Enhanced Instructional Transition Guide

Transcription

1 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Unit 03: Linear Equations, Inequalities, and Applications (12 days) Possible Lesson 01 (12 days) POSSIBLE LESSON 01 (12 days) This lesson is one approach to teaching the State Standards associated with this unit. Districts are encouraged to customize this lesson by supplementing with districtapproved resources, materials, and activities to best meet the needs of learners. The duration for this lesson is only a recommendation, and districts may modify the time frame to meet students needs. To better understand how your district is implementing CSCOPE lessons, please contact your child s teacher. (For your convenience, please find linked the TEA Commissioner s List of State Board of Education Approved Instructional Resources and Midcycle State Adopted Instructional Materials.) Lesson Synopsis: Students solve equations and inequalities using various methods, including concrete models, tables, graphs, and algebraic methods. Students formulate equations and inequalities to represent problem situations, solve the equations and inequalities, and justify the final solutions in terms of the problem situation. TEKS: The Texas Essential Knowledge and Skills (TEKS) listed below are the standards adopted by the State Board of Education, which are required by Texas law. Any standard that has a strike-through (e.g. sample phrase) indicates that portion of the standard is taught in a previous or subsequent unit. The TEKS are available on the Texas Education Agency website at A.1 Foundations for functions.. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to: A.1C A.1D Describe functional relationships for given problem situations and write equations or inequalities to answer questions arising from the situations. Supporting Standard Represent relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities. Readiness Standard A.3 Foundations for functions.. The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. The student is expected to: A.3A Use symbols to represent unknowns and variables. Supporting Standard A.4 Foundations for functions.. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is page 1 of 110

2 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days expected to: A.4A Find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations. Readiness Standard A.7 Linear functions.. The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to: A.7A A.7B A.7C Analyze situations involving linear functions and formulate linear equations or inequalities to solve problems. Supporting Standard Investigate methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, select a method, and solve the equations and inequalities. Readiness Standard Interpret and determine the reasonableness of solutions to linear equations and inequalities. Supporting Standard Performance Indicator(s): High School Mathematics AlgebraI Unit03 PI01 Analyze problem situations that can be represented algebraically by an equation such as the following: Ellie is making an abstract triangular collage for the wall. She has 14 feet of edging to go around the collage. If the longest side is twice the smallest and the other side is four less than three times the smallest, find the length of each side of the triangular collage. Sad Sack is peeling potatoes at a rate of 3 potatoes per minute. Four minutes later, Beetle Bailey joins him and peels potatoes at a rate of 5 potatoes per minute. If the total number of potatoes needed to be peeled is 36, how long must each peel potatoes? If the total number of potatoes peeled is 36, how many potatoes did each peel? Create a graphic organizer for each problem situation that includes formulating a representative equation, selecting two different solution methods (concrete model, table, graph, or algebraic method), solving the equation by each method, and interpreting the reasonableness of the solution in terms of the problem situation. Standard(s): A.1C, A.1D, A.3A, A.4A, A.7A, A.7B, A.7C ELPS ELPS.c.1C, ELPS.c.3H, ELPS.c.5B page 2 of 110

3 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days High School Mathematics AlgebraI Unit03 PI02 Analyze a problem situation that can be represented algebraically by an inequality such as the following: Cat found an online dealer, Tidy Paws, who sells flea dip for $3.00 a quart bottle plus a $21.00 handling charge per order. If Cat can pay at most $95.00 for flea dip at Tidy Paws, how many bottles could she purchase? Create a graphic organizer for the problem situation that includes formulating a representative inequality, selecting two different solution methods (concrete model, table, graph, or algebraic method), solving the inequality by each method, and interpreting the reasonableness of the solution in terms of the problem situation. Standard(s): A.1C, A.1D, A.3A, A.4A, A.7A, A.7B, A.7C ELPS ELPS.c.1C, ELPS.c.3H, ELPS.c.5B Key Understanding(s): Equations can be written to represent problem situations algebraically and solved using a variety of methods. The reasonableness of the solution can be justified in terms of the problem situation. Inequalities can be written to represent problem situations algebraically and solved using a variety of methods. The reasonableness of the solution can be justified in terms of the problem situation. Misconception(s): Some students may think that answers to both equations and inequalities are exact answers instead of correctly identifying the solutions to equations as exact answers and the solutions to inequalities as range of answers. Some students may think that anytime a negative is involved, the inequality switches and not just when multiplying or dividing by a negative. Vocabulary of Instruction: associative property commutative property consecutive numbers distributive property equation evaluate expression graphic solution inequality inverse operation numeric solution proportions rates ratios simplify solving an equation solving an inequality terms page 3 of 110

4 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Materials List: algebra tiles (1 set per student) cardstock (2 sheets per 2 students) cardstock (4 sheets per 2 students) glue (1 per 2 students) graphing calculator (1 per student) graphing calculator with display (1 per teacher) plastic zip bag (quart size) (1 per 2 students) plastic zip bag (quart size) (1 per 2 students) scissors (1 per 2 students) scissors (1 per teacher) Attachments: All attachments associated with this lesson are referenced in the body of the lesson. Due to considerations for grading or student assessment, attachments that are connected with Performance Indicators or serve as answer keys are available in the district site and are not accessible on the public website. Multi-Step Equations KEY Multi-Step Equations Math Detective KEY Math Detective Math Detective Cards Scrambled Egg Quations KEY Scrambled Egg Quations page 4 of 110

5 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Multi-Step Equations Practice Problems KEY Multi-Step Equations Practice Problems Which is Greater? Are You Sure? KEY Which is Greater? Are You Sure? Solving Inequalities KEY Solving Inequalities Unbalanced Inequalities Recording Sheet KEY Unbalanced Inequalities Recording Sheet Cards Unbalanced Inequalities Inequality Connection KEY Inequality Connection Seeing Inequalities KEY Seeing Inequalities Equations and Inequalities with Consecutive Numbers KEY Equations and Inequalities with Consecutive Numbers Problem Solving with Ratios KEY Problem Solving with Ratios Problem Solving with Proportions KEY Problem Solving with Proportions page 5 of 110

6 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Problem Solving with Rates KEY Problem Solving with Rates Tying Up Equations and Inequalities KEY Tying Up Equations and Inequalities PI GETTING READY FOR INSTRUCTION Teachers are encouraged to supplement and substitute resources, materials, and activities to meet the needs of learners. These lessons are one approach to teaching the TEKS/Specificity as well as addressing the Performance Indicators associated with each unit. District personnel may create original lessons using the Content Creator in the Tools Tab. All originally authored lessons can be saved in the My CSCOPE Tab within the My Content area. Suggested Day Suggested Instructional Procedures Notes for Teacher 1-2 Topics: Engage 1 One- and two-step equations Multi-step equations Students use prior knowledge to compare one- and two-step equations of the previous unit with the multi-step equations addressed in this unit. Students predict how the properties for simplifying expressions can be applied to solve multi-step equations. Instructional Procedures: MATERIALS algebra tiles (1 set per student) TEACHER NOTE Students continue to use algebra tiles during this lesson. Plastic models can be used or students can be given the cardstock models to cut and keep as their own personal set. 1. Distribute a set of algebra tiles to each student. 2. Display the following equation and instruct students to think about how they could solve the equation 3x + (2x - 5) = 13-2(x + 2). Allow students time to think through a process. Facilitate a class discussion of student page 6 of 110

7 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher responses. Ask: How does this equation compare to the one- and two-step equations from the previous lesson? Answers may vary. The equation is harder. The equation has more parts; etc. How many x-terms are in the equation? (Two x-terms on one side and one x-term on the other side.) How does each side of the equation compare to the expressions from the previous lesson? Answers may vary. Each side looks like an expression that needs to be simplified; etc. What properties could be used to simplify each side of the equation? (Distributive, Commutative, and Associative Properties) Why could this equation be considered a multi-step equation? Answers may vary. It takes more than one or two steps to solve the equation algebraically; etc. What methods could be used to solve this equation? Answers may vary. Concrete models, algebraic, graphs, tables; etc. Topics: ATTACHMENTS Multi-step equations Solution methods (concrete models, tables, graphs, algebraic methods) Explore/Explain 1 Students solve multi-step equations using concrete models, tables, graphs, and algebraic methods. Instructional Procedures: Day 1 1. Place students in pairs. Distribute handout: Multi-Step Equations and a set of algebra tiles to each student. Teacher Resource: Multi-Step Equations KEY (1 per teacher) Teacher Resource: Multi-Step Equations (1 per teacher) Handout: Multi-Step Equations (1 per student) Teacher Resource: Math Detective KEY (1 per teacher) Handout: Math Detective (1 per student) Card Set: Math Detective Cards (1 set per group) page 7 of 110

8 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher 2. Refer students to Sample problem 1 on handout: Multi-Step Equations. Display teacher resource: Multi- Step Equations, and facilitate a class discussion on using algebra tiles, pictorial models, graphs and tables in the graphing calculator, and algebraic methods to model and solve a multi-step equation. Model the various methods for solving a multi-step equation using Sample problem Instruct students to work with their partner to complete Sample problems 2 3. Allow students time to complete the problems, and monitor students to check for understanding. Using teacher resource: Multi-Step Equations, facilitate a class discussion of student results, clarifying any misconceptions. 4. Distribute handout: Math Detective to each student. Instruct students to work with their partner to complete the handout. This may be completed as homework, if necessary. MATERIALS algebra tiles (1 set per student) graphing calculator (1 per student) graphing calculator with display (1 per teacher) cardstock (4 sheets per 2 students) scissors (1 per teacher) plastic zip bag (quart size) (1 per 2 students) Day 2 5. Prior to instruction, create card set: Math Detective Cards by copying on cardstock, laminating, cutting apart, and placing in plastic zip bags. One card set will be given to each pair of students. 6. Place students with the same partner from the previous day. Distribute card set: Math Detective Cards to each pair of students. Instruct students to use the cards to check and correct their answers on handout: Math Detective. Facilitate a class discussion to summarize student results. Ask: TEACHER NOTE Only one example of concrete models is used with multiple step equations because that method gets very cumbersome. Algebraic methods are mainly used when solving multiple-step equations, and tables and graphs are used to verify solutions. What is done first when solving a multi-step problem? (Clear the parentheses by distribution.) What is done next when solving a multi-step problem? (Combine like terms.) What is done next when solving a multi-step problem? (Adding or subtracting the constant from the side with the variable.) What is done next when solving a multi-step problem? (Multiplying or dividing by the coefficient to find the solution.) How can the solution to the equation be checked? Answers may vary. The solution can be page 8 of 110

9 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher substituted back into the problem. The solution can be verified with the graphing calculator by analyzing the graph and/or the table; etc. 7. Refer students to Sample problem 4 on handout: Multi-Step Equations. Display teacher resource: Multi- Step Equations, and facilitate a class discussion on formulating an equation for a problem situation. Model formulating an equation using Sample problem Instruct students to work with their partner to complete Sample problems 5 7. Allow students time to complete the problems, and monitor students to check for understanding. Using teacher resource: Multi-Step Equations, facilitate a class discussion of student results, clarifying any misconceptions. 9. Instruct students to work independently to complete the Practice problems on handout: Multi-Step Equations. This may be completed as homework, if necessary. 3 Topics: Multi-step equations Elaborate 1 Students continue to solve multi-step equations using algebraic methods. Instructional Procedures: 1. Place students in pairs. Distribute handout: Scrambled Egg Quations, scissors, and glue to each pair of students. 2. Instruct students to work with their partner to complete the activity, putting both names on the answer sheets on pages 1 2. The remaining pages are cut up to glue in the appropriate places on the answer sheets. Allow ATTACHMENTS Teacher Resource: Scrambled Egg Quations KEY (1 per teacher) Handout: Scrambled Egg Quations (1 per 2 students) Teacher Resource: Multi-Step Equations Practice Problems KEY (1 per teacher) Handout: Multi-Step Equations Practice Problems (1 per student) MATERIALS page 9 of 110

10 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher students time to complete the activity, and monitor students to check for understanding. Facilitate a class discussion to summarize the steps involved in solving a multi-step equation, clarifying any misconceptions. 3. Distribute handout: Multi-Step Equations Practice Problems. Instruct students to work independently to complete the handout. This may be completed as homework, if necessary. scissors (1 per 2 students) glue (1 per 2 students) graphing calculator (1 per student) TEACHER NOTE In the previous activity: Sorting It Out with Properties, students used a specific order to simplify expressions. In this activity, students use a specific order to solve multiple-step equations. 4 5 Topics: Engage 2 Inequality symbols Properties of inequalities Students use prior knowledge to review inequalities and their properties. Instructional Procedures: 1. Place students in pairs. Distribute handout: Which is Greater? Are You Sure? to each student. 2. Instruct students to work with their partner to complete problems Allow students time to complete the problems. Display teacher resource: Which Is Greater? Are You Sure?, and facilitate a class discussion of student results, clarifying any misconceptions. Stress the effect of multiplying and dividing by a negative on an inequality statement. Ask: ATTACHMENTS Teacher Resource: Which is Greater? Are You Sure? KEY (1 per teacher) Teacher Resource: Which is Greater? Are You Sure? (1 per teacher) Handout: Which is Greater? Are You Sure? (1 per student) TEACHER NOTE Handout: Which is Greater? Are You Sure? is a review of inequalities and their properties. Do not spend too much time on this activity. The remainder of class is to teach students how to solve inequalities using a variety of methods. page 10 of 110

11 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher What caused the statement to become false? (Multiplying or dividing both sides of an inequality statement by a negative.) How could the statement be corrected to make the statement true? (Switch the inequality sign around.) Topics: ATTACHMENTS Inequalities Solution methods (concrete models, tables, graphs, algebraic methods) Explore/Explain 2 Students solve inequalities using concrete models, tables, graphs, and algebraic methods. Teacher Resource: Solving Inequalities KEY (1 per teacher) Teacher Resource: Solving Inequalities (1 per teacher) Handout: Solving Inequalities (1 per student) Instructional Procedures: Day 4 MATERIALS 1. Place students in pairs. Distribute handout: Solving Inequalities and a set of algebra tiles to each student. Refer students to page 1. Display teacher resource: Solving Inequalities, and facilitate a class discussion on methods of solving inequalities and their solutions. 2. Refer students to Sample problems 1 4 on handout: Solving Inequalities. Prior to filling in the tables on the handout, instruct students to solve the inequality using algebra tiles, then put the symbolic solution in the table, and finally enter functions into y 1 and y 2 of the graphing calculator to analyze the graphs and tables. Allow students time to complete the problems, and monitor students to check for understanding. Using teacher resource: Solving Inequalities, facilitate a class discussion of student results, clarifying any misconceptions. algebra tiles (1 set per student) graphing calculator (1 per student) graphing calculator with display (1 per teacher) TEACHER NOTE Unless otherwise noted, the Standard Window {x: 10 x 10} {x: 10 y 10} is used on the graphing calculator. page 11 of 110

12 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher 3. Instruct students to work with their partner to complete the Guided Practice problems 1 2 on handout: Solving Inequalities. These problems may be completed as homework, if necessary. Day 5 4. Facilitate a class discussion to debrief Guided Practice problems on handout: Solving Inequalities. Clarify any misconceptions. 5. Place students in pairs. Refer students to Sample problems 5 6 on handout: Solving Inequalities. Display teacher resource: Solving Inequalities, and facilitate a class discussion on special cases of inequalities. Model solving inequalities involving special cases using Sample problems 5 6. Instruct students to independently answer the questions after problem 6, and share answers with their partner. Allow students time to complete answering and sharing questions, and monitor students to check for understanding. Facilitate a class discussion of student responses, clarifying any misconceptions. TEACHER NOTE Some students have difficulty remembering if and when to switch the inequality sign when negative values are involved. In this curriculum as students are solving inequalities, they keep variables positive in order to avoid multiplying or dividing by negative values. It is a teacher choice, if another method is preferred. 6. Instruct students to work independently to complete the Practice problems on handout: Solving Inequalities. This may be completed as homework, if necessary. 6 Topics: Solving inequalities Explore/Explain 3 Students continue to solve inequalities using a card game. Instructional Procedures: 1. Prior to instruction, create card set: Unbalanced Inequalities by copying on cardstock, laminating, cutting out, and placing in plastic zip bags. One card set will be needed for each pair of students. ATTACHMENTS Teacher Resource: Unbalanced Inequalities Recording Sheet KEY (1 per teacher) Handout: Unbalanced Inequalities Recording Sheet (1 per student) Card Set: Unbalanced Inequalities (1 set per pair) MATERIALS page 12 of 110

13 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher 2. Facilitate a class discussion to debrief the Practice problems from handout: Solving Inequalities. Clarify any misconceptions. 3. Place students in pairs. Distribute handout: Unbalanced Inequalities Recording Sheet to each student. Distribute a card set: Unbalanced Inequalities to each pair of students. Instruct students to work with their partner using the card set to complete the table on the handout. graphing calculator (1 per student) cardstock (2 sheets per 2 students) scissors (1 per teacher) plastic zip bag (quart size) (1 per 2 students) 4. Allow students time to complete the activity, and monitor students to check for understanding. Facilitate a class discussion of student results, clarifying any misconceptions. Instruct students to make corrections as necessary. 7 Topics: Inequalities Verbal representations Symbolic representations Explore/Explain 4 Students explore and explain verbal statements and inequalities. Instructional Procedures: 1. Distribute handout: Inequality Connection to each student. Instruct students to work independently to complete the handout. Display teacher resource: Inequality Connection, and facilitate a class discussion to debrief and check for student understanding. 2. Place students in pairs. Distribute handout: Seeing Inequalities to each student. Refer students to page 1 of ATTACHMENTS Teacher Resource: Inequality Connection KEY (1 per teacher) Teacher Resource: Inequality Connection (1 per teacher) Handout: Inequality Connection (1 per student) Teacher Resource: Seeing Inequalities KEY (1 per teacher) Teacher Resource: Seeing Inequalities (1 per teacher) Handout: Seeing Inequalities (1 per student) MATERIALS page 13 of 110

14 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher the handout. Display teacher resource: Seeing Inequalities, and facilitate a class discussion of verbal expressions. Instruct students to work with their partner to complete problems A G. Allow students time to complete the problems, and monitor students to check for understanding. Facilitate a class discussion of student results, clarifying any misconceptions. 3. Divide the class into 8 groups, and assign each group 1 of the Guided Practice problems 1 9 on handout: Seeing Inequalities to each group. Instruct each group to write an expression to represent their assigned problem situation. Allow groups time to complete their expressions. Using teacher resource: Seeing Inequalities, instruct groups to record their expression on the display, as students complete their handout. Clarify any misconceptions. graphing calculator (1 per student) graphing calculator with display (1 per teacher) TEACHER NOTE Students connect verbal statements and symbolic inequalities. This skill is used in applications of equations and inequalities. 4. Instruct students to work independently on Practice problems 1 10 on handout: Seeing Inequalities. This may be completed as homework, if necessary. 8 Topics: Applications of equations and inequalities Consecutive numbers Elaborate 2 Students set up verbal problems involving consecutive integers into equations and inequalities. Students solve the equations and inequalities using methods of choice. Students justify answers in terms of the problem situation. ATTACHMENTS Teacher Resource: Equations and Inequalities with Consecutive Numbers KEY (1 per teacher) Teacher Resource: Equations and Inequalities with Consecutive Numbers (1 per teacher) Handout: Equations and Inequalities with Consecutive Numbers (1 per student) Instructional Procedures: 1. Facilitate a class discussion to debrief Practice problems on handout: Seeing Inequalities. Clarify any misconceptions. 2. Place students in pairs. Distribute handout: Equations and Inequalities with Consecutive Numbers to MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) page 14 of 110

15 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher each student. 3. Refer students to page 1 of the handout. Display teacher resource: Equations and Inequalities with Consecutive Numbers, and facilitate a class discussion on consecutive numbers. Instruct students to work with their partner to complete problems 1 2. Allow students time to complete the problems, and monitor students to check for understanding. Using teacher resource: Equations and Inequalities with Consecutive Numbers, facilitate a class discussion of student results, clarifying any misconceptions. 4. Instruct students to work with their partner to complete problems 3 7 on handout: Equations and Inequalities with Consecutive Numbers. Allow students time to complete the problems, and monitor students to check for understanding. Facilitate a class discussion of student results, clarifying any misconceptions. 5. Instruct students to work independently to complete Practice problems 1 6 on handout: Equations and Inequalities with Consecutive Numbers. This may be completed as homework, if necessary. 9 Topics: Applications of equations and inequalities Ratios Elaborate 3 Students set up verbal problems involving ratios into an algebraic equation. Students solve the equations using methods of choice. Students justify answers in terms of the problem situation. ATTACHMENTS Teacher Resource: Problem Solving Ratios KEY (1 per teacher) Teacher Resource: Problem Solving with Ratios (1 per teacher) Handout: Problem Solving with Ratios (1 per student) Instructional Procedures: 1. Facilitate a class discussion to debrief Practice problems on handout: Equations and Inequalities with Consecutive Numbers. Clarify any misconceptions. MATERIALS graphing calculator (1 per student) page 15 of 110

16 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher 2. Place students in pairs. Distribute handout: Problem Solving with Ratios to each student. Instruct students to work with their partner to complete the Sample problem. Allow students time to complete the problem. Display teacher resource: Problem Solving with Ratios, and facilitate a class discussion of student results, clarifying any misconceptions. graphing calculator with display (1 per teacher) 3. Instruct students to work with their partner on Guided Practice problems 1 3 on handout: Problem Solving with Ratios. Allow students time to complete the problems, and monitor students to check for understanding. Facilitate a class discussion of student results, clarifying any misconceptions. 4. Instruct students to work independently on Practice problems 1 8 on handout: Problem Solving with Ratios. This may be completed as homework, if necessary. 10 Topics: Applications of equations and inequalities Proportions Elaborate 4 Students set up verbal problems involving proportions into an algebraic equation. Students solve the equations using methods of choice. Students justify answers in terms of the problem situation. ATTACHMENTS Teacher Resource: Problem Solving with Proportions KEY (1 per teacher) Teacher Resource: Problem Solving with Proportions (1 per teacher) Handout: Problem Solving with Proportions (1 per student) Instructional Procedures: 1. Facilitate a class discussion to debrief Practice problems on handout: Problem Solving with Ratios. Clarify any misconceptions. 2. Place students in pairs. Distribute handout: Problem Solving with Proportions to each student. Refer students to page 1 of the handout. Display teacher resource: Problem Solving with Proportions, and facilitate a class discussion on proportions. Instruct students to work with their partner to complete Sample MATERIALS graphing calculator (1 per student) graphing calculator with display (1 per teacher) page 16 of 110

17 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher Problem 1. Allow students time to complete the problems, and monitor students to check for understanding. Using teacher resource: Problem Solving with Proportions, facilitate a class discussion of student results, clarifying any misconceptions. 3. Instruct students to work with their partner to complete Sample problem 2 on handout: Problem Solving with Proportions. Allow students time to complete the problems, and monitor students to check for understanding. Using teacher resource: Problem Solving with Proportions, facilitate a class discussion of student results, clarifying any misconceptions. 4. Instruct students to work with their partner to complete Guided Practice problem 1 on handout: Problem Solving with Proportions. Allow students time to complete the problems, and monitor students to check for understanding. Facilitate a class discussion by having a volunteer pair display the problem and solution. 5. Instruct students to work independently to complete Practice problems 1 8 on handout: Problem Solving with Proportions. This may be completed as homework, if necessary. 11 Topics: Applications of equations and inequalities Rates Elaborate 5 Students set up verbal problems involving rates into an algebraic equation. Students solve the equations using methods of their choice. Students justify answers in terms of the problem situation. ATTACHMENTS Teacher Resource: Problem Solving with Rates KEY (1 per teacher) Teacher Resource: Problem Solving with Rates (1 per teacher) Handout: Problem Solving with Rates (1 per student) Instructional Procedures: 1. Facilitate a class discussion to debrief Practice problems on handout: Problem Solving with Proportions. Clarify any misconceptions. MATERIALS graphing calculator (1 per student) page 17 of 110

18 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher 2. Place students in pairs. Distribute handout: Problem Solving with Rates to each student. Instruct students to work with their partner to complete Sample problems 1 7. Allow students time to complete the problems, and monitor students to check for understanding. Display teacher resource: Problem Solving with Rates, and facilitate a class discussion of student results, clarifying any misconceptions. graphing calculator with display (1 per teacher) 3. Instruct students to work independently on Practice problems 1 6 on handout: Problem Solving with Rates. This may be completed as homework, if necessary. 12 Evaluate 1 Instructional Procedures: 1. Facilitate a class discussion to debrief Practice Problems on handout: Problem Solving with Rates. Clarify any misconceptions. 2. Assess student understanding of related concepts and processes by using the Performance Indicator(s) aligned to this lesson. ATTACHMENTS Teacher Resource (optional): Tying Up Equations and Inequalities KEY (1 per teacher) Handout (optional): Tying Up Equations and Inequalities PI (1 per student) MATERIALS Performance Indicator(s): graphing calculator (1 per student) page 18 of 110

19 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher High School Mathematics AlgebraI Unit03 PI01 Analyze problem situations that can be represented algebraically by an equation such as the following: Ellie is making an abstract triangular collage for the wall. She has 14 feet of edging to go around the collage. If the longest side is twice the smallest and the other side is four less than three times the smallest, find the length of each side of the triangular collage. Sad Sack is peeling potatoes at a rate of 3 potatoes per minute. Four minutes later, Beetle Bailey joins him and peels potatoes at a rate of 5 potatoes per minute. If the total number of potatoes needed to be peeled is 36, how long must each peel potatoes? If the total number of potatoes peeled is 36, how many potatoes did each peel? TEACHER NOTE As an optional assessment tool, use handout (optional): Tying Up Equations and Inequalities PI. Create a graphic organizer for each problem situation that includes formulating a representative equation, selecting two different solution methods (concrete model, table, graph, or algebraic method), solving the equation by each method, and interpreting the reasonableness of the solution in terms of the problem situation. Standard(s): A.1C, A.1D, A.3A, A.4A, A.7A, A.7B, A.7C ELPS ELPS.c.1C, ELPS.c.3H, ELPS.c.5B page 19 of 110

20 Enhanced Instructional Transition Guide High School Courses/Algebra I Unit 03: Suggested Duration: 12 days Suggested Day Suggested Instructional Procedures Notes for Teacher High School Mathematics AlgebraI Unit03 PI02 Analyze a problem situation that can be represented algebraically by an inequality such as the following: Cat found an online dealer, Tidy Paws, who sells flea dip for $3.00 a quart bottle plus a $21.00 handling charge per order. If Cat can pay at most $95.00 for flea dip at Tidy Paws, how many bottles could she purchase? Create a graphic organizer for the problem situation that includes formulating a representative inequality, selecting two different solution methods (concrete model, table, graph, or algebraic method), solving the inequality by each method, and interpreting the reasonableness of the solution in terms of the problem situation. Standard(s): A.1C, A.1D, A.3A, A.4A, A.7A, A.7B, A.7C ELPS ELPS.c.1C, ELPS.c.3H, ELPS.c.5B 03/27/13 page 20 of 110

21 Multi-Step Equations KEY Remember, the goal of solving an equation is to isolate the variable using inverse operations. To solve multi-step equations algebraically, follow the steps listed below: Distribute to clear grouping symbols. Multiply by common denominator to clear fractions. Combine like terms. Use addition/subtraction properties to move variable terms to one side and numeric values to the other side. Use multiplication/division properties to isolate the variable. Sample Problems I. Solve the following equation A. Using concrete models. B. Using algebraic methods. C. Using the graphing calculator. D. Check the solution for reasonableness. 1. 2(3x + 4) = -4 Concrete Model Solution Algebraic Solution Graphic Solution, Table Solution Check = (3x + 4) = -4 6x + 8 = -4 6x = x = x = -2 y 1 = 2(3x + 4) y 2 = -4 2(3(-2) + 4) = -4 2(-6 + 4) = -4 2(-2) = -4-4 = = , TESCCC 07/10/12 page 1 of 5

22 Multi-Step Equations KEY II. Solve the following equation A. Using algebraic methods. B. Using the graphing calculator. C. Check the solution for reasonableness (5x + 3) = 13x Algebraic Solution 9 + 2(5x + 3) = 13x x + 6 = 13x 10x + 15 = 13x 10x x = 13x 10x 15 = 3x = x Graphic Solution, Table y 1 = 9 + 2(5x + 3) y 2 = 13x Solution Check 9 + 2(5(5) + 3) = 13(5) 9 + 2(25 + 3) = (28) = = = 65 III. Solve the following equation by any method. 3. 3x + (2x 5) = 13 2(x + 2) x = , TESCCC 07/10/12 page 2 of 5

23 Multi-Step Equations KEY IV. For each word problem, formulate an equation, solve the equation, and justify the solution. 4. Thirty plus twice a larger number equals 100. Find the number x = 100 x = (35) = Two coats and four dresses cost $420. If each dress cost $50, how much do the coats cost each? 2x + 4(50) = 420 x = $110 Two coats at $110 each would be $220. Four dresses at $50 would be $200. The total would be $ Mr. Powell travels 24 miles to work each day. The difference between the distance Mr. Clay travels in six days and the distance that Mr. Powell travels in five days is 96 miles. How many miles does Mr. Clay travel to work? 6x 5(24) = 96 x = 36 miles If Mr. Clay travels 36 miles for 6 days he has traveled 216 miles. In 5 days Mr. Powell will travel 120 miles. The difference would be 96 miles. 7. In his science class, Andrew is required to convert temperatures he collected in degrees Fahrenheit into degrees Celsius. The teacher has said that degrees Celsius are equal to 5 9 of the difference between degrees Fahrenheit and 32. a. What is the general equation that represents degrees Celsius as a function of degrees Fahrenheit? C o = 5 9 (Fo 32) b. If Andrew determined the temperature of his solution is 113 o F, what would be the temperature of his solution in degree Celsius? C o = 5 9 ((113) 32) Co = 45 o c. If Jaylee measured the temperature of her solution to be 55 o C, what would be the temperature of her solution in degrees Fahrenheit? 55 = 5 9 (Fo 32) F o = 131 o 2012, TESCCC 07/10/12 page 3 of 5

24 Multi-Step Equations KEY Practice Problems 1. 3(2x + 5) = 3 Algebraic Solution 3(2x + 5) = 3 6x + 15 = 3 6x = x = x = -2 y 1 = 3(2x + 5) y 2 = 3 Graphic Solution 2. 2(x + 1) 3x = 3(3 + 2x) Algebraic Solution 2(x + 1) 3x = 3(3 + 2x) 2x + 2 3x = 9 + 6x -1x + 2 = 9 + 6x -1x x = 9 + 6x + 1x 2 = 9 + 7x 2 9 = 9 + 7x 9-7 = 7x = x y 1 = 2(x + 1) 3x y 2 = 3(3 + 2x) Graphic Solution 2012, TESCCC 07/10/12 page 4 of 5

25 Multi-Step Equations KEY Solve the following equation by any method. 3. 2(x 3) 17 = 13 3(x + 2) x = 6 Formulate an equation for the word problems. Solve the equation. Justify the solution. 4. The larger of two numbers is 23. If three times the larger is 5 more than eight times the smaller, find the smaller number. 3(23) = 5 + 8x x = 8 Five more than eight times eight is 69. That is equal to three times twenty-three. 5. If four times Catherine s age is subtracted from 5 times Jose s age, the difference is 32 years. Jose is sixteen. Find Catherine s age. 5(16) 4x = 32 x = 12 Catherine is 12 years old. Five times Jose s age is 80. Four times twelve is 48. The difference is Charles has been asked to insert a trapezoid in the floor tiling of the rotunda of the courthouse. One base must be 8 feet and the height must be 10 feet. a. What general equation can be used to represent the area of the trapezoid as a function of the second base? A = 1 2 (10)(8 + b 2) A = 5(8 + b 2 ) b. If Charles wants the area of the trapezoid to be 65 square feet, what should be the measure of the second base? 65 = 5(8 + b 2 ) b 2 = 5 feet 2012, TESCCC 07/10/12 page 5 of 5

26 Multi-Step Equations Remember, the goal of solving an equation is to isolate the variable using inverse operations. To solve multi-step equations algebraically, follow the steps listed below: Distribute to clear grouping symbols. Multiply by common denominator to clear fractions. Combine like terms. Use addition/subtraction properties to move variable terms to one side and numeric values to the other side. Use multiplication/division properties to isolate the variable. Sample Problems I. Solve the following equation A. Using concrete models. B. Using algebraic methods. C. Using the graphing calculator. D. Check the solution for reasonableness. 1. 2(3x + 4) = -4 Concrete Model Solution Algebraic Solution Graphic Solution, Table Solution Check 2012, TESCCC 07/11/12 page 1 of 5

27 Multi-Step Equations II. Solve the following equation A. Using algebraic methods. B. Using the graphing calculator. C. Check the solution for reasonableness (5x + 3) = 13x Algebraic Solution Graphic Solution, Table Solution Check III. Solve the following equation by any method. 3. 3x + (2x 5) = 13 2(x + 2) 2012, TESCCC 07/11/12 page 2 of 5

28 Multi-Step Equations IV. For each word problem, formulate an equation, solve the equation, and justify the solution. 4. Thirty plus twice a larger number equals 100. Find the number. 5. Two coats and four dresses cost $420. If each dress cost $50, how much do the coats cost each? 6. Mr. Powell travels 24 miles to work each day. The difference between the distance Mr. Clay travels in six days and the distance that Mr. Powell travels in five days is 96 miles. How many miles does Mr. Clay travel to work? 7. In his science class, Andrew is required to convert temperatures he collected in degrees Fahrenheit into degrees Celsius. The teacher has said that degrees Celsius are equal to 5 9 of the difference between degrees Fahrenheit and 32. a. What is the general equation that represents degrees Celsius as a function of degrees Fahrenheit? b. If Andrew determined the temperature of his solution is 113 o F, what would be the temperature of his solution in degree Celsius? c. If Jaylee measured the temperature of her solution to be 55 o C, what would be the temperature of her solution in degrees Fahrenheit? 2012, TESCCC 07/11/12 page 3 of 5

29 Multi-Step Equations Practice Problems 1. 3(2x + 5) = 3 Algebraic Solution Graphic Solution 2. 2(x + 1) 3x = 3(3 + 2x) Algebraic Solution Graphic Solution 2012, TESCCC 07/11/12 page 4 of 5

30 Multi-Step Equations Solve the following equation by any method. 3. 2(x 3) 17 = 13 3(x + 2) Formulate an equation for the word problems. Solve the equation. Justify the solution. 4. The larger of two numbers is 23. If three times the larger is 5 more than eight times the smaller, find the smaller number. 5. If four times Catherine s age is subtracted from 5 times Jose s age, the difference is 32 years. Jose is sixteen. Find Catherine s age. 6. Charles has been asked to insert a trapezoid in the floor tiling of the rotunda of the courthouse. One base must be 8 feet and the height must be 10 feet. a. What general equation can be used to represent the area of the trapezoid as a function of the second base? b. If Charles wants the area of the trapezoid to be 65 square feet, what should be the measure of the second base? 2012, TESCCC 07/11/12 page 5 of 5

31 Math Detective KEY Answer Sheet Problem 1 st Step 2 nd Step 3 rd Step Check Solution 1. 4 x x -12 = -24 4x = -12 x = x x = 18-3x = 3 x = x 2 x 4 9 3x + 6 = 9 3x = 3 x = 1 4. x x 3 2x x 2 = 10 4x = 12 x = x 2 6 3x + 6 = 6 3x = 0 x = o x x 1-44 = 9x = 9x -5 = x 7. 3 x 2 6 3x + 6 = -6 3x = -12 x = x x 8 = 16 4x = 24 x = x 1 2x x + 2 = -12 7x = -14 x = x 4x x = 7 3x = -21 x = x 1 4x 3 2 3x 4 = 2 3x = 6 x = x 4x x = -4-3x = -21 x = , TESCCC 07/11/12 page 1 of 1

32 Answer Sheet Math Detective Problem 1 st Step 2 nd Step 3 rd Step Check Solution 1. 4( x 3) (5 x) x 2 x x x 3 2x ( x 2) x x ( x 2) ( x 2) x 1 2x x 4x x 1 4x x 4x , TESCCC 07/11/12 page 1 of 1

33 Problem 4 x 3 24 Math Detective Cards 3 5 x 18 2x 2 x 4 9 Math Detective x x 3 2x 5 10 Math Detective 3 x 2 6 Math Detective 44 10x x 1 Math Detective 3 x 2 6 Math Detective 4 x 2 16 Math Detective 5x 1 2x 3 12 Math Detective 28 x 4x 7 Math Detective 7x 1 4x 3 2 Math Detective 17 x 4x 4 Math Detective Math Detective Math Detective 2012, TESCCC 07/12/12 page 1 of 4

34 1 st Step 4x Math Detective Cards 15 3x 18 3x 6 9 Math Detective 4x 2 10 Math Detective 3x 6 6 Math Detective 44 9x 1 Math Detective 3x 6 6 Math Detective 4x 8 16 Math Detective 7x 2 12 Math Detective 28 3x 7 Math Detective 3x 4 2 Math Detective 17 3x 4 Math Detective Math Detective Math Detective 2012, TESCCC 07/12/12 page 2 of 4

35 2 nd Step 4x 12 Math Detective Cards 3x 3 3x 3 Math Detective 4x 12 Math Detective 3x 0 Math Detective 45 9x Math Detective 3x 12 Math Detective 4x 24 Math Detective 7x 14 Math Detective 3x 21 Math Detective 3x 6 Math Detective 3x 21 Math Detective Math Detective Math Detective 2012, TESCCC 07/12/12 page 3 of 4

36 3 rd Step x 3 Math Detective Cards x 1 x 1 Math Detective x 3 Math Detective x 0 Math Detective 5 x Math Detective x 4 Math Detective x 6 Math Detective x 2 Math Detective x 7 Math Detective x 2 Math Detective x 7 Math Detective Math Detective Math Detective 2012, TESCCC 07/12/12 page 4 of 4

37 Scrambled Egg Quations KEY Cut out each of the scrambled problems and glue or tape it to the appropriate place on the answer sheets. Cut out each of the equations After Distribution and glue or tape it under the appropriate problem on the answer sheet. Continue doing the same for combining like terms, adding/subtracting, and multiplying/dividing. Problem 1 (3 + x)5 2 = ANSWER SHEETS Problem 2 7x + 2(4 2x) = After distribution A x 2 = After combining like terms L 5x + 13 = -17 After add/subtract AA 5x = -30 After divide/multiply x = -6 After distribution G 7x + 8 4x = After combining like terms O 3x + 8 = -19 After add/subtract CC 3x = -27 After divide/multiply x = -9 Problem 3 5x (7 x) = (-13) After distribution F 5x 7 + x = After combining like terms N 6x 7 = -49 After add/subtract W 6x = -42 After divide/multiply KK x = -7 Problem 4 (-x + 2) = After distribution D -4x = After combining like terms R -4x + 20 = -4 After add/subtract V -4x = -24 After divide/multiply II x = , TESCCC 07/12/12 page 1 of 2

38 Scrambled Egg Quations KEY Problem 5-5(x + 4) = x ANSWER SHEETS Problem (-3 x) = After distribution I -5x 20 = x After combining like terms K -5x 20 = -28 7x After add/subtract Z 2x = -8 After divide/multiply FF x = -4 After distribution J x = After combining like terms T -24 4x = -60 After add/subtract U -4x = -36 After divide/multiply MM x = 9 Problem 7-3(4 - x) = x After distribution E x = x After combining like terms M x = x After add/subtract BB 4 = 2x After divide/multiply HH x = 2 Problem (-x + 5) = x After distribution C x 15 = x After combining like terms Q 3x = x After add/subtract DD 21 = 3x After divide/multiply EE x = 7 Problem 9 5x + 2(x + 6) = After distribution H 5x + 2x + 12 = After combining like terms S 7x + 12 = 40 After add/subtract Y 7x = 28 After divide/multiply LL x = 4 Problem 10-3x 10 = 6 5(4 + x) After distribution B -3x 10 = x After combining like terms P -3x 10 = -14 5x After add/subtract X 2x = -4 After divide/multiply JJ x = , TESCCC 07/12/12 page 2 of 2

39 Scrambled Egg Quations Cut out each of the scrambled problems and glue or tape it to the appropriate place on the answer sheets. Cut out each of the equations After Distribution and glue or tape it under the appropriate problem on the answer sheet. Continue doing the same for combining like terms, adding/subtracting, and multiplying/dividing. ANSWER SHEETS Problem 1 Problem 2 After distribution After distribution After combining like terms After combining like terms After add/subtract After add/subtract After divide/multiply After divide/multiply Problem 3 Problem 4 After distribution After distribution After combining like terms After combining like terms After add/subtract After add/subtract After divide/multiply After divide/multiply 2012, TESCCC 07/12/12 page 1 of 4

40 Scrambled Egg Quations ANSWER SHEETS Problem 5 Problem 6 After distribution After distribution After combining like terms After combining like terms After add/subtract After add/subtract After divide/multiply After divide/multiply Problem 7 Problem 8 After distribution After distribution After combining like terms After combining like terms After add/subtract After add/subtract After divide/multiply After divide/multiply Problem 9 Problem 10 After distribution After distribution After combining like terms After combining like terms After add/subtract After add/subtract After divide/multiply After divide/multiply 2012, TESCCC 07/12/12 page 2 of 4

41 Scrambled Egg Quations SCRAMBLED PROBLEMS (-3 x) = x (7 x) = (-13) 1 (3 + x)5 2 = (x + 4) = x 7-3(4 x) = x (-x + 5) = x 2 7x + 2(4 2x) = x 10 = 6 5(4 + x) 4 (-x + 2) = A x 2 = AFTER DISTRIBUTION 9 5x + 2(x + 6) = B -3x 10 = x C x 15 = x E x = x G 7x + 8 4x = D -4x = F 5x 7 + x = H 5x + 2x + 12 = I -5x 20 = x AFTER COMBINING LIKE TERMS K -5x 20 = -28 7x J x = L 5x + 13 = -17 M x = x O 3x + 8 = -19 Q 3x = x S 7x + 12 = 40 N 6x 7 = -49 P -3x 10 = -14 5x R -4x + 20 = -4 T -24 4x = , TESCCC 07/12/12 page 3 of 4

42 Scrambled Egg Quations AFTER ADDING AND SUBTRACTING U -4x = -36 V -4x = -24 W 6x = -42 Y 7x = 28 AA 5x = -30 CC 3x = -27 X 2x = -4 Z 2x = -8 BB 4 = 2x DD 21 = 3x SOLUTION AFTER DIVISION OR MULTIPLICATION EE x = 7 FF x = -4 GG x = -6 II x = 6 KK x = -7 MM x = 9 HH x = 2 JJ x = -2 LL x = 4 NN x = , TESCCC 07/12/12 page 4 of 4

43 Multi-Step Equations Practice Problems KEY 1. 2(5x 3) = 4 Algebraic Solution Graphic Solution 10x 6 = 4 10x = x = x = 1 y 1 = 2(5x 3) y 2 = (x 2) + 2x = 4(2x+3) Algebraic Solution 3(x 2) + 2x = 4(2x+3) 3x 6 + 2x = 8x x 6 = 8x x 6 5x = 8x x -6 = 3x = 3x = 3x = x y 1 = 3(x 2) + 2x y 2 = 4(2x + 3) Graphic Solution 2012, TESCCC 07/12/12 page 1 of 2

44 Multi-Step Equations Practice Problems KEY Formulate an equation for the word problems. Solve the equation. Justify the solution. 3. Two times the difference of a number decreased by five is fourteen. What is the number? 2(x 5) = 14 x = 12 The difference of 12 and 5 is 7. Two times seven equals Five times the difference of a number subtracted from ten is forty. What is the number? 5(10 x) = 40 x = 2 The difference of 10 2 is 8. Eight times five equals forty. 5. Four times the sum of three times a number increased by two is equal to three times the difference of two times the number decreased by eight. What is the number? 4(3x + 2) = 3(2x 8) x = (3( ) + 2) = 3(2( ) 8) 4( ) = 3( ) 4(-14) = 3( ) -56 = A box contains red and blue chips. There are 2 more blue chips than red chips. If the number of red chips is tripled and the number of blue chips is doubled, there will be an equal number of red and blue chips. How many blue chips does the original box contain? A. 10 B. 6 C. 1 D , TESCCC 07/12/12 page 2 of 2

45 1. 2(5x - 3) = 4 Algebraic Solution Multi-Step Equations Practice Problems Graphic Solution 2. 3(x 2) + 2x = 4(2x+3) Algebraic Solution Graphic Solution 2012, TESCCC 07/12/12 page 1 of 2

46 Multi-Step Equations Practice Problems Formulate an equation for the word problems. Solve the equation. Justify the solution. 3. Two times the difference of a number decreased by five is fourteen. What is the number? 4. Five times the difference of a number subtracted from ten is forty. What is the number? 5. Four times the sum of three times a number increased by two is equal to three times the difference of two times the number decreased by eight. What is the number? 6. A box contains red and blue chips. There are 2 more blue chips than red chips. If the number of red chips is tripled and the number of blue chips is doubled, there will be an equal number of red and blue chips. How many blue chips does the original box contain? A. 10 B. 6 C. 1 D , TESCCC 07/12/12 page 2 of 2

47 Which is Greater? Are You Sure? KEY Fill in the following chart. Beginning Statement Transformation Resulting Statement Is the Resulting Statement True or False? 1. 7 = 7 Add 3 to both sides 10 = 10 True 2. 7 = = = 7 Subtract 13 from both sides Multiply both sides by -8 Divide both sides by 21-6 = -6 True -56 = -56 True 1 3 = 1 3 True 5. 7 > 6 Add 3 to both sides 10 > 9 True 6. 6 < Subtract 13 from both sides Multiply both sides by -8 Divide both sides by 21-7 < -6 True If the Resulting Statement is False, rewrite the statement to make it true FALSE True > -6 Add 3 to both sides 10 > -3 True < > < Subtract 13 from both sides Multiply both sides by -8 Divide both sides by 21 Add -5 to both sides Subtract -11 from both sides Multiply both sides by 9 Divide both sides by < -6 True FALSE True > 1 True -5 < -4 True True 3 1 FALSE During which operations did the resulting statement become false? What can you do to make the resulting statement true? Answers will vary. Sample: When multiplying or dividing by a negative number, the statement became false. You would have to turn the inequality sign around. 2012, TESCCC 07/12/12 page 1 of 1

48 Which is Greater? Are You Sure? Fill in the following chart. Beginning Statement 1. 7 = = = = > < > < > < Transformation Add 3 to both sides Subtract 13 from both sides Multiply both sides by -8 Divide both sides by 21 Add 3 to both sides Subtract 13 from both sides Multiply both sides by -8 Divide both sides by 21 Add 3 to both sides Subtract 13 from both sides Multiply both sides by -8 Divide both sides by 21 Add -5 to both sides Subtract -11 from both sides Multiply both sides by 9 Divide both sides by -14 Resulting Statement Is the Resulting Statement True or False? If the Resulting Statement is False, rewrite the statement to make it true. During which operations did the resulting statement become false? What can you do to make the resulting statement true? 2012, TESCCC 07/12/12 page 1 of 1

49 Solving Inequalities KEY Inequalities are solved for x using the same steps as equations. Clear parenthesis using distribution. Combine variable terms. To keep from having to multiply or divide by a negative number, make sure the final variable term has a positive coefficient. Add/subtract to get constant terms isolated. Multiply/divide to isolate the variable. If you multiply or divide by a negative number you must flip the inequality sign. For final numeric solutions to match the direction of the graphic solution, the variable should be on the left side of the inequality. Flip the inequality if needed. Make sure to keep the correct order of the inequality. If 3 x, then x 3. 3 Solutions can be written verbally, symbolically, and graphically. Justify solutions algebraically by testing a value in the selected interval solution or by using the graphing calculator to analyze tables and graphs. Sample Solutions Verbal Symbolic Graphic x is less than two x 2 2 x is less than or equal to two x 2 2 x is greater than two x is greater than or equal to two x 2 x Symbols used in Special Case examples means all real numbers work, shown graphically as means no solution, no graph since no numbers work 2012, TESCCC 07/12/12 page 1 of 6

50 Solving Inequalities KEY Sample Problems Solve the following inequalities, giving both a symbolic and number line solution. Use algebraic methods. Check solutions for reasonableness 1. 2(2 x) < x 2 Algebraic Method 2(2 x) < x 2 4 2x < x 2 4 2x + 2x < x 2 + 2x 4 < 3x < 3x < 3x < x x > 2 Justification by Graphing and Table Equation(s): y 1 = 2(2 x) y 2 = x 2 2 Check: Choosing a value in the solution interval of 3 2(2 (3)) < (3) 2 2(-1) < 1-2 < 1 True statement, therefore the solution is justified. The inequality requires that 2(2 x) or the red line to be less than or below x 2 or the blue line. Notice in the graph the red line is below the blue line only when x > 2. x y 1 y y 1 is greater than y Point of Intersection y 1 is less than y y 1 < y 2 when x > , TESCCC 07/12/12 page 2 of 6

51 2. 5x 4 6 Algebraic Method 5x + 4 < -6 5x < x < -10 x < -2 Solving Inequalities KEY Justification by Graph and Table Equation(s): y 1 = 5x+4 y 2 = -6-2 Check: Choosing a value in the solution interval of -3 5(-3) + 4 < < 1-11 < 1 True statement, therefore the solution is justified. x y 1 y y 1 < y 2 when x < (x + 2) 4x 8 Algebraic Method 6(x + 2) 4x 8 6x x 8 2x x x x -2-2 Justification by Graph and Table Equation(s): y 1 = 6(x+2)- 4x y 2 = 8 x y 1 y y 1 y 2 when x (9 x) 4( x + 18) Window: {x: -10, 10, 1} {y: 0, 80, 10} Algebraic Method 5(9 - x) 4(x + 18) 45 5x 4x x + 5x 4x x 45 9x x x x x -3-3 Justification by Graph and Table Equation(s): y 1 = 5(9-x) y 2 = 4(x+18) x y 1 y y 1 y 2 when x , TESCCC 07/12/12 page 3 of 6

52 Solving Inequalities KEY 5. 4x 7 4x Special Case Algebraic Method 4x x 4x 4x x 4x 0-7 Zero is never less than -7, therefore this is never true, and the answer is empty set,. Justification by Graph and Table Equation(s): y 1 = 4x y 2 = x Red line is never less than blue line. x y 1 y y 1 is never less than y x 7 5x 8 Special Case Algebraic Method 5x + 7 > 5x - 8 5x + 7 5x > 5x - 8 5x 7 > -8 Seven is always greater than negative eight; therefore this is true for all real numbers,. Justification by Graph and Table Equation(s): y 1 = 5x + 7 y 2 = 5x 8 Red line is always greater than blue line. x y 1 y y 1 is always greater than y 2. What do both special cases have in common? The x terms cancel out. How do you know if the answer is empty set or all real numbers? If the remaining inequality after canceling the x terms is true, the answer is all real numbers. If it is false the answer is empty set. 2012, TESCCC 07/12/12 page 4 of 6

53 Solving Inequalities KEY Guided Practice Solve the following inequalities, giving both a symbolic and number line solution. Solve using algebraic methods. Use the graphing calculator to justify the solution by graphs and tables x Algebraic Method x > -4 6 > 5 2 x 4 10 > 5 2 x Justification by Graph and Table Equation(s): y 1 = x y 2 = - 4 (10) 2 5 > 5 2 x > x Flip to put x on the left. x < 4 4 x y 1 y y 1 > y 2 when x < x 3 5x 9 Algebraic Method Justification by Graph and Table 7x + 3 5x + 9 7x + 3 5x 5x + 9 5x 2x x x x 3 3 Equation(s): y 1 = 7x + 3 y 2 = 5x + 9 x y 1 y y 1 y 2 when x , TESCCC 07/12/12 page 5 of 6

54 Solving Inequalities KEY Practice Problems Directions: Solve each inequality as indicated, giving the solution in both symbolic and number line form. Show all work and solutions on paper. Use a graph and table to solve. 1. 3( x 4) 5( x 1) 5 x 6 Graph and Table 6 Equation(s): y 1 = 3(x + 4) 5(x 1) y 2 = 5 Red line is below blue line when x 6. x y 1 y y 1 y 2 when x 6. Use algebraic properties to solve. (Justify solutions by graphs and tables.) x x x x x 5x 11 empty set x x < ( x 3) 2x 21 x (3x 1) ( x 1) 6( x 10) x (2x 1) 2( x 1) 6( x 2) : all real numbers 2012, TESCCC 07/12/12 page 6 of 6

55 Solving Inequalities Inequalities are solved for x using the same steps as equations. Clear parenthesis using distribution. Combine variable terms. To keep from having to multiply or divide by a negative number, make sure the final variable term has a positive coefficient. Add/subtract to get constant terms isolated. Multiply/divide to isolate the variable. If you multiply or divide by a negative number you must flip the inequality sign. For final numeric solutions to match the direction of the graphic solution, the variable should be on the left side of the inequality. Flip the inequality if needed. Make sure to keep the correct order of the inequality. If 3 x, then x 3. Solutions can be written verbally, symbolically, and graphically. Justify solutions algebraically by testing a value in the selected interval solution or by using the graphing calculator to analyze tables and graphs. Sample Solutions Verbal Symbolic Graphic x is less than two 3 x 2 2 x is less than or equal to two x 2 2 x is greater than two x is greater than or equal to two x 2 x Symbols used in Special Case examples means all real numbers work, shown graphically as means no solution, no graph since no numbers work 2012, TESCCC 07/12/12 page 1 of 6

56 Solving Inequalities Sample Problems Solve the following inequalities, giving both a symbolic and number line solution. Use algebraic methods. Check solutions for reasonableness 1. 2(2 x) < x 2 Algebraic Method Justification by Graphing and Table 2012, TESCCC 07/12/12 page 2 of 6

57 Solving Inequalities 2. 5x 4 6 Algebraic Method Justification by Graph and Table 3. 6(x + 2) 4x 8 Algebraic Method Justification by Graph and Table x 4 x 18 Window: {x: -10, 10, 1} {y: 0, 80, 10} Algebraic Method Justification by Graph and Table 2012, TESCCC 07/12/12 page 3 of 6

58 Solving Inequalities 5. 4x 7 4x Special Case Algebraic Method Justification by Graph and Table 6. 5x 7 5x 8 Special Case Algebraic Method Justification by Graph and Table What do both special cases have in common? How do you know if the answer is empty set or all real numbers? 2012, TESCCC 07/12/12 page 4 of 6

59 Solving Inequalities Guided Practice Solve the following inequalities, giving both a symbolic and number line solution. Solve using algebraic methods. Use the graphing calculator to justify the solution by graphs and tables x Algebraic Method Justification by Graph and Table 2. 7x 3 5x 9 Algebraic Method Justification by Graph and Table 2012, TESCCC 07/12/12 page 5 of 6

60 Solving Inequalities Practice Problems Directions: Solve each inequality as indicated, giving the solution in both symbolic and number line form. Show all work and solutions on paper. Use a graph and table to solve. 1. 3(x + 4) 5(x 1) 5 Use algebraic properties to solve. (Justify solutions by graphs and tables.) x x 4. 5x 5x x 6. 5(x + 3) 2x (3x + 1) (x 1) 6(x + 10) 8. 4(2x + 1) < 2( x 1) + 6(x + 2) 2012, TESCCC 07/12/12 page 6 of 6

61 Unbalanced Inequalities Recording Sheet KEY Solve the inequality by any method. Show all work. Match cards to check. Fill in the correct numeric answer and graph for each inequality. Numeric Number Line Problem Solution Graphic Solution x 54 x > x 4 30 x x 4 32 x > x x< x 7 x< x 11 x < x 3 75 x x x < x 22 x < x 4 2x 6 x x 7 2x 25 x x 2x 46 x > x 10 x < x 4x 136 x x 5x 1 x < x 2x 88 x x x 40 x > x 2x 11 x < x 9 4x 9x 39 x < (4x 8) < 1 (3x + 9) x < , TESCCC 07/12/12 page 1 of 1

62 Unbalanced Inequalities Recording Sheet Solve the inequality by any method. Show all work. Match cards to check. Fill in the correct numeric answer and graph for each inequality x x 4 30 Problem 3. 12x x x x x x x x 4 2x x 7 2x x 2x x x 4x x 5x x 2x x x x 2x x 9 4x 9x (4x 8) < 1 (3x + 9) 3 Numeric Solution Number Line Graphic Solution 2012, TESCCC 07/12/12 page 1 of 1

63 Cards: Unbalanced Inequalities x > 9 x 17 x > -3 x < 20 x < 5 x < -2 x 9 x < 11 x < 6 x -10 x 4 x > 8 x < 0 x 25 x < -38 x -33 x > -29 x < 33 x < -16 x < , TESCCC 07/12/12 page 1 of 2

64 Cards: Unbalanced Inequalities , TESCCC 07/12/12 page 2 of 2

65 Inequality Connection KEY Sue Ann had three boxes of coins labeled A, B, and C. For each verbal statement below indicate which box or boxes fit the statement and what symbolic notation can be used to indicate the inequality. A 1 10 B 10 C Verbal Statement Boxes that can be defined by the statement Fill in the Inequality Sign for each statement The box contains at least 16. A, C t 16 The box contains less than 16. B t < 16 The box contains at most 16. A, B t 16 The box contains greater than 16. C t > 16 The box contains no more than 16. A, B t 16 The box contains no less than 16. A, C t , TESCCC 07/12/12 page 1 of 1

66 Inequality Connection Sue Ann had three boxes of coins labeled A, B, and C. For each verbal statement below indicate which box or boxes fit the statement and what symbolic notation can be used to indicate the inequality. A 1 10 B 10 C Verbal Statement Boxes that can be defined by the statement Fill in the Inequality Sign for each statement The box contains at least 16. t 16 The box contains less than 16. t 16 The box contains at most 16. t 16 The box contains greater than 16. t 16 The box contains no more than 16. t 16 The box contains no less than 16. t , TESCCC 07/12/12 page 1 of 1

67 Seeing Inequalities KEY Compare the three statements below. What is different about each? Answers will vary. What is the same? Answers will vary. Write an algebraic expression in terms of x to represent each verbal expression in the box below the statement. Seven less than five times a number Seven is five times a number Seven is less than five times a number 5x 7 7 = 5x 7 < 5x Using the algebraic expression for seven less than five times a number : A. Evaluate the expression for x = B. Evaluate the expression for x = C. Solve for x if the expression is equal to 7. 5x 7 = 7 5x = x = x = 2.8 D. Solve for x if the expression is more than 8. 5x 7 > 8 5x > x > x > 3 E. Solve for x if the expression is at most 17. 5x x x x 4.8 F. Solve for x if the expression is at least 18. 5x x x x 5 G. Solve for x if the expression is less than 22. 5x 7 < -22 5x < x < x < , TESCCC 07/12/12 page 1 of 3

68 Seeing Inequalities KEY Guided Practice Write and solve inequalities to represent the following situations. 1. The tallest tree in Carin s yard is an elm, which is 16 feet tall. There is also a cherry tree in the yard. C = height of cherry tree in feet. C < In football, the place kicker can score points two different ways. Coach Hill expects his kicker to score at least 50 points during the season. C = coach s expectation C Cameron s parents have limited Back to School shopping to no more than $150. C = Cameron s spending limit. C Chico is five feet tall. When he stands on a chair, he can reach at most 16 feet. C = height of the chair in feet. C The price of a pair of sandals is reduced by $5. Cora has $16, but that is not enough to buy the sandals. C = price of sandals before the discount. C > Canae is making candy. Her brother ate 5 ounces of the candy, and now there is less than a pound. C = Original weight of the candy in ounces. C < The driving age in the state of Texas is at least 15 with a driving permit. If Cass were five times as old as he is now, he would be old enough to drive. C = Cass s age now. C 3 8. Research indicates that 95% of people have a palm width that is at least 3.12 inches long but no more than This information is used when companies design control panels, keyboards, and gloves. P 3.12 and P 3.86 Combined into one 3.12 P For what values of x will the angle be obtuse? x > 90 and x < 180 Combined into one 90 < x < 180 (x) o 2012, TESCCC 07/12/12 page 2 of 3

69 Seeing Inequalities KEY Practice Problems Directions: Write an inequality for each of the problem situations below. Solve each inequality, giving the solution in both symbolic and number line form. 1. A number decreased by 4 is at least 9. x (-4) 9 x The sum of twice a number and 5 is less than 17. 2x + 5 < 17 x < Five times a number decreased by 10 is greater than 5. 5x 10 > -5 x > Four times a number is at least x -48 x A number divided by negative six is no less than 5. x/-6 5 x Charlotte must have at least 320 points in her science class to get a B or better. She needs at least a B so she can play on the softball team. Charlotte currently has 168 points from four tests and 79 points from her quizzes. Charlotte will be taking her final exam worth 100 points. What must Charlotte score on the final exam to reach her goal? x 320 x What is the width of a rectangle whose length is 4 feet and perimeter is at most 20 feet? 2(4) + 2w 20 w Triangle FGH is an obtuse triangle. The greatest angle in the triangle has a measure of (6d). What are the possible values of d? 6d > 90 and 6d < < d < Susie has a budget of $92 to spend on clothes. The shorts she wants to buy are on sale for $14 each. The shirts are on sale for $12 each. If Susie purchases four shorts, what is the maximum number of shirts she can buy to go with the shorts? 14(4) + 12s 92 s 3 Three shirts is the maximum number she can purchase. 10. For what values of x will the angle be acute? 2x 6 > 0 and 2x 6 < 90 x > 3 and x < 48 Combined: 3 < x < (2x 6) o 2012, TESCCC 07/12/12 page 3 of 3

70 Seeing Inequalities Compare the three statements below. What is different about each? What is the same? Write an algebraic expression in terms of x to represent each verbal expression in the box below the statement. Seven less than five times a number Seven is five times a number Seven is less than five times a number Using the algebraic expression for seven less than five times a number : A. Evaluate the expression for x = 6. B. Evaluate the expression for x = 2. C. Solve for x if the expression is equal to 7. D. Solve for x if the expression is more than 8. E. Solve for x if the expression is at most 17. F. Solve for x if the expression is at least 18. G. Solve for x if the expression is less than , TESCCC 07/12/12 page 1 of 3

71 Seeing Inequalities Guided Practice Write and solve inequalities to represent the following situations. 1. The tallest tree in Carin s yard is an elm, which is 16 feet tall. There is also a cherry tree in the yard. C = height of cherry tree in feet. 2. In football, the place kicker can score points two different ways. Coach Hill expects his kicker to score at least 50 points during the season. C = coach s expectation. 3. Cameron s parents have limited Back to School shopping to no more than $150. C = Cameron s spending limit. 4. Chico is five feet tall. When he stands on a chair, he can reach at most 16 feet. C = height of the chair in feet. 5. The price of a pair of sandals is reduced by $5. Cora has $16, but that is not enough to buy the sandals. C = price of sandals before the discount. 6. Canae is making candy. Her brother ate 5 ounces of the candy, and now there is less than a pound. C = Original weight of the candy in ounces. 7. The driving age in the state of Texas is at least 15 with a driving permit. If Cass were five times as old as he is now, he would be old enough to drive. C = Cass s age now. 8. Research indicates that 95% of people have a palm width that is at least 3.12 inches long but no more than This information is used when companies design control panels, keyboards, and gloves. 9. For what values of x will the angle be obtuse? (x) o 2012, TESCCC 07/12/12 page 2 of 3

72 Seeing Inequalities Practice Problems Directions: Write an inequality for each of the problem situations below. Solve each inequality, giving the solution in both symbolic and number line form. 1. A number decreased by 4 is at least The sum of twice a number and 5 is less than Five times a number decreased by 10 is greater than Four times a number is at least A number divided by negative six is no less than Charlotte must have at least 320 points in her science class to get a B or better. She needs at least a B so she can play on the softball team. Charlotte currently has 168 points from four tests and 79 points from her quizzes. Charlotte will be taking her final exam worth 100 points. What must Charlotte score on the final exam to reach her goal? 7. What is the width of a rectangle whose length is 4 feet and perimeter is at most 20 feet? 8. Triangle FGH is an obtuse triangle. The greatest angle in the triangle has a measure of (6d). What are the possible values of d? 9. Susie has a budget of $92 to spend on clothes. The shorts she wants to buy are on sale for $14 each. The shirts are on sale for $12 each. If Susie purchases four shorts, what is the maximum number of shirts she can buy to go with the shorts? 10. For what values of x will the angle be acute? (2x 6) o 2012, TESCCC 07/12/12 page 3 of 3

73 Equations and Inequalities with Consecutive Numbers KEY Within a problem situation, verbal phrases identify if the problem is an equation or inequality. Fill in the symbol used to represent the verbal phrases below. Verbal Phrase Symbol is = is equal to = is less than < is greater than > is at most is at least is no more than is no less than Because consecutive integers follow specific patterns, they can be used to solve problems. Type Numeric Representation Algebraic Generalization Consecutive integers {5, 6, 7, 8} {x, x+1, x+2, x+3} Consecutive even integers {2, 4, 6, 8} {x, x+2, x+4, x+6} Consecutive odd integers {3, 5, 7, 9} {x, x+2, x+4, x+6} What do you notice about consecutive even or odd integers? They are set up the same way by skipping one number in-between. Write an equation or inequality for each of the problem situations below. Solve each equation or inequality and give the final solution in set form. 1. The sum of three consecutive integers is 24. What are the integers? x + (x + 1) + (x + 2) = 24 3x + 3 = 24 3x = 21 x = 7 {7, 8, 9} 2. The sum of four consecutive integers is at least 126. What is the set of the smallest four consecutive integers that fits this situation? x + (x + 1) + (x + 2) + (x + 3) 126 x 30 {30, 31, 32, 33} 2012, TESCCC 04/07/13 page 1 of 3

74 Equations and Inequalities with Consecutive Numbers KEY 3. The sum of three consecutive odd integers is less than 54. What is the set of the largest three consecutive odd integers that fits this situation? x + (x + 2) + (x + 4) < 54 x < 16 {15, 17, 19} 4. Four more than five times a number is less than 54. Evaluate this problem situation for values of the number. 5x + 4 < 54 x < Two times the sum of a number and four increased by three is at most six less than the number. Evaluate this problem situation for values of the number. 2(x + 4) + 3 x 6 x Three times the sum of a number and two is greater than or equal to 33 but less than or equal to 66. Which interval represents the values of the number in this situation? 3(x + 2) 33 and 3(x + 2) 66 x 9 and x 20 9 x Change the following mathematical sentences into verbal sentences using consecutive integer terminology = Three consecutive integers whose sum is equal to seventy-eight. Three consecutive even integers whose sum is at most seventy-nine. Three consecutive even integers whose sum is not more than seventynine. 2012, TESCCC 04/07/13 page 2 of 3

75 Equations and Inequalities with Consecutive Numbers KEY Practice Problems Directions: Write an equation or inequality for each of the problem situations below. Use the equation or inequality to solve the problem situation. 1. The sum of three consecutive integers is 48. What is the set of the largest three consecutive integers that fits this situation? x = 15 x + x x + 2 = 48 {15, 16, 17} 2. The sum of three consecutive even integers is greater than 54. What is the set of the smallest three consecutive even integers that fits this situation? x > 16 x + x x + 4 > 54 {18, 20, 22} 3. Five decreased by eleven times a number is greater than 71. Evaluate this problem situation for values of the number. 5 11x > 71 x < Negative four times the difference of six and a number is at least seven times the number. Evaluate this problem situation for values of the number. -4(6 x) 7x x Four times a number increased by seven must be at least 31 but at most 55. Which interval represents the values of the number in this situation? 31 4x x 12 x 6 and x Change the math sentence, , into a verbal phrase using the terminology of consecutive numbers. Is there more than one way to write the phrase? If so, give another example. Three consecutive odd integers whose sum is at least eighty-one. Three consecutive odd integers whose sum is no less than eighty-one. 2012, TESCCC 04/07/13 page 3 of 3

76 Equations and Inequalities with Consecutive Numbers Within a problem situation, verbal phrases identify if the problem is an equation or inequality. Fill in the symbol used to represent the verbal phrases below. Verbal Phrase is is equal to is less than is greater than is at most is at least is no more than is no less than Symbol Because consecutive integers follow specific patterns, they can be used to solve problems. Type Numeric Representation Consecutive integers {5, 6, 7, 8} Consecutive even integers {2, 4, 6, 8} Consecutive odd integers {3, 5, 7, 9} Algebraic Generalization What do you notice about consecutive even or odd integers? Write an equation or inequality for each of the problem situations below. Solve each equation or inequality and give the final solution in set form. 1. The sum of three consecutive integers is 24. What are the integers? 2. The sum of four consecutive integers is at least 126. What is the set of the smallest four consecutive integers that fits this situation? 2012, TESCCC 04/07/13 page 1 of 3

77 Equations and Inequalities with Consecutive Numbers 3. The sum of three consecutive odd integers is less than 54. What is the set of the largest three consecutive odd integers that fits this situation? 4. Four more than five times a number is less than 54. Evaluate this problem situation for values of the number. 5. Two times the sum of a number and four increased by three is at most six less than the number. Evaluate this problem situation for values of the number. 6. Three times the sum of a number and two is greater than or equal to 33 but less than or equal to 66. Which interval represents the values of the number in this situation? 7. Change the following mathematical sentences into verbal sentences using consecutive integer terminology = , TESCCC 04/07/13 page 2 of 3

78 Practice Problems Equations and Inequalities with Consecutive Numbers Directions: Write an equation or inequality for each of the problem situations below. Use the equation or inequality to solve the problem situation. 1. The sum of three consecutive integers is 48. What is the set of the three consecutive integers that fits this situation? 2. The sum of three consecutive even integers is greater than 54. What is the set of the smallest three consecutive even integers that fits this situation? 3. Five decreased by eleven times a number is greater than 71. Evaluate this problem situation for values of the number. 4. Negative four times the difference of six and a number is at least seven times the number. Evaluate this problem situation for values of the number. 5. Four times a number increased by seven must be at least 31 but at most 55. Which interval represents the values of the number in this situation? 6. Change the math sentence, , into a verbal phrase using the terminology of consecutive integers. Is there more than one way to write the phrase? If so, give another example. 2012, TESCCC 04/07/13 page 3 of 3

79 Problem Solving with Ratios KEY Sample Problem Excellent Environmental estimates that a pond contains 3000 fish. Some are catfish, some are perch, and the rest are bass. After testing samples by dragging a net through the pond, they determine that the ratio of catfish to perch to bass is 2:5:3. Scale Factor x Catfish x Perch x Bass x Total Fish x + 5x + 3x a. How can the catfish be represented using x? 2x b. How can the perch be represented using x? 5x c. How can the bass be represented using x? 3x d. Write an equation showing the total number of fish in the pond. 2x + 5x + 3x = 3000 e. Find the number of each type of fish in the pond. Catfish 600, Perch 1500, Bass , TESCCC 03/27/13 page 1 of 4

80 Problem Solving with Ratios KEY Guided Practice: 1. The ratio of two integers is 17:13. Their sum is 390. Scale Factor x Large number x Small number x Total x + 13x a. How can the smallest integer be represented by using x? 13x b. How can the largest integer be represented by using x? 17x c. Write an equation showing how to find the two integers whose sum is x + 13x = 390 d. Find the two integers. Small integer 169, Large integer Aunt Aggie never had children so she has decided to divide her upper ranch of 7,200 acres between her sister s 18 children and her brother s 12 children. If Aunt Aggie uses the ratio of her sister s children and brother s children to divide up the 7200 acres of land, how much land will her sister s children receive, and how much land will her brother s children receive? Sister s children 4320 acres, Brother s children 2880 acres 3. The sides of a triangle have lengths in the ratio 3:5:7. The perimeter of the triangle is ninety centimeters. Find the lengths of the sides. 3x + 5x + 7x = 90 x = 6 sides are 18 cm, 30 cm, 42 cm 2012, TESCCC 03/27/13 page 2 of 4

81 Problem Solving with Ratios KEY Practice Problems 1. Jason is in charge of the juice department at Bob s Grocery. He is preparing to stock the juice display. The orange juice display holds a total of 160 half-gallon containers. Bob s Grocery carries a national brand of orange juice and a local brand. The local brand outsells the national brand by 7:3. a. How can the local brand be represented using x? 7x b. How can the national brand be represented using x? 3x c. Write an equation for the total number of half-gallon juice containers in the display. 7x + 3x = 160 d. How many containers of each kind should he put in the case in order to have the correct ratio? 112 gal of local brand orange juice, 48 gal of national brand orange juice 2. Bob s Grocery also carries three types of cereal on the cereal shelves. The shelves can hold 510 boxes of cereal. An analysis of sales shows that sugared cereals, granola cereals, and other cereals sell in the ratio of 5:3:9. a. How can the sugared cereals be represented using x? 5x b. How can the granola cereal be represented using x? 3x c. How all other cereals be represented using x? 9x d. Write an equation for the total number of cereal boxes on the shelves. 5x + 3x + 9x = 510 e. How many boxes of each kind of cereal should be put on the shelves so that when full, they will have cereal boxes in this ratio? 150 sugared cereals, 90 granola cereals, 270 other cereals 3. Ruth Ann is creating a scale model of a sculpture to be placed in front of Redding High School. The scale ratio comparing the model to the actual object is 1 inch: 8 inches. a. What generalized formula could be used to find the scale model compared to the actual? scale = 1 8 actual b. What generalized formula could be used to find the actual compared to the scale model? actual = 8 1 scale c. If the actual sculpture is to be 14 feet high, how tall will the scale model be in inches? 21 in 4. Merry is making an abstract triangular collage for the wall. She has 14 feet of edging to go around the collage. If the longest side is twice the smallest and the other side is four less than three times the smallest, find the length of each side of the triangular collage. 3 ft, 6 ft, 5 ft 2012, TESCCC 03/27/13 page 3 of 4

82 Problem Solving with Ratios KEY 5. Tomas has decided to build a triangular garden and fence it in using all of a 52-foot board. If the largest side must be 4 more than four times the smallest and the other side three times the smallest, how long will each side be? 6 ft, 28 ft, 18 ft 6. Jackson is replacing the wood frame around the rectangular window in her kitchen. The width of the window is eight less than three times the height. The total amount of wood needed to go around the window frame is 272 inches. What is the width and height of the window? 36 in, 100 in 7. Alex was hired by Nathan s Nursery to shovel 100 tons of fertilizer. He has been shoveling fertilizer at a rate of 16 tons per day. After 3 days, Nathan s Nursery hires Manny to help Alex, so they can finish the job faster. Manny can shovel 10 tons per day. a. Let x be the number of days Alex has shoveled fertilizer. How would the number of days Manny has been shoveling fertilizer be represented? x 3 b. Write an expression for the number of tons of fertilizer Alex has shoveled. 16x c. Write an expression for the number of tons of fertilizer Manny has shoveled. 10(x 3) d. Write an equation for the total number of tons Alex and Manny must shovel. 16x + 10(x 3) = 100 e. Find the amount of the 100 tons shoveled by Alex and Manny and the amount of time each works shoveling fertilizer. Alex 5 days and 80 tons, Manny 2 days and 20 tons 8. Park Place s stock is valued at $18.20 per share. The earnings of Park Place for one year amounts to $2.80 per share. a. Find the value to earnings ratio in reduced form. b. What generalized formula could be used to find earnings given value? x c. What generalized formula could be used to find value given earnings? 13 2 x d. If Park Place earned $3,360,000 in that year, what was the total value of its stock? $21, 840, , TESCCC 03/27/13 page 4 of 4

83 Problem Solving with Ratios Sample Problem Excellent Environmental estimates that a pond contains 3000 fish. Some are catfish, some are perch, and the rest are bass. After testing samples by dragging a net through the pond, they determine that the ratio of catfish to perch to bass is 2:5:3. Scale Factor x Catfish Perch Bass Total Fish a. How can the catfish be represented using x? b. How can the perch be represented using x? c. How can the bass be represented using x? d. Write an equation showing the total number of fish in the pond. e. Find the number of each type of fish in the pond. 2012, TESCCC 03/27/13 page 1 of 4

84 Problem Solving with Ratios Guided Practice 1. The ratio of two integers is 17:13. Their sum is 390. Scale Factor x Large number Small number Total a. How can the smallest integer be represented by using x? b. How can the largest integer be represented by using x? c. Write an equation showing how to find the two integers whose sum is 390. d. Find the two integers. 2. Aunt Aggie never had children so she has decided to divide her upper ranch of 7,200 acres between her sister s 18 children and her brother s 12 children. If Aunt Aggie uses the ratio of her sister s children and brother s children to divide up the 7200 acres of land, how much land will her sister s children receive, and how much land will her brother s children receive? 3. The sides of a triangle have lengths in the ratio 3:5:7. The perimeter of the triangle is ninety centimeters. Find the lengths of the sides. 2012, TESCCC 03/27/13 page 2 of 4

85 Problem Solving with Ratios Practice Problems 1. Jason is in charge of the juice department at Bob s Grocery. He is preparing to stock the juice display. The orange juice display holds a total of 160 half-gallon containers. Bob s Grocery carries a national brand of orange juice and a local brand. The local brand outsells the national brand by 7:3. a. How can the local brand be represented using x? b. How can the national brand be represented using x? c. Write an equation for the total number of half-gallon juice containers in the display. d. How many containers of each kind should he put in the case in order to have the correct ratio? 2. Bob s Grocery also carries three types of cereal on the cereal shelves. The shelves can hold 510 boxes of cereal. An analysis of sales shows that sugared cereals, granola cereals, and other cereals sell in the ratio of 5:3:9. a. How can the sugared cereals be represented using x? b. How can the granola cereal be represented using x? c. How all other cereals be represented using x? d. Write an equation for the total number of cereal boxes on the shelves. e. How many boxes of each kind of cereal should be put on the shelves so that when full, they will have cereal boxes in this ratio? 3. Ruth Ann is creating a scale model of a sculpture to be placed in front of Redding High School. The scale ratio comparing the model to the actual object is 1 inch: 8 inches. a. What generalized formula could be used to find the scale model compared to the actual? b. What generalized formula could be used to find the actual compared to the scale model? c. If the actual sculpture is to be 14 feet high, how tall will the scale model be in inches? 4. Merry is making an abstract triangular collage for the wall. She has 14 feet of edging to go around the collage. If the longest side is twice the smallest and the other side is four less than three times the smallest, find the length of each side of the triangular collage. 2012, TESCCC 03/27/13 page 3 of 4

86 Problem Solving with Ratios 5. Tomas has decided to build a triangular garden and fence it in using all of a 52-foot board. If the largest side must be 4 more than four times the smallest and the other side three times the smallest, how long will each side be? 6. Jackson is replacing the wood frame around the rectangular window in her kitchen. The width of the window is eight less than three times the height. The total amount of wood needed to go around the window frame is 272 inches. What is the width and height of the window? 7. Alex was hired by Nathan s Nursery to shovel 100 tons of fertilizer. He has been shoveling fertilizer at a rate of 16 tons per day. After 3 days, Nathan s Nursery hires Manny to help Alex, so they can finish the job faster. Manny can shovel 10 tons per day. a. Let x be the number of days Alex has shoveled fertilizer. How would the number of days Manny has been shoveling fertilizer be represented? b. Write an expression for the number of tons of fertilizer Alex has shoveled. c. Write an expression for the number of tons of fertilizer Manny has shoveled. d. Write an equation for the total number of tons Alex and Manny must shovel. e. Find the amount of the 100 tons shoveled by Alex and Manny and the amount of time each works shoveling fertilizer. 8. Park Place s stock is valued at $18.20 per share. The earnings of Park Place for one year amounts to $2.80 per share. a. Find the value to earnings ratio in reduced form. b. What generalized formula could be used to find earnings given value? c. What generalized formula could be used to find value given earnings? a. If Park Place earned $3,360,000 in that year, what was the total value of its stock? 2012, TESCCC 03/27/13 page 4 of 4

87 Problem Solving with Proportions KEY Proportion An equation that shows that two ratios are equivalent. If a problem situation is proportional, the Graph is linear Line passes through the origin y is always constant (Constant of proportionality) x Equation is in the form y = kx (k = m) Proportions can be solved by two methods, cross-multiplication or using a function rule. Cross Product If b 0, d 0 a c = ad = bc b d Function Rule y kx Where k is the constant of proportionality Sample Problem 1. Use the table below to answer the following questions. Computers Students A. What is the ratio of computers to students? 3 10 B. Use the cross product method to answer the following. If there are 1,250 students in the school, how many computers would be needed to meet the projected ratio? 3 x computers 2012, TESCCC 03/27/13 page 1 of 4

88 Problem Solving with Proportions KEY C. Use the function rule to create a table and answer the following. What generalized formula could be used to represent number of students (y) versus computers (x)? Table varies. x y y 10 3 x Hint: Number of students must be greater, so multiply by fraction greater than one. D. Use the function rule to create a table and answer the following. What generalized formula could be used to represent number of computers (y) versus students (x)? Table varies. x y y 3 10 x Hint: Number of computers must be less, so multiply by fraction less than one Notice that the ratio is always found in the generalized formula of a proportion. This value is the constant of proportionality. These generalized formulas can be used to find missing values. 2. A real estate agent sold a house for $84,000. The agent s commission was $5040. A. What was the commission to selling price ratio in lowest terms? B. What would the commission be for a house that sells for $278,000 if the commission to selling price ratio remains the same? 3 (278,000) $16, The commission would be $16, , TESCCC 03/27/13 page 2 of 4

89 Problem Solving with Proportions KEY Guided Practice 1. The freshmen at Eagle Nest High School outnumber the sophomores 7:5. a. What generalized formula could be used to find the number of sophomores given the number of freshmen? 5 7 x Fewer sophomores so fraction has to less than 1 b. What generalized formula could be used to find the number of freshmen given the number of sophomores? 7 5 x More freshmen so fraction has to be greater than 1 c. If there are 231 freshmen, how many sophomores are in the high school? 5 (231) 165 sophomores , TESCCC 03/27/13 page 3 of 4

90 Problem Solving with Proportions KEY Practice Problems Define your variables. Write an equation. Solve the equation. Answer the question or questions being asked by the problem. 1. The ratio of two integers is 9:7. Their sum is Find the two integers. 9x + 7x = 1024 Large integer is 576, Small integer is The sides of a triangle are in the ratio of 5:6:7. The perimeter of the triangle is 234 centimeters. How long is each side? 65 cm, 78 cm, 91 cm 5x + 6x + 7x = 234 x = The length and width of a rectangle are in the ratio of 5:4. The perimeter of the rectangle is 126 inches. What are the length and width of the rectangle? length is 35 in. and width is 28 in. Length = 5x, width = 4x P = 2(5x + 4x) 126 = 2(9x) x = 7 4. At Schilling s Gas and Shop unleaded gas outsells regular gas in the ratio of 9:4. The monthly quota for gas is a total of 26,000 gallons. How many gallons of each kind should be ordered so that the quota will have the same ratio? Unleaded = 18,000 gal, regular = 8,000 gal 9x = unleaded, 4x = regular 9x + 4x = x = Dottie has 32 feet of chain link fence to enclose a play area for her dog. She wants the play area to be in the shape of a rectangle with the length two feet more than the width. What should be the length and width of the play area? Width = 7 feet, length = 9 feet Width = x, length = x + 2 P = 2(x) + 2(x + 2) 32 = 2x + 2x + 4 x = 7 6. The ratio of two integers is 6:13. The smaller integer is 54. Find the larger integer. 13 (54) 117, The larger integer is The blueprint dimensions for a newly constructed house are proportional to the house s actual dimensions. On the blueprints the house s foundation measures 75 cm in length by 40 cm in width. The house s actual foundation measures 15 meters in length. What is the foundation s actual width? 15m (40 cm) 8m 75cm The actual width of the foundation is 8 meters. 8. A real-estate agent earned a commission of $3870 on the sale of an $86,000 house. What was the selling price of a different home, if the real-estate agent was paid the same rate of commission and earned $4950? (4950) $110, The selling price of the home would be $110, , TESCCC 03/27/13 page 4 of 4

91 Problem Solving with Proportions An equation that shows that two ratios are equivalent. If a problem situation is proportional, the Graph is linear Line passes through the origin y is always constant (Constant of proportionality) x Equation is in the form y = kx (k = m) Proportions can be solved by two methods, cross-multiplication or using a function rule. Cross Product If b 0, d 0 a c = ad = bc b d Function Rule y kx Where k is the constant of proportionality Sample Problem 1. Use the table below to answer the following questions. Computers Students A. What is the ratio of computers to students? B. Use the cross product method to answer the following. If there are 1,250 students in the school, how many computers would be needed to meet the projected ratio? 2012, TESCCC 03/27/13 page 1 of 4

92 Problem Solving with Proportions C. Use the function rule to create a table and answer the following. What generalized formula could be used to represent number of students (y) versus computers (x)? x y D. Use the function rule to create a table and answer the following. What generalized formula could be used to represent number of computers (y) versus students (x)? x y Notice that the ratio is always found in the generalized formula of a proportion. This value is the constant of proportionality. These generalized formulas can be used to find missing values. 2. A real estate agent sold a house for $84,000. The agent s commission was $5040. A. What was the commission to selling price ratio in lowest terms? B. What would the commission be for a house that sells for $278,000 if the commission to selling price ratio remains the same? 2012, TESCCC 03/27/13 page 2 of 4

93 Problem Solving with Proportions Guided Practice 1. The freshmen at Eagle Nest High School outnumber the sophomores 7:5. a. What generalized formula could be used to find the number of sophomores given the number of freshmen? b. What generalized formula could be used to find the number of freshmen given the number of sophomores? c. If there are 231 freshmen, how many sophomores are in the high school? 2012, TESCCC 03/27/13 page 3 of 4

94 Problem Solving with Proportions Practice Problems Define your variables. Write an equation. Solve the equation. Answer the question or questions being asked by the problem. 1. The ratio of two integers is 9:7. Their sum is Find the two integers. 2. The sides of a triangle are in the ratio of 5:6:7. The perimeter of the triangle is 234 centimeters. How long is each side? 3. The length and width of a rectangle are in the ratio of 5:4. The perimeter of the rectangle is 126 inches. What are the length and width of the rectangle? 4. At Schilling s Gas and Shop unleaded gas outsells regular gas in the ratio of 9:4. The monthly quota for gas is a total of 26,000 gallons. How many gallons of each kind should be ordered so that the quota will have the same ratio? 5. Dottie has 32 feet of chain link fence to enclose a play area for her dog. She wants the play area to be in the shape of a rectangle with the length two feet more than the width. What should be the length and width of the play area? 6. The ratio of two integers is 6:13. The smaller integer is 54. Find the larger integer. 7. The blueprint dimensions for a newly constructed house are proportional to the house s actual dimensions. On the blueprints the house s foundation measures 75 cm in length by 40 cm in width. The house s actual foundation measures 15 meters in length. What is the foundation s actual width? 8. A real-estate agent earned a commission of $3870 on the sale of an $86,000 house. What was the selling price of a different home, if the real-estate agent was paid the same rate of commission and earned $4950? 2012, TESCCC 03/27/13 page 4 of 4

95 Problem Solving with Rates KEY Sample Problem The Gooding High School French Club needs to raise money so that they could attend a French foreign film in Austin. The club has decided to sponsor a pet-dipping day. Cat, the club president, has a price quote from Barking Lot Grooming for the cost of the flea dip. Barking Lot Grooming will sell the flea dip for $5.00 a bottle plus a shipping charge per order of $ Complete the table on cost versus number of bottles for Barking Lot Grooming Bottles of Flea Dip Process Cost (1) (2) (3) (4) (5) (6) 33 x 3 + 5(x) 5x Write an expression that represents the amount of money charged by Barking Lot Grooming for an order of flea dip. y = 5x Evaluate the cost for 9 bottles, 15 bottles, and 25 bottles of shampoo (9) = (15) = (25) = , TESCCC 03/27/13 page 1 of 4

96 Problem Solving with Rates KEY 4. If the Gooding French Club paid $88.00 for flea dip, how many bottles did they buy? How did you find the answer? 3 + 5(x) = 88 x = The Gooding French Club can spend no more than y dollars on x bottles of flea dip. a. What general inequality can be used to represent the amount the Gooding French Club can spend? 3 + 5x y b. Cat has determined that they can spend no more than $50 on flea dip. Write an inequality that represents this situation x Solve the inequality algebraically. Actual answer is 9.4 bottles but store will not ship a partial bottle so answer is 9 bottles. 7. Solve the inequality using the table on the graphing calculator. 2012, TESCCC 03/27/13 page 2 of 4

97 Problem Solving with Rates KEY Practice Problems 1. Susie s Sweets charges 20 cents for each donut and 15 cents for a box to carry them. a. Use a table to determine an expression that can be used to represent the cost of the donuts in a box. 0.20x b. What would be the cost of one dozen donuts in a box? $2.55 c. What would be the cost of 100 donuts in a box? $20.15 d. Formulate and solve an equation to determine the number of donuts in a box you could buy for $ x = 3.55 You could buy 17 donuts in a box. e. The principal can spend no more than $10.00 on donuts for refreshments at the teacher s meeting. How many donuts in a box can he buy? 0.20x x He can buy at most 49 donuts in a box. 2. In Silver Gate, Montana, when the temperature is 0 o Celsius, people drink 3600 cups of hot chocolate per day. Consumption decreases by 30 cups per day for each rise of 1 o C. a. Use a table to determine an expression that can be used to represent the number of cups of hot chocolate drunk per day as a function of degrees Celsius x b. How many cups would they drink at 5 o C? 3450 c. How many cups would they drink at 25 o C? 2850 d. Formulate and solve an equation to determine the temperature if 3000 cups of hot chocolate were consumed in a day. Is this answer reasonable? Explain x = 3000 The temperature would be 20 o C. Sample: It is not reasonable that they would drink that much hot chocolate when the temperature is 20 o C. 3. Speedy is at a point three miles from his home. He rides his bike at a quarter of a mile per minute away from home. a. Make a table to show Speedy s distance from home after each elapsed minute. b. Use the table to write an expression for the number of miles traveled versus time. y = x c. How far is Speedy from home after 20 minutes? 8 miles d. How long will it take Speedy to be 10 miles from home? 28 min 2012, TESCCC 03/27/13 page 3 of 4

98 Problem Solving with Rates KEY 4. Jackie is three miles from her home. She starts riding her bike at a quarter of a mile per minute toward home. a. Make a table to show Jackie s distance from home after each elapsed minute. b. Use the table to write an expression for the number of miles traveled versus the time. y = x c. How far is she from home after 8 minutes? 1 mi d. How long will it take her to be half a mile from home? 10 min e. How long will it take her to get home? 12 min 5. James Millhouse is renting a car at a local rental agency. The cost of renting the car is $38 per day, plus $0.35 for each mile driven over 150 miles. James rented this car for one day and drove it over 150 miles. a. What equation can be used to determine y, the total cost of the rental, if it is driven y miles? y = 0.35(x 150) + 38 b. James drove from Las Vegas to the south rim of the Grand Canyon, a distance of 280 miles. He then drove back to Las Vegas the same day along the same route to turn in his rental car. How much was the bill for his rental car? y = 0.35( ) + 38; The total cost on the rental car bill is $ c. If James only had $150 to spend on the rental car, how many miles could he drive? 0.35(x 150) x 470. He could drive 470 or less miles in order to not be charged more than his $ The owners of a new sporting goods store want to spend no more than $1000 for at least ten minutes of advertising. The cost to advertise on a local television station is $200 per minute. The cost to advertise on a local radio station is $50 per minute. If the owners want the maximum exposure time for their $1000 using both television and radio, which of the following is a reasonable solution? Justify your reasoning. a. Advertise 12 minutes on television and 2 minutes on radio No. Although is greater than ten minutes, the cost would be 12(200) + 2(50) = $2500. This is over the $1000 advertising budget. b. Advertise 3 minutes on television and 8 minutes on radio Yes is greater than ten minutes. The cost would be 3(200) + 8(50) = $1000. This is exactly the $1000 advertising budget. c. Advertise 1 minute on television and 18 minutes on radio No. Although is greater than ten minutes, the cost would be 1 (200) + 18(50) = $1100. This is over the $1000 advertising budget. d. Advertise 4 minutes on television and 4 minutes on radio No. Although the cost is exactly $1000, the total time of advertising is only 8 minutes which is under the target of 10 minutes. 2012, TESCCC 03/27/13 page 4 of 4

99 Problem Solving with Rates Sample Problem The Gooding High School French Club needs to raise money so that they could attend a French foreign film in Austin. The club has decided to sponsor a pet-dipping day. Cat, the club president, has a price quote from Barking Lot Grooming for the cost of the flea dip. Barking Lot Grooming will sell the flea dip for $5.00 a bottle plus a shipping charge per order of $ Complete the table on cost versus number of bottles for Barking Lot Grooming. Bottles of Flea Dip Process Cost x 2. Write an expression that represents the amount of money charged by Barking Lot Grooming for an order of flea dip. 3. Evaluate the cost for 9 bottles, 15 bottles, and 25 bottles of shampoo. 2012, TESCCC 03/27/13 page 1 of 4

100 Problem Solving with Rates 4. If the Gooding French Club paid $88.00 for flea dip, how many bottles did they buy? How did you find the answer? 5. The Gooding French Club can spend no more than y dollars on x bottles of flea dip. a. What general inequality can be used to represent the amount the Gooding French Club can spend? b. Cat has determined that they can spend no more than $50 on flea dip. Write an inequality that represents this situation. 6. Solve the inequality algebraically. 7. Solve the inequality using the table on the graphing calculator. 2012, TESCCC 03/27/13 page 2 of 4

101 Problem Solving with Rates Practice Problems 1. Susie s Sweets charges 20 cents for each donut and 15 cents for a box to carry them. a. Use a table to determine an expression that can be used to represent the cost of the donuts in a box. b. What would be the cost of one dozen donuts in a box? c. What would be the cost of 100 donuts in a box? d. Formulate and solve an equation to determine the number of donuts in a box you could buy for $3.55. e. The principal can spend no more than $10.00 on donuts for refreshments at the teacher s meeting. How many donuts in a box can he buy? 2. In Silver Gate, Montana, when the temperature is 0 o Celsius, people drink 3600 cups of hot chocolate per day. Consumption decreases by 30 cups per day for each rise of 1 o C. a. Use a table to determine an expression that can be used to represent the number of cups of hot chocolate drunk per day as a function of degrees Celsius. b. How many cups would they drink at 5 o C? c. How many cups would they drink at 25 o C? d. Formulate and solve an equation to determine the temperature if 3000 cups of hot chocolate were consumed in a day. Is this answer reasonable? Explain. 3. Speedy is at a point three miles from his home. He rides his bike at a quarter of a mile per minute away from home. a. Make a table to show Speedy s distance from home after each elapsed minute. b. Use the table to write an expression for the number of miles traveled versus time. c. How far is Speedy from home after 20 minutes? d. How long will it take Speedy to be 10 miles from home? 2012, TESCCC 03/27/13 page 3 of 4

102 Problem Solving with Rates 4. Jackie is three miles from her home. She starts riding her bike at a quarter of a mile per minute toward home. a. Make a table to show Jackie s distance from home after each elapsed minute. b. Use the table to write an expression for the number of miles traveled versus the time. c. How far is she from home after 8 minutes? d. How long will it take her to be half a mile from home? e. How long will it take her to get home? 5. James Millhouse is renting a car at a local rental agency. The cost of renting the car is $38 per day, plus $0.35 for each mile driven over 150 miles. James rented this car for one day and drove it over 150 miles. a. What equation can be used to determine y, the total cost of the rental, if it is driven y miles? b. James drove from Las Vegas to the south rim of the Grand Canyon, a distance of 280 miles. He then drove back to Las Vegas the same day along the same route to turn in his rental car. How much was the bill for his rental car? c. If James only had $150 to spend on the rental car, how many miles could he drive? 6. The owners of a new sporting goods store want to spend no more than $1000 for at least ten minutes of advertising. The cost to advertise on a local television station is $200 per minute. The cost to advertise on a local radio station is $50 per minute. If the owners want the maximum exposure time for their $1000 using both television and radio, which of the following is a reasonable solution? Justify your reasoning. a. Advertise 12 minutes on television and 2 minutes on radio b. Advertise 3 minutes on television and 8 minutes on radio c. Advertise 1 minute on television and 18 minutes on radio d. Advertise 4 minutes on television and 4 minutes on radio 2012, TESCCC 03/27/13 page 4 of 4

103 Tying Up Equations and Inequalities KEY 1. Solve the equation using concrete models. Show an illustration and explain the process symbolically. 2(x 1) = 4 Illustration Symbolic Explanation + - = (x 1) = x 2 = = x = x = x x = 3 = , TESCCC 03/27/13 page 1 of 4

104 Tying Up Equations and Inequalities KEY 2. Solve the equation and inequality using the graphing method and the graphing calculator. Show the graph and justify the solution using the table. a) 2(x 1) + x = 6 (2x + 3) Graph Table x Y1 Y x = 1 b) 5(x + 1) 3x 5(x + 1) 3 Graph Table x Y1 Y Y1 Y2 x , TESCCC 03/27/13 page 2 of 4

105 Tying Up Equations and Inequalities KEY 3. Solve the equations and inequalities using algebraic properties. Show all steps. a) 28x 6(3x 5) = 60 b) 3(x 2) 8x < 44 28x 6(3x 5) = 60 3(x 2) 8x < 44 28x 18x + 30 = 60 3x 6 8x < 44 10x + 30 = x < 44 10x = 30-5x < 50 x = 3 x > -10 Solve the following problems using any method. 4. The length and width of a rectangle are in the ratio of 3:2. The perimeter is seventy-three centimeters. a. How can the length of the rectangle be represented using x? 3x b. How can the width of the rectangle be represented using x? 2x c. Write an equation to calculate the perimeter. P = 2(3x + 2x), 73 = 2(5x) d. Find the length and the width of the rectangle. x = 7.3 cm Length 3(7.3) = 21.9 cm, width 2(7.3) = 14.6 cm 5. Ellie is making an abstract triangular collage for the wall. She has 14 feet of edging to go around the collage. If the longest side is twice the smallest and the other side is four less than three times the smallest, find the length of each side of the triangular collage. x + (3x 4) + 2x = 14; x = 3 ft Shortest side= 3 ft, longest side = 6 ft, other side= 5 ft. 2012, TESCCC 03/27/13 page 3 of 4

106 Tying Up Equations and Inequalities KEY 6. Eddie and Marsha are pulling KP duty. The master sergeant has assigned them to peeling potatoes. Eddie is peeling potatoes at a rate of 3 potatoes per minute. Four minutes later, Marsha joins him and peels potatoes at a rate of 5 potatoes per minute. a. If x represents the number of minutes Eddie has been peeling potatoes, how could you represent the number of minutes Marsha has been peeling potatoes? x 4 b. Write an expression for the number of potatoes Eddie has peeled. 3x c. Write an expression for the number of potatoes Marsha has peeled. 5(x 4) d. Write an expression for the total number of potatoes that have been peeled. 3x + 5(x 4) e. If the master sergeant wants a total of 36 potatoes to be peeled, how long must each peel potatoes? How many potatoes must each peel? Eddie 7 min and Marsha 3 min Eddie 21 potatoes and Marsha 15 potatoes 7. Cat is wondering if they can save money by shopping around. She found another local dealer, Tidy Paws, who sells the flea dip for $3.00 a quart bottle plus a $21.00 handling charge per order. a. Complete the table on cost versus number of bottles for Tidy Paws Bottles of Flea Dip Process Cost (1) (2) (3) (4) 33 x (x) 3x + 21 b. Write an expression that represents the amount of money charged by Tidy Paws for an order of flea dip. y = 3x + 21 c. Evaluate the cost for 9 bottles, 15 bottles, and 25 bottles of shampoo. 48, 66, 96 d. If Gooding French Club paid $75.00 for flea dip at Tidy Paws, how many bottles did they buy? How did you find the answer? 18, Answers will vary. e. The Gooding French Club can pay at most $95.00 for flea dip at Tidy Paws. Write an inequality to represent the situation. Solve the inequality to determine the number of bottles. 3x x They can purchase at most 24 bottles. 2012, TESCCC 03/27/13 page 4 of 4

107 Tying Up Equations and Inequalities PI 1. Solve the equation using concrete models. Show an illustration and explain the process symbolically. 2(x 1) = 4 Illustration Symbolic Explanation 2012, TESCCC 03/27/13 page 1 of 4

108 Tying Up Equations and Inequalities PI 2. Solve the equation and inequality using the graphing method and the graphing calculator. Show the graph and justify the solution using the table. a) 2(x 1) + x = 6 (2x + 3) Graph Table b) 5(x + 1) 3x 5(x + 1) 3 Graph Table 2012, TESCCC 03/27/13 page 2 of 4