Paperfolding

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In this lesson, folded and cut paper is used to develop the concept of reflectional symmetry and explore the angle relationships inherent in certain geometric shapes. For example, it we fold a sheet of paper in half twice and cut the folded corner with one straight cut, various unfolded shapes can be produced:

What observations can we make about the unfolded shape?

• The fold lines, m₁ amd m₂, form lines of symmetry for the shape.
• The shape is a quadrilateral.
• There are two obtuse and two acute angles.
• The lines of symmetry intersect at right angles.
• The shape has two pair of congruent sides.
• Varying 'x', the angle of cut, will affect the resulting shape. What angle of cut will produce a rhombus? Is there anything else you'd notice about the rhombus you'd cut out?

By exploring this type of activity, many intuitions and concepts relative to symmetry can be discussed and clarified. As the lesson progresses, the activities become more complex, and a variety of shapes are discovered:

Try the above activity...answer the following questions before you unfold the final shape:

• What relationship do m₁ and m₂, the lines of symmetry, have with each other?
• What do you notice about the sides of the shape?
• What kind of angles will the shape have?
• What kinds of shapes can you cut out?

Now consider the cut angle, x (see diageam below). How does varying the cut angle affect the shape that results when the cut off piece is unfolded? What about angle a. Is there a range of angles that will produce various kinds of shapes? Is there a relationship between the angle of fold m₂ and the cut angle? How does varying both affect the outcome?

Further on in the lesson, the students experiment with other types of folding and cutting activity. Throughout the discussions and explorations, appropriate geometry vocabulary is brought to the fore, and symmetry and congruence concepts are developed. Symmetry is further explored with mirrors in the next lesson.