Transcript Chapter 1

Chapter 1
Using Geogebra
Doing Mathematics
• Geogebra
 Dynamic
 Easier to see what happens to geometric
objects
• Make observations
• Formulate conjectures
 If … [hypothesis] … then … [conclusion]…
Conjectures
• Hypothesis
 Assumptions
• Conclusion
 What happens when assumptions hold
• Often expressed using
 If … then …
Conjectures
• Ask yourself
 What is happening
 Why is it happening
 What properties of figures made it happen
Geogebra Tips
• Straight edge tools
Geogebra Tips
• Polygon tools
Geogebra Tips
• Circle tools
Geogebra Tips
• Changing object property – color
Geogebra Tips
• Using arrow tool
 Drag a point, view results
Geogebra Tips
• The Point tool
 Use to establish an intersection
Observing, Conjecturing
• Inductive reasoning
 Examine examples
 Discover patterns
• Deductive reasoning
 Justify the conjecture
 Find explanation why conjecture holds
 Proof
Language of Geometry
• Formal definitions
 Precision necessary
 Math definitions are prescriptive
• Definition different from properties
 Example: define rectangle
 Construct figure only with definition
Polygons
• Closed figure
 In the plane
 Formed by set of line segments
 Each segment shares each endpoint with
exactly 1 other line segment
• Poly -  many
• - gons  angles
Quadrilateral
• Joining midpoints of arbitrary
quadrilateral?
Quadrilateral
• Self intersecting polygon
Quadrilateral
• Concave polygon
Viviani’s Theorem
• What was the sum of l + m + n?
 What if P was outside the triangle?
 What if the triangle was not equilateral?
 What if it
is a square?
P
Family of Quadrilaterals
• Identify the categories
Angles Inscribed in Circles
• What relationships did you find?
Rules of Logic
• Statements … open? closed? true? false?
 All cars are blue.
 The car is red.
 Yesterday was Sunday.
 Rectangles have four interior angles.
 Construct the perpendicular bisector.
Rules of Logic
• Use logical operators
 and, or
• Evaluate truth of logical combinations
 P and Q
Rules of Logic
• Combining with or

P or Q
Rules of Logic
• Negating a statement

not P
Chapter 1
Using Geogebra