VIEWS: 30 PAGES: 11 CATEGORY: Current Events POSTED ON: 5/2/2010 Public Domain
Solving and Graphing Inequalities – Grade Seven Ohio Standards Lesson Summary: Connection In this lesson, students solve inequalities and represent the Patterns, Functions and solution on a number line. They write and solve inequalities Algebra representing problem situations, graph the solutions on number lines and explain the results in the solutions. Benchmark H Solve linear equations and Estimated Duration: Three to four hours inequalities symbolically, graphically and numerically. Commentary: Indicator 4 Two key components of this lesson include the use of inverse Create visual operations to solve equations and inequalities and representing representations of the solution on a number line with appropriate symbols. equation-solving processes Through the use of real-world contexts, in which students can that model the use of inverse operations. use a variety of strategies to solve, they better understand the algorithm, the use of inverses and the symbolic representation Benchmark I of the solution. Use writing tasks to have students explain their Explain how inverse understanding of inverse operations in solving equations and operations are used to inequalities and describing given number line graphs solve linear equations. representing solutions. Indicator 4 Create visual representations of equation-solving processes Pre-Assessment: that model the use of This assesses students’ prior knowledge of inequalities and inverse operations. solving equations. Students write inequalities to represent Benchmark K problem situations that include words such as greater than, less Graph linear equations and than, at least, at most, etc. Students write the meaning of inequalities. inequalities. Students write and solve equations from problem situations and represent the solutions on number lines. Indicator 6 Represent inequalities on a • Have students complete the Pre-Assessment, Attachment A, number line or a coordinate individually. Collect and evaluate for understanding. Score plane. using the following rubric. Mathematical Processes Scoring Guidelines: Benchmark I Use the following guidelines to score the assessment. Select, apply, and translate among mathematical representations to solve problems; e.g., representing a number as a fraction, decimal or percent as appropriate for the problem. 1 Solving and Graphing Inequalities – Grade Seven Meets Expectations • Represents each situation correctly as an inequality with a variable and describes what the variable represents in each situation. • Provides an adequate explanation of each inequality. • Represents each situation correctly as an equation with a variable • Correctly solves each equation and represents the solution on a number line. Adequate Understanding • Represents more than two, but not all of the situations correctly as inequalities with variables and describes what the variables represent in most of the situations. • Provides an explanation of each inequality, perhaps with minor errors. • Represents the situations as equations, perhaps with minor errors. • Solves each equation, perhaps with minor errors, with or without representing the solution on a number line. Needs Intervention • Represents one or two of the situations correctly as inequalities with variables and describes what the variables represent in these situations. • Provides little or no explanation of the inequalities. • Represents one of the situations correctly as an equation. • Makes numerous errors in solving the equation, but does not represent the solution on a number line. Post-Assessment: Students write and solve inequalities from problem situations and represent the solution on number lines. Students provide explanations of the results of the inequalities. Scoring Guidelines: Use the rubric as an example of scoring the assessment. Meets Expectations • Represents each situation correctly as an inequality. • Correctly solves each inequality. • Provides an adequate explanation of each solution. Adequate Understanding • Represents more than two, but not all of the situations correctly as an inequalities. • Solves each inequality, perhaps with minor errors. • Provides an explanation of each inequality, perhaps with minor errors. Needs Intervention • Represents one or two of the situations correctly as inequalities. • Makes numerous errors in solving the inequalities. • Provides little or no explanation of the inequalities. 2 Solving and Graphing Inequalities – Grade Seven Instructional Procedures: Instructional Tip: Solving equations or reviewing how to solve equations is a prerequisite to solving inequalities. 1. Review the pre-assessments with students, noting and correcting any misconceptions. 2. Present other problems similar to those in the pre-assessment as needed for additional practice. 3. Have students create a number line showing numbers greater than 4. Circulate the room to view students’ number lines. Make anecdotal notes, ask questions and make comments to clear up any misconceptions. Fro example, students may have shown 5 and numbers greater than 5, or 4 and numbers greater than 4, but may have shaded only the whole numbers. 4. Ask students to represent y > 4, where y represents any number, on another number line. Ask students to compare the two number lines. Ask students to choose a number from both number lines that makes a true statement. Instructional Tip: It is not necessary for students to include many numbers on the number line. Lead them to identify the given number with a circle around the number. In this and the following tasks, they should shade to the left or right of the number and when appropriate include the number by shading the circle. 5. Have students create a new number line and show the numbers less than or equal to -3. (Fill in the circle around -3 and shade to the left of -3.) Next, have students represent the inequality x -3 on a different number line and compare the two number lines. Ask questions ≤ to get at students understanding. 6. Present as many of these types of problems as needed for understanding. Have students correct questions 4 through 7 on the pre-assessment. 7. Have students write in their journals about how to represent an inequality on a number line. Check students’ entries. 8. Present the following scenario to students: Amy bought four scarves for less than $100. What could be the price of one of the scarves if the scarves are of equal value? The variable x represents the price in dollars of one scarf. 4x < 100 4x < 100 4 4 x < 25 3 Solving and Graphing Inequalities – Grade Seven The price of one scarf is anything less than $25, such as $2.00, $8.48 or $24.99. a. Have each student write an inequality to solve. Tell students to share their inequalities with partners. Select students to present their inequalities to the class. Record the responses on the board or overhead projector. Discuss the inequalities with the class, addressing any misconceptions. Ask questions such as: • What could be the total cost of the four scarves? • Could the price of one scarf be more or less than $100? Why? b. Have students solve the inequality and represent the solution on a number line. Circulate the room answering any questions that may arise. Select students to present showing to the class solution steps and using the number line on the board or overhead projector. Ask questions about the solution such as: • What could one scarf cost? • Is this the only price that one scarf could cost? Why? Are there other prices for the cost of one scarf? • What are some possible prices for each of the four scarves? Is the total of the prices of the four scarves more or less than $100? c. Remind students to make sure the solution represents the original statement in the problem. d. Have students check their answers. 9. Present this scenario: Katie is taking some friends to the movies to celebrate her 14th birthday. Her mother said she could spend no more than $40. Tickets to movies are $8.50 each for Katie and her friends. How many tickets can Katie buy for her friends and herself? Let x represent the number of friends for whom Katie can buy tickets. Since she also buys a ticket for herself, 8.5x + 8.5 40 ≤ 8.5x + 8.5 -8.5 40 – 8.5 ≤ 8.5 x 31.5 ≤ 8.5 8. 5 ≤x 3.71 31.5 (Note: is rounded to 3.71, since it is more than 3.70588.) 8. 5 Therefore Katie can take three friends since she has enough money for three friends and herself.) a. Ask questions before students write the inequality such as: • What does Katie’s mother mean when she says, “spend no more than $40?” • How would you represent the amount of Katie’s ticket in the inequality? How would you represent the number of tickets for Katie’s friends with a variable? b. Have students write the inequality to represent this problem. Select students to share their inequality with the class. Record the responses and discuss the inequalities presented, addressing any misconceptions. 4 Solving and Graphing Inequalities – Grade Seven Instructional Tip: In the explanation of the solution, students need to be mindful of the reasonableness of the solution. For example, if a solution is a decimal and the question pertains to the number of people, books, tickets, etc., the number needed has to be a whole number since it represents whole parts and not pieces. Graphs with these types of solutions should reflect whole number solutions. For example, in Katie’s problem, the graph would be , c. Have students solve and graph the solution of the inequality. Have students share and discuss their work with a partner. Select students to solve the problem on the board or overhead projector. Allow students to ask questions about the steps and the solution. Select other students to explain the solution. 10. Present these problems as homework and/or practice in class. Have students write, solve and represent the solutions to the inequalities on number lines. Have students work the problems individually before sharing ideas with partners. Collect and evaluate the papers to conference with students individually. a. The sum of Dan’s age and his sister’s age is greater than 45. Dan’s age is three years more than twice his sister’s age. What is the youngest age could Dan be? (Dan’s age can be no less than 33, assuming the ages are in whole number of years.) b. The combined weights of two packages are at most 300 pounds. The first package weighs125 pounds more than the second package. What is the most that the first package could weigh? (The first package could weigh at most 212.5 pounds.) c. Mary and her mother went to the theater. The mother’s ticket cost three times as much as Mary’s ticket. The total cost of the tickets was less than $68. What could be the cost of the mother’s ticket? (The mother’s ticket costs $50.97 or less. Mary’s ticket must cost less than $17. If Mary paid $17 for her ticket, the mother’s ticket would cost $51, and the combined ticket cost would be exactly $68.) d. The Music Club needs to sell at least 500 bakery items in a week in a bake sale. How many bakery items does the club need to sell by the end of each school day to meet the goal? (The Music Club needs to sell 100 or more bakery items each school day to meet its goal.) 11. Have students write in their journal the similarities and differences in equations and inequalities and the relationship between solving equations and inequalities. Select students to share their journal entries. Differentiated Instructional Support: Instruction is differentiated according to learner needs, to help all learners either meet the intent of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified indicator(s). • Solve simple linear equations and inequalities with only one step involved. Have students talk through the process of finding the solutions. • Ask students to give numbers greater than and less than a given number. • Review the translation of written phrases to algebraic expressions. • Present an inequality. Have students write a situation for the inequality. 5 Solving and Graphing Inequalities – Grade Seven Extensions: • Introduce graphing linear functions on a coordinate plane. • Have students graph compound sentences (containing conjunctions or disjunctions) and explain the meaning of the graph. An example of a compound sentence is x > 5 and x < 12. This sentence represents a conjunction since and connects the two inequalities. The graphs show that the number x is between 5 and 12, not including 5 or 12. An example of a disjunction is x < -4 or x >1. A disjunction is two sentences connected with or. The graph shows shading to the left of -4, which includes all numbers less than -4 and shading to the right of 1, which includes all numbers greater than 1. Materials and Resources: The inclusion of a specific resource in any lesson formulated by the Ohio Department of Education should not be interpreted as an endorsement of that particular resource, or any of its contents, by the Ohio Department of Education. The Ohio Department of Education does not endorse any particular resource. The Web addresses listed are for a given site’s main page, therefore, it may be necessary to search within that site to find the specific information required for a given lesson. Please note that information published on the Internet changes over time, therefore the links provided may no longer contain the specific information related to a given lesson. Teachers are advised to preview all sites before using them with students. For the teacher: Overhead projector or board, overhead transparencies, calculators (optional) For the student: Pencil, paper, journals, calculators (optional) Vocabulary: • at least ( ) ≥ • at most ( ) ≤ • greater than (>) • greater than or equal to ( )≥ • less than (<) • less than or equal to ( ) ≤ Technology Connections: Allow students to use calculators for computations. Research Connections: Arter, Judith and Jay McTighe. Scoring Rubrics in the Classroom: Using Performance Criteria for Assessing and Improving Student Performance. Thousand Oaks, Calif.: Corwin Press, 2001. Cawletti, Gordon. Handbook of Research on Improving Student Achievement. Arlington, Va.: Educational Research Service, 1999. 6 Solving and Graphing Inequalities – Grade Seven Attachments: Attachment A, Pre-Assessment Attachment B, Pre-Assessment Key Attachment C, Post-Assessment Attachment D, Post-Assessment Key 7 Solving and Graphing Inequalities – Grade Seven Attachment A Pre-Assessment Name_______________________________ Date__________________________ Directions: Express each situation as an inequality using a variable. Describe what each variable represents in each situation. 1. Mary has more than $30 in her purse. 2. John scored fewer than six points in the basketball game. 3. The book will cost at least $16.99. Directions: Explain what each inequality means. 4. a > 9 5. c < -4 6. y ≤ 12 7. n ≥ 0 Directions: Write and solve an equation for each problem using a variable Describe what each variable represents in each situation . Show steps for solving the equation. Represent the solution for each equation on a number line. 8. Sam earned $135 for mowing 5 lawns. How much did Sam earn for mowing each lawn if he charged the same amount for each lawn? 9. John purchased four books at the same price and a $12 pen at the bookstore. The total cost of the purchases was $80. How much did each book cost? 8 Solving and Graphing Inequalities – Grade Seven Attachment B Pre-Assessment Key 1. Mary has more than $30 in her purse. n > 30, where n represents the amount of money in Mary’s purse. 2. John scored fewer than six points in the basketball game. s< 6, where s represents the number of points scored. 3. The book will cost at least $16.99. b ≥ 16.99, where b represents the cost of the book. 4. a > 9 The number a is greater than 9. 5. c < -4 The number c is less than -4. 6. y ≤ 12 The number y is less than or equal to 12. 7. n ≥ 0 The number n is greater than or equal to 0. 8. Sam earned $135 for mowing 5 lawns. How much did Sam earn for mowing each lawn if he charged the same rate? Let x = the number of dollars charged for each lawn 5x = 135 5x = 135 5 5 x = 27 Sam earned $27 for mowing each lawn. 9. John purchased four books at the same price and a $12 pen at the bookstore. The total cost was $80 for the purchases. How much did each book cost? The variable x represents the cost of each book. 4x + 12 = 80 4x + 12 −12 = 80 − 12 4x = 68 x = 17 9 Solving and Graphing Inequalities – Grade Seven Attachment C Post-Assessment Name_______________________________ Date__________________________ Directions: Read each problem. Write and solve an inequality for each problem. Describe what each variable represents in each problem. Represent the solution for each inequality on a number line. Explain what the solution indicates for each problem. 1. Raegan works at a restaurant. He earns $4 an hour, plus tips. He made $20 in tips one day. How many hours could Raegan have worked if he earned less than $52? 2. A building has at least 150 windows on two floors. How many windows could be on each floor? 3. John needs to buy gasoline for his car and cannot spend more than $25. On the day John buys gasoline, the cost is $1.78 per gallon. How many gallons of gasoline could John buy? 4. The cost of a concert ticket is $25. This does not include parking which costs $5. How many tickets can Robb purchase without spending more than $200? 10 Solving and Graphing Inequalities – Grade Seven Attachment D Post-Assessment Key 1. Let h represent the number of hours Raegan worked at the restaurant. 4h + 20 < 52 4h < 32 h<8 Raegan worked less than 8 hours. 2. Let f represent the number of windows on each floor. 2f 150 ≥ f 75 ≥ Each floor could have at least 75 windows. 3. Let g represent the number of gallons of gasoline John purchases. 1.78 g ≤ 25 g ≤ 14.04 John could buy 14.04 or less gallons without spending more than $25. 4. Let t represent the number of tickets Robb purchases. 25t + 5 200 ≤ 25t 195 ≤ t 7.8 ≤ Robb can purchase 7 tickets and not spend more than $200 and pay for parking. 11