1. 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

3. Polygon Task: Triangles Lined Up in A Row Learning and Teaching Linear Functions Nanette Seago, Judith Mumme and Nicholas Branca Shapes, Functions, & Patterns Name: ____________________________________

4. Polygon Task: Triangles Lined Up in A Row WORKSPACE

5. Shapes Functions, & Patterns Predict the volume of building 100. Find a rule for any building (n).

6. Block Structure: What is the volume of building 10? WORKSPACE

7. 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. 4.5. 6. Name: ____________________________________ Volume: _______ _______ _______ _______ _______ _______ # of Faces: _______ _______ _______ _______ _______ _______ # of Edges: _______ _______ _______ _______ _______ _______ # of Vertices: ______ _______ _______ _______ _______ _______

8. 1. 2.3. 4.5. 6.

9. 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

10. 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.

11. Trim the paper to the grid and cut out a 4 x 4 square from each corner.

12. Trim the paper to the grid and cut out a 3 x 3 square from each corner.

13. Trim the paper to the grid and cut out a 5 x 5 square from each corner.

14. Trim the paper to the grid and cut out a 6 x 6 square from each corner.

15. Trim the paper to the grid and cut out a 2 x 2 square from each corner.

16. Investigating Nets Cube pattern follows…See word document for Cylinder pattern

19. Investigating Nets Name: ____________________________________

20. Investigating Nets

22. Name: ____________________________________

23. Investigating Nets Tetra- hedron Hexa- hedron Octa- hedron Dodeca- hedron Icosa- hedron What shape are all the faces?trianglesquaretrianglepentagontriangle How many faces?4681220 How many edges?612 30 Each face touches how many vertices?34353 Each edge joins how many faces?22222 Each vertex touches how many faces?33435 If the edge measures one linear unit, find the approximate surface area of the polyhedron. 1.72 un 2 6 un 2 3.44 un 2 20.64 un 2 8.60 un 2 ANSWER KEY

24. 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.

25. 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.Label the base area amounts on the shapes you drew on the grid paper. 2.List any two solids which have the same base.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?Which rectangular prism has a base congruent to the base of the square pyramid AND has the same height as the square pyramid? Name: ________________________________________

26. 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? Volume of 3-D Shapes Use Power Solids to compare the volume of 3-D shapes.

27. 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.

29. Geometry/Science Connection Finding the Formula for the Surface Area of a Sphere

30. 1. 2. 3. 4.

31. 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. ________________________

32. 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. ________________________ Great Circle The inside of the sphere sliced through the Great Circle or the base of a hemisphere A = ∏ r 2 It takes twice as much string to cover the outside of the hemisphere as it does to cover the base of the hemisphere. SA sphere = 4 ∏ r 2