Chapter One

1. Chapter One Getting Started…

2. 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

3. 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

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

5. Fun Patterns Find the next character in the sequenceJ, F, M, A, . . . January, February, March, April, May Find the next character in the sequenceS, M, T, W, . . . Sunday, Monday, Tuesday, Wednesday, Thursday Find the next character in the sequenceZ, O, T, T, F, F, S, S, . . . Zero, One, Two, Three, Four, Five, Six, Seven, Eight Find the next character in the sequence3, 3, 5, 4, 4, . . . One has 3 letters, Two has 3, Three has 5, Four has 4, Five has 4, Six has 3

6. Lesson 1-1 Points, Lines, and Planes • Essential Understandings: • The characteristics and properties of 2 and 3 dimensional geometric shapes can be analyzed to develop mathematical arguments about geometric relationships. • Essential Questions: • How do algebraic concepts relate to geometric concepts? • How do patterns and functions help us represent data and solve real-word problems?

7. Points • A point names a location and has no size. It is represented by a dot. • Always use a CAPITAL letter to name a point. • Never name two points with the same letter (in the same sketch). A B A C

8. A B C Lines • Use a lowercase italicized letter or two points on the line. • A straight path that has no thickness and extends forever. • Never name a line using three points. m

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

10. Planes • Usually represented by a rectangle or parallelogram. • Use an italicized CAPITAL letter or any three non-collinear points. (Sometimes four are used.) • A plane is a flat surface that has no thickness and extends forever. R B I K E CANNOT name BIK as these points are collinear. Plane R, or IKE, KEB, BIE, BKE, IKE, KIE, etc.

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

12. 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

13. Postulate An accepted statement or fact.

14. l B A Postulate 1-1 Through any two points there is exactly one line. (Say what? Two points make a line.)

15. Postulate 1-2 Through any three noncollinear points there is exactly one plane. (Say what? Three non-collinear points make a plane.) Plane AFGD Plane ACGE Plane ACH Plane AGF Plane BDG Etc.

16. Postulate 1-3 If two points lie on a plane, then the line containing those points lies on the plane. P A B

17. B P A C D Postulate 1-4 If two lines intersect, then they intersect in exactly one point. (Say what? Two lines intersect at a point.)

18. Postulate 1-5 If two planes intersect, then they intersect in exactly one line. (Say what? Two planes intersect in a line.) B P A Plane Pand Plane R intersect at the line R

19. 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.

20. Segments (line segments) • Use the two endpoints to name a segment. • Part of a line consisting of two points (endpoints) and all the points inbetween. Q • Do not show the endpoints in the name. R

21. Rays • The arrow in the name always goes left to right regardless of the physical ray. • Opposite rays form a line and share an endpoint. • Part of a line that starts at an endpoint and extends forever in one direction. • Use the endpoint as the first letter and any other point on the ray. A X Y G E F

22. E F A B G H D C Parallel/Skew and Coplanar/Non-Coplanar • Parallel lines are coplanar lines that do not intersect. • Skew lines are non-coplanar lines which do not intersect. • Parallel planes are planes that do not intersect.

23. Congruence and Tick Marks PQ and RS represent numbers. Use equality for numbers. vs and represent geometric figures. Use congruence for figures. • Congruent segments are segments that have the same length. • Tick marks are used in diagrams to show congruence. Q P R S

24. Segment Bisector • The midpoint of a segment bisects the segment into two congruent segments. • A segment bisector is a ray, segment, or line that intersects a segment at a midpoint. C C M A M A

25. Segment Addition Postulate • If B is between A and C, then A B C

26. Ruler Postulate • If the coordinates of points A and B are the numbers a and b, then the distance AB is written as: A B