More Origami Solutions for
Teaching
Selected Topics in Geometry
Deborah A. McAllister,
Professor
The
Shirley A. McDonald, Grade 7
Mathematics Teacher
Web Site:
http://oneweb.utc.edu/~deborah-mcallister/camta04origami.html
(See link for GCTM 2006)
Georgia Council of Teachers
of Mathematics
Georgia Mathematics
Conference
Friday, October 20, 2006, 1:15
p.m. – 3:30 p.m.
Session Description
Participants will learn basic origami folds that will be useful for teaching geometry concepts, including reflections, rotations, and geometric solids. Figures to be created will be selected from modular and action origami, and activities will be correlated to standards. Projects completed by middle grades students will be displayed. A list of origami Web sites will be provided.
We will be using some of the same and some different origami figures from the 2005 session.
Origami Figures
Sailboat
Samurai Helmet Pyramid
Box Tessellation
Hexahedron Pointed
Cube
Wreath and Pinwheel
Star
Sailboat
1. Begin with the colored side of the paper facing up.
2. Orient the paper so that one corner is pointed toward you.
3. Fold the bottom corner to the top corner (taco fold). Crease and unfold.
4. Rotate the paper 90 degrees, and fold the bottom corner to the top corner. Crease and unfold.
5. Turn the paper over, so that the white side of the paper is facing up.
6. Make two folds, bottom edge to top edge and left edge to right edge. Crease and unfold each.
7. Orient the paper so that one corner is pointed toward you.
8. Bring the left and right corners of the square together, so that they meet at the bottom corner. The top corner will also meet at the bottom corner.
9. Press flat. This forms the preliminary base.
10. Open the paper, with the white side facing up.
11. Fold the top and bottom corners to the center.
12. As you fold the top and bottom edges together, fold the sails to the center.
13. Fold the bottom corner to the center line of the back of the boat. This will form a flat base for the boat.
14. Fold the right-hand sail straight downward.
15. Fold the sail upward, making it shorter than the left-hand sail.
16. Tuck the leftover sail into the base of the boat.
17. Note the similar triangles.
The Sailboat, p. 52-55, from:
Morin, J. (1998). The ultimate origami book.
Samurai Helmet
1. Begin with the white side of the paper facing up.
2. Fold the paper in half, diagonally.
3. Place the isosceles triangle on the table with the right angle at the top.
4. Fold each base angle to the right angle.
5. Rotate the square 180 degrees.
6. Fold one of the bottom points to the top corner.
7. To make the horn, fold the top layer of the corner outward, so that the top edge is parallel to the center line.
8. Repeat the previous two steps to make the other horn.
9. Fold a single layer of the paper at the bottom to a point approximately halfway between the top of the helmet and the center line.
10. Fold up the lower edge of the same layer to form a horizontal band.
11. Flatten the helmet.
12. Fold the remaining lower corner to meet the top of the helmet.
13. Tuck the lower corner into the helmet.
14. This piece can be used as a corner bookmark.
Samurai Helmet, p. 49, from:
Beech, R. (2003). The origami handbook.
Box
1. Begin with the colored side of the paper facing up.
2. Make a book fold, and unfold, in both directions.
3. Turn the paper to the white side facing up.
4. Fold each corner to the center.
5. Cupboard fold, and unfold, the top edge and the bottom edge to the center.
6. Cupboard fold, and unfold, the left edge and the right edge to the center.
7. Unfold the top and bottom triangular flaps.
8. Turn up and crease the left and right cupboard folds to 90 degrees to form two sides of the box.
9. Push inward on the small creases where the colored and white sides of the paper meet. Make a fold on the crease that connects these two folds. This will form the third side of the box.
10. Pull the tab down and inside the box to line the box.
11. Repeat the previous two steps to form the fourth side of the box.
12. Reinforce the creases on the box.
13. To make the box cover, repeat the above directions, leaving a small gap at the center when making the cupboard folds.
Masu Box, p. 56-57, from:
Gross, G. M. (2001). Origami, Easy to make paper creations.
Hexahedron
1. Begin with a square sheet of paper, with the white side of the paper facing upward.
2. Book fold one vertical crease at center to form halves. Open.
3. Cupboard fold two vertical creases to center to form fourths. Open.
4. With the white side of the paper facing upward, fold dog ears at the upper left and the lower right of white side (isosceles right triangles with the corner touching the vertical line at the ¼ crease line).
5. Fold the sides to center (close the cupboard), with the open collar at the upper right and lower left.
6. Make a horizontal crease at center. Open that crease.
7. Fold the upper right corner to the mid-point of left-hand side. Open that crease.
8. Fold lower left corner to the mid-point of right-hand side. Open that crease.
9. Tuck the upper right corner into the left-hand flap.
10. Tuck the lower left corner into the right-hand flap.
11. Flip the parallelogram so that the smooth side is facing upward.
12. Fold the acute vertices to the opposite endpoint on the midline (forms a square).
13. Flip the square so that the rough/tucked side is facing upward.
14. Fold the piece into a triangle, along the tucked midline.
15. Open this piece to see four triangles.
16. A total of three strips are needed (fold two additional strips).
-----
17. Work with the tucked sides facing upward.
18. Place one strip vertically.
19. Place and tuck one strip horizontally at the middle.
20. Place and tuck one strip vertically at the center.
21. Push the unit up into a three-sided pyramid.
22. Hold the unit together as three-sided pyramid.
23. Tuck the loose ends into the pockets to form a hexahedron.
Cube
1. Begin with a square sheet of paper, with the white side of the paper facing upward.
2. Book fold one vertical crease at center to form halves. Open.
3. Cupboard fold two vertical creases to center to form fourths. Open.
4. With the white side of the paper facing upward, fold dog ears at the upper left and the lower right of white side (isosceles right triangles with the corner touching the vertical line at the ¼ crease line).
5. Fold the sides to center (close the cupboard), with the open collar at the upper right and lower left.
6. Make a horizontal crease at center. Open that crease.
7. Fold the upper right corner to the mid-point of left-hand side. Open that crease.
8. Fold lower left corner to the mid-point of right-hand side. Open that crease.
9. Tuck the upper right corner into the under left-hand flap.
10. Tuck the lower left corner into the right-hand flap.
11. Flip the parallelogram so that the smooth side is facing upward.
12. Fold the acute vertices to the opposite endpoint on the midline (forms a square).
13. Flip the square so that the rough/tucked side is facing upward.
14. Fold the piece into a triangle, along the tucked midline.
15. Open this piece to see four triangles.
16. A total of six strips are needed (fold five additional strips).
-----
17. Work with the tucked sides facing upward.
18. Place one strip vertically.
19. Place and tuck one strip horizontally at the middle.
20. Place and tuck one strip vertically at the center.
21. Push the left flap under the uppermost piece to form a corner of the cube.
22. Hold the unit together as a corner.
23. Tuck additional pieces into the pockets to form a cube. Each face of the cube will contain three colors, with one color on one pair of opposite sides, and two different colors on the other pair of opposite sides.
Wreath and Pinwheel
Materials: Eight paper squares, same size (two colors)
Do the following to
each square, with the white side of the paper facing up:
- Fold the square to create four creases: vertical in center, horizontal in center, two diagonals.
- Fold the top corners to the center to make the roof of a house.
- Fold the house in half such that the flaps are on the inside.
- Hold the half-house by the acute angle at the bottom left.
- Push in the bottom right corner to form a parallelogram.
Making the wreath:
- Position one piece with the folded edge to the left, and the acute angle at the bottom left.
- Position the next piece with the folded edge at the top and the acute angle at the upper left.
- Slide the acute angle on the right piece into the fold pocket of the left piece.
- Fold down the tips of the left piece into the valley of the right piece.
- Attach the remaining pieces.
- Connect the last piece to the first piece.
Making the pinwheel:
Gently slide the sides of the wreath toward the center.
Questions regarding
symmetry:
1. Describe the reflectional and rotational symmetries of the following:
- Square
- House
- Half-house
- Parallelogram
- Wreath
- Pinwheel
2. Slide the pinwheel to the wreath. Push on a pair of opposite sides to get a pinwheel with only two wings. What are the reflectional and rotational symmetries of this figure?
3. Slide the pinwheel to the wreath. Push on opposite sides to produce other shapes. What shapes can you create in the center opening?
Making a Wreath and a Pinwheel,
p. 73-75, from:
Lappan, G.,
Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998). Connected mathematics: Kaleidoscopes, hubcaps,
and mirrors.
Star
1. Begin with two squares of coordinating and/or contrasting paper.
2. Book fold, and unfold, each square in half once.
3. For each piece of paper, cut along the crease to form rectangles.
4. You will use one rectangle of each design to form one star.
5. The two rectangles will be folded as mirror images of each other.
6. With the colored side of the paper facing up, fold, and unfold, each rectangle so the short ends meet (hamburger fold).
7. With the white side of the paper facing up, fold each rectangle so the long ends meet (hot dog fold). Crease well.
8. Place each piece on the table with the open end facing away from you, and with one piece above the other.
9. On the upper piece, fold the upper left corner to the crease, forming a flap in the shape of an isosceles right triangle.
10. On the upper piece, fold the bottom right corner to the open edge, forming a flap in the shape of an isosceles right triangle.
11. On the bottom piece, fold the lower left corner to the open edge, forming a flap in the shape of an isosceles right triangle.
12. On the bottom piece, fold the upper right corner to the crease, forming a flap in the shape of an isosceles right triangle.
13. Note the line symmetry (reflection) of the two pieces.
14. Place your fingernail at the top edge of the center crease of the upper piece. Fold the right side downward to form an arrow pointing to the right.
15. Place your fingernail at the bottom edge of the center crease of the lower piece. Fold the right side upward to form an arrow pointing to the right.
16. Place your fingernail at the bottom edge of the center crease of the upper piece. Fold the left side upward to form an arrow pointing to the left.
17. Place your fingernail at the top edge of the center crease of the lower piece. Fold the left side downward to form an arrow pointing to the left.
18. Turn the upper piece over. Do not turn the lower piece.
19. Orient the upper piece so that it looks like a “Z” that has been rotated 45 degrees clockwise. Fold and crease both small triangles (delineated by an open midline on each larger triangle), top downward and bottom upward. Unfold.
20. Orient the lower piece so that it looks like a “Z” that has been rotated 45 degrees clockwise. There are no open midlines on this piece. Fold and crease each triangle into halves, top downward and bottom upward. Unfold.
21. Place the lower piece on top of the upper piece, with the centers aligned, and so that the pieces are perpendicular.
22. Fold the bottom point of the bottom piece up to tuck into the top pocket of the top piece.
23. Fold and tuck the top point of the bottom piece into the bottom pocket of the top piece.
24. Turn the joined pieces over.
25. Tuck the top point of the bottom piece into the left pocket of the top piece.
26. Tuck the bottom point of the bottom piece into the right pocket of the top piece.
Traditional Star, p. 8-12, from:
Gleason, K. (2003). Christmas origami. Back Pack Books.
(Hexahedron with one sheet of paper)
1. Begin with the white side of the paper facing up.
2. Fold the paper in half, diagonally, in both directions.
3. Place the isosceles triangle on the table with the right angle at the top.
4. To divide the hypotenuse into thirds:
a. Estimate a point on the hypotenuse as one-third of the distance from the bottom left vertex to the bottom right vertex. Make a small crease (point A).
b. Fold the bottom right vertex to point A. Make a small crease (point B).
c. Fold the bottom left vertex to point B. Make a small crease (point C).
d. Fold the bottom right vertex to point C. Make a small crease (point D).
5. Fold the bottom right vertex to point C. Crease.
6. Fold the bottom left vertex to point D, placing the vertex inside of the previously-formed triangle. Crease. (The figure will look like a house.)
7. Fold the triangles at the top (original right angle of the triangle), making creases toward both the front and back of the figure.
8. Mountain fold the top layer of the triangle so that it is tucked into the figure (into the house).
9. Crease each diagonal of the square with a mountain fold and a valley fold.
10. Open the figure by pinching the triangles.
11. Fold the lower triangle to the right, over one face of the figure.
12. To close the figure, tuck the upper half of the upper triangle into the lower triangle.
13. A garland can be made with several hexahedra, thread, and a long needle.
Boursin, D. (2005). Easy origami.
Pyramid
Bottom
1. Begin with the bird base, which begins with the preliminary base.
2. To make the preliminary base:
a. With the white side of the paper facing up, fold the paper in half, through the midpoints of a pair of opposite sides. Unfold. Repeat with the other pair of sides. (From the perspective of the printed side of the paper facing up, these are mountain folds.)
b. With the printed side of the paper facing up, fold one diagonal. Unfold. Fold the other diagonal. (From the perspective of the printed side of the paper facing up, these are valley folds.)
c. Press the center of the paper upward.
d. Push in on horizontal, opposite mountain folds.
e. Flatten the figure to make a small square, or the preliminary base.
3. To make the bird base:
a. With the open end at the bottom, fold the top layer of each side (bottom vertex to side vertex) to the center line, forming right triangles.
b. Fold down the upper triangle.
c. Unfold the three triangles.
d. Open the upper layer by pulling the vertex up and flattening the sides of the new parallelogram along the folds of the triangles.
e. Repeat steps a through d on the other side of the figure.
f. Flatten the figure to make a parallelogram, or the bird base.
4. Fold down the two top points, one to the back and one to the front. You will form a kite.
5. Pull apart the two bottom points to make the square bottom of the pyramid.
6. Shape the figure into a pyramid.
Top
1. Begin with the preliminary base, in a different size and color.
2. With the open end at the bottom, fold the top layer of each side (top vertex to side vertex) to the center line, forming right triangles on all four sides.
3. Unfold the sides.
4. Raise one side to vertical. Open and flatten it along the midline of the side, now at the center.
5. Repeat for the other three sides.
6. Crease the four printed faces.
To assemble the pyramid, put the inner folds of the top between the edges of the pyramid.
Pyramid, p. 60-61, from:
Boursin, D. (2005). Easy origami.
Tessellation
1. The color that is face-up will be the design. The color that is face-down will be the background.
2. With the background side face-down, make a diagonal fold. Unfold. Make the other diagonal fold.
3. Fold the bottom right corner to the center of the square. Repeat for the bottom left corner. Unfold each.
4. Fold each bottom corner to meet the crease mark to form a small triangle.
5. Fold each triangle over the crease mark to form a trapezoid.
6. Fold the top edge down to the center line. Unfold.
7. Fold the top edge to meet the crease mark.
8. Fold the top right corner to meet the center line. Repeat for the top left corner.
9. Turn the module over. The small horizontal segment should be at the top.
10. Fold the bottom corner to the center.
11. Fold the left corner to the center.
12. Fold the top edge down, reversing the crease.
13. Fold the right corner to the center. Unfold.
14. Fold the corner to meet the crease mark.
15. Fold along the crease mark.
16. You should have a square. Make three more of the same units.
To assemble the tessellation as a 2 x 2 array:
.................................................. Fold the left triangular flap of piece 1 over the right blunt
/ / / flap of piece 2.
/ 2 / 1 /
/ / / Fold the bottom triangular flap of piece 2 over the top blunt
.................................................. flap of piece 3.
/ / /
/ 3 / 4 / Fold the left triangular flap of piece 4 over the right blunt
/ / / flap of piece 3.
..................................................
Fold the bottom triangular flap of piece 1 over the top blunt flap of piece 4. Glue, tape, or staple (optional).
Tessellation, p. 8-11, from:
Gleason, K.A. (2006). Geogami.
Pointed
1. The color that is face-up will be the outer ring. The color that is face-down will be the inner ring. You will need eight small squares (cut/tear two large squares into four pieces each).
2. Make one diagonal fold.
3. Fold the upper layer of the right angle of the right triangle down to the left of the midpoint of the base to from two, noncongruent, right triangles. Unfold.
4. Repeat step 3, folding to the right of the midpoint of the base. Unfold.
5. Pull the edges of the top layer toward each other. Follow the creases to flatten at the bottom. Pinch the top into a triangle.
6. Fold the triangle to the right.
7. Turn the module over.
8. Fold the left vertex (larger, upper layer) to the right vertex. The shape of the piece will be a chevron.
9. Fold the upper layer back so that the two larger triangles are symmetrical and have parallel sides.
10. Turn the module over.
11. Make the additional seven pieces.
12. Connect the pieces as the horn of one into the pocket of the other.
13. Fold the central point to secure the horn in the pocket.
14. Continue, until the first piece connects to the last piece.
Pointed
Gleason, K.A. (2006). Geogami.
Georgia
From http://www.georgiastandards.org/math.aspx
Grade 3
M3M. Measurement
M3M3. Students will understand and measure
the perimeter of simple geometric figures (squares and rectangles).
b. Understand the concept of perimeter as being the boundary of a simple geometric figure.
M3G. Geometry
M3G1. Students will further develop their
understanding of geometric figures by drawing them. They will also state and
explain their properties.
a. Draw and classify previously learned fundamental geometric figures and scalene, isosceles and equilateral triangles.
b. Identify and explain the properties of fundamental geometric figures.
M3P. Process Skills
M3P3. Students will communicate
mathematically.
d. Use the language of mathematics to express
mathematical ideas precisely.
M3P4. Students will make connections among mathematical ideas and to other
disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of
mathematics.
M3P5. Students will represent mathematics in
multiple ways.
a. Create and use representations to organize, record, and
communicate mathematical ideas.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
Terms/Symbols: isosceles triangle.
Grade 4
M4M. Measurement
M4M2. Students will understand the concept of
angles and how to measure it.
a. Use tools, such as a protractor or angle ruler, and other methods such as paper folding, drawing a diagonal in a square, to measure angles.
b. Understand the meaning and measure of a half rotation (180o) and a full rotation (360o).
M4G. Geometry
M4G1. Students will define and identify the
characteristics of geometric figures
through examination and construction.
a. Examine and compare angles in order to classify and identify triangles by their angles.
b. Describe parallel and perpendicular lines in plane geometric figures.
c. Examine and classify quadrilaterals (including parallelograms, squares, rectangles, trapezoids, and rhombi).
d. Compare and contrast the relationships among quadrilaterals.
M4G2. Students will understand fundamental
solid figures.
a. Compare and contrast a cube and a rectangular prism in terms of the number and shape of their faces, edges, and vertices.
b. Describe parallel and perpendicular lines and planes in connection with the rectangular prism.
c. Construct/collect models for solid geometric figures (cube, prisms, cylinder, etc.).
M4P. Process Skills
M4P3. Students will
communicate mathematically.
d. Use the language of mathematics to express
mathematical ideas precisely.
M4P4.
Students will make connections among mathematical ideas and to other
disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of
mathematics.
M4P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and
communicate mathematical ideas.
c. Use representations to model and interpret
physical, social, and mathematical phenomena.
Terms/Symbols: parallel, perpendicular, diagonal line, plane, degree, rotation, parallelogram, trapezoid, rhombus, quadrilateral, congruent, cube.
Grade 5
M5P. Process Skills
M5P3. Students will
communicate mathematically.
d. Use the language of mathematics to express
mathematical ideas precisely.
M5P4.
Students will make connections among mathematical ideas and to other
disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of
mathematics.
M5P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and
communicate mathematical ideas.
c. Use representations to model and interpret
physical, social, and mathematical phenomena.
Terms/Symbols: irregular polygon, polygon.
Grade 6
Geometry
M6G1. Students will further develop their
understanding of plane figures.
a. Determine and use lines of symmetry.
b. Investigate rotational symmetry, including degree of rotation.
c. Use the concepts of ratio, proportion and scale factor to demonstrate the relationships between similar plane figures.
Process Standards
M6P3. Students will communicate
mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M6P4. Students will make connections among
mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M6P5. Students will represent mathematics in
multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
Terms/Symbols: proportional relationships, right rectangular prism, pyramid, geometric solid, geometric figures, line symmetry, rotational symmetry, similar plane figures, scale factor.
Grade 7
Geometry
M7G2. Students will demonstrate understanding
of transformations.
a. Demonstrate understanding of translations, dilations, rotations, reflections, and relate symmetry to appropriate transformations.
M7G3. Students will use the properties of
similarity and apply these concepts to geometric figures.
a. Understand the meaning of similarity, visually compare geometric figures for similarity, and describe similarities by listing corresponding parts.
c. Understand congruence of geometric figures as a special case of similarity: The figures have the same size and shape.
Process Standards
M7P3. Students will communicate
mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M7P4. Students will make connections among
mathematical ideas and to other
disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M7P5. Students will represent mathematics in
multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
Terms/Symbols: polyhedron, translation, rotation, reflection, dilation, symmetry, bisector, parallel lines, perpendicular lines, similar, congruent, point, line, plane, line segment, endpoints, diagonal
Grade 8
Geometry
M8G1. Students will understand and apply the
properties of parallel and perpendicular lines and understand the meaning of
congruence.
a. Investigate characteristics of parallel and perpendicular lines both algebraically and geometrically.
b. Apply properties of angle pairs formed by parallel lines cut by a transversal.
c. Understand the properties of the ratio of segments of parallel lines cut by one or more transversals.
d. Understand the meaning of congruence: that all corresponding angles are congruent and all corresponding sides are congruent.
Process Standards
M8P3. Students will communicate
mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M8P4. Students will make connections among
mathematical ideas and to other
disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M8P5. Students will represent mathematics in
multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
Terms/Symbols: transversal, vertical angles, complementary angles, supplementary angles, alternate interior angles, alternate exterior angles, corresponding angles