Changing Both Dimensions

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Author: J Tynon
School District: Changing Both Dimensions
Subject: Mathematics
Grade Range: 7-8
Duration/Time Required:

  • Number of days: 1
  • Length of period: 52 minutes

Standard:

  • Content - Algebra
    • 8.A.5 Use physical models to perform operations with polynomials
    • 8.A.11 Factor a trinomial in the form ax2 + bx + c; a=1 and c having no more than three sets of factors
  • Process -
    Problem Solving

    • 8.PS.1 Use a variety of strategies to understand new mathematical content and to develop more efficient methods

    Connections

    • 8.CN.1 Understand and make connections among multiple representations of the same mathematical idea

    Representation

    • 8.R.2 Explain, describe, and defend mathematical ideas using representations

Objective: Students will use the area model to represent the area of a modified square with unknown length, x, in two forms (factored and expanded form) to develop an understanding of equivalent expressions, that is, of two expressions that represent the same area.

Instructional Plan: Students are given a square with an unknown length, x, then are instructed to modify the square’s dimensions in some form such as increasing the length by two and increasing the width by 3 to form a rectangle that is now made up of four parts, the original square plus three rectangles. Students are then asked to find the area of the new rectangle formed by using the new length multiplied by the new width i.e. (x + 2)(x + 3). Next students are asked to find the area of the new rectangle using the sum of the parts of the new rectangle i.e. (x2 + 2x + 3x + 6 or x2 + 5x + 6). Once both forms of the area are found students are asked to graph both functions on their graphing calculators to compare their graphs and make conjectures about the two expressions found. Through this activity students develop and understanding of equivalent quadratic expressions through a model that they know and understand, area of a rectangle, since they know that the two expressions found represent the area of the same figure and through the graphing exercise they see the same relationship among the two functions of area graphically. Students also start to develop procedures for factoring a quadratic expression in which the leading coefficient is one.

Assessment: Homework, quiz, test.

Resources:

  • Teacher Resources: Teacher manual of CMP “Frogs, Fleas, and Painted Cubes”
  • Student resources: Notes, student book of CMP “Frogs, Fleas, and Painted Cubes”
  • Materials: Transparencies of the activity in student book.
  • Technology: Graphing Calculator