Modeling with Expressions

Transcription

1 Locker LESSON.1 Modeling with Expressions Common Core Math Standards The student is expected to: A-SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. Also A-SSE.1b, N-Q. Mathematical Practices MP.6 Precision Language Objective Explain the meaning of each term of an algebraic expression that models a real-world situation. Name Class Date.1 Modeling with Expressions Essential Question: How do you interpret algebraic expressions in terms of their context? Explore Interpreting Parts of an Expression Resource Locker An expression is a mathematical phrase that contains operations, numbers, and/or variables. The terms of an expression are the parts that are being added. A coefficient is the numerical factor of a variable term. There are both numerical expressions and algebraic expressions. A numerical expression contains only numbers while an algebraic expression contains at least one variable. Identify the terms and the coefficients of the expression 8p + q + 7r. terms: 8p, q, 7r ; coefficients: 8,, 7 Identify the terms and coefficients of the expression 18 - x - 4y. Since the expression involves subtraction rather than addition, rewrite the expression as the sum of the terms: 18 - x - 4y = 18 + (-x) + (-4y). So, the terms of the expression are 18, -x, -4y and the coefficients are -, -4. ENGAGE Essential Question: How do you interpret algebraic expressions in terms of their context? Identify the meaning of the individual parts of the expression and their relationships to each other. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss what makes a good deal when purchasing a car. Then preview the Lesson Performance Task. Identify the terms and coefficients in the expression x + 3y - 4z Since the expression involves both subtraction and addition, rewrite the expression as the sum of the terms: x + 3y - 4z + 10 = x + 3y + (-4z) So, the terms of the expression are x, 3y, -4z, 10 and the coefficients are, 3, -4. Tickets to an amusement park are $60 for adults and $30 for children. If a is the number of adults and c is the number of children, then the cost for a adults and c children is 60a + 30c. What are the terms of the expression? 60a, 30c What are the factors of 60a? What are the factors of 30c? 60, a 30, c 60, 30 What are the coefficients of the expression? Interpret the meaning of the two terms of the expression. 60a is the cost for a adults. 30c is the cost for c children. The price of a case of juice is $ Fred has a coupon for 0 cents off each bottle in the case. The expression to find the final cost of the case of juice is 15-0.b, wherein b is the number of bottles. What are the terms of the expression? 15, -0.b Module 45 Lesson 1 Name Class Date.1 Modeling with Expressions Essential Question: How do you interpret algebraic expressions in terms of their context? Explore Interpreting Parts of an Expression Resource A-SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. Also A-SSE.1b, N-Q. An expression is a mathematical phrase that contains operations, numbers, and/or variables. The terms of an expression are the parts that are being added. A coefficient is the numerical factor of a variable term. There are both numerical expressions and algebraic expressions. A numerical expression contains only numbers while an algebraic expression contains at least one variable. Identify the terms and the coefficients of the expression 8p + q + 7r. terms: ; coefficients: Identify the terms and coefficients of the expression 18 - x - 4y. Since the expression involves rather than addition, rewrite the expression as the of the terms: 18 - x - 4y =. So, the terms of the expression are and the coefficients are. Identify the terms and coefficients in the expression x + 3y - 4z Since the expression involves both and addition, rewrite the expression as the of the terms: x + 3y - 4z + 10 =. So, the terms of the expression are and the coefficients are. Tickets to an amusement park are $60 for adults and $30 for children. If a is the number of adults and c is the number of children, then the cost for a adults and c children is 60a + 30c. 8p, q, 7r What are the terms of the expression? What are the factors of 60a? What are the factors of 30c? What are the coefficients of the expression? Interpret the meaning of the two terms of the expression. The price of a case of juice is $ Fred has a coupon for 0 cents off each bottle in the case. The expression to find the final cost of the case of juice is 15-0.b, wherein b is the number of bottles. 8,, 7 subtraction sum 18 + (-x) + (-4y) 18, -x, -4y -, -4 subtraction x + 3y + (-4z) + 10 x, 3y, -4z, 10, 3, -4 60a, 30c 60, a 30, c 60, 30 for a adults. 30c is the cost for c children. What are the terms of the expression? 15, -0.b 60a is the cost Module 45 Lesson 1 sum HARDCOVER PAGES Turn to these pages to find this lesson in the hardcover student edition. 45 Lesson.1

2 What are the factors of each term? 15 is the only factor of the first term and -0. and b are the factors of the second term. Do both terms have coefficients? Explain. No, the term 15 has no variable part, so 15 is not a coefficient. What are the coefficients? -0. What does the expression 15-0.b mean in the given situation? If Fred has a coupon, 0 cents is subtracted from the price of every bottle in the case. If Fred does not have a coupon, then the case costs $15. Reflect 1. Sally identified the terms of the expression 9a + 4b - 18 as 9a, 4b, and 18. Explain her error. 9a + 4b + (-18), so the terms are 9a, 4b, and What is the coefficient of b in the expression b + 10? Explain. 1; b = 1(b), so the coefficient of b is 1. Explain 1 Interpreting Algebraic Expressions in Context In many cases, real-world situations and algebraic expressions can be related. The coefficients, variables, and operations represent the given real-world context. Interpret the algebraic expression corresponding to the given context. Example 1 Curtis is buying supplies for his school. He buys p packages of crayons at $1.49 per package and q packages of markers at $3.49 per package. What does the expression 1.49p q represent? Interpret the meaning of the term 1.49p. What does the coefficient 1.49 represent? The term 1.49p represents the cost of p packages of crayons. The coefficient represents the cost of one package of crayons, $1.49. Interpret the meaning of the term 3.49q. What does the coefficient 3.49 represent? The term 3.49q represents the cost of q packages of markers. The coefficient represents the cost of one package of markers, $3.49. Interpret the meaning of the entire expression. The expression 1.49p q represents the total cost of p packages of crayons and q packages of markers. Jill is buying ink jet paper and laser jet paper for her business. She buys 8 more packages of ink jet paper than p packages of laser jet paper. Ink jet paper costs $6.95 per package and laser jet paper costs $8 per package. What does the expression 8p (p + 8) represent? Interpret the meaning of the first term, 8p. What does the coefficient 8 represent? The term 8p represents the cost of p packages of laser jet paper. The coefficient represents the cost of one package of laser jet paper, $8. Interpret the meaning of the second expression, 6.95(p + 8). What do the factors 6.95 and (p + 8) represent? Module 46 Lesson 1 PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices This lesson provides an opportunity to address Mathematical Practice MP.6, which calls for students to communicate with precision. Students learn to interpret the meaning of a mathematical expression in reference to a real-world context rather than seeing the expression as numbers alone. This encourages students to think about how variables represent real-world quantities as well as how various words and phrases can represent mathematical operations. EXPLORE Interpreting Parts of an Expression INTEGRATE TECHNOLOGY Students may complete the Explore activity either online or in the book. CONNECT VOCABULARY Distinguish the terms expression, term, and coefficient from one another. Point out that an expression is composed of one or more terms. A term can be a variable, a constant, or a variable that is multiplied by a constant, which is called a coefficient. AVOID COMMON ERRORS Note that in middle school, students may have seen terms of an expression defined as parts of an expression that are added or subtracted. Point out that in this course, terms are defined as parts of an expression that are added. EXPLAIN 1 Interpreting Algebraic Expressions in Context Focus on Communication MP.3 Stress that to understand the meaning of a mathematical expression, students must consider the meaning of every term and operation. Have students make a list of the words that represent each mathematical operation. Modeling with Expressions 46

3 QUESTIONING STRATEGIES When you multiply the price of an item by the number of those items purchased, what does the product represent? the cost of buying the given number of items If each term of an expression represents the cost of an item purchased, what does the whole expression represent? the total cost of all items purchased The term 6.95(p + 8) represents the cost of the ink jet paper represents the cost of one package of ink jet paper. (p + 8) represents the number of ink jet paper packages that Jill bought. Interpret the expression 8p (p + 8). The expression represents the total cost of packages of ink jet and laser jet paper that Jill bought. Your Turn Interpret the algebraic expression corresponding to the given context. 3. George is buying watermelons and pineapples to make fruit salad. He buys w watermelons at $4.49 each and p pineapples at $5 each. What does the expression 4.49w + 5p represent? The expression 4.49w + 5p represents the total cost of w watermelons and p pineapples 4. Sandi buys 5 fewer packages of pencils than p packages of pens. Pencils costs $.5 per package and pens costs $3 per package. What does the expression 3p +.5(p - 5) represent? The expression 3p +.5(p - 5) represents the total cost of pens and pencils. EXPLAIN Comparing Algebraic Expressions QUESTIONING STRATEGIES The real-world meaning of a variable may limit the numerical values it can have. What do you know about the possible value of a variable that represents the population of a city? It must be a positive integer. What do you know about the possible value of a variable that represents a tax rate? It must be a positive number less than one. Focus on Critical Thinking MP.3 When comparing two algebraic expressions to determine which is greater, students can check their answers by choosing numbers to substitute for the variables, then evaluating each expression and comparing the results. Explain Comparing Algebraic Expressions Given two algebraic expressions involving two variables, we can compare whether one is greater or less than the other. We can denote the inequality between the expressions by using < or > symbols. If the expressions are the same, or equivalent expressions, we denote this equality by using =. Suppose x and y give the populations of two different cities where x > y. Compare the expressions and tell which of the given pair is greater. Example 1 x + y and x The expression x is greater. Putting the lesser population, y, together with the greater population, x, gives a population that is less than double the greater population. x_ y and _ y x Since x > y, x_ y will be than So x_ > _ y y x. greater 1 and y _ x will be less than 1. Your Turn Suppose x and y give the populations of two different cities where x > y and y > 0. Compare the expressions and tell which of the given pair is greater. 5. x_ x + y and _ x + y x 6. (x + y) and (x + y) Since x < x + y, the first ratio must be (x + y) represents doubling (x + y) less than 1. Since x < x + y, the second (x + y) represents squaring (x + y) Since x and y are the populations of two ratio must be greater than 1. cities, their sum is much greater than. So _ x + y x > x_ x + y. Thus, squaring the sum produces a greater number. So, (x + y) > (x + y). Module 47 Lesson 1 COLLABORATIVE LEARNING Peer-to-Peer Activity Have students work in pairs, taking turns writing situations and trying to translate each other s situation into an algebraic expression. Have students compare their expressions and discuss why they wrote them in the way that they did. 47 Lesson.1

4 Explain 3 Modeling Expressions in Context The table shows some words and phrases associated with the four basic arithmetic operations. These words and phrases can help you translate a real-world situation into an algebraic expression. Addition Subtraction Multiplication Division Operation Words Examples the sum of, added to, plus, more than, increased by, total, altogether, and less than, minus, subtracted from, the difference of, take away, taken from, reduced by times, multiplied by, the product of, percent of divided by, division of, quotient of, divided into, ratio of, 1. A number increased by. The sum of n and 3. n + 1. The difference of a number and. less than a number 3. n - 1. The product of 0.6 and a number. 60% of a number n 1. The quotient of a number and 5. A number divided by 5 3. n 5 or n 5 EXPLAIN 3 Modeling Expressions in Context QUESTIONING STRATEGIES How is 3 less a number translated into an algebraic expression? How is 3 less than a number translated? Are they equivalent expressions? 3 n and n 3; no, the order matters in subtraction. Example 3 Write an algebraic expression to model the given context. Give your answer in simplest form. the price of an item plus 6% sales tax Reflect Price of an item + 6% sales tax p p The algebraic expression is p p, or 1.06p. 7. Use the Distributive Property to show why p p = 1.06p. p p = 1p p = p( ) = 1.06p the price of a car plus 8.5% sales tax The price of a car p % sales tax 0.085p The algebraic expression is p p, or p p = 1.085p. 8. What could the expression 3(p p) represent? Explain. Sample Answer: 3(p p) is 3 times the expression p p. It could represent the cost of 3 identical items with a 6% sales tax. Your Turn Write an algebraic expression to model the given context. Give your answer in simplest form. 9. the number of gallons of water in a tank, that already has 300 gallons in it, after being filled at 35 gallons per minute for m minutes m Focus on Modeling MP.4 Encourage students to follow the pattern Quantity Operation Quantity when modeling a real-world situation. Once students identify the quantities and choose an operation, the writing of an algebraic expression becomes a straightforward process. Quantity + - Quantity 10. the original price p of an item less a discount of 15% p p = 0.85p Module 48 Lesson 1 DIFFERENTIATE INSTRUCTION Multiple Representations Students with conceptual processing difficulties may struggle with the many different verbal descriptions and symbolic notations for expressing the same operation. For example, k divided by or the quotient of k and can both be written as k or k. Resource sheets listing a variety of possible descriptions and notations for each standard operation may be helpful to such students. Modeling with Expressions 48

5 ELABORATE QUESTIONING STRATEGIES What is the difference between a variable and a constant? A variable represents a value that can change. A constant is a numerical value that does not change. What are like terms in an expression? Like terms are terms that contain the same variables raised to the same power. AVOID COMMON ERRORS When modeling situations that involve subtraction or division, some students may write the terms in the incorrect order. Tell students to double-check that the order makes sense within the context of the verbal expression when they write a subtraction or division expression. Elaborate 11. When given an algebraic expression involving subtraction, why is it best to rewrite the expression using addition before identifying the terms? Subtraction is defined as adding a negative term. Rewriting all subtraction as addition keeps the negative sign with the term and helps avoid errors. 1. How do you interpret algebraic expressions in terms of their context? In order to interpret algebraic expressions in terms of their context, identify the meaning of the individual parts of the expression and their relationships to each other. 13. How do you simplify algebraic expressions? You can simplify algebraic expressions by using the properties of real numbers and combining like terms. 14. Essential Question Check In How do you write algebraic expressions to model quantities? First identify the descriptive words that indicate which basic arithmetic operations are involved. Then create a verbal model relating the quantities. Finally, choose a variable to represent the unknown quantity and write the expression. Evaluate: Homework and and Practice Practice Identify the terms and the coefficients of the expression p - 7z. 8x - 0y p - 7z = (-0) + 5p + (-7z) terms: -0, 5p, and -7z; coefficients: 5 and -7 Online Homework Hints and Help Extra Practice 8x - 0y - 10 = 8x + (-0y) + (-10) terms: 8x, -0y, and -10; coefficients: 8 and -0 SUMMARIZE THE LESSON How can you translate a real-world situation into an algebraic expression? First look for key words that indicate which operation or operations are involved. Identify the quantities that you are dealing with, and choose a variable to represent the unknown. Write each quantity and operation in algebraic notation. Simplify, if necessary. Identify the factors of the terms of the expression a + 11b 4. 13m - n 5; 6 and a; 11 and b 13 and m; - and n 5. Erin is buying produce at a store. She buys c cucumbers at $0.99 each and a apples at $0.79 each. What does the expression 0.99c a represent? The expression represents the total cost of c cucumbers and a apples. 6. The number of bees that visit a plant is 500 times the number of years the plant is alive, where t represents the number of years the plant is alive. What does the expression 500t represent? The expression represents the total number of bees that will visit the plant in its lifetime. Module 49 Lesson 1 LANGUAGE SUPPORT Visual Cues Have students copy and complete the graphic organizer. Next to each operation, write a word phrase in the left box and its corresponding algebraic symbol or expression in the right box. Words Addition Subtraction Multiplication Division Algebra 49 Lesson.1

6 7. Lorenzo buys 3 shirts at s dollars apiece and pairs of pants at p dollars a pair. What does the expression 3s + p represent? The expression represents the total cost of 3 shirts and pairs of pants. EVALUATE 8. If a car travels at a speed of 5 mi/h for t hours, then travels 45 mi/h for m hours, what does the expression 5t + 45m represent? The expression represents the total distance traveled by the car. 9. The price of a sandwich is $1.50 more than the price of a smoothie, which is d dollars. What does the expression d represent? The expression represents the price of a sandwich. 10. A bicyclist travels 1 mile in 5 minutes. If m represents minutes, what does the expression m 5 represent? The expression represents the number of miles the bicyclist travels in m minutes. ASSIGNMENT GUIDE Concepts and Skills Explore Interpreting Parts of an Expression Practice Exercises 1 4, What are the factors of the expression (y - )(x + 3)? The factors of the expression are (y - ) and (x + 3). 1. Explain the Error A student wrote that there are two terms in the expression 3p - (7-4q). Explain the student s error. There are three terms, 3p, -7, and 4q in the expression: 3p - (7-4q) = 3p q = 3p + (-7) + 4q 13. Yolanda is buying supplies for school. She buys n packages of pencils at $1.40 per package and m pads of paper at $1.0 each. What does each term in the expression 1.4n + 1.m represent? What does the entire expression represent? The term 1.4n represents the cost of n packages of pencils at $1.40 per package. The term 1.m represents the cost of m pads of paper at $1.0 per pad. The expression represents the total cost of n packages of pencils and m pads of paper. 14. Chris buys p pairs of pants and 4 more shirts than pairs of pants. Shirts cost $18 each and pair of pants cost $5 each. What does each term in the expression 5p + 18(p + 4) represent? What does the entire expression represent? The term 5p represents the cost p pairs of pants at $5 each. The term 18(p + 4) represents the cost of the shirts at $18 per shirt. The entire expression represents the total cost of the shirts and pants that Chris bought. Module 50 Lesson 1 Image Credits: Leonard Lenz/Corbis Example 1 Interpreting Algebraic Expressions in Context Example Comparing Algebraic Expressions Example 3 Modeling Expressions in Context VISUAL CUES Exercises 5 10, Exercises 15 19, 5 Exercises 0 4, 6 7 English language learners may have difficulty with the word term because it has so many different meanings. Explain that in mathematics a term is a part of an expression that is added. You may wish to have students highlight each term in a different color, then circle the addition signs between the terms. Focus on Communication MP.3 In the algebraic expression x 9x, 9x is listed as a term rather than 9x. By definition, a term is added, so the subtraction is rewritten as addition of the inverse: x 9x = x + ( 9x ). Exercise Depth of Knowledge (D.O.K.) Mathematical Practices Recall of Information MP.4 Modeling Skills/Concepts MP. Reasoning 0 4 Skills/Concepts MP.4 Modeling Strategic Thinking MP.3 Logic Modeling with Expressions 50

7 GRAPHIC ORGANIZERS Present a table like the one below to help students organize information for writing expressions. In the first column, they can write words or phrases. They can use the second column for the corresponding variables and operation symbols. Quantity 1 Quantity Quantity 3 Operation Operation Words Symbols Suppose a and b give the populations of two states where a > b. Compare the expressions and tell which of the given pair is greater or if the expressions are equal. 15. b_ and 0.5 a + b Since a > b, 16. a + 13c and b + 13c, where c is the population of a third state 17. _ a - b and a - _ b _ a - b = _ a - _ b. Since _ a < a, the expression _ a - _ b < a - _ b. Therefore, _ a - b < a - _ b. 18. a + b and b b_ a + b < b_ b + b = 1_. So 0.5 > b_ a + b. In both the expressions, the same value,13c, is being added to the population. Since a > b, adding 13c to a results in a greater value than adding 13c to b. a + 13c > b + 13c b = b + b; This is less than a + b because a > b, which implies that adding b to b is less than adding a to b. So, a + b > b. Point out that students should customize their tables for each problem. For example, this organizer is used for a problem that has three different quantities and two operations. Other problems will have different numbers of quantities and operations. Image Credits: Jessie Walker/Lived In Images/Corbis 19. 5(a + b) and (a + b)5 The expression 5(a + b) = (a + b) + (a + b) + (a + b) + (a + b) + (a + b), which is adding the total populations 5 times. The expression (a + b)5 = (a + b) + (a + b) + (a + b) + (a + b) + (a + b), which also is adding the total populations 5 times. Therefore, 5(a + b) = (a + b)5. Write an algebraic expression to model the given context. Give your answer in simplest form. 0. the price s of a pair of shoes plus 5% sales tax. 1. the original price p of an item less a discount of 0% s s = 1.05s p - 0.0p = 0.80p. the price h of a recently bought house plus 10% property tax h + 0.1h = 1.1h 3. the principal amount P originally deposited in a bank account plus 0.3% interest P P = 1.003P Module 51 Lesson 1 51 Lesson.1

8 4. Match each statement with the algebraic expression that models it. A. the price of a winter coat and a 0% discount B x + 0.0x = 1.0x B. the base salary of an employee and a % salary increase A x - 0.0x = 0.80x C. the cost of groceries and a % discount with coupons D x + 0.0x = 1.0x D. the number of students attending school last year and a 0% increase from last year C x - 0.0x = 0.98x AVOID COMMON ERRORS Some students may write division expressions in the wrong order. Explain that they need to pay attention to the order of the numbers and variables in the verbal description. For example, the quotient of 3 and a number p is 3 p, not p 3. H.O.T. Focus on Higher Order Thinking 5. Critique Reasoning A student is given the rectangle and the square shown. The student states that the two figures have the same perimeter. Is the student correct? Explain your reasoning. x + 4 x 6. Multi-Step Yon buys tickets to a concert for himself and a friend. There is a tax of 6% on the price of the tickets and an additional booking fee of $0 for the transaction. Write an algebraic expression to represent the price per person. Simplify the expression if possible. x + x + Yes, the student is correct. For the rectangle, the perimeter is: (x + 4) + x + (x + 4) + x = 4x + 8 For the square, the perimeter is: (x + ) + (x + ) + (x + ) + (x + ) = 4x + 8 Because (x + 4) + x + (x + 4) + x = 4x + 8 = (x + ) + (x + ) + (x + ) + (x + ), the perimeters are equal. Price of tickets in dollars: t Tax on the tickets: 0.06t Additional fee: 0 Total price for both Yon and his friend: t t + 0 t t + 0 Since there are two people, divide the total price by to get the price per person: t t + 0 The price per person is, where the variable t represents the price of the two tickets in dollars. 1.06t + 0 After simplifying the expression, it becomes _, or 0.53t Focus on Communication MP.3 Stress that students should strive to be precise with language and should look for key words and phrases that determine how to translate words into mathematical expressions. When students hear the phrase the quotient of 5 and 3 more than a number n, they should immediately break it down into component quantities 5 and 3 more than a number n. Quotient indicates division of the first quantity, 5, by the second quantity, 3 more than a number n. Three more than a number n indicates that the second quantity is the sum of 3 and the variable n. VISUAL CUES Encourage students to circle or underline the key words that represent mathematical operations in a verbal description before translating it to an algebraic expression. Module 5 Lesson 1 Modeling with Expressions 5

9 JOURNAL Have students describe quantities they encounter in their own lives, and then write algebraic expressions that model them. 7. Persevere in Problem Solving Jerry is planting white daisies and red tulips in his garden and he wants to choose a pattern in which the tulips surround the daisies. He uses tiles to generate patterns starting with two rows of three daisies. He surrounds these daisies with a border of tulips. The design continues as shown. Border 1 Border Border 3 a. Jerry writes the expression 8(b - 1) + 10 for the number of tulips in each border, wherein b is the border number and b 1. Explain why Jerry s expression is correct. For Border 1, tulips are added above and below the daisies, which gives a total of six tulips. A tulip is then added to the end of each row of daisies, which gives four tulips. So, there is a total of ten tulips in Border 1. For Border, there are eight additional tulips added to the garden plus the ten original tulips from Border For Border 3, there are eight additional tulips added to the garden plus the ten original tulips from Border 1. 8() + 10 So, the expression for the number of tulips in the garden with Border b is 8(b - 1) So, Jerry s expression is correct. b. Elaine wants to start with two rows of four daisies. Her reasoning is that Jerry started with two rows of three daisies and his expression was 8(b - 1) + 10, so if she starts with two rows of four daisies, her expression will be 10(b - 1) Is Elaine s statement correct? Explain. In part a, the number 10 comes from the top and bottom rows of 3 tulips in Border 1 and the 4 tulips on the ends of the two original rows of daisies. If Elaine starts with two rows of four tulips, then the number of tulips in Border 1 would change to 1 since a daisy in each row is being added. So, Elaine s expression is not correct. The correct expression for starting with two rows of four daisies would be 8(b - 1) + 1. Module 53 Lesson 1 53 Lesson.1

10 Lesson Performance Task Becky and Michele are both shopping for a new car at two different dealerships. Dealership A is offering $500 cash back on any purchase, while Dealership B is offering $1000 cash back. The tax rate is 5% at Dealership A but 8% at Dealership B. Becky wants to buy a car that is $15,000, and Michele is planning to buy a car that costs $0,000. Use algebraic expressions to help you answer the following questions. a. At which dealership will Becky get the better deal? How much does she save? b. At which dealership will Michele get the better deal? How much does she save? c. What generalization can you make that would help any shopper know which dealership has the better deal? [Hint: At what price point would the two deals be equal?] a. First write an algebraic expression to represent each dealership s special. Dealership A p p = 1.05p Dealership B p p = 1.08p Now substitute the price that Becky plans to pay to see which deal is better for her. Dealership A: 1.05(15,000) = 15, = 15,50 Dealership B: 1.08(15,000) = 16, = 15,00 Dealership A > Dealership B So, Becky will get a better deal with Dealership B, since it is $50 cheaper. b. Substitute the price that Michele plans to pay into each expression. Dealership A: 1.05(0,000) = 1, = 0,500 Dealership B: 1.08(0,000) = 1, = 0,600 Dealership A < Dealership B So, Michele will get a better deal with Dealership A, since it is $100 cheaper. c. The two deals are equivalent when the additional 3% tax equals $500 (0.03p = 500), which occurs when the purchase price is $16,667. Students generalizations may suggest that any shopper planning to buy a car with a price less than $16,667 will get the better deal at Dealership B, while anyone planning to buy a car at a price higher than $16,667 will get a better deal at Dealership A. Image Credits: Tom Grill/ Corbis QUESTIONING STRATEGIES Why is the dealership that is offering the greatest amount of cash back not necessarily the best deal? The dealership that gives the greatest amount of cash back has a higher tax rate. The greater the cost of the car, the more the buyer must pay in tax. Focus on Modeling MP.4 Before students can model the situation in the Lesson Performance Task, they need to understand how tax is calculated. Review with students their experience with sales tax when purchasing an item at a store. The rate of the tax, written as a decimal, is multiplied by the price, p, of the item to find the amount of tax owed. The cost of the item is then added to the tax amount to find the total cost. For a tax rate of 5% this can be written as p p or as 1.05p. Module 54 Lesson 1 EXTENSION ACTIVITY Have students research the various incentives offered by car dealers, such as cash rebates, financing offers, down payments, leasing deals, special warranties, and tune-up packages. Then have students write a few paragraphs describing how comparison shopping and understanding dealers offers could save them thousands of dollars when buying a new car. They should include one numerical example that shows how the same car could cost them different amounts with two different sets of incentives. Scoring Rubric points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Modeling with Expressions 54