Patterns and Inductive Reasoning Lesson 1-1 Patterns and Inductive Reasoning

Inductive Reasoning Making conclusions/predictions based on patterns and examples. Find the next two terms: 3, 9, 27, 81, . . . Draw the next picture: Find the next two terms: 384, 192, 96, 48, . . . 243, 729 24, 12

Making a Conjecture Make a conclusion based on inductive reasoning. Use the table to make a conjecture about the sum of the first six positive even numbers. 2 = 2 = 1·2 2 + 4 = 6 = 2·3 2 + 4 + 6 = 12 = 3·4 2 + 4 + 6 + 8 = 20 = 4·5 2 + 4 + 6 + 8 + 10 = 30 = 5·6 = 6·7 = 42

Counterexample (like a contradiction) An example for which the conjecture is incorrect. Conjecture: the product of two positive numbers is greater than either number. counterexample

Fun Patterns Find the next character in the sequence J, F, M, A, . . . January, February, March, April, May Find the next character in the sequence S, M, T, W, . . . Sunday, Monday, Tuesday, Wednesday, Thursday Find the next character in the sequence Z, O, T, T, F, F, S, S, . . . Zero, One, Two, Three, Four, Five, Six, Seven, Eight Find the next character in the sequence 3, 3, 5, 4, 4, . . . One has 3 letters, Two has 3, Three has 5, Four has 4, Five has 5, Six has 3

Points, Lines, and Planes Lesson 1-2 Points, Lines, and Planes

Point A point does not have an actual size; it represents a location. How to Sketch: Use dots How to label: Use CAPITAL letters Never name two points with the same letter (in the same sketch). A B C A

Line A set of points that extends infinitely in opposite directions and has no thickness or width. How to sketch : use arrows at both ends. How to name: 2 ways (1) small italics letter — line m (2) any two points on the line — Never name a line using three points — m A B C

Collinear Points Collinear points are points that lie on the same line. (The line does not have to be visible.) E A B C D F Collinear Non collinear

Planes A plane is a flat surface that extends indefinitely in all directions. How to sketch: Use a parallelogram (four sided figure) How to name: 2 ways (1) Capital italics letter — Plane M (2) Any 3 (or more) noncollinear points in the plane — Plane: ABC/ ACB / BAC / BCA / CAB / CBA A M B C Horizontal Plane Vertical Plane Other

Different planes in a figure: B Plane ABCD Plane EFGH Plane BCGF Plane ADHE Plane ABFE Plane CDHG Etc. D C E F H G

Coplanar Objects Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible. Are the following points coplanar? A, B, C ? Yes A, B, C, F ? No H, G, F, E ? Yes E, H, C, B ? Yes A, G, F ? Yes C, B, F, H ? No

Postulate An accepted statement or fact. You accept a postulate as true without proof; you try to determine if a conjecture is true or false.

Postulate 1-1 Through any two points there is exactly one line. t B A

Postulate 1-2 If two lines intersect, then they intersect in exactly one point. C D B A P

Postulate 1-3 If two planes intersect, then they intersect in exactly one line. B P A Plane P and Plane R intersect at the line R

Postulate 1-4 Plane AFGD Plane ACGE Plane ACH Plane AGF Plane BDG Etc. Through any three noncollinear points there is exactly one plane. Plane AFGD Plane ACGE Plane ACH Plane AGF Plane BDG Etc.

3 Possibilities of Intersection of a Line and a Plane (1) Line passes through plane — intersection is a point. (2) Line lies on the plane — intersection is a line. (3) Line is parallel to the plane — no common points.