1-3. Literal Equations. UNPACKING the TEKS. Objective. content Student expectations. Preparing to Teach. Teacher Resources

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1 UNPACKING the TEKS IN this LESSON ELPS content Student expectations In this lesson you will focus on TEKS (12)(E) and (1)(A). (12)(E) Solve mathematic and scientific formulas, and other literal equations, for a specified variable. Students are already familiar with solving multi-step equations with a variable on each side of the equation. In this lesson, students will solve literal equations for a variable. Mathematical and scientific formulas use literal equations that are essential in many applications. Rewriting a literal equation in terms of one of the variables may simplify calculations when substituting numbers for the other variables. Students are expected to use the properties of equality and inverse operations to solve literal equations for one variable. When solving an application problem, students must choose the appropriate formula and then solve the formula for the variable they need to find. This skill shows a deeper understanding of the information and parameters given in the problem. Encourage students to explain their choice of formula(s). Process Student expectations (1)(A) Apply mathematics to problems arising in everyday life, society, and the workplace. Throughout the lesson, students will write formulas as literal equations to solve problems. When the literal equation contains only variables, encourage students to treat the variables they are not solving for as if they were constants. Have students verbalize what the variable represents instead of using the letter (such as radius instead of r). Remind students to describe what they know in the problem, determine what they need to know, and then write a plan for solving the problem. english Language Proficiency Standards READING (4)(C) Develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials. Teaching with Ti Objective Students will be able to rewrite and use literal equations and formulas. Preparing to Teach BIG ideas: equivalence Solving equations and inequalities ESSENTIAL UNDERSTANDING A literal equation is an equation that involves two or more variables. The solution of a literal equation can be found using the properties of equality and inverse operations to form a series of simpler equations. The properties of equality can be used repeatedly to isolate any particular variable. Teacher Resources All of the reproducible write-in activities and the Student Companion pages can be accessed at the Teacher Resource Download Center. Lesson Overview When solving literal equations, students can use the same properties of equality and inverse operations that they have used throughout this chapter. Students can treat all variables that they are not solving for as constants. This skill is particularly useful in working with math and science formulas. For example, the uniform motion formula d = rt (distance = rate * time) can be solved for r, yielding a specific formula for speed (rate = distance/time). These Teaching with TI activities are available at the Teacher Resource Download Center. TI-Nspire Content Exploration, Practice, Quiz, and Standardized Test Prep documents See Problem 3 for support for English Language Learners at all four proficiency levels. college & career Readiness Standards Algebraic Reasoning, II.C.1; Problem Solving and Reasoning, VIII.A.5; Communication and Representation, IX.A.1; Connections, X.A.2 Topic 1 45 Lesson 1-3

2 SUPPORTING ALL LEARNERS Vocabulary Builder Review Content Vocabulary variable / variable Reproducible Write-in activity New Content Vocabulary formula / fórmula literal equation / ecuación literal Download Additional Vocabulary Support from the Teacher Resource Download Center. Answers are found in the Teacher Resource Download Center. Review terms presented in earlier lessons. Then introduce the new terms literal equation and formula. Emphasize that a literal equation involves two or more variables. The equation is called literal because the coefficients and constants are replaced with letters. As needed, have students complete the following sentences. 1. When you rewrite literal equations, you may have to divide by a coefficient, a, or a variable expression. [variable] 2. A formula is a special type of because it involves two or more variables. [literal equation] Pre-AP english Language Learners Assess Understanding Write the word circumference on the board and have students repeat the term as you say it aloud. Ask: What is circumference? [distance around a circle] Draw a circle on the board. Trace the circle with your finger as you say: The circumference of a circle is the total distance around the outside, or the edge of the circle. Use your finger to trace a circle on your desk. Have students trace the circumference of real objects, such as cans, CDs, or cups. enrichment/advanced Each lesson offers opportunities for enrichment and extensions. Look for an enrichment strategy for Problem 4. This strategy also aligns to Pre-AP expectations. In Step 5, there is an Enrichment activity. 1 RESPONSE TO INTERVENTION Each lesson has teaching strategies and resources to support struggling students. Look for this icon throughout the lesson for these strategies. TieR 1 Ongoing support can be found with Problem 2 on page TieR 2 Strategic Intervention is available in Step 5, Assess and Intervene. 3 TieR 3 Use MathXL for School to generate Intense Intervention opportunities. Topic 1 46 Lesson 1-3

3 ENGAGE & EXPLORE INTERACTIVE LEARNING SOLVE IT! Go to PearsonTEXAS.com and click on the Solve It for Lesson 1-3. Project the animation on an interactive whiteboard or other screen. PURPOSE Students see how literal equations can be used to model real-world situations. PROMPT QUESTIONS Q: What is the relationship between the number of pizzas and the number of sandwiches you can buy? [As the number of pizzas increases, the number of sandwiches decreases.] ANSWER 8 sandwiches; 6 sandwiches; explanations may vary. You are ordering pizzas and sandwiches. You have a budget of $80. How many sandwiches can you buy if you buy 4 pizzas? 5 pizzas? Explain your answer. Digital INTERACTIVE Concept Exploration MENU Pizza $10 Sandwich $5 Student Companion, page 13 FOCUS ON PROCESS TEKS ANALyzE MAThEMATICAL RELATIONShIPS (1)(F) Have students complete the writing prompt in the Student Companion to indicate the mathematical relationships they identified in the problem. CONNECT ThE MATh In this Solve It, the amount of money spent depends on two variables, the number of pizzas and the number of sandwiches. In this lesson, students will learn to use algebra to model situations with multiple variables. EXPLAIN GUIDED PROBLEM SOLVING Homework Helper Take Note Boxes and Stepped-Out Problems are also found in the Homework Helper. Digital Key Concept TAKE NOTE Show Formulas before Problem 3. Allow students to continue the practice of rewriting literal equations using the formulas in the Take Note box. For example, have them solve A = 1 2 bh for h. Students will find that h = 2A b. Topic 1 47 Lesson 1-3

4 PROBLEM 1 Rewriting a Literal Equation Use the questions below to help build students understanding. Have students complete the Got It in the Student Companion. Digital Stepped-Out Learning Animation ScaffoLd LEaRning Q: Think Why should you rewrite the equation? [If you rewrite the equation, you have to isolate y only once. Then substitute for x. If you substitute for x first, you must isolate y twice (once for each x-value).] Q: How does knowing how to solve a one-variable linear equation help you solve a literal equation? [Solving the literal equation involves the same steps as solving a one-variable, linear equation.] got it? check UndERSTanding Q: In part (b), why might it be more efficient to solve for y before substituting a value for x? [The variable y can be found for any value of x without having to rewrite the equation each time.] answers a. m = 4 + 5n ; -3; 2; 7 2 b. y = 10; y = 4; check students work. Student Companion, page 14 focus on process TEkS apply mathematics (1)(a) Q: How does the equation 10x + 5y = 80 model the problem in the Solve It? [The expression 10x represents the cost of x pizzas and the expression 5y represents the cost of y sandwiches. The total cost, or the sum of 10x and 5y, must be equal to the budget of $80.] Topic 1 48 Lesson 1-3

5 PROBLEM 2 Rewriting a Literal Equation With Only Variables Use the questions below to help build students understanding. Have students complete the Got It in the Student Companion. ScaffOLd LEaRning Q: Think How can you solve a literal equation for a variable? [When a literal equation contains only variables, treat the variables you are not solving for as constants.] Q: Factor each of the following expressions: -3x - 9 = (x + 3) [ 3(x + 3)] 2xy + 6y = (x + 3) [2y(x + 3)] 9x - 18 = (x - 2) [9(x 2)] ax - bx = ( - ) [x(a b)] Q: How can you use this work to solve ax - bx = c? [Factor ax bx as x(a b), then divide by a b.] Q: Describe the operations performed on the left side of the equation. In what order are they performed? [First b is subtracted from a, then the result is multiplied by x.] Q: What should you do to isolate x? What is the result? [Divide by (a b). The result is x = c a b.] got it? check UndERSTanding Q: Which variables can be thought of as constants to solve the equation? [t, r, and p] answer x = -t - r p Digital Stepped-Out Learning Animation Student Companion, page 14 1 RESPONSE TO INTERVENTION For students needing additional support when solving -t = r + px for x, ask them to compare the equation to -3 = 1 + 2x. You may wish to use one color to write -t and -3, use a second color to write r and 1, and use a third color to write p and 2. Then ask students to solve the equation that contains only the variable x and apply each step to the literal equation. Topic 1 49 Lesson 1-3

6 PROBLEM 3 Rewriting a Geometric Formula Use the questions below to help build students understanding. Have students complete the Got It in the Student Companion. Digital Stepped-Out Learning Animations SCAFFolD learning Q: What does circumference mean? [the distance around the circle] Q: The formula C = 2pr relates circumference and radius. Which variable do you need to solve for? Which property of equality should you use? [r; Division Property of Equality] GoT IT? CHECk understanding Q: What formula is needed to solve this problem? [A = 1 2 bh] Q: What does the b stand for? What does the h stand for? [length of the base and height] Q: What is the variable that you are solving for? [h] Student Companion, page 15 ANSwER 6 in. ELPS ENGLISH LANGUAGE PROFICIENCY STANDARDS READING (4)(C) BEGINNING Explain to students that question words such as what, how many, and how much can help them figure out what they need to find in a math problem. Point to the word what as you read Problem 3 together. Provide the sentence frame: In this problem we need to find the [radius]. Students must also determine the given information that will help them solve the problem. Read the Got It together and ask students what quantity they must find [height h]. Ask what information is known [Area A = 24 in. 2 and base b = 8 in.]. Then ask what formula relates the known and unknown quantities [A = 1 2bh] and solve the problem together. INTERMEDIATE Complete the Beginning activity so students are clear about what is known and what they need to find in the Got It. Then explain that they should also look for the form their answer should take. Have students read the problem again. Ask: What are the units for the answer? [in.] In order to be completely correct, the answer should be correctly calculated and give the correct units. Then have students solve the problem and read each other s solutions. ADVANCED In addition to the question words, it is also important to understand the verbs that tell you what to do in a problem. In some cases students will only need to find the number answer (or choose from possible answers), but math problems often require more extended responses. Show homework assignments on the screen and have students take turns reading problems that use words like the following; classify, determine, estimate, explain, illustrate, interpret. Discuss the meanings of these words. [Example: The term explain requires that they tell how they found their answer or why it is correct.] Then have students read the Got It and Solve It. Finally, ask students to write a follow-up problem using one of the previously discussed words. [Illustrate your answer using both an acute and an obtuse triangle.] ADVANCED HIGH Go over some of the words in the Advanced activity and then have students read problems containing some of the following words: analyze, compare, define, describe, infer, justify, predict, prove, relate. Help students understand what these words are telling them to do so that they can give complete answers. [Example: If they see the word prove, they should think about how to develop a step-by-step argument to lead logically from the given information to the conclusion.] Then have students read the Got It and Solve It. Finally, ask students to write a follow-up problem using one of the previously discussed words. [Analyze the change in height of this triangle if the base length is doubled.] Topic 1 50 Lesson 1-3

7 PROBLEM 4 Rewriting a Formula Use the questions below to help build students understanding. Have students complete the Got It in the Student Companion. Digital Stepped-Out Learning Animation ScaFFold learning Q: Think How do you know which formula to use? [Read the information given in the problem. This problem gives you a measure of time and a distance. You need to find the rate, so use d = rt.] Q: Which formula involves what you know and what you are looking for? [d = rt] Q: How is the solution to this formula similar to the solution to the formula from Problem 3? [Both solutions use the Division Property of Equality.] got it? check understanding Q: What do you know? [The distance is 5,000 miles and the average rate is 91 mi/day.] Q: What do you want to find out? [the number of days it takes the whales to migrate one way] Q: What formula can you use? [d = rt] Q: What variable do you need to solve for? [t] Student Companion, page 15 answer about 55 days FocuS on PRoceSS TekS create RePReSenTaTionS To communicate mathematical ideas (1)(e) Q: Why is the formula d = rt the appropriate formula to model this problem? [The problem includes the distance the butterfly travels and the length of time it takes. This formula relates distance, rate, and time.] Pre-AP ENRICHMENT/ADVANCED Have advanced students research and write their own word problem about the distance a migrating animal travels. Then have them exchange and solve their classmate s problem. Topic 1 51 Lesson 1-3

8 ELABORATE LESSON CHECK close Q: What situations can best be modeled by literal equations? [Literal equations can be used to model situations where one variable depends on other variables. For example, the distance a runner travels depends on how fast and for how long she runs, so d = rt.] Student Companion, pages ANsWeRs p 1. a. x = m + 2n b. F = 9 5 C (60 - v) yd 3. A = w 2 (1 - p 4 ) 4. a. literal equation b. literal equation c. both d. both 5. Answers may vary. Sample: They are the same in each case since you are isolating a variable by using inverse operations. They are different because, in an equation in one variable, to isolate the variable, inverse operations are used on numbers only. In a literal equation, inverse operations are used on variables as well as numbers. 6. Formulas * The Student Companion has write-in spaces for students to enter answers. Topic 1 52 Lesson 1-3

9 ELABORATE PRACTICE Practice and Application ASSIGNMENT GUIDE Homework Helper, pages ONLINE PRACTICE Some students may prefer to do their homework online. Assign the exercises through PearsonTEXAS.com. Homework and Tutorials App. Students can access these same exercises as well as instructional tutorials on the Homework and Tutorials app. Virtual Nerd Tutorials Students can also access the instructional tutorials from PearsonTEXAS.com. Homework Quick Check Use the exercises with an asterisk (*) as a quick check of students understanding of the TEKS Content Standards addressed in this lesson. Exercise Depth of Knowledge TEKS Standards Content Process Skill/Concept Understanding (12)(E) 5* 3 Strategic Thinking (12)(E) (1)(A) Skill/Concept Understanding (12)(E) 10* 3 Strategic Thinking (12)(E) (1)(G) 11 3 Strategic Thinking (12)(E) (1)(A) 12 3 Strategic Thinking (12)(E) (1)(A) 13 3 Strategic Thinking (12)(E) (1)(A) Skill/Concept Understanding (12)(E) 20 3 Strategic Thinking (12)(E) (1)(A) 21* 3 Strategic Thinking (12)(E) (1)(B) 22 3 Strategic Thinking (12)(E) (1)(A) Skill/Concept Understanding (12)(E) 32 3 Strategic Thinking (12)(E) (1)(A) 33* 3 Strategic Thinking (12)(E) (1)(D) 34* 3 Strategic Thinking (12)(E) (1)(E) 35 3 Strategic Thinking (12)(E) (1)(A) ANSWERS Use the Answers and Solutions CD-ROM to access stepped-out solutions and choose formats for printing or for whiteboard display. 1. y = -2x + 5; 7; 5; y = -2x + 4; 8; 2; y = 3x - 9 ; ; -9 5 ; y = 4x + 7 ; ; 9; yd m in ft 9. 7 cm Topic 1 53 Lesson 1-3

10 ELABORATE PRACTICE answers continued was added to the left side of the equation instead of subtracted; 2m - 3 = -6n, 2m = n. 11. h = V pr2; 10 cm 12. h = an; 87 hits 13. h = n 7/ ; 8 ft 14. x = 2m x = ay 1 + n b + a -ax y = 17. h = 3V 2x pr g = 2A - f 19. a = 2b - x h F sides 22. a. m = 2E v 2 b. 50 kg p 23. x = m + n 24. x = c a x = by + v 27. x = S - C 28. x = ay C b 29. x = A - C 30. x = 4b Bt x = 2y cm Check students work. 25. x = t r + s 34. a. h = A - 2s2 ; 14 cm 4s b. A = 6s Homework Helper, pages Topic 1 54 Lesson 1-3

11 ELABORATE PRACTICE Texas End-of-Course Practice Homework Helper, page p = chirps Exercise Depth of Knowledge TEKS Standards Content Process Student Companion, page Skill/Concept Understanding (12)(E) 2 2 Skill/Concept Understanding (12)(E) 3 2 Skill/Concept Understanding (12)(E) 4 2 Skill/Concept Understanding (12)(E) 5 2 Skill/Concept Understanding (12)(E) 6 3 Strategic Thinking (12)(E) (1)(F) ANSWERS 1. B 2. F 3. D 4. G 5. A 6. a. r = c 2p b. 6m [2] Both parts answered correctly. [1] One part answered correctly. [0] Neither part answered correctly. Topic 1 55 Lesson 1-3

12 EVALUATE ASSESS AND INTERVENE Lesson Quiz Reproducible Write-in activity ONLINE ASSESSMENT You may opt to have students take the Lesson Quiz with Advance, Pearson s online assessment service on PearsonTEXAS.com. The Lesson Quiz will be automatically scored and appropriate remediation, practice, or enrichment will be assigned based on student performance. Download Lesson Quiz 1-3 from the Teacher Resource Download Center. The quiz is in an editable format. answers 1. y = 5-2x; y = 1; y = x = s q + r 3. g = 11 5 t 4. A r 2 = C 2r 5. The formula solved for the width of the garden is w = 40 - /. This means as the value of the length increases, the value of the width decreases. DIFFERENTIATION PLAN The table below suggests a Prescription for Differentiation based on a student s score on the Lesson Quiz. Students scoring 2 points or fewer may need more practice or reteaching, while those scoring 5 points will benefit from extension or enrichment. PRESCRIPTION FOR DIFFERENTIATION REMEDIATION ON-LEVEL EXTENSION 0 2 points 3 4 points 5 points Topic 1 56 Lesson 1-3

13 EVALUATE ASSESS AND INTERVENE Differentiated Intervention Use students scores on the Lesson Quiz to prescribe differentiated assignments. Download Differentiated Resources from the Teacher Resource Download Center. Most of these pages are available in editable formats. Answers can also be found in the Teacher Resource Download Center. Reteaching Reproducible Write-In Activities Reteaching 2 REMEDIATION 0-2 Points RETEACHING (2 pages) Provides reteaching and practice exercises for the key lesson concepts. ADDITIONAL VOCABULARY SUPPORT Helps students develop and reinforce understanding of mathematical vocabulary terms and key concepts. Practice Think About a Plan ON-LEVEL 3-4 Points PRACTICE (2 pages) Provides extra practice for each lesson. For simpler practice exercises, use the Form K Practice pages. THINK ABOUT A PLAN Helps students develop specific problem-solving skills and strategies by providing scaffolded guiding questions; also provides focused practice of TEKS Math Process Standards. Enrichment Activity/Game/Puzzle EXTENSION 5 Points ENRICHMENT Presents interesting problems and activities that extend the concepts of the lesson. ACTIVITIES, GAMES, AND PUZZLES Offer additional opportunities for concept development, fluency practice, enrichment, and fun! ToPic 1 57 LEsson 1-3