Algebraic Transformations

  • Algebraic Transformations

    Grade: 6th to 8th, High School

    This applet allows you to transform a plus sign in eight different ways: rotate it ¼, ½, ¾, or a full turn, or flip it vertically, horizontally, or diagonally.

    Activity

    Instructions

    Free Mode

    In free mode, you can investigate the results each time you perform a transformation.

    • Click on any of the eight transformation buttons to rotate the shape. Which letter appears at the top after each transformation?
    • The Reset button will reset the figure so that the A returns to the top.

    Double Transformation Mode

    In double transformation mode, you can investigate what happens to the shape when two movements are performed one after the other.

    • Click on any transformation along the left side of the table, and then click on any transformation along the top. After both transformations, which letter appears at the top?
    • The Clear button will remove all results from the table.
    • The Show All button will not be active until results have been determined for at least 5 squares. After that, the button can be pressed to reveal the results for the entire chart. (But it is recommended that you attempt to complete the chart on your own and identify any patterns before using this button.)

      Exploration

      Is the act of double transformations commutative? That is, does the order in which you perform two actions influence the result?

      • Perform a ¾ turn, followed by a ½ turn. Which letter appears at the top?
      • Then, reverse the order of the moves. That is, perform a ½ turn, followed by a ¾ turn. Now which letter appears at the top? Is it the same letter?

      Now, try this.

      • Perform a vertical reflection, followed by a diagonal reflection. Which letter appears at the top?
      • Then, reverse the order of the moves. That is, perform a diagonal reflection, followed by a vertical reflection. Now which letter appears at the top? Is it the same letter?

      Investigate other double transformations. When the order of two transformations is reversed, is the result the same or different?

       

      Objectives and Standards

      NCTM Standards and Expectations
      • Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
      • 6-8
      • High School (9-12)
      • Algebra
      • Geometry