# Geometry of Minerals

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1 Geometry of Minerals Objectives Students will connect geometry and science Students will study 2 and 3 dimensional shapes Students will recognize numerical relationships and write algebraic expressions Suggested Grade Levels 5 th 8 th Grade Subject areas Language Arts Science Technology Timeline Approximately four, 45 minute periods Background Knowledge Students need to have studied the following in other classes. Science - rocks, minerals, and characteristics of minerals. Technology - Internet surfing, saving images, basic desktop publishing in technology. Math - basic use of variables and writing expressions. Materials Rocks with visible crystals 3-dimensional geometric models would be helpful Rulers Microscope or magnifying glass Polygon vocabulary sheet Student activity sheets Lesson See activity sheets below

2 Vocabulary Polygon a closed shape with all straight edges Face flat surface of a polygon Vertex corner point where two sides intersect Vertices plural for vertex Edge on a polygon, the segment that connects two vertices on a pyramid or prism, the segment where two faces meet Diagonal the segment from one vertex to any other nonadjacent vertex Prism Three-dimensional shape with parallel top and bottom faces Pyramid three-dimensional shape with one base and the top comes to a point Polygons Name Number of sides Triangle 3 Quadrilateral 4 Pentagon 5 Hexagon 6 Heptagon 7 Octagon 8 Nonagon 9 Decagon 10 Undecagon 11 Dodecagon 12 n-gon n sides

3 Rocks and Geometry Activity Sheet 1 Purpose: Investigate the mathematical properties of the shapes, designs, and structures in rocks. Supplies: rocks, microscope or magnifying glass, ruler or straight edge Procedure: 1. Select a rock. 2. Use a microscope or magnifying glass to locate a crystal in the rock. 3. Examine 1 face of the crystal. 4. Count the number of edges on 1 face of the crystal. 5. Using the straightedge, draw the shape in the space provided. 6. Fill in the number of edges and vertices. 7. Write the name of the polygon. 8. Repeat steps 2 7 to find 2 different shapes. 9. Complete all questions. Let s study the geometry in the crystals! 1. What type of rock do you have? 2. What characteristics did you use to classify your rock? Hint for drawing a polygon. First draw a circle. Then determine the number of edges of the polygon. Next, place that many dots on the circle. Finally, use a straight edge to connect the dots. Use a straight edge to draw your shapes below. Look for a pattern Polygon 1 Number of edges Name of polygon (from the polygon vocabulary sheet) Number of faces

4 Polygon 2 Number of edges Name of polygon (from the polygon vocabulary sheet) Number of faces Polygon 3 Number of edges Name of polygon (from the polygon vocabulary sheet) Number of faces Questions to ponder 1. If a polygon had 15 sides, how many vertices would it have? 2. If a polygon had 28 sides, how many vertices would it have? 3. If a polygon had 100 sides, how many vertices would it have? 4. Use a complete sentence to describe how the number of edges compares to the number of vertices. 5. Write a math equation the express the relationship between the number of edges and the number of vertices. 6. Joe says his rock had a polygon with 6 edges and 7 vertices but he already took his rock back home with him. What would you say to him about his polygon and how would you back up your statement?

5 Rocks and Geometry Activity Sheet 2 Let s look for more patterns in polygons. On each shape below draw all the diagonals possible but only from 1 vertex. Determine the number of edges, the number of diagonals you drew, and the number of triangles formed. Name of polygon Number of edges = Number of diagonals = Number of triangles formed= Name of polygon Number of edges = Number of diagonals = Number of triangles formed= Name of polygon Number of edges = Number of diagonals = Number of triangles formed= Name of polygon Number of edges = Number of diagonals = Number of triangles formed=

6 Questions to ponder Look for patterns in the numbers on your polygons to try to answer these questions. Consider a decagon. How many diagonals could you draw from 1 vertex? How many triangles would be formed? 1. Consider a polygon with 22 sides. How many diagonals could you draw from 1 vertex? How many triangles would be formed? 2. Consider a triangle with 53 sides. How many diagonals could you draw from 1 vertex? How many triangles would be formed? 3. Use a complete sentence to describe how the number of diagonals you can draw from 1 vertex compares to the number of edges on any polygon. 5. Write an algebraic equation to show the above relationship. 6. Describe how the number of triangles formed compares to the number of edges when the diagonals are drawn from 1 vertex on any polygon. 7. Write an algebraic equation to show this relationship. 8. Can a polygon with 33 sides, have 30 diagonals from 1 vertex, and form 30 triangles? Explain your answer.

7 Rocks and Geometry Activity Sheet 3 Here s a Challenge! How many diagonals can you draw in any polygon? (You might want to use a calculator.) Before you start Try this method to draw the diagonals on the hexagon below. At point A use dotted lines to draw the ` diagonals. Count how many diagonals ` you drew. Write that number above the letter A. A 3 3 B At point B use a dashed line to draw the Try diagonals. this Count how many you drew. Write that number above B. At point C, use a different kind of line and draw all the diagonals that haven t already been drawn. Count the new diagonals you drew and write the number below the C. Repeat Try these! the same procedure at each vertex. To find the total number of diagonals, add the numbers you wrote at each vertex. F 0 E 0 Total number of diagonals is = 9 D 1 C 2 Try this method to determine the number of diagonals in a pentagon, an octagon, and a decagon. Total number of diagonals =

8 Total number of diagonals = Total number of diagonals = Look for a pattern in the number of diagonals drawn from consecutive vertices. Try these! 1. How many diagonals would a 15-gon have? 2. How many diagonals would a 25-gon have? 3. Use complete sentences to tell someone on the phone how to calculate the number of diagonals in a polygon with 40 sides.

9 Rocks and Geometry Activity Sheet 4 Let s Go 3-D! Look at your rock again with your microscope or magnifying glass. This time look for the whole crystal with its flat sides and top and bottom surfaces. Describe what you see using the geometry terms of faces, edges, and vertices. Tell how many faces you see on 1 crystal and give the shape of the faces on the crystal. Try to draw what the crystal would look like if you took it out of the rock. Trade rocks with someone. Examine this rock, too. Write a description of the crystals you see in this rock. Write a brief comparison of the similarities and differences between the crystals in the 2 rocks.

10 Did some of the shapes look a little like these? How were your crystal shapes similar to these? How were your crystal shapes different from these? Let s look at some geometric characteristics. 1. What is the shape of the base? 2. How many edges does the base have? 3. How many faces does this shape have? (Be sure to count the ones on the back that you can t see) 4. How many edges does the shape have? (Be sure to count the ones on the back that you can t see) 5. How many vertices does the shape have? (Be sure to count the ones on the back that you can t see) 6. How does the top of the shape compare to the bottom?

11 The Crystal Systems of Minerals The crystals in minerals all grow in one of these six shapes on the microscopic level. Remember, however, that minerals can grow in many different ways. Cubic System Tetrahedral System Monoclinic or Rhombohedral System Triclinic System Orthorhombic System Hexagonal System People who work in geology, gemology, and crystallography classify crystals in different ways. Some people classify them by the way light bends through it, others classify them completely by points and lines of symmetry and rotation. All classifications are based on the various axis of rotation of the basic crystal shapes. An imaginary line runs through a perfect crystal so that during a single rotation around this line the outline of the crystal form appears identically 2, 3, 4 or 6 times. Look on the Internet at for the web site Crystallography and Minerals Arranged by Crystal Form. You will see the axis of symmetry for different crystal classifications. Notice the crystals aren t the basic shapes you see above. Remember, crystals can form in many ways. Check the Internet again, at for the web site Virtual Crystallography. The animations demonstrate how the crystal shapes could morph in their development.

12 Rocks and Geometry Activity Sheet 5 In 3-dimensional geometry there are 2 kinds of solids: Prisms and Pyramids. Pyramids come to a point at the top like the Egyptian pyramids. Prisms have flat and parallel tops and bottoms like the crystals in minerals. Let s look for some patterns in Prisms and Pyramids. Fill in the chart. You might want to make or use some models to help you count. Shape of Type of pyramid Number of Number base vertices faces Triangle Triangular pyramid of Number edges of Square Square pyramid Pentagon Pentagonal pyramid Hexagon Hexagonal pyramid Shape base Triangle of Type of prism Number of vertices Triangular prism Number faces of Number edges of Square Square prism Pentagon Pentagonal prism Hexagon Hexagonal prism

13 Let s look for some patterns and relationships in Pyramids. Vertices 1. If the base of a pyramid were a decagon, how many vertices would it have? 2. If the base of a pyramid had 28 sides, how many vertices would it have? 3. If the base of a pyramid had 100 sides, how many vertices would it have? 4. What is the relationship between the number of edges of the base and the number of vertices? Faces 1. If the base of a pyramid were a nonagon, how many faces would it have? 2. If the base of a pyramid had 38 sides, how many faces would it have? 3. If the base of a pyramid had 100 sides, how many faces would it have? 4. What is the relationship between the number of edges of the base and the number of faces? 5. Write an algebraic equation for that relationship. 5. Write an algebraic equation for that relationship. Edges 1. If the base of a pyramid had 23 sides, how many edges would it have? 2. If the base of a pyramid had 45 sides, how many edges would it have? 3. If the base of a pyramid had 100 sides, how many edges would it have? 4. What is the relationship between the number of edges of the base and the number of faces? 5. Write an algebraic equation for that relationship.

14 Let s look for some patterns and relationships in Prisms. Vertices 1. If the base of a prism were a decagon, how many vertices would it have? 2. If the base of a prism had 28 sides, how many vertices would it have? 3. If the base of a prism had 100 sides, how many vertices would it have? 4. What is the relationship between the number of edges of the base and the number of vertices? 5. Write an algebraic equation for that relationship. Faces 1. If the base of a prism were a nonagon, how many faces would it have? 2. If the base of a prism had 38 sides, how many faces would it have? 3. If the base of a prism had 100 sides, how many faces would it have? 4. What is the relationship between the number of edges of the base and the number of faces? 5. Write an algebraic equation for that relationship. Edges 1. If the base of a prism had 23 sides, how many edges would it have? 2. If the base of a prism had 45 sides, how many edges would it have? 3. If the base of a prism had 100 sides, how many edges would it have? 4. What is the relationship between the number of edges of the base and the number of faces? 5. Write an algebraic equation for that relationship.

15 Minerals from Mars! / Extension We know some of the minerals on Mars because meteorites from Mars have been recovered from Antarctica. The following minerals are found on Mars: Hematite Calcite Olivine Plagioclase Clinopyroxene Pyrite Orthopyroxene Magnesite Jarosite Magnetite We can look up the properties of these minerals and find their crystal class on the Internet. Go to again, Crystallography. Select A to Z List in the top navigation bar. Use the alphabetical listings to look up the characteristics of the minerals and their characteristics in the list above.

17 Extensions In Geometry, students can study the 5 platonic solids, the special properties, tessellating characteristics, variations, and find nets to construct them. In Geometry, students can design the structure of the museum using the shapes of the prisms and pyramids. The web site Nets of Crystals provides links to patterns you can cut out and tape together to make paper models of the crystal shapes. The address is Evaluation/Assessment Museum Curator Project. Resources Relational GeoSolids. Learning Resources, Inc. Vernon Hills, IL. Geometry of Crystals. Ward s Natural Science Establishment, Inc. Rochester, NY. Geometry in the Middle Grades. NCTM. Reston, VA

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