Basics of Geometry Chapter 1

Chapter Objectives Sketch points, lines, planes, and their intersections. Measure segments and add their lengths. Measure angles and classify them based on their measures.

Chapter Goals Introduction to the basic elements of geometry: points, lines, planes, segments, and angles. Shown reasoning skills which you will use throughout the course, specifically making predictions and conjectures based on patterns you observe.

Content 1.1 Finding and Describing Patterns 1.2 Inductive Reasoning 1.3 Points, Lines, and Planes 1.4 Sketching Intersections 1.5 Segments and Their Measures 1.6 Angles and Their Measures

Finding and Describing Patterns Section 1.1

Objectives: Find patterns and use them to make predictions.

Key Vocabulary Pattern Prediction

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.

Definitions Pattern Prediction A set of numbers or objects in which all the members are related with each other by a specific rule or rules. Prediction Process or act of foretelling future events, conditions, outcomes, or trends on the basis of current information or patterns.

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

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

Example 2: Describe a Visual Pattern Describe a pattern in the figures. SOLUTION The figures have 3 sides, 4 sides, 5 sides, and 6 sides. ANSWER The number of sides increases by one. 12

Your Turn: Describe a pattern. 1. ANSWER The number of equal parts into which the circle is divided increases by one. 2. ANSWER The shaded triangle moves one space in a counterclockwise direction.

Example 3: 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.

Example 3: 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

Example 4: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. -5 -2 4 13 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!!!

Example 5: Describing a Number Pattern Describe a pattern in the numbers. a. 3, 6, 9, 12, 15, 18, . . . b. 1, 4, 9, 16, 25, 36, . . . SOLUTION a. Each number after the first is 3 more than the previous one. b. The numbers are squares of consecutive numbers. 1 4 9 16 25 36 12 22 32 42 52 62 17

Your Turn: Describe a pattern. 1. 4, 8, 12, 16, 20, 24, . . . ANSWER Each number after the first is 4 more than the previous number. 2. 35, 30, 25, 20, 15, 10, . . . ANSWER Each number after the first is 5 less than the previous number.

Example 6: Make a Prediction Sketch the next figure you expect in the pattern. 19

Example 6: Make a Prediction SOLUTION The arrow’s color changes back and forth between green and red. The arrow makes a quarter turn each time. ANSWER The fourth figure is red. Its arrow points to the right. 20

Your Turn – Make a Prediction: Sketch the next two figures you expect in the pattern. ANSWER 1. ANSWER 2.

Your Turn – Make a Prediction: Write the next two numbers you expect in the pattern. 7. –2, –5, –8, –11, . . . ANSWER –14; –17 8. 4, 10, 16, 22, . . . ANSWER 28; 34

Joke Time Where do you find a dog with no legs? Right where you left him. What do you call a cow with no legs? Ground beef. What do you call a cow with 2 legs? Lean beef.

Assignment 1.1 Exercises Pg. 5 – 7: #1 – 25 odd, 35, 37