Name Class Date. Essential question: How do you interpret, evaluate and write algebraic expressions that model real-world situations?

Transcription

1 Name Class Date Variables and Expressions Going Deeper Essential question: How do you interpret, evaluate and write algebraic expressions that model real-world situations? A-SSE.1.1a ENGAGE Interpreting Expressions An expression is a mathematical phrase that contains operations, numbers, and/or variables. A numerical expression contains only numbers and operations, while an algebraic expression contains at least one variable. A term is a part of an expression that is added. The coefficient of a term is the numerical factor of the term. A numerical term in an algebraic expression is referred to as a constant term. Algebraic Expression Terms Coefficients 2 x 2-16x x 2, -16x, constant term 32 2 is the coefficient of 2x. -16 is the coefficient of -16x. Recall that the order of operations is a rule for simplifying a numerical expression: 1. Parentheses (simplify inside parentheses) 1-6 (7-4) = Exponents (simplify powers) = Multiplication and Division (from left to right) = Addition and Subtraction (from left to right) = 8 REFLECT 1a. Write the expression 3m - 4n - 8 as a sum. How does this help you identify the terms of the expression? Identify the terms. 1b. Explain and illustrate the difference between a term and a coefficient. 1c. What is the coefficient of x in the expression x - 2? Explain your reasoning. 1d. What is the value of if you subtract then add? If you add then subtract? Why is the order of operations necessary? Chapter 1 5 Lesson 1

2 To evaluate an algebraic expression, substitute the value(s) of the variable(s) into the expression and simplify using the order of operations. 2 A-SSE.1.1b EXAMPLE Evaluating Algebraic Expressions Evaluate the algebraic expression x(4x - 1 0) 3 for x = 2. A Substitute 2 for x in the expression. ( 4-10 ) 3 B Simplify the expression according to the order of operations. Multiply within parentheses. Subtract within parentheses. Simplify powers. Multiply. REFLECT 2a. Explain why x and 4x - 10 are factors of the expression x(4x - 10 ) 3 rather than terms of the expression. What are the terms of the factor 4x - 10? 2b. Evaluate 5a + 3b and (5 + a)(3 + b) for a = 2 and b = 4. How is the order of the steps different for the two expressions? 2c. In what order would you perform the operations to correctly evaluate the expression 2 + (3 4) 9? What is the result? 2d. Show how to move the parentheses in the expression 2 + (3-4) 9 so that the value of the expression is 9. Chapter 1 6 Lesson 1

3 3 A-SSE.1.1 ENGAGE Writing Algebraic Expressions The table shows some words associated with the four arithmetic operations. They can help you translate verbal phrases into algebraic expressions. Operation Words Examples addition subtraction multiplication division plus, the sum of, added to, more than, increased by, how many altogether minus, less, less than, the difference of, subtracted from, reduced by, how many more, how many less times, multiply, the product of, twice, double, triple, percent of divide, divided by, divide into, the quotient of, half of, one-third of, the ratio of n + n - n n n REFLECT The verbal phrase the quotient of 3 more than a number and 5 can be modeled as follows: Quantity 1 Quantity 2 3a. What words in the phrase represent Quantity 1? Translate these words into an algebraic expression using n for the variable. 3b. Write an algebraic expression to represent the overall phrase. Explain why you have to use some sort of grouping symbol. 3c. Show two ways to rewrite the verbal phrase so that it could be represented by the algebraic expression 5 (n + 3). Chapter 1 7 Lesson 1

4 You can create a verbal model to help you translate a verbal expression into an algebraic expression. 4 A-SSE.1.1 EXAMPLE Modeling with Algebraic Expressions Write an algebraic expression to model the following phrase: the price of a meal plus a 15% tip for the meal. A Complete the verbal model. Price of meal (dollars) 15% Price of meal (dollars) B Choose a variable for the unknown quantity. Include units. Let represent the. C Write an algebraic expression for the situation. Simplify, if possible. REFLECT 4a. A 15% tip represents the ratio 15 cents to 100 cents. Why does this make 15% a unit-less factor? 4b. What units are associated with the total cost? Explain. 4c. What could the expression p p represent, including units? 2 4d. What if the tip is 20% instead of 15%? How can you represent the total cost with a simplified algebraic expression? Identify the units for the expression. 4e. What if the tip is 20% instead of 15% and 3 people are sharing the cost evenly? How can you represent the amount that each person pays with a simplified algebraic expression? Identify the units for the expression. Chapter 1 8 Lesson 1

5 PRACTICE Identify the terms of each expression and the coefficient of each term. 1. 7x + 8y 2. a - b 3. 3 m 2-6n Evaluate each expression for a = 2, b = 3, and c = a - 5b b 2 (c + 4) ab 7. a 2 + b 2 - c 2 8. (a - c)(c + 5) (a - b ) a + (b - c) (a + b) - ab a 2 + bc Alex purchased a 6-hour calling card. He has used t minutes of access time. Write an algebraic expression to represent how many minutes he has remaining and identify the units for the expression. 14. A store is having a sale on used video games. Each game costs $12. Write an algebraic expression to represent the cost of buying v video games. Identify the units for the expression. 15. Sara is driving home from college for the weekend. The average speed of her car for the trip is 45 miles per hour. Write a verbal model and algebraic expression to represent the distance Sara s car travels in h hours. Identify the units for the expression. 16. It costs $20 per hour to bowl and $3 for shoe rental. Write a verbal model and an algebraic expression to represent the cost for n hours and identify the units for the expression. 17. Jared earns 0.25 vacation days for every week that he works in a calendar year. He also gets 10 paid company holidays per year. Write a verbal model and an algebraic expression to represent the amount of time he gets off from work in a year after working for w weeks and identify the units for the expression. Chapter 1 9 Lesson 1

6 18. Sam collects baseball cards. He currently has 112 cards in his collection. He plans to buy 5 new cards every month. Write a verbal model and algebraic expression to represent the total number of cards Sam has after m months. Identify the units for the expression. 19. There are 575 fireworks to be shot off in a fireworks display. Every minute 12 new fireworks are shot off for the display. Write a verbal model and algebraic expression to represent the number of fireworks left to be shot off after t minutes. Identify the units for the expression. 20. Lindsay gets paid a base salary of $400 per week plus a 0.15 commission on each sale she makes. Write a verbal model and algebraic expression to represent Lindsay s total salary for the week if she makes d dollars worth of sales during the week. Identify the units for the expression. Chapter 1 10 Lesson 1

7 Name Class Date 1-1 Additional Practice Give two ways to write each algebraic expression in words b 2. x x (2)(t) 5. z y 7. Sophie s math class has 6 fewer boys than girls, and there are g girls. Write an expression for the number of boys. 8. A computer printer can print 10 pages per minute. Write an expression for the number of pages the printer can print in m minutes. Evaluate each expression for r = 8, s = 2, and t = st 10. r s 11. s + t 12. r t 13. r s 14. t s 15. Paula always withdraws 20 dollars more than she needs from the bank. a. Write an expression for the amount of money Paula withdraws if she needs d dollars. b. Find the amount of money Paula withdraws if she needs 20, 60, and 75 dollars. Chapter 1 11 Lesson 1

8 Problem Solving Write the correct answer. 1. For her book club, Sharon reads for 45 minutes each day. Write an expression for the number of hours she reads in d days. 3. According to the 2000 census, the number of people per square mile in Florida was about 216 more than the number of people per square mile in Texas. Write an expression for the number of people per square mile in Florida if there were t people per square mile in Texas. 2. The minimum wage in 2003 was $5.15. This was w more than the minimum wage in Write an expression for the minimum wage in The cost of a party is $550. The price per person depends on how many people attend the party. Write an expression for the price per person if p people attend the party. Then find the price per person if 25, 50, and 55 people attend the party. Use the table below to answer questions 5 6, which shows the years five states entered the Union. Select the best answer. 5. North Carolina entered the Union x years after Pennsylvania. Which expression shows the year North Carolina entered the Union? A x C x State B 1845 x D 1787 x 6. The expression f 26 represents the year Alabama entered the Union, where f is the year Florida entered. In which year did Alabama enter the Union? F 1819 H 1837 G 1826 J The number of states that entered the Union in 1889 was half the number of states s that entered in Which expression shows the number of states that entered the Union in 1889? A 2s C s + 2 B s 2 D 2 s Year Entered into Union Florida 1845 Indiana 1816 Pennsylvania 1787 Texas 1845 West Virginia 1863 Chapter 1 12 Lesson 1