1.1 Patterns and Inductive Reasoning Geometry

Objectives/Assignment: Find and describe patterns. Use inductive reasoning to make real-life conjectures.

Finding & Describing Patterns Geometry, like much of mathematics and science, developed when people began recognizing and describing patterns. In this course, you will study many amazing patterns that were discovered by people throughout history and all around the world. You will also learn how to recognize and describe patterns of your own. Sometimes, patterns allow you to make accurate predictions.

Ex. 1: Describing a Visual Pattern Describe the next figure in the pattern. 4 5 1 2 3

Ex. 1: Describing a Visual Pattern - Solution The sixth figure in the pattern has 6 squares in the bottom row. 5 6

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. 1, 4, 16, 64 Many times in number pattern, it is easiest listing the numbers vertically rather than horizontally.

Ex. 1: Describing a Number Pattern How do you get to the next number? That’s right. Each number is 4 times the previous number. So, the next number is 256, right!!! Describe a pattern in the sequence of numbers. Predict the next number. 1 4 16 64

Ex. 1: Describing a Number Pattern How do you get to the next number? That’s right. You add 3 to get to the next number, then 6, then 9. To find the fifth number, you add another multiple of 3 which is +12 or 25, That’s right!!! Describe a pattern in the sequence of numbers. Predict the next number. -5 -2 4 13

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. 5,10,20,40… 80

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. 1, -1, 2, -2, 3… -3

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. 15,12,9,6,… 3

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. 1,2,6,24,120… 720

Ex. 1: Describing a Number Pattern Describe a pattern in the sequence. Predict the next letter. O,T,T,F,F,S,S,E,… 1,2,3,4,5,6,7,8, N 9

Goal 2: Using Inductive Reasoning Much of the reasoning you need in geometry consists of 3 stages: Look for a Pattern: Look at several examples. Use diagrams and tables to help discover a pattern. Make a Conjecture. Use the example to make a general conjecture. Okay, what is that? A conjecture is an unproven statement that is based on observations. Discuss the conjecture with others. Modify the conjecture, if necessary. 3. Verify the conjecture. Use logical reasoning to verify the conjecture is true IN ALL CASES. (You will do this in Chapter 2 and throughout the book).

Ex. 2: Making a Conjecture Complete the conjecture. Conjecture: The sum of the first n odd positive integers is ________?_________. How to proceed: List some specific examples and look for a pattern.

Ex. 2: Making a Conjecture First odd positive integer: 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 The sum of the first n odd positive integers is n2. =12 =22 =32 =42

Ex. 2: Making a Conjecture First odd positive integer: 2 = 2 2 + 4 = 6 2 + 4 + 6= 12 2 + 4 + 6 + 8 = 20 2 + 4 + 6 + 8 + 10 = 30 The sum of the first 6 odd positive integers is what? = 1x2 = 2x3 = 3x4 = 4x5 = 5x6 6x7 = 42

Note: To prove that a conjecture is true, you need to prove it is true in all cases. To prove that a conjecture is false, you need to provide a single counter example. A counterexample is an example that shows a conjecture is false.

Ex. 3: Finding a counterexample Show the conjecture is false by finding a counterexample. Conjecture: For all real numbers x, the expressions x2 is greater than or equal to x.

Ex. 3: Finding a counterexample- Solution Conjecture: For all real numbers x, the expressions x2 is greater than or equal to x. The conjecture is false. Here is a counterexample: (0.5)2 = 0.25, and 0.25 is NOT greater than or equal to 0.5. In fact, any number between 0 and 1 is a counterexample.

Ex. 3: Finding a counterexample Conjecture: The sum of two numbers is greater than either number. The conjecture is false. Here is a counterexample: -2 + -5 = -7 -7 is NOT greater than -2 or -5. In fact, the sum of any two negative numbers is a counterexample.

Ex. 3: Finding a counterexample Conjecture: The difference of two integers is less than either integer. The conjecture is false. Here is a counterexample: -2 - -5 = 3 3 is NOT less than -2 or -5. In fact, the difference of any two negative numbers is a counterexample.

Ex. 4: Using Inductive Reasoning in Real-Life Moon cycles. A full moon occurs when the moon is on the opposite side of Earth from the sun. During a full moon, the moon appears as a complete circle.

Ex. 4: Using Inductive Reasoning in Real-Life Use inductive reasoning and the information below to make a conjecture about how often a full moon occurs. Specific cases: In 2005, the first six full moons occur on January 25, February 24, March 25, April 24, May 23 and June 22.

Ex. 4: Using Inductive Reasoning in Real-Life - Solution A full moon occurs every 29 or 30 days. This conjecture is true. The moon revolves around the Earth approximately every 29.5 days.

Ex. 4: Using Inductive Reasoning in Real-Life - NOTE Inductive reasoning is very important to the study of mathematics. You look for a pattern in specific cases and then you write a conjecture that you think describes the general case. Remember, though, that just because something is true for several specific cases does not prove that it is true in general.

Ex. 4: Using Inductive Reasoning in Real-Life Use inductive reasoning and the information below to make a conjecture about the temperature. The speed with which a cricket chirps is affected by the temperature. If a cricket chirps 20 times in 14 seconds, what is the temperature? 5 chirps 45 degrees 10 chirps 55 degrees 15 chirps 65 degrees The temperature would be 75 degrees.

Find a pattern for each sequence. Use the pattern to show the next two terms or figures. 1. 3, –6, 18, –72, 360 2. Use the table and inductive reasoning. 3. Find the sum of the first 10 counting numbers. 4. Find the sum of the first 1000 counting numbers. Show that the conjecture is false by finding one counterexample. 5. The sum of two prime numbers is an even number. –2160; 15,120 55 500,500 Sample: 2 + 3 = 5, and 5 is not even -1

1.2 Drawing, Nets & Other Models Geometry

Objectives/Assignment: Make isometric and orthographic drawings Use nets and cross sections to analyze 3-dimensional figures. (G.GMD.4)

Draw the next figure in each sequence. GEOMETRY LESSON 1-2 Draw the next figure in each sequence. 1. 2.

1. The block is rotating counterclockwise about its base. GEOMETRY LESSON 1-2 Solutions 1. The block is rotating counterclockwise about its base. The next figure is: 2. The block is rotating clockwise about its front face. The next figure is:

There are many ways to represent a three dimensional object There are many ways to represent a three dimensional object. An isometric drawing is a way to show three sides of a figure from a corner view. You can use isometric dot paper to make an isometric drawing. This paper has diagonal rows of dots that are equally spaced in a repeating triangular pattern.

Make an isometric drawing of the cube structure below. GEOMETRY LESSON 1-2 Make an isometric drawing of the cube structure below. You will draw all the edges of the figure that you can see. Start by drawing the front face of the figure. Next, draw the back edges of the figure. Finally, fill in the right face, top faces, and left edges. Step 1: Draw the front. Step 2: Draw the back. Step 3: Complete.

An orthographic drawing shows six different views of an object: top, bottom, front, back, left side, and right side. (In your book, we only use the top, front, and right side views for an orthographic drawing.)

Make an orthographic drawing of the isometric drawing below. Orthographic drawings flatten the depth of a figure. An orthographic drawing shows three views. Because no edge of the isometric drawing is hidden in the top, front, and right views, all lines are solid.

A foundation drawing shows the base of a structure and the height of each part. A foundation drawing of the Sears Tower is shown below. The building is made up of nine sections. The numbers tell how many stories tall each section is. 49 89 65 109

Create a foundation drawing for the isometric drawing below. To make a foundation drawing, use the top view of the orthographic drawing.

(continued) Because the top view is formed from 3 squares, show 3 squares in the foundation drawing. Identify the square that represents the tallest part. Write the number 2 in the back square to indicate that the back section is 2 cubes high. Write the number 1 in each of the two front squares to indicate that each front section is 1 cube high.

A net is a diagram of the surfaces of a three-dimensional figure (a two-dimensional pattern) that can be folded to form the three-dimensional figure. To identify a three-dimensional figure from a net, look at the number of faces and the shape of each face.

A and C are the back and front faces. GEOMETRY LESSON 1-2 Is the pattern a net for a cube? If so, name two letters that will be on opposite faces. The pattern is a net because you can fold it to form a cube. Fold squares A and C up to form the back and front of the cube. Fold D up to form a side. Fold E over to form the top. Fold F down to form another side. After the net is folded to form a cube, the following pairs of letters are on opposite faces: A and C are the back and front faces. B and E are the bottom and top faces. D and F are the right and left side faces.

Think of the sides of the square base as GEOMETRY LESSON 1-2 Draw a net for the figure with a square base and four isosceles triangle faces. Label the net with its dimensions. Think of the sides of the square base as hinges, and “unfold” the figure at these edges to form a net. The base of each of the four isosceles triangle faces is a side of the square.

Use the figure at the right for Exercises 1–2. 1. Make an isometric drawing of the cube structure. 2. Make an orthographic drawing.

Drawings, Nets, and Other Models 3. Is the pattern a net for a cube? If so, name two letters that will be on opposite faces. yes; A and C, B and D, E and F 4. Draw a net for the figure. Sample:

ASSIGNMENT: Textbook: 1.1 page 6 #2-12 even page 6 # 20-22 even

Page 716 (#1-17)