 Scoring Rubric Copyright © McGraw-Hill Education 036_037_ALG1_C1PT1_ISG_672788.indd 37 Common Errors Some students may have difficulty writing a function rule in Part D. In particular, students might write f(x) = 3x + 1.5 instead of f(x) = 1.5x + 3. Encourage students to check their rule by seeing if it works for some of the ordered pairs on the graph. For instance, the point (4, 9) lies on the graph, so students should be able to evaluate their function for x = 4 and find that f(4) = 9. If they get a different result, have students revisit the steps they used to develop the function rule. 22/08/14 9:10 PM Part Max Points A 2 The domain is 0 ≤ x ≤ 6 because it is impossible to purchase a negative amount of cherries and a customer cannot purchase more than 6 pounds. The range is 0 ≤ y ≤ 12 because it is impossible to spend a negative amount of money and if a customer purchases the maximum 6 pounds of cherries he or she will spend $12. B 2 The line y = 2.5x represents the cost of buying 3 or fewer pounds of cherries. y = 2.5(2.5) or y = 6.25 represents the cost of buying 2.5 pounds of cherries. So 2.5 pounds of cherries cost $6.25. You can check your answer by making sure the graph passing through the point (2.5, 6.25). C 2 The graph represents a relation, because it is a set of ordered pairs. The graph also represents a function, because each x-value (number of pounds) has a unique corresponding y-value (cost). D 2 f(x) = 3 + 1.5x; Find two points on the graph when 3 < x ≤ 6. Use these two points to find the slope of the line passing through them. Use the slope and one point to find the y-intercept. Then use the slope and y-intercept to write the function. Total 8 Full Credit Response CHAPTER 1 Performance Task 37 1 Performance Task Performance Task At the Box Office At the Box Office Students will write expressions and write and solve an equation to determine a cost-efficient way to buy tickets. Provide a clear solution to the problem. Be sure to show all of your work, include all relevant drawings, and justify your answers. Toby and his friends want to go see a play in the new theater downtown. He has the following options to save money on a purchase of several tickets. Each ticket agent sells the tickets at the same full-price before any discounts are applied. Ticket-Time SUPERSTUB Bill’s Box Office For every two full-price tickets purchased, receive one free ticket! 20% off any order of 4 or more tickets! Get $10 off your total order price. STANDARDS A.CED.1, F.IF.2 Part A Standards for Mathematical Practice: The Chapter 1 Performance Task reinforces Mathematical Practices SMP 1, SMP 2, SMP 3, SMP 4, and SMP 7. Define a variable and write expressions to represent the cost of 10 tickets if purchased at Ticket-Time, Superstub, or Bill’s Box Office. Simplify each expression. Based on your expressions, is it possible to determine whether the total price for 10 tickets is less at one ticket agent than at another? Justify your answer. Copyright © McGraw-Hill Education Jump Start • What is the given information in Part A? the number of tickets purchased • Aside from the discount information, what other information is missing? the cost of a full-price ticket Next, have students read Part A. If students have difficulty determining the Ticket-Time expression, use a table to consider how many tickets are actually free for 2, 3, 4, 5, 6, 7, and 8 tickets. 38 CHAPTER 1 Expressions, Equations, and Functions Emphasizing the Standards for Mathematical Practice This Performance Task is closely aligned to SMP 1 (Make sense of problems and persevere in solving them). The piecewise behavior of the cost for tickets purchased at Ticket-Time may present a challenge to some students. Students will write and graph such functions later in the course. Part C of the Performance Task is a multi-part question that requires several steps before arriving at the required answer. In Part C, students must make the connection that purchasing tickets in multiples of 3 means that for each grouping of 3 tickets, 2 are paid for and 1 is free. Hence, dividing n by 3 actually gives the number of free tickets. 038_039_ALG1_C1PT2_ISG_672788.indd 38 Copyright © McGraw-Hill Education 38 CHAPTER 1 Expressions, Equations, and Functions 2/20/15 5:22 PM SMP 2 Teaching Tip Part B provides an opportunity to connect to SMP 2 (Reason abstractly and quantitatively). In this case, students must reason both abstractly and quantitatively. Students must reason abstractly to make the connection between the expressions and the cost, and that 8b > 7b for b > 0. Quantitatively, since b is a positive whole number and the expressions represent a dollar amount, Ticket-Time will always cost less than Superstub for 10 tickets. Students may make comparisons between other ticket agents, but only this comparison is independent of the value of b. Part B Suppose a school buys 34 tickets for $500 using Bill’s Box Office. Could the school have saved more money if it bought the tickets at Superstub? Show your work. Part C Suppose a full-price ticket costs $15. A customer buys n tickets at Ticket-Time, where n is a multiple of 3. Write a function C(n) that gives the cost of the tickets with the discount. Evaluate C(6) and describe its meaning. Part D Copyright © McGraw-Hill Education Explain which ticket agent gives the worst or the best deal, as the number of tickets purchased increases. CHAPTER 1 Performance Task 39 Scoring Rubric 038_039_ALG1_C1PT2_ISG_672788.indd 39 Part Copyright © McGraw-Hill Education A B 2/20/15 5:22 PM Max Points Full Credit Response 2 Let b = the original price of a ticket; Ticket-Time: 7b, Superstub: 8b, Bill’s Box Office: 10b - 10. You can only tell that the price for Ticket-Time will be less than Superstub because for any positive whole number value of b, 8b > 7b. 2 Yes, they could have saved $92; 500 + 10 = 510, 510 ÷ 34 = 15, so the original price of a ticket is $15. 0.8(15)(34) = 408, so the cost after the 20% discount at Superstub is $408. $500 - $408 = $92 Common Errors Make sure students are correctly interpreting each discount. For example, at Superstub, purchasing less than 4 tickets involves no discount. Purchasing 4 or more results in a 20% discount. Students should know that multiplying by 0.8 (or taking 80% of a number) is equivalent to finding 20% of the original price and then subtracting to find the cost after discount. C(n) = 10n; For every 3 tickets purchased, one will be free, 2 C 2 D 2 Total 8 so ___ n is the number of tickets that will be paid for if 3 ticketes are purchased in multiples of 3. C(6) = 10(6) or 60, so 6 tickets will cost $60. As the number of tickets increases, the worst deal is Bill’s Box Office, because the discount is $10 regardless of the number of tickets purchased. The best deal is Ticket-Time, if tickets are purchased in multiples of 3, because the free tickets equate to about a 33% discount, and 33% > 20%. CHAPTER 1 Performance Task 39 Diagnosing Errors Students who answer Item 4 incorrectly may be confusing x-values and y-values when locating points on the graph. Remind them that the input values of the function are the x-values, and the outputs are the y-values. 1. Six expressions are shown. Select all the expressions that are equivalent to 3x - 12y. 4. The graph shows part of the function y = f(x). A.SSE.2 y 4 2 4(x - 3y) −2 2x - 6y - (x + 6y) 2 O x −2 3(x - 4y) x + 4y + 2(x − 8y) 6x - 6y - (3x - 6y) 4(x − 3y) − x 2. Last year, Hector downloaded x songs per month. His friend Britney downloaded 60 more songs last year than Hector. Write an equation that represents the number of songs y Britney downloaded per month last year, in terms of the number that Hector downloaded. A.CED.1 y=x+5 3. Theo is solving the equation 3x - 2 = -4. He adds 2 to both sides of the equation and then he divides both sides of the equation by 3. Select all the properties shown below that allow Theo to justify these steps. A.REI.1 Distributive Property Addition Property of Equality Complete the following. F.IF.1 f(2) = 3 . f(1) = 2 . f( -1 ) = 0 f( 0 ) = 1 5. Consider the following expression. 2(x − 1) + 4x2 + 2(x2 − 1) When the expression is completely simplified, the coefficient of the x2-term is 6 . A.SSE.1a 6. In the graph, each ordered pair gives the low temperature and the high temperature for each day of a recent winter week in Pinewood. Draw a vertical line on the graph that can be used to show why this relation is not a function. F.IF.1 Temperatures in Pinewood y 8 6 4 2 x Commutative Property Multiplication Property of Equality Division Property of Equality 0 2 4 6 8 Low Temperature (°F) x(x + 1) Copyright © McGraw-Hill Education Students who have difficulty with Item 6 may not understand the various ways to identify a function. This is an opportunity to discuss multiple representations. For example, students may understand that in a function each input value must be paired with exactly one output value, and they may be able to recognize a function when it is presented in a mapping diagram or table of values, but they may not understand how the definition of a function can be displayed as a graph. Be sure students recognize that any vertical line can only pass through one point of a function. If there is a vertical line that passes through more than one point, the relation cannot be a function because the input value (x-value) corresponding to the line is paired with more than one output value (y-value). Standardized Test Practice High Temperature (°F) Standardized Test Practice 7. For the function f(x) = ___________ with domain 2 {1, 2, 3, 4, 5}, what is the range? F.IF.1 Range: {1, 3, 6, 10, 15} 40 CHAPTER 1 Expressions, Equations, and Functions Test-Taking Strategy 040_041_ALG1_C1STP_ISG_672788.indd 40 2/20/15 5:22 PM 40 CHAPTER 1 Expressions, Equations, and Functions Copyright © McGraw-Hill Education Some students may have trouble with Item 2 because they are not asked to write the complete equation, but instead are only asked to write the values of a and b. Point out to students that they should be sure to write down all the intermediate steps in their solution process. Once students have written the final equation y = x + 5 or y = 1x + 5, they can draw boxes around the values of a and b in the equation. This will help ensure that they write these values correctly in the answer boxes. Diagnosing Errors 8. Consider each product or sum. Select Rational or Irrational for each row. Then explain why you selected Rational or Irrational for each product or sum. N.RN.3 Product or Sum Rational Irrational ___ ___ The product of √2 and √7 . ___ ___ The product of √2 and √18 . ___ 3 The sum of √17 and ___. 5 __ √2 · √18 = 6 The sum of π and 19. ___ The sum of √16 and 0.9. __ √14 is irrational. __ In Item 8, students who incorrectly ___ _____ identify the product of √2 and √18 as irrational may not be using properties of square roots to multiply the given values. Be _ sure ___ students _____ understand that √a √b = √ab . Explanation The sum of a rational number and an irrational number is irrational. The sum of a rational number and an irrational number is irrational. __ √16 + 0.9 = 4.9 9. Consider each relation. Is each relation a function? Select Yes or No in each row. For any row in which you select No, explain why the relation is not a function. F.IF.1 Is the relation a function? Yes Students who incorrectly answer the last part of Item 9 may not understand that the given relation maps every positive integer greater than 1 to at least two outputs. It may be helpful for students to write down some sample input/output combinations. For instance, the integer 8 maps to 1, 2, 4, and 8. Seeing a specific example like this may help students recognize that the relation is not a function. No {(-3, 2), (3, 2), (2, 2)} {(-2, -5), (- 1, 0), (0, -3), (1, 4), (1, 6)} The relation that maps each real number to 2 times the number The relation that maps every real number to 0 The relation that maps each positive integer to its factors For the relation in the second row the input value 1 is assigned to two output values, 4 and 6. For the relation in the fifth row, each positive integer other than one has at least two factors, so for each input other than 1 there would be multiple outputs. 10. Solve the equation 0.2(4t + 10) = 6.8. Show your work and justify each step of the solution process. A.REI.1 4t + 10 = 34 (Division Prop. of =); 4t = 24 (Subtraction Prop. of =); t = 6 (Division Prop. of =) Copyright © McGraw-Hill Education 11. Aya plans on buying a new television with a retail price of x dollars. The store is having a sale, and there is also a rebate. The function p(x) = 0.8x - 150 gives the price p that Aya will end up paying for the television. a. Interpret the meaning of the terms 0.8x and 150 in the function, in terms of the sale and the rebate. Rubrics A.SSE.1a 0.8x represents the price after a 20% discount; 150 represents a $150 rebate. b. Evaluate p(1500) and describe what it means. Item 10 [2] Correct answer of t = 6 and at least two correct properties for the solution [1] Correct answer of t = 6 only [0] no response OR incorrect answer and reasoning F.IF.1, F.IF.2 p(1500) = 0.8(1500) - 150 = 1050; Aya will pay $1050 for a television that retails for $1500. c. Aya plans to end up paying between $1400 and $1800 for the television. Find the retail prices that she can shop for when at the store. How do these values relate to the domain and range of the function? F.IF.1, F.IF.5 $1937.50 to $2437.50; domain: 1937.5 ≤ x ≤ 2437.5, range: 1400 ≤ p ≤ 1800 CHAPTER 1 Standardized Test Practice 41 Test-Taking Strategy 040_041_ALG1_C1STP_ISG_672788.indd 41 22/08/14 9:12 PM Copyright © McGraw-Hill Education Some students may find the tables in Items 8 and 9 overwhelming. Suggest that students focus on a single row at a time. In order to help them do this, students may want to use a sheet of blank paper to cover all the rows below the one they are working on. This will also help ensure that students place a check mark in the appropriate cell in the correct row. Item 11 [4] All 3 parts answered correctly and completely [3] 2 parts answered correctly and completely [2] 2 parts answered correctly, but incompletely [1] 1 part answered correctly [0] no response OR incorrect answers and reasoning CHAPTER 1 Standardized Test Practice 41