1. Unwrapping Solids Living on Flatland
How do you make a flat picture of something that exists in three dimensions?
Anticipatory set for M&G 3.5 up to slide 14. After that it is part of instruction
Ceci n’est pas une pipe is the painting by Rene Magritte - “This in not a pipe”.
M&G 3.5 Anticipatory Set

2. A Picture of a Pipe M&G 3.5 Anticipatory Set
How does the picture of something represent it? Can we understand all of an object’s qualities by looking only at a picture of it?
M&G 3.5 Anticipatory Set

3. Which way is up? M&G 3.5 Anticipatory Set
Where does our depth perception do to this 3-D picture in 2-D space.
M&G 3.5 Anticipatory Set
Which way is up?

4. Is it moving? M&G 3.5 Anticipatory Set
What are the visual effects you experience when staring at this picture?
Is it moving?
M&G 3.5 Anticipatory Set

5. Map Projections M&G 3.5 Anticipatory Set
Well, so what does this have to do with nets?
M&G 3.5 Anticipatory Set

6. Gerardus Mercator
Gerardus Mercator ( ). Frontispiece to Mercator's Atlas sive Cosmographicae, Courtesy of the Library of Congress, Rare Book Division, Lessing J. Rosenwald Collection.
M&G 3.5 Anticipatory Set

7. Map Projections
A map projection is used to portray all or part of the round Earth on a flat surface. This cannot be done without some distortion.
Every projection has its own set of advantages and disadvantages. There is no "best" projection.
The mapmaker must select the one best suited to the needs, reducing distortion of the most important features.
Map projections are nets.
M&G 3.5 Anticipatory Set

8. Mapmakers and mathematicians have devised almost limitless ways to project the image of the globe onto paper. Scientists at the U. S. Geological Survey have designed projections for their specific needs—such as the Space Oblique Mercator, which allows mapping from satellites with little or no distortion.
Why it is important.
M&G 3.5 Anticipatory Set

9. Projections of the Earth onto Geometric Solids
M&G 3.5 Anticipatory Set

10. A Mercator Projection M&G 3.5 Anticipatory Set
M&G 3.5 Anticipatory Set

11. http://www. colorado. edu/geography/gcraft/notes/mapproj/mapproj_f
M&G 3.5 Anticipatory Set

12. http://www. colorado. edu/geography/gcraft/notes/mapproj/mapproj_f
M&G 3.5 Anticipatory Set

13. http://www. colorado. edu/geography/gcraft/notes/mapproj/mapproj_f
M&G 3.5 Anticipatory Set

14. Buckminster Fuller’s Dymaxion Projection of the Earth
This mapping best preserves area.
M&G 3.5 Anticipatory Set

15. Unfolded Dymaxion map with nearly-contiguous land masses.
M&G 3.5 Anticipatory Set

16. What are we learning today?
What is a net?
Why do we need nets?
How to create nets for common solids.
Prisms
Cylinders
Pyramids
Cones
How do we create a solid from its net?
M&G 3.5 Anticipatory Set

17. Nets
In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron.
Polyhedral nets are a useful aid to the study of polyhedra and solid geometry in general, as they allow for models of polyhedra to be constructed from material such as thin cardboard.
When the Earth is mapped on a polyhedron, its net is a flat world map, e.g. the Dymaxion map.
Beginning of lesson.
Have geometric solids that can be unfolded available for student use.
Net of a dodecahedron
M&G 3.5 Anticipatory Set

18. Hypercube - A representation of four dimensions
Net of a tesseract
Hypercube - A representation of four dimensions
The net is on the left. Since the tesseract is a 4 diminsional figure, its net is in 3 dimensions.
Make this with multilink cubes.
M&G 3.5 Anticipatory Set

19. The Lateral Surface of a cylinder and its net.
Have a model of this made from a piece of paper.
The Lateral Surface of a cylinder and its net.
M&G 3.5 Anticipatory Set

20. Net for a Cylinder
Show model from solids kit
M&G 3.5 Anticipatory Set

21. Net for a Rectangular Solid
Show physical model
M&G 3.5 Anticipatory Set

22. Faces, Edges, and Vertices
Face - the polygonal sides of a polyhedron
Edge - the line segment formed by two adjacent faces of a polyhedron.
Vertices - The points on a polyhedron where three or more edges meet.
M&G 3.5 Anticipatory Set

23. Faces, edges and vertices
How many faces in total?
How many vertices in total?
How many edges in total? Remember that some edges are shared, but this does not show up on the net.
edges
edges
M&G 3.5 Anticipatory Set

24. Net for a Square Pyramid
Show students model
M&G 3.5 Anticipatory Set

25. Net for a Rectangular Prism
M&G 3.5 Anticipatory Set

26. Net for a Hexagonal Prism
M&G 3.5 Anticipatory Set

27. What polyhedron does this net make?
Triangular Prism
M&G 3.5 Anticipatory Set

28. What Three - Dimensional shape does this net make?
A cone
M&G 3.5 Anticipatory Set

29. What polyhedron does this net make?
Rectangular Prism
M&G 3.5 Anticipatory Set

30. What polyhedron does this net make?
Pentagonal Pyramid
M&G 3.5 Anticipatory Set