Diagrams in Geometry Unit 0 Introduction to Geometry

Diagrams in Geometry Unit 0: Introduction to Geometry and Technology

Challenge in Geometry • Coordinate the verbal meanings captured by language with the spatial ones represented in diagrams. • Diagrams are strongly related to visual imagery. • Working with diagrams can help develop spatial thinking. 1/4/2022 Geometry Institute 2

A diagram is a sophisticated mathematical device for thinking and communicating. • There is no geometry without diagrams. • Diagrams communicate something that verbal language cannot, and vice versa. • Making diagrams can help with the problemsolving process. • Producing diagrams can also promote a better understanding of the diagram structures, and renders coherent the meanings that different elements of the diagram are intended to convey. 1/4/2022 Geometry Institute 3

Activity 1 Draw a circle and a line. How many times do they intersect? How many times could they intersect? How many times must they intersect? 1/4/2022 Geometry Institute 4

Activity 1 (Cont) • The circle, – you had to think where to put the center and what the radius would be. – By paying attention to how you are drawing you are becoming aware of properties of the circle. • The line, – How many different possibilities of positions are there? Is the line intersecting the circle? In how many points? Did the prompt suggest intersection? 1/4/2022 Geometry Institute 5

Secant and Tangent Lines • The secant, or “cut-line”, comes from the Latin verb seco, meaning “I cut”. Could a secant go through the center of the circle? Will it have a special name if it does? • Tangent or “touch-line” comes from the Latin verb tango, meaning “I touch”. 1/4/2022 Geometry Institute 6

Activity 1 – Part 2 • Draw a secant line with movable points A and B on the circumference of the circle. • Draw a radius to point A. • Move point B all around the circumference, tilting the secant right or left of the radius. • When point B overlaps point A, what is the name of the line? What is the relationship of this new line with the radius of the circle? 1/4/2022 Geometry Institute 7

Activity 2 Illustrate the claim that two circles can intersect zero, one, two, or infinitely many times. 1/4/2022 Geometry Institute 8

A diagram is a “built” geometric artifact, with both a narrative of successive construction and a purpose. Very often diagrams are given, pre-made. They are used to illustrate a geometric configuration for a specific problem. Often the construction of the diagram is not obvious or explained, but the construction steps involved deep geometric knowledge. 1/4/2022 Geometry Institute 9

Activity 3 Construct a kite in as many different ways as you can. • How is your construction showing the properties of a kite? • Share one of your constructions with the whole class. 1/4/2022 Geometry Institute 10

Learning how to read a diagram requires some knowledge Diagrams may not appear visually to represent specific figures, but they can include markers that assert the desired properties. 1/4/2022 Geometry Institute 11

Activity 4 Given a right triangle BAC with right angle at A, let P be a point on segment BC and I be a point on segment AB such that segment PI is perpendicular to segment AB, and J be a point on segment BC such that segment JP is perpendicular to segment AC. Where should P lie on segment BC to minimize the length of segment IJ? 1/4/2022 Geometry Institute 12

Activity 5 In Activity 4, when would J be at the midpoint of AC in order for IJ to be minimized? Under what conditions is the location of P such that PIAJ is a square? 1/4/2022 Geometry Institute 13

Conclusion When working with Geometric diagrams, we sometimes need to be able to see them in different ways – sometimes as drawings of a particular situation, but sometimes as visual statements of generality. 1/4/2022 Geometry Institute 14