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# GEOMETRY FOR MIDDLE SCHOOL TEACHERS MA 241

UK

GPA 3.54

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This 3 page Class Notes was uploaded by Kennith Herman on Friday October 23, 2015. The Class Notes belongs to MA 241 at University of Kentucky taught by Staff in Fall. Since its upload, it has received 22 views. For similar materials see /class/228147/ma-241-university-of-kentucky in Mathematics (M) at University of Kentucky.

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Date Created: 10/23/15

Shapes and Designs Extensions 3 H Write a clear explanation of the measurement of angles in radians and why this is a more natural notion than measurement in degrees F Propose a de nition of the measure of a solid angle where three four or more planes meet at common vertex of a polyhedron and explain why your de nition is reasonable In particular your de nition should be compatible with a three dimensional analog of the Angle Addition Postulate CliffsQuickReview Geometry p 12 9 Describe all possible cases when two sets 730 r of polar coordinates actually correspond to the same point 4 Review the de nitions of trigonometric functions from the unit circle a Drawing on this sketch the graphs of the functions ft9 sin0 and ft9 cos 0 and explain how you can deduce these naturally from the unit circle de nition b Continuing to think about the unit circle de nition complete the following for mulas and give brief explanations for each sin70 7 sin6 ii cos70 iii sin 7139 9 cos7r 9 v sin 7 t9 vi cos7r 7 t9 vii sin7r2 9 cos7r2 9 ix sin7r2 7 x cos 7r2 7 xi sin26 cos26 t9 t9 5 Describe a procedure to determine the rectangular coordinates Ly of a point from its polar coordinates r and justify why it works 6 Cylindrical and Spherical Coordinates a Justify the following conversion from cylindrical coordinates 7quot t9 2 to rectangular coordinates x y z rcos0 y rsin0 2 2 b Justify the following conversion from spherical coordinates 7quot 0 b to rectangular coordinates Luz z rcos6sin y rsin6sin z rcos 7 Read Chapter 2 of CIiffsQuickReview Geometry on Parallel Lines Prove Theorerns 177 24 8 It turns out that without assurning Postulates 11 and 12 of CIiffsQuickReview one can prove that the sum of the measures of the angles of any triangle cannot exceed 180 degrees a Learn the proof of this angle surn theorern See7 for exarnple7 the proof of the Saccheri Legendre Theorem in Kay7 College Geometry A Discovery Approach b Use this result to prove Postulate 127 thereby showing that it was not necessary to assume this as a postulate after all MA241 Exam 3 Review You should be able to H F 9 7 U 03 5 00 Recognize whether or not two given sets are congruent or similar Give the de nition7 via functions7 for two sets to be congruent or similar Understand the results of applying a function of the form f Ly a cw b0z d to a set in the plane Given a function mapping the plane to itself7 determine the image of a given set under action of the function Given two similar sets in the plane7 nd at least one one to one onto function from one set onto the other that exhibits the required scaling factor Given a function mapping one region onto another7 determine whether or not that the ratios of corresponding line segments are equal Prove that any two circles are congruent Explain how the notion of congruence can be used to de ne the notion of symmetry of a set Derive the formulas for the surface areas of rectangular boxes7 cylinders7 cones7 and pyramids Demonstrate how the formulas for areas of various regions or surface area and volume for various solids exhibit show by what factors these areas and volumes change under scaling of the object Give the de nition of reptile and predict how many copies of smaller reptiles are nec essary to comprise a larger one with a given scaling factor Prove or disprove Every triangle is a reptile Use the scale of a map to determine the distance between two locations on the map Given two similar sets7 determine missing measurements from given measurements7 and be familiar with some applications to physical measurements Explain how to use a mirror to estimate the height of an object Explain how to use shadows to estimate the height of an object

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