Folding to Construct Shapes: Math Task

Use a pencil, straightedge (if you use a ruler as your straightedge do not measure!), and patty paper to complete the constructions described in the problems below. You can fold the patty paper to create creases and to place segments or angles on top of each other.

  1. Draw a line segment on the patty paper, making sure the line segment is not parallel to the edges of the paper
    1. Construct a line segment that is perpendicular to your original segment.  A perpendicular line segment should form 90-degree angles with your original segment at the point of intersection.
      1. Use pictures (a fold could be represented by a dotted line ) and words to describe how you constructed the perpendicular line segment.
      2. Your classmate Jared doesn’t think your new segment is perpendicular to your original line segment.  Write a convincing mathematical explanation that would convince him.
         
    2. Now construct a line segment that is a perpendicular bisector to your original segment. A perpendicular bisector should
      • form 90-degree angles with your original segment at the point of intersection, and
      • intersect your original segment at its midpoint.
         
      1. Use pictures (a fold could be represented by a dotted line ) and words to describe how you constructed the perpendicular bisector.
      2. Your classmate Jared doesn’t think your new segment is a perpendicular bisector.  Write a convincing mathematical explanation that would  convince him.
         
    3. Using the same piece of patty paper, now construct a line segment that is parallel to your original segment.
      1. What does it mean for one segment to be parallel to another?
      2. Use pictures and words to describe how you constructed the parallel line segment.
      3. Again, Jared needs convincing.  Write a convincing mathematical explanation that would convince him that the new segment is parallel to your original.  How do you know the segment you’ve constructed has the properties you listed in part i?
  2. For each of the following, start with a freshly drawn segment on a clean piece of patty paper, making sure the line segment is not parallel to the edges of the paper.  Then construct the shape.
    1. a non-square rectangle, with your segment as one of its sides (a)
    2. a square, with your segment as one of the sides (b)
      • What are the properties of a square?;
      • Use pictures and words to describe how you constructed the square.
      • How do you know your shape has each of the properties you listed for a square?  Write a convincing mathematical explanation.
      • Before you tried your method, why did you think it would work?
      • Were there methods you tried that didn’t work?  If so, what were those methods?
    3. an isosceles triangle, with your segment as one of the two equal sides (c, following from b above)
    4. an isosceles triangle, with your segment as the base (d)
    5. an equilateral triangle, with your segment as one of its sides (e)
    6. choose one of the triangles you just constructed and answer the following (f)
      • What are the properties of the triangle you chose?
      • Use pictures and words to describe how you constructed the triangle.
      • How do you know your shape has each of the properties you listed for that type of triangle?  Write a convincing mathematical explanation.
      • Before you tried your method, why did you think it would work?
      • Were there methods you tried that didn’t work?  If so, what were those methods?

© 2008 by Education Development Center, Inc. from The Fostering Geometric Thinking Toolkit. Portsmouth, NH: Heinemann. Reproduced with permission.