Why is the use of inductive reasoning important to understanding mathematics?

Make a conclusion about each of the following: Scenario #1: You see a dark cloud in the sky… Scenario #2: You hear a fire truck coming down the road… Scenario #3: You hear fans cheering at a football game…

When you make a conclusion based on a pattern of examples or past events.

The conclusion made based on inductive reasoning. CONJECTURES ARE NOT ALWAYS TRUE!!!

Find the next three terms of the sequence: 33, 39, 45, ____, ____, ____

Find the next three terms of the sequences: 1.25, 1.45, 1.65, ____, ____, ____ 13, 8, 3, ____, ____, ____ 1, 3, 9, ____, ____, ____ 32, 16, 8, ____, ____, ____

Find the next three terms of the sequence: 1, 3, 7, 13, 21, ____, ____, ____

Find the next three terms of the sequences: 10, 12, 15, 19, ____, ____, ____ 1, 2, 6, 24, ____, ____, ____

Sometimes there may be more than one pattern to study. What is the next figure?

Find the next figure in the sequence:

If six points on a circle are joined by as many segments as possible, how many non-overlapping pieces are created? 2 Points3 Points4 Points

If six points on a circle are joined by as many segments as possible, how many non-overlapping pieces are created? 6 Points

Find the sum of the first 20 odd numbers: 1 = = = =

Assignment on Moodle Homework: Lesson #1 – Inductive Reasoning

Compare and contrast a line, line segment, and a ray.

The basic unit of Geometry A point has NO size Points are named using capital letters We call them “point A” A B

A series of points that extends without end in two directions A line is made up of an infinite number of points The arrows show that the line extends in both directions forever. A line is named with a single lower case letter or by two points on the line. A B l AB

Name two points on line m. Give any three names for the line. P R m Q Hint: The order of the letters DOES NOT matter when naming a line.

Three or more points that lie on the same line. P R Q S T Points that are NOT on the same lines are said to be NONCOLLINEAR. S, Q, T are collinear points. S, Q, P are noncollinear points.

Name three OTHER points that are collinear. P R Q S T Name three OTHER points that are noncollinear.

A flat surface that extends without end in all directions For any three NONCOLLINEAR points, there is only one plane. Named by a single uppercase script letter or by three or more NONCOLLINEAR points.

Points and lines that are in the same plane. Points and lines that are NOT on the same plane are said to be NONCOPLANAR. A, B, C are coplanar. A, B, F are noncoplanar points. D C E F BA

Not all planes CREATE the figure, some planes pass through the figure.

How many planes are in the figure? Name the “front” plane in 3 different ways? Name a point that is coplanar with E, F, and G? Name a point that is coplanar with D, C, and F? Name two lines that are coplanar with AB and DC? AB C D EF G H

Assignment on Moodle Homework: Lesson #2a – Points, Lines, Planes, Segments, and Rays

Has a definite starting point and extends without end in one direction. The starting point is called the ENDPOINT. A ray is named using the endpoint first then another point in the ray. A B AB

Part of a line that has a definite beginning and ending Named using the endpoints. A B AB

Name two segments and two rays. Hint: Be sure to use the correct symbols. P R Q S

Parallel Lines Coplanar lines that do not intersect. Skew Lines Noncoplanar lines that are neither parallel or intersecting.

The foundations of Geometry. Facts about Geometry that we accept are true.

1-1: Two points determine a unique line. 1-2: If two lines intersect, they intersect at exactly one point. 1-3: Three noncollinear points determine a plane. 1-4: Two planes intersect at exactly one line.

Points D, E, and F are noncollinear. Name all of the different lines that can be drawn through these points. D F E What postulate did you use?

Points Q, R, S, and T are noncollinear. Name all of the different lines that can be drawn through these points. Q S R T

Name the intersection of plane CGA and plane HCA. C G H A What postulate did you use?

Name the intersection of plane ABC and plane DFC. D C E F BA

Page 15 #22-26 and Page 20 #1-8, 13-30

Make a distinction between lines or angles being congruent and lines or angles being equal.

You and your family are driving on I-80 through Nebraska. You entered the interstate at mile marker 126. You decide to drive as far as you can before stopping for breakfast within 1.5 hours. Assume that on the highway you drive an average speed of 60 mph. How far will you travel in 1.5 hours? At what mileage marker will you exit to get to breakfast? Does the direction you travel affect the distance you travel?

Two segments with the same length are CONGRUENT ( ≌ ). If AB ≌ CD, then AB = CD The segment AB The length of the segment AB A B C D

What is AC? AB 2 in CB 3 in If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC

If DT = 60 find the value of x, then find DS and ST. DS 2x - 8 T 3x - 12

If F is between E and G, and EG = 94, EF = 4x – 20, and FG = 2x + 30 find the value of x, EF, and FG.

What can you say about point B? CB 3 in A point that divides a segment in half is called a MIDPOINT. BA 3 in

A segment bisector, bisects a segment at the midpoint, therefore, cutting it into two equal parts. CB 3 in BA

If TQ bisects AB at M and MB = 8x + 7 and AB = 126, find x. B M A Q T

If M is the midpoint of TQ and TM = 3x + 5 and MQ = x + 17, find x. B M A Q T

Page 28 #13, 14, 27, 28 Page 36 #35-40

Defined by two noncollinear rays Side Vertex

Angles could be named in three ways: 1.By the vertex angle 2.By the number inside the angle 3.By three points on the angle (vertex point must be in the middle) P Q R 1 A B C D

Refer to the figure to answer each question What other names could be used to identify ∠ BCD? Name the vertex of ∠ 1. What are the sides of ∠ 2? A B C D 1 2

If m ∠ PQR = 30 m ∠ RQS = 25 what is the m ∠ PQS? P Q S R

If R is in the interior of ∠ PQS, then m ∠ PQR + m ∠ RQS = m ∠ PQS P Q S R

If m ∠ AEG = 75, m ∠ 1 = 25 – x, and m ∠ 4 = 5x + 20, find the measure of x E F C B A D G

If m ∠ 3 = 32, find the m ∠ CED E F C B A D G

If m ∠ 2 = 6x - 20, m ∠ 4 = 3x + 18, and m ∠ CED = 151, find the value of x E F C B A D G

If m ∠ 1 = 49 – 2x, m ∠ 4 = 4x + 12, and m ∠ 2 = 15x, find x E F C B A D G

Acute Angle Right Angle Obtuse Angle Angles whose measure is < 90 ° Angles whose measure is = 90 ° Angles whose measure is > 90 ° Straight Angle Angles whose measure is = 180 °

Divides an angle into two congruent angles. P Q R S

AB bisects ∠ CAD. Solve for x and find m ∠ CAD. C A D B (7x + 4) ° (10x - 20) °

BX bisects ∠ ABC. If the m ∠ ABC = 5x + 18, m ∠ CBX = 2x + 12, find m ∠ ABC. C B A X

Two lines that intersect to form a right angle. CBA

A segment, line, or ray that is perpendicular to a line at its midpoint. CB 3 in A

If AC = 7, then find AB. _____ is the angle bisector of ______. m ∠ ACG = ______ CG = 2x + 2, DC = 5x – 1, Find x and CG. BCA D G E F

Page 30 #32-35 Page 37 #48-51

Explain in words what vertical angles are and WHY they are congruent.

What conclusion can you make about ∠ 1 and ∠ 2? B A Q T ° 2

Angles across from one another at an intersection. B A Q T ° 2 VERTICAL ANGLES ARE CONGRUENT

Find the value of the variables B A Q T (4x + 1) ° (7x + 3) ° 65 °

B A Q T Two coplanar angles with a common side and vertex, but no common interior points.

B AQ T Two angles whose sum is 90 °

Find the value of z: (4z - 10) ° z°z°

B A Q T Two angles whose sum is 180 °

Find the value of y: (6y – 10) ° (6y + 10) °

Page 50 #13-18 and 21