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Variables on Both Sides

Lesson Plan

Variables on Both Sides

Objectives

Students will learn to solve multistep linear equations and inequalities involving variables on both sides of the relation symbol. Students will be able to:

  • understand the similarities and differences in solution techniques for equations vs. inequalities.
  • combine like terms.
  • understand the concepts of “no solution” and “infinitely many solutions” and be able to identify certain equations/inequalities as “always true” or “never true.”
  • represent real-world situations as equations/inequalities and express the solution in terms of the original situation.

Essential Questions

  • How can mathematics support effective communication?
  • How are relationships represented mathematically?
  • How can expressions, equations and inequalities, be used to quantify, solve, model, and/or analyze mathematical situations?
  • How can data be organized and represented to provide insight into the relationship between quantities?

Vocabulary

  • Coefficient: The coefficient of a term in an expression is the number that is multiplied by one or more variables or powers of variables in the term.
  • Function: A relation in which every input value has a unique output value.
  • Linear Equation: An equation of the form Ax + By = C, where A ≠ 0 and B ≠ 0. The graph of a linear equation is a straight line.
  • Rate of Change: The ratio of the change in the output value and change in the input value of a function.
  • Simultaneous Linear Equations: Two or more linear equations that share a set of variables and either exactly one solution, no solutions, or infinitely many solutions; often called a “system of equations.”
  • Slope: The measure of the steepness of a line; the ratio of the change in the y-values to the x-values of the linear function; rate of change.
  • y-intercept: The y-coordinate of the point where a line intersects the y-axis; the value of the function when x = 0.

Duration

90–120 minutes

Prerequisite Skills

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Materials

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Related Materials & Resources

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Formative Assessment

  • View
    • The Entrance Ticket evaluates students’ skills in combining like terms; translating between worded statements, mathematical inequalities, and their graphical representation; and solving inequalities.
    • The Babysitting Activity provides scenarios that students must represent with an appropriate equation or inequality.

Suggested Instructional Supports

  • View
    Active Engagement, Modeling, Explicit Instruction
    W: Students learn how to solve equations and inequalities with variables on both sides of the relation symbol by combining like terms, and using the addition, subtraction, multiplication, and division properties of equality and inequality. The lesson focuses on how real-world situations can be represented as equations and inequalities, which can then be solved using the algebraic techniques learned in the lesson. 
    H: The Entrance Ticket emphasizes the similarities between equation solutions and inequality solutions. When students are finished, review problems 1–4 together. Then ask students, “What algebraic technique did we use to simplify these expressions?” If necessary, review to make sure students are competent at combining like terms. 
    E: Using the Lesson Transparency as guided instruction, students will learn how to process each step in the solution. Encourage students to copy and recreate each step on their own after they have followed the procedures. Remind students that it is easier to watch the process than to do it alone, and they should practice it on their own to test their understanding. 
    R: Tic-tac-toe equations/inequalities is an activity that invites strategizing about which equation/inequality to solve to advance one’s likelihood of winning against another competitor. Students must also weigh the selection of easier or more difficult items to attempt and must consider the reasons they find certain items easier or more difficult. 
    E: The Babysitting Activity reinforces and assesses students’ understanding of the solution techniques and the degree to which their results make sense in the real-world applications that uses familiar quantities of money. 
    T: Emphasis is placed on checking the solution. Students learn to ask whether their solutions are reasonable as they relate to the original situation by substituting the solution for the variable in the original equation/inequality and identifying if there is a solution for the variable, no solution, or infinite solutions. By having students routinely check their answers, these concepts are reinforced and students see how the algebraic techniques they learned help them solve equations and inequalities that represent real-world situations. 
    O: The combination of like terms, isolation of the variable, and addition/subtraction and multiplication/division properties of equations and inequalities are reinforced and used to solve equations and inequalities with variables on both sides of the relation symbol. The lesson focuses on using the algebraic techniques students have learned to represent real-world situations as equations or inequalities and to solve for the variable. The similarities and differences in solution techniques for equations vs. inequalities are again emphasized.  

Instructional Procedures

  • View

    Distribute the Entrance Ticket (M-8-2-2_Entrance Ticket.docx and M-8-2-2_Entrance Ticket KEY.docx). Instruct students to complete this individually.

    As students complete the Entrance Ticket, discuss problems 1–4 as a class. For problems 5 and 6, randomly choose two students to share their answers on the board with the class. Discuss the answers with the class, clearing up any questions before going further.

    For problems 7–10, randomly call on students to share their responses. Discuss the answers as a class.

    “What does it mean to ‘solve’ an equation? An inequality? How does this solution relate to the original equation or inequality?” (Solutions are values which, when substituted for the variable, make the equation or inequality true.)

     

    Distribute the Lesson Transparency (M-8-2-2_Lesson Transparency.docx and M-8-2-2_Lesson Transparency KEY.docx) and problem 1 for the class:                

    • 12x + 30 = 9x

     

    “Let’s work together to solve some equations that have a variable on both sides. First of all, who can remind us of our goal when solving an equation?” (To isolate the variable!)

    “Think about the goal when solving this equation for x and the steps needed to find a solution:

    12x + 30 = 9x

    12x + 30 − 9x = 9x − 9x         Remember, we need to do the same thing to both sides of the equation to preserve the balance.”

    (12x − 9x) + 30 = 0                 “Combine like terms.”

    3x + 30 = 0

    3x + 30 − 30 = 0 − 30

    3x = −30

    3x / 3 = -30 / 3

    x = −10

    Work through the preceding problem as a class, discussing how the addition/subtraction and multiplication/division properties of equality and the concept of combining like terms help us to solve this equation.

    Instruct students to complete problem 2 on their own, and then review it as a class:

    • 4y − 2 = 8y + 10          (y = −3)

     

    Instruct students to complete problem 3 on their own, and then review it as a class:

    • 3y > 10 − 2y                (y > 2)

     

    Instruct students to complete problem 4 on their own, and then review it as a class:

    • 7x + 2 < x − 10            (x  −2)

     

    Discuss the similarities and differences in solution techniques for equations vs. inequalities. “For an inequality, why do we need to reverse the relation sign when we multiply or divide both sides by a negative number?” (Possible student responses may range from simply stating “because of the multiplication property of inequalities” to a more detailed analysis/explanation. Whatever the response, make sure students understand what to do when multiplying or dividing both sides of an inequality by a negative number. If they are unsure, work through a few examples to refresh this skill.)

    For problems 5–8 on the Lesson Transparency, have students fold a paper into four sections and label them as shown in the following diagram.

    Original Problem:

     

     

     

    Solution:

    Work:

     

     

     

     

    Notes:

    Display problems 5–8 and have students work in pairs, putting their work and notes for each example on a separate section of the paper. This will give them room to make notes, corrections, and adjustments as each problem is worked through and discussed with the class. Advise students that if their answers look a bit different than usual, they need not worry; the answers will be explained in class.

    “Let’s look at a few more examples.”


    Example 1

     

     

    Example 2

     

     

    Example 3

     

     

    Activity 1: Real-World Inequalities

    “Now that you are becoming proficient at working with and solving inequalities, it is important to see how they may be used in real life.”

    Distribute the Babysitting Activity (M-8-2-2_Babysitting.docx and M-8-2-2_Babysitting KEY.docx). Have students complete the activity in groups of four or five. Ask them to be prepared to share their work and answers with the class.

    Walk around the room to assess student understanding and make sure all students are participating. Pick one group to discuss and show their solutions to problem 1 on the board, and choose another group to do problem 2. Encourage students to make notes, corrections, and adjustments on their Babysitting Activity sheets.

     

    Activity 2: No Solution/All Real Numbers

    Write the following inequality on the board. Instruct students to solve it..

    3y – 6 – 5y > 2 + 4y – 6y

    As students work, many will likely get stuck or become confused. This is intentional as this inequality has no solution! When all students have attempted the problem and seem to be coming to the general consensus that something is out of the ordinary, explain to the class that this is an example of an inequality that has no real solution. Observe:

                3y – 6 – 5y > 2 + 4y – 6y

                2y – 6 > 2 – 2y

                2y – 6 + 2y > 2 – 2y + 2y

    6 > 2              This statement will never be true. Because the variable terms cancelled out, we were left with a numerical statement that, in this case, is not true. Therefore, the inequality has no solution.

     

    “Now try this example.”

    8(x + 7) < 2(4x) – 9

    Again, students will recognize that something is out of the ordinary in this inequality, too. They may be tempted to state “no solution” because of the previous example. Discuss with the class, however, that this is actually an example of an inequality that has a solution of all real numbers.

    Observe:

    8(x + 7) < 2(4x) – 9

    8x – 56 < 8x – 9

    8x – 56 + 8x < 8x – 9 + 8x

    56 < 9         This statement is always true. Because the variable terms cancelled out, we were left with a numerical statement that, in this case, is always true. Therefore, any real number may be a solution for this inequality.

     

    Distribute the Special Cases Worksheet (M-8-2-2_Special Cases.docx and M-8-2-2_Special Cases KEY.docx) and instruct students to work through the problems. Walk around the room as students work to troubleshoot questions and keep them on task.

     

    Activity 3: Tic-Tac-Toe Equations

    Students will play a variation of the original tic-tac-toe game. Instead of simply choosing boxes to place Xs and Os in, students must correctly solve the equation or inequality in order to claim the box. The first student to get three Xs or Os in any row, column, or diagonal is the winner.

     

     

     

     

     

     

     

    Sample game board:

    (Note: Some answers are in fraction form. Make sure you specify that all fractions should be reduced to lowest terms.)

     

    Extension:

    • As a challenge to students who show proficiency, ask them to work as a small group (or pair, depending on the number of students). They should be prepared to present to the class a discussion of the multiplication and division properties of inequalities and how, by using our knowledge of inverse operations and reciprocals, the division property can be explained in terms of the multiplication property. Provide students with chart paper and markers.
    • Refer students back to the Special Cases Worksheet. problems 1 and 2 deal with situations that have no solution. Randomly pick a pair of students to go to the board and share their work with the class. Discuss with students how, after applying algebraic techniques, the variable drops away, leaving a false statement for the equation and inequality. This means that there is no solution.

    Ask students if an answer of “no solution” seems reasonable. Do some guess-and-check work and analyze 4x + 1= 4x − 6. “Can we think of any real number to make this true?” When we say there is “no solution,” we mean there is no real number that can be substituted for the variable x that would make the equation/inequality true.

    Problems 3 and 4 deal with statements that are true for all real numbers x.

    Again, randomly pick a pair to share their work with the class. Say, “Again when we applied algebraic techniques, the variable dropped out. But this time, we ended with a statement that is true! That means our equation/inequality is true for any real number that we substitute for our variable x. How many solutions do we have then?” (All real numbers.) Have students check this by substituting different values for x and seeing for themselves whether the equation/inequality is true.

    • Partner Activity: With students in pairs, have them practice identifying different types of solutions by having one partner make up an equation or inequality with a variable on each side of the relation symbol and the other partner solve it. Together they should determine if there is a solution for x, no solution, or if it is true for all real numbers. Then have them switch roles, with the other partner making up the equation/inequality. Walk around the room to assist and evaluate student understanding, having students who need more reinforcement repeat the exercise as necessary.
    • Small Groups: Each of the paired and group activities can be repeated if necessary for students who need additional practice. A small group of approximately four or five students can work as a team; one student makes up an equation or inequality with the variable on both sides of the relation symbol and the rest of the team solves for the variable. Guide the group members to check their solution by substituting the solution for the variable in the original equation/inequality. Ask them what it means to have “no solution” and “infinite solutions.”

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Final 04/12/13
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