Lesson 2.1 Inductive Reasoning in Geometry HOMEWORK: Lesson 2.1/1-15 odds, 22-24, 31-40, 42 EC: Due Wednesday Page 104 – “Improve Reasoning Skills” #1-8

Vocabulary Inductive reasoning: make conclusions based on patterns you observe Conjecture: conclusion reached by inductive reasoning based on evidence Geometric Pattern: arrangement of geometric figures that repeat

We will be doing this ALOT this year!! Objectives: Use inductive reasoning to find the next term in a number or picture pattern To use inductive reasoning to make conjectures. Mathematicians use Inductive Reasoning to find patterns which will then allow them to conjecture. We will be doing this ALOT this year!!

A generalization made with inductive reasoning (Drawing conclusions) Conjectures A generalization made with inductive reasoning (Drawing conclusions) EXAMPLES: Bell rings M, T, W, TH at 7:40 am Conjecture about Friday? Chemist puts NaCl on flame stick and puts into flame and sees an orange-yellow flame. Repeats for 5 other substances that also contain NaCl also producing the same color flame. Conjecture?

reasoning that is based on patterns you observe. Inductive Reasoning – reasoning that is based on patterns you observe. Ex. 1: Find the next term in the sequence: A) 3, 6, 12, 24, ___, ___ B) 1, 2, 4, 7, 11, 16, 22, ___, ___ C) ,___, ___

Solutions Ex. 1: Find the next term in the sequence: A) 3, 6, 12, 24, ___, ___ B) 1, 2, 4, 7, 11, 16, 22, ___, ___ C) 48 96 Rule: x2 29 37 Rule: +1, +2, +3, +4, … Rule: divide each section by half

Steps of Inductive Reasoning 1. Process of observing data 2. Recognizing patterns 3. Making generalizations based on those patterns

Identify the pattern and find the next 3 numbers: 1, 4, 9, 16, ____, ____, ____    2) 1, 3, 6, 10, ____, ____, ____ 3) 1, 1, 2, 3, 5, 8, ____, ____, ____

Solutions Identify the pattern and find the next 3 numbers: 1, 4, 9, 16, ____, ____, ____    sequence of perfect squares 2) 1, 3, 6, 10, ____, ____, ____ +2, +3, +4, +5, … 3) 1, 1, 2, 3, 5, 8, ____, ____, ____ Fibonacci – add the 2 previous numbers to get the next. 25 36 49 15 21 28 13 21 34

An example of inductive reasoning Suppose your history teacher likes to give “surprise” quizzes. You notice that, for the first four chapters of the book, she gave a quiz the day after she covered the third lesson. Based on the pattern in your observations, you might generalize …

Solution Based on the pattern in your observations, you might generalize … that you will have a quiz after the third lesson of every chapter.

Identifying a Pattern Identify the pattern and find the next item in the pattern. January, March, May, ... Observe the data.. Identify the pattern.. Make a generalization

Solution January, March, May, ... Alternating months of the year make up the pattern. (skip every other month) The next month is July.

Identifying a Pattern Identify the pattern and find the next item in the pattern. 7, 14, 21, 28, … Observe the data.. Identify the pattern.. Make a generalization

Solution 7, 14, 21, 28, … Multiples of 7 make up the pattern. (add 7 to each term to get the next) The next multiple is 35.

Identifying a Pattern Identify the pattern and find the next item in the pattern.

Solution In this pattern, the figure rotates 90° counter-clockwise each time. The next figure is .

Inductive reasoning can be used to make a conjecture about a number sequence Consider the sequence 10, 7, 9, 6, 8, 5, 7, . . . Make a conjecture about the rule for generating the sequence. Then find the next three terms.

Solution 10, 7, 9, 6, 8, 5, 7, . . Look at how the numbers change from term to term The 1st term in the sequence is 10. You subtract 3 to get the 2nd term. Then you add 2 to get the 3rd term.

10, 7, 9, 6, 8, 5, 7, . . You continue alternating between subtracting 3 and adding 2 to generate the remaining terms. The next three terms are 4, 6, and 3.

Be very careful with the wording/terms you use to describe the pattern Identifying a Pattern Find the next item in the pattern 0.4, 0.04, 0.004, … Be very careful with the wording/terms you use to describe the pattern

Solution 0.4, 0.04, 0.004, … Rules & descriptions can be stated in many different ways: Multiply each term by 0.1 to get the next. Divide each term by 10 to get the next. The next item would be 0.0004.

Writing Conjectures Complete the conjecture. The sum of two positive numbers is ? . List some examples and look for a pattern. 1 + 1 = 2 3.14 + 0.01 = 3.15 3,900 + 1,000,017 = 1,003,917 The sum of two positive numbers is positive.

Writing Conjectures Complete the conjecture. The product of two odd numbers is ? . (what kind of numbers are ‘odd numbers’?) List some examples and look for a pattern. 1  1 = 1 3  3 = 9 -5  7 = -35 The product of two odd numbers is odd.

Geometric Patterns Arrangement of geometric figures that repeat Use inductive reasoning and make conjecture as to the next figure in a pattern Use inductive reasoning to describe the pattern and find the next two figures in the pattern.

Solution Following the pattern: blue L, red +, green T… the next figures would be the red + and the green T

Geometric Patterns Use inductive reasoning to describe the pattern and find the next two figures in the pattern.

Green triangle on the bottom and then two sides CCW Solution Following the pattern: green triangle is moving CCW 120° (or rotating CCW every other side of the hexagon)… the next figures would be Green triangle on the bottom and then two sides CCW

Geometric Patterns Describe the figure that goes in the missing boxes. Describe the next three figures in the pattern below.

Solutions

Make a conjecture about the sum of the first 30 odd numbers. = 12 = 22 = 32 = 42 = 52 .. = 302 1 = 1 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25 .. 1 + 3 + 5 +...+ 61 = 900

cont.: Make a conjecture about the sum of the first 30 odd numbers. 𝒏 = the amount of numbers added Sum of the first 𝒏 odd numbers = 𝒏 𝟐

Truth in Conjectures To show that a conjecture is always true, you must prove it. To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample. A counterexample can be a drawing, a statement, or a number.

Inductive Reasoning assumes that an observed pattern will continue. This may or may not be true. Ex: x = x • x This is true only for x = 0 and x = 1 Conjecture – A conclusion you reach using inductive reasoning.

The first 3 odd prime numbers are Counter Example – To a conjecture is an example for which the conjecture is incorrect. The first 3 odd prime numbers are 3, 5, 7. Make a conjecture about the 4th. 3, 5, 7, ___ One would think that the rule is add 2, but that gives us 9 for the fourth prime number. Is that true? What is the next odd prime number? 11 No

For every integer n, n3 is positive. Finding a Counterexample Show that the conjecture is false by finding a counterexample. For every integer n, n3 is positive. Pick integers and substitute them into the expression to see if the conjecture holds. Let n = 1. Since n3 = 1 and 1 > 0, the conjecture holds. Let n = –3. Since n3 = –27 and –27  0, the conjecture is false. n = –3 is a counterexample.

Two complementary angles are not congruent. Finding a Counterexample Show that the conjecture is false by finding a counterexample. Two complementary angles are not congruent. 45° + 45° = 90° If the two congruent angles both measure 45°, the conjecture is false.

Monthly High Temperatures (ºF) in Abilene, Texas Finding a Counterexample Show that the conjecture is false by finding a counterexample. The monthly high temperature in Abilene is never below 90°F for two months in a row. Monthly High Temperatures (ºF) in Abilene, Texas Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 88 89 97 99 107 109 110 106 103 92 The monthly high temperatures in January and February were 88°F and 89°F, so the conjecture is false.

Planets’ Diameters (km) Finding a Counterexample Show that the conjecture is false by finding a counterexample. The radius of every planet in the solar system is less than 50,000 km. Planets’ Diameters (km) Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune 4880 12,100 12,800 6790 143,000 121,000 51,100 49,500 Since the radius is half the diameter, the radius of Jupiter is 71,500 km and the radius of Saturn is 60,500 km. The conjecture is false.

Supplementary angles are adjacent. Finding a Counterexample Show that the conjecture is false by finding a counterexample. Supplementary angles are adjacent. 23° 157° The supplementary angles are not adjacent, so the conjecture is false.