Presented by:
CATHY JONES
Secondary Math Instruction Specialist
Center for Mathematics and Science Education
Arkansas NASA Education Resource Center
WAAX 202 #1 University of Arkansas
Fayetteville, Arkansas 72701
(479) 575-3875
(479) 575-5680 (FAX)
e-mail: cej001@uark.edu
http://www.uark.edu/~k12 info/
Download all materials from this session at www.cmasemath.pbwiki.com
Name: ____________________________________
Shapes, Functions, & Patterns
Polygon Task: Triangles Lined Up in A Row
Learning and Teaching Linear Functions
Nanette Seago, Judith Mumme and Nicholas
Branca
Polygon Task: Triangles Lined Up in A Row
WORKSPACE
Predict the volume of building 100.
Find a rule for any building (n).
Block Structure: What is the volume of building 10?
WORKSPACE
Name: ____________________________________
A) Build the exact models using the appropriate colored
linking cubes. We use the linking cubes so we can keep it
together when picking it up and moving it around.
Remember the cubes cannot hang in space.
B) Draw the design on Grid paper or Isometric Dot paper.
C) Determine the volume, number of faces, number of
edges, and number of vertices of each model.
1.
2.
3.
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# of Faces: _______
_______
# of Edges: _______
# of Vertices: ______
Volume:
4.
5.
6.
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1.
2.
3.
4.
5.
6.
Designing the Largest Box
Functions from Formulas
The owner of a large factory has 100
rectangular sheets of metal 11 feet by 8.5 feet.
These sheets need to be made into vats that
will hold the greatest amount possible of the
waste from the plant. These vats must all be of
equal size and will be constructed by turning up
the sides and welding. You own a metal shop
and can get this job if you convince the owner
that you can build the vat with the greatest
volume.
Designing the Largest Box
Functions from Formulas
• Begin with a rectangular sheet of 8 ½” x 11”
cardstock.
• From each corner, cut a square of assigned size.
• Fold up the four resulting flaps, and tape them
together to form an open box.
• The volume of the box will vary, depending on the
size of the squares. Write a formula that gives the
volume of the box as a function of the size of the
cutout squares.
• Use the function to determine what size the
squares should be to create the box with the
largest volume.
Trim the paper to the grid and cut out a 4 x 4 square from each corner.
Trim the paper to the grid and cut out a 3 x 3 square from each corner.
Trim the paper to the grid and cut out a 5 x 5 square from each corner.
Trim the paper to the grid and cut out a 6 x 6 square from each corner.
Trim the paper to the grid and cut out a 2 x 2 square from each corner.
Investigating Nets
Cube pattern follows…See word document for Cylinder pattern
Name: ____________________________________
Investigating Nets
Investigating Nets
Name: ____________________________________
Investigating Nets
Investigating Nets
ANSWER KEY
Tetrahedron
Hexahedron
Octahedron
Dodecahedron
Icosahedron
triangle
square
triangle
pentagon
triangle
How many faces?
4
6
8
12
20
How many edges?
6
12
12
30
30
Each face touches how many vertices?
3
4
3
5
3
Each edge joins how many faces?
2
2
2
2
2
Each vertex touches how many faces?
3
3
4
3
5
1.72 un2
6 un2
3.44 un2
20.64 un2
8.60 un2
What shape are all the faces?
If the edge measures one linear unit, find the
approximate surface area of the polyhedron.
Area and Volume of 3-D Shapes
Use Power Solids to compare the area and volume of 3-D shapes.
Investigating Using Power Solids
Ordering By Area of the Base
Working with your group, put your power solids in order from smallest to
largest by the area of their bases. Check the order by tracing the shape onto grid
paper and counting the squares or by measuring the dimensions and computing the
base area of each solid.
Investigating Using Power Solids…Ordering by Volume
Working with your group, put your power solids in order from smallest to
largest by their volume. Check the order by filling and pouring rice or sand.
Name: ________________________________________
Area of 3-D Shapes
Use Power Solids to compare the area of 3-D shapes.
Investigating Using Power Solids
Ordering By Area of the Base
Working with your group, put your power solids in order from smallest to
largest by the area of their bases. Check the order by tracing the shape onto grid
paper and counting the squares or by measuring the dimensions and computing the
base area of each solid.
1. Label the base area amounts on the shapes you drew on the grid paper.
2. List any two solids which have the same base.
3. Which rectangular prism has a base congruent to the base of the square pyramid AND has
the same height as the square pyramid?
Volume of 3-D Shapes
Use Power Solids to compare the volume of 3-D shapes.
Investigating Using Power Solids…Ordering by Volume
Working with your group, put your power solids in order from smallest to largest by their
volume. Check the order by filling and pouring rice or sand.
1.
How many square pyramids does it take to fill the rectangular prism with the same base? __________
2.
How many triangular pyramids does it take to fill the triangular prism with the same base? _________
3.
What could we write about the volumes of pyramids and prisms that have congruent bases and the same heights?
4.
How many cones does it take to fill the cylinder? ____________
5.
What could we write about the volumes of a cone and a cylinder having the same height and congruent bases?
6.
How many cones does it take to fill the hemisphere? _________
7.
What could we write about the volumes of a cone and a hemisphere if the base of the cone is congruent to the great
circle of the hemisphere and the height of the cone is the diameter of the sphere?
8.
What do we know about the diameter of the great circle of the sphere and the height of the cone?
9.
How many hemispheres does it take to fill the sphere? _________
10. How many cones does it take to fill the sphere? ______________
11. What could we write about the volumes of a cone and a sphere when the height of the cone is congruent to the
diameter of the great circle of the sphere?
Activity: Tripyramidal Box
1. Cut out the patterns on the solid lines.
2.Fold back on the scored lines.
3. Close the nets to make 3 pyramids.
4. Now put the 3 pyramids together to make a box or
“tripyrmidal”.
5. Discuss…Volume of CUBE = lwh.
The area of the Base can be B=lw, so the Volume of the CUBE
could also be written as V = Bh
6. We can now develop a formula for the Volume a Pyramid? The
volume of one of the pyramids is given by the formula
V = (1/3)lwh = (1/3)Bh.
Finding the Formula
for the
Surface Area of a Sphere
Geometry/Science Connection
1.
3.
2.
4.
Find the Formula for Surface Area of a Sphere
Name:___________________________
Circumference, Area, Surface Area
1. What part of a planet or sun would the circular ring represent? __________________
2. When we look at a 2-dimensional picture of a planet or sun what does the circle represent? __________________________________
_________________________________________________________________________________________________________________
3. What is the formula for the area of a flat circular surface? ______________________
INVESTIGATE
Use string, a nail, and a styrofoam hemisphere and cover the flat surface with the string. Mark off the amount required to cover. Now cover
the outside of the hemisphere (not including the flat surface). Compare your measures.
4. Describe what you found. ________________________________________________________________________________
_______________________________________________________________________________________________________
5. From what you have found write a formula for covering the entire surface area of the sphere. ________________________
Find the Formula for Surface Area of a Sphere
Circumference, Area, Surface Area
Name:___________________________
Great Circle
1. What part of a planet or sun would the circular ring represent? __________________
2. When we look at a 2-dimensional picture of a planet or sun what does the circle represent? __________________________________
The inside of the sphere sliced through the Great Circle or the base of a hemisphere
_________________________________________________________________________________________________________________
A = ∏ r2
3. What is the formula for the area of a flat circular surface? ______________________
INVESTIGATE
Use string, a nail, and a styrofoam hemisphere and cover the flat surface with the string. Mark off the amount required to cover. Now cover
the outside of the hemisphere (not including the flat surface). Compare your measures.
It takes twice as much string to cover the outside of the
4. Describe what you found. ________________________________________________________________________________
hemisphere as it does to cover the base of the hemisphere.
_______________________________________________________________________________________________________
SA sphere = 4 ∏ r2
5. From what you have found write a formula for covering the entire surface area of the sphere. ________________________