Solving and Graphing Inequalities - Grade Seven by qym17251

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									                      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.




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                    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.


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                    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


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                    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.




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                     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.




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                    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.




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                  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




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                    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.




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