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Modeling Situations With Linear Equations
  • Introduction: Mathematical Goals
    • This lesson unit is intended to help you assess how well students use algebra in context, and in particular, how well students:

      1. Explore relationships between variables in everyday situations.
      2. Find unknown values from known values.
      3. Find relationships between pairs of unknowns, and express these as tables and graphs.
      4. Find general relationships between several variables, and express these in different ways by rearranging formulae.
  • Formative Assessment Task: The Guitar Class
    • A music teacher runs a guitar class for 20 weeks. The class meets each week in a rented music studio. Suppose that it costs the teacher c dollars to rent the studio for 20 weeks. The class contains n students. Each student pays the teacher a single fee of f dollars for the course.

      The teacher makes a profit of p dollars at the end of the course.

      1. Suppose that c = 400 and f = 70. Write an equation to show how the profit made, p, depends on n, the number of students attending.
      2. Graph your equation, and explain the significance of the point where the graph crosses the horizontal axis.
      3. Write a formula for calculating p when you know c, n, and f.
      4. Write a formula for calculating f when you know c, n, and p.
  • Common Core State Standards
    • Common Core State Standards for Mathematical Practices Common Core State Standards for Content

      This lesson involves a range of mathematical practices from the standards, with emphasis on:

      1. Reason abstractly and quantitatively.
      1. Model with mathematics.

      This lesson involves mathematical content in the standards from across the grades, with emphasis on the following Domain.Cluster:

      • 8.F: Construct a function to model a linear relationship between two quantities.
  • Lesson Structure
    • This is an activity that links several aspects of algebra. A situation is presented, and relationships between the variables are explored in depth. Letters are used to represent unknowns, generalized numbers, and variables. The problem-solving context gives you a chance to assess how well students are able to combine and apply different aspects of their algebra knowledge and skills.

      • Before the lesson, students work individually on the assessment task, The Guitar Class. You then review their work and create questions for students to answer at the end of the lesson, to help them to improve their solutions.
      • During the lesson,
        • Students translate between words, algebraic formulae, tables, and graphs in an interactive whole-class discussion. The intention is that you focus students on making sense of the context using algebra, rather than just the routine use of techniques and skills.
        • Students then work in pairs to graph the relationship between two of the variables.
        • In a final whole-class discussion, students identify general formulae showing the relationships between all the variables.
      • At the end of the lesson, students review and improve their individual work.
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