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1.1 Patterns and Inductive Reasoning

1.1 Patterns and Inductive Reasoning. Day 1 part 1 CA Standards 1.0, 3.0. Subtract the integers. 17 – 9 9 – 17 5 – ( -3) -5 – (-6). Evaluate the sum 2 2 + 4 2 5 2 + (-2) 2 (-1) 2 + 1 2 (-5) 2 + 0 2. Warmup. Geometry Recognizing and describing patterns Example

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1.1 Patterns and Inductive Reasoning

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  1. 1.1 Patterns and Inductive Reasoning Day 1 part 1 CA Standards 1.0, 3.0

  2. Subtract the integers. 17 – 9 9 – 17 5 – ( -3) -5 – (-6) Evaluate the sum 22 + 42 52 + (-2)2 (-1)2 + 12 (-5)2 + 02 Warmup

  3. Geometry • Recognizing and describing patterns • Example • 1, 4, 16, 64, …. • 5, 0, -5, -10, … • 1, 3, 6, 10, 15, …

  4. Example • Describe a pattern in the sequence of numbers. Predict the next number. • 1, 4, 16, 64, …. Each number is 4 times the previous number. • -5, -2, 4, 13, … 256 25 +3 +6 +9 +12

  5. Definition • Conjecture • An unproven statement that is based on observations. • Inductive reasoning • Looking for patterns and making conjectures

  6. Complete the conjecture. • Conjecture: The sum of the first n odd positive integer is ______ 1 1 + 3 1 + 3 + 5 1 + 3 + 5 + 7 n2 = 1 = 12 = 4 = 22 = 9 = 32 = 16 = 42

  7. Complete the conjecture . • Conjecture: The sum of the first 5 odd positive integer is _____.

  8. On your own • Conjecture : The sum of any two odd numbers is ____

  9. Finding and Describing Patterns • Sketch the next figure in the pattern. . . . . . .

  10. Finding and Describing Patterns • Describe a pattern in the sequence of numbers. Predict the next number. • 4, 16, 36, 64, … • 4.4, 40.4, 400.4, 4000.4, …

  11. Pg. 6 # 3 – 15

  12. 1.2 Points, Lines, and Planes Day 1 Part 2 CA Standard 1.0

  13. Definitions • Point has no dimension. It is usually represented by a small dot. • Line extends in one dimension. It is usually represented by a straight line with two arrowheads. • Plane extends in two dimensions. It is usually represented by a shape that looks like a tabletop or wall.

  14. Line AB (line consists of the endpoints) • Segment AB • Ray AB A B A B B A

  15. more definitions… • Collinear points are points that lie on the same line. • Coplanar points are points that lie on the same plane. • Ray consists of the initial point. • Intersect means to have one or more points in common.

  16. Decide whether the statement is true or false * True • Point A lines on line l. • Point C lies on line m. • A,B, and C are collinear. • D,E, and B are coplanar. • A,B, and C are coplanar. m . l . False D A False True . E True . C . B .

  17. Draw three noncollinear points J, K and L. • Then draw JK, KL and LJ.

  18. Name a point that is coplanar with the given points. • D, C & F • E, F, & G • B, C, & F G B C D A H E F H G E

  19. Fill in each blank with the appropriate response. Point B • AB and BC intersect at _____ • AD and AH intersect at _____ • Plane ABHE and ABCD intersect at _____ B C D A E F H G Point A Side AB

  20. Sketch the figure described • Three points that are coplanar but not collinear. • Three lines that intersect in a point and all lie in the same plane. • Two planes that do not intersect. • Three planes that intersect in a line. • Two lines that lie in a plane but do not intersect.

  21. CW/HW • Pg. 6 # 16 – 38 • Pg. 13 # 2 – 66 even

  22. 1.3 Segments and their measures Day 2 Part 1 CA Standards 1.0, 15.0, 17.0

  23. Warmup • Simplify the expression. • 1. • 2. • 3. • 4. • 5.

  24. Definitions • Postulates: rules that are accepted without proof. • Theorems: rules that are proved. • Axioms: rules that are accepted without proof.

  25. Segment Addition Postulate * If B is between A and C, then AB + BC = AC. If AB + BC = AC, then B is between A and C. . . . C A B

  26. Examples • In the diagram of the collinear points, PT=20, QS=6, and PQ = QR = RS. Find each length. • QR • RS • RT • RP • SP T . S . . Q R P

  27. 1. AB = 6x + 16 BC = 2x – 5 AC = 46 2. AB = ½ k + 4 BC = 3k – 4 AC = 6k – 18 Suppose B is between A and C. Use the segment addition postulate to solve for the variable.

  28. Congruent? AB = AD AB is equal to AD Segment AB is congruent to segment AD

  29. The distance formula The distance formula is formula for computing the distance between two points in a coordinate plane. Pythagorean Theorem c2 = a2 + b2 c a b

  30. Examples • Find the distance between each pair of points • A(-4, 7) B(6,2) • C(-3, 6) D(0,2) • E(1,4) F(4, -4)

  31. Find the distance between each pair of points. • 1. X(0,0), Y(2,8), Z( -1, 2) • 2. L(-2,0), M(5,1), N(4, -4)

  32. 1.4 Angles and their measures Day 2 Part 2 CA Standard 4.0,12.0, 13.0

  33. Review • Determine if the statement is true or false. • E lies on BD. • A, B, and D are collinear. • B lies in plane ADC. • The intersection of DE and AC is B. . D . . B . C A . E

  34. Definition of an Angle • Angle consists of two different rays that have the same initial point . C Sides Vertex . . A B

  35. Angle addition postulate • If P is in the interior of <RST, then m<RSP + m<PST = m<RST . R P . m<RSP . S m<PST . T

  36. Example • Use the Angle Addition Postulate to find the measure of the unknown angle. m<ABC= . D C . . A 45° 60° . B

  37. Different types of angles * Supplementary angles Straight angle Acute angle Obtuse angle Right angle Complementary angles

  38. State whether the angle appears to be acute, right, obtuse, or straight. . . . E . A C F . D . B . G . H

  39. Exterior . . Interior

  40. Review • Plot the given points and classify the given angles as acute, right, obtuse, or straight. A(-2,4), B(-5,1), C(0,0), and D(3,0) • <ACB • <BCD • <ACD

  41. Draw a sketch that uses all of the following information. • D is in the interior of <BAE. • E is in the interior of <DAF. • F is in the interior of <EAC. • m<BAC = 130° • m<EAC = 100° • m<BAD = m<EAF = m<FAC • Now find m<FAC • Find m<BAD • Find m<BAE

  42. Pg. 21 # 4 – 12 even, 19 – 22, 40 – 43 • Pg. 29 # 1 – 16, 26 – 39

  43. 1.5 Segment and Angle Bisectors Day 3 Part 1 CA Standards 16.0, 17.0

  44. Warmup • Fill in the blanks. • The end point of RQ is _______. • Line segments with equal measures are ___________. • A rule of geometry that is accepted without proof is a _______________. • A pair of opposite rays form a ________. Q congruent postulate or axiom line

  45. Midpoint Formula x2 x1 Example: Find the midpoint of (-2,3) and (5,-2) y2 y1

  46. More examples… • Find the coordinates of the midpoint. 1. A(-7, 2), B(3,0) 2. K(7,3), L(2,1)

  47. Challenge question • The midpoint of RP is M(2,4). One endpoint is R(-1,7). Find the coordinates of the other endpoint.

  48. PT is the angle bisector of <RPS. Find the two angle measures not given in the diagram. . R T . 37° . . P S

  49. PT is the angle bisector of <RPS. Find the value of x. . . R T (3x + 13) (5x – 7) . . P S

  50. Segment Bisector and Midpoint • Draw a segment AB. • Place the compass point at A. Use a compass setting greater than half the length of AB. Draw an arc. • Keep the same compass setting. Place the compass point at B. Draw an arc. It should intersect the other arc in two places. • Use a straightedge to draw a segment through the points of intersection. This segment bisects AB at M, the midpoint of AB.

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