1. Elementary Mathematics Institute August 18, 2004 Two Dimensional Shapes: Triangles and Quadrilaterals

2. Day 2 Agenda  Exploring Triangles (naming triangles by angle and by side)  Measuring Angles in Polygons  Informal Proof: Sum of the Angles in Polygons  Quadrilaterals Quad Nets Pinpointing Properties  Perimeter and Area Bumper-Cars (constant area) Wreck Tangles (constant perimeter)

3. Can a triangle be made with three sides of any length? abcYes or No 6820 1210 6186 81218 121410 2088 14816 686 61218

4. Can a triangle be made with three sides of any length? abcYes or No 6820 no 201210 yes 6186 no 81218 yes 121410 yes 2088 no 14816 yes 686 61218 no

5. Draw a conclusion about the lengths of the sides of a triangle. In a triangle the sum of the length of the two shorter sides must be greater than the length of the third side.

6. Angles in Polygons

7. Developing an Informal Proof for the Sum of Interior Angles of Polygons Use the geoboards to make different polygons. Divide the polygon into triangles. Measure the angles. Find a pattern. Number of SidesNumber of TrianglesSum of Interior Angles 3 4 5 6 7 8 9 10

8. Developing an Informal Proof for the Sum of Interior Angles of Polygons Use the geoboards to make different polygons. Divide the polygon into triangles. Measure the angles. Find a pattern. Number of SidesNumber of TrianglesSum of Interior Angles 3 1180 4 2360 5 3540 6 4720 7 5900 8 61080 9 71260 10 8

9. What can you say about the number of sides of a polygon and the sum of its interior angles?

10. The sum of all the angles in a polygon is equal to the number of sides minus two times 180 degrees. S = (n-2)(180) S=sum of angles in a polygon; n=number of angles This works because we know that the sum of all the angles in any triangle equals 180° and we found out when we drew them that there are two less triangles than the number of sides in the polygon.

11. Is it possible for two shapes to have the same area but different perimeters? Explain your answer by using words and drawings. Is it possible for two shapes to have the same perimeter but different areas? Explain your answer by using words and drawings. Can you figure out a perimeter if you know its area? Why or why not? Perimeter and Area

12. Designing Bumper-Car Rides Each tile represents one square meter. A bumper car ride design that consists of only one square meter would require 4 meters of bumper rail to surround it. How many meters of railing are needed for this floor plan? Begin a table for recording data. Number of TilesNumber of Rails 14 2?

13. Designing Bumper-Car Rides Begin a table for recording data. Number of TilesNumber of Rails 14 2? 3 4 5

14. Designing Bumper-Car Rides Design a bumper-car floor plan with an area of 24 square meters and a perimeter of 22 meters. Design a bumper-car floor plan with an area of 24 square meters and many rail sections. What is the floor plan with the most rails? With the fewest number of rails?

15. Wreck-Tangles How do areas of rectangles with equal perimeters compare? Complete this activity using the push pins, 30 cm string loop, cardboard and worksheet. Conclusion: