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Connecting Geometric and Algebraic Representations

Connecting Geometric and Algebraic Representations. Cheryl Olsen Shippensburg University October 11, 2005. Objectives. Geometric figures: Understand the concepts of length, area, volume & surface area Recognize/represent 3-dimensional figures

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Connecting Geometric and Algebraic Representations

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  1. Connecting Geometric and Algebraic Representations Cheryl Olsen Shippensburg University October 11, 2005

  2. Objectives • Geometric figures: • Understand the concepts of length, area, volume & surface area • Recognize/represent 3-dimensional figures • Understand formulas for area and volumes of 2-dimensional & 3-dimensional figures • Measurement • Solve problems involving length, area, & volume of geometric objects

  3. Question 1: Imagine a cone inscribed in a cylinder of the same size, so that the base of the cone is the same as the base of the cylinder and the top of the cone touches the top of the cylinder. Imagine also a sphere inscribed in a cylinder so that the sphere touches the cylinder at the north and south pole and all the way around the equator. What is the ratio of the volumes of a cone, sphere and cylinder?

  4. Question 2: Tennis balls are often packed snugly three to a can. What percent of the volume of the can do the tennis balls occupy?

  5. Area of Rectangle Area of rectangle = base * height

  6. Area of Parallelogram Area of parallelogram = base * height

  7. Area of Triangle Area of triangle = ½ * area of parallelogram Area of triangle = ½ * base * height

  8. Area of Trapezoid base of parallelogram = b1+b2 Area of trapezoid = ½ * area of parallelogram Area of trapezoid = ½ * (b1+b2) * h

  9. Area of a Circle

  10. Area of a Circle Cut apart your pizza and rearrange the slices so that it is a figure that we know how to find the area.

  11. Area of a Circle height = radius of circle base = ½ of circumference of circle Area of Circle = ½ * circumference * radius Area of Circle = ½ * ( 2 * Pi * radius ) * radius Area of Circle = Pi * radius2

  12. Surface Area & Volume of 3-dimensional shapes • Cylinder • Cone • Sphere

  13. Activities A,B,C • Groups of 4 • 3 groups start with Activity A, 3 groups start with Activity B, & 3 groups start with Activity C • After 15 minutes we’ll rotate

  14. Activity ARelationship between Volume of Cylinder and Volume of Sphere • Volume of sphere is 2/3 of the volume of the cylinder.

  15. Activity BRelationship between Volume of Cylinder and Volume of Cone • Volume of cone is 1/3 of the volume of the cylinder.

  16. Activity CSurface Area of a Sphere • Surface Area = 4*Pi*radius2

  17. Volume of Cylinder(in which a cone & sphere fit inside it) Volume =

  18. Volume of Sphere(which fits inside previous cylinder) Volume = 2/3 * volume of cylinder • = 2/3 * • = 2/3 *

  19. Volume of Cone(which fits inside previous cylinder) Volume = 1/3 * volume of cylinder • = 1/3 * • = 1/3 *

  20. Volume Cylinder vs. Volume Cone vs. Volume Sphere • Volume of Cylinder • Volume of Sphere • Volume of Cone Ratios of the volumes are (Cylinder : Sphere : Cone ) 6:4:2 OR 3:2:1

  21. Tennis balls are often packed snugly three to a can. What percent of the volume of the can do the tennis balls occupy?

  22. Case Study Video on Volume of Cylinder

  23. Cylinder Surface Area • What does a net of a cylinder look like? How does this help determine the surface area of a cylinder? • Surface Area =

  24. Milk Tanker • A stainless steel milk tanker in the shape of a right circular cylinder is 38 feet long and 5 feet in diameter. • Determine the amount of stainless steel material needed to construct the tanker. • Assume that 12% of the material you start with will be wasted in the construction process.

  25. Milk Tanker Circumference = Pi * diameter 5 ft circumference 38 ft Surface Area =(38)(circumference) + 2(Pi * radius2) = (38)(Pi * 5) + 2(Pi * 2.52) = (190*Pi) + (50*Pi/4) = 202.5 Pi square feet

  26. Milk Tanker • Assume that 12% of the material you start with will be wasted in the construction process. Surface Area =202.5 Pi square feet ~ 636.17 square feet Since 12% of the original material will be wasted we can think of 88% of the original material = 636.17 sq ft .88 * original material = 202.5 Pi ~636.17 sq ft original material = 636.17 sq ft ~ 723 sq ft .88

  27. Name that Common Solid • Side view and front view are triangles. Top view is a circle. • Side view and front view are rectangles. Top view is a circle. • Side view and front view are triangles. Top view is a square. Cone Cylinder Square pyramid

  28. Name that Common Solid 4. Side view and front view are triangles. Top view is a rectangle. 5. Side view and front view are rectangles. Top view is a rectangle. Rectangular pyramid Rectangular Prism

  29. Name that Common Solid 6. Side view, front view, and top view are all congruent squares. 7. Side view, front view, and top view are all congruent, and all triangles. Cube Triangular Pyramid

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