Download Assign 2 - MathExperiences

SED ME 559
Paper Folding
Spring 2010
Day 3 Classwork
I. Folding the Circle
Adapted from: Sobel, Max A. and Maletsky, Evan M.,Teaching Mathematics a Sourcebook of Aids, Activities, and
Strategies. Englewood Cliffs, NJ: Prentice Hall , 1975.
A lot of different shapes can be made by folding a circle. Along the way, many geometric and
arithmetic concepts can be discussed.
Begin with a circle. Discuss the following terms: circle, point, line, segment, circumference,
chord, radius, diameter, center, and symmetry. If the circle does not have the center marked,
have students find the center.
Ask questions! Is a diameter a chord? (yes) Is a chord a diameter? (not necessarily) If you
were to take a piece of string the length of the diameter and wrap it around the circumference,
would it go all the way around? How many pieces that size would it take to go around? (a
little more than three pieces) Does a circle have sides? Is it symmetrical? How many lines of
symmetry?
Pick any point of
the circle and fold
to the center.
Is the new figure symmetrical? (yes)
How many lines of symmetry? (one)
Fold again so it forms a sharp point.
Discuss: vertex, angle, acute angle, right angle, obtuse angle, straight angle, and rays.
Is the new figure symmetrical? (yes) How many lines of symmetry? (one)
Fold a third time to form two more vertices so a point on the circle will still pass through
the center of the circle.
Discuss: triangle (could it be called trilateral or trivertex?), equilateral, isosceles,
equiangular, scalene, polygon, regular (all sides and angles have the same measure) polygon,
perimeter, area, and, of course, symmetry.
Find the midpoint of one side by creasing lightly. Visualize what figure will show if the
opposite vertex is folded to the midpoint. Do the fold. Were you right?
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SED ME 559
Paper Folding
Spring 2010
Discuss: quadrilateral, trapezoid, isosceles trapezoid, symmetry, and maybe area formula if it
is review. Is it possible to have a right (base angle) trapezoid? an obtuse (base angle)
trapezoid? Is it regular?
If you fold over one of the triangles, what will you get? Visualize! Now do it.
Discuss: Parallelogram, rhombus, why it is not a square, area, symmetry, and if it is regular.
Fold over the other triangle and hold it tightly.
Discuss: how this triangle is related to the original triangle and congruent triangles. What part
of the original triangle is this one? What part of the trapezoid? What part of the rhombus?
Check out your answers.
Set the small triangle in the palm of your hand and let the triangles pop up and form a
vertex.
Unfold the tetrahedron to the large triangle. Fold one vertex to the center of the circle.
What is this figure called? (polygon - quadrilateral - trapezoid - isosceles trapezoid) Is it
regular? Symmetrical? How many lines?
Fold in another vertex.
What is this called? (polygon - pentagon) Is it regular? Symmetrical?
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SED ME 559
Paper Folding
Spring 2010
Fold in the third vertex.
What is this called?(polygon - hexagon) Is it regular? (yes!) How do you know?
Symmetrical?
Lay the hexagon in the palm of your hand and gently squeeze until the triangles overlap.
What is this shape? A truncated tetrahedron. It is also called a frustum - look it up!!
Go back to the large triangle. Fold in the trapezoid, then fold back on the crease obtained
when you made the hexagon.
Discuss: convex versus concave. What is it? (nonagon) Symmetrical?
Do the same fold for another of the triangles that made up the trapezoid.
What is it? (concave 11-gon - undecagon) Symmetrical?
Do the same fold with the third triangle.
What is it? (dodecagon) A star! (Star of David)
Lots of other polygons can be made by folding in along folds (convex polygons) or by folding
under on existing folds (concave polygons). How many can you find? Classify them.
Fraction work can be done by letting the original triangle have an area of one unit.
Then find the surface area of the isosceles trapezoid. …the rhombus . …the hexagon .
Find the areas of all the polygons formed.
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Paper Folding
Spring 2010
II. Paper Folding Geometry
1. Construct a straight line. Mark a point on your paper. Construct a line through the point.
Draw a line on your paper. Construct a perpendicular to the line. Construct a
perpendicular to a point on the line. Construct a perpendicular to a point not on the line.
2. Draw a line on your paper. Construct a parallel to the line. Construct a parallel to a point
not on the line. Can you do it without constructing two perpendiculars?
3. Draw an acute angle. Construct the angle bisector. Draw an obtuse angle. Construct the
angle bisector.
4. Construct a 45 degree angle. Construct a 135 degree angle. Construct a 22.5 degree
angle. Construct a 60 degree angle. Construct a 30 degree angle. Construct a 15 degree
angle. Can you name all the angles you could easily construct with paperfolding?
Assume that you can bisect an angle twice (that is, divide it into four equal parts), but
after that the measures wouldn't be precise enough.
5. Draw a line segment AB. Construct the midpoint M. Construct the perpendicular
bisector of AB.
6. Draw a horizontal line segment. Call it AB.
a. Construct an isosceles triangle with AB as one of the two equal sides.
b. Construct an isosceles triangle with the vertex angle greater than 90 degrees (an
obtuse triangle).
c. Construct an isosceles triangle with the vertex angle less than 90 degrees (an acute
triangle).
7. Draw a line segment.
a. Construct an acute isosceles triangle with this line as the base.
b. Construct an obtuse isosceles triangle with this line as the base.
8. Take any of the above isosceles triangles and fold one base angle onto the other. What do
you notice?
9. Construct a right triangle. Construct a right isosceles triangle. Construct a 30-60-90 right
triangle.
10. Draw two straight lines that intersect a point. Show, by construction, that the vertical
angles formed are equal.
11. Draw a right triangle with the right angle constructed. Construct the midpoint of the
hypotenuse. Draw the median from this midpoint. How does the length of the median
compare to the length of the hypotenuse? Why?
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