1-5: Inductive Reasoning Homework 5: p.35 3,5, 6-9, 15-17 Learning Objectives: Apply inductive reasoning to a variety of situations Learn that inductive reasoning does not always lead to a conclusion Entry Task: Find the next three terms of each sequence A 0, 1, 1, 2, 3, 5, 8, 13, 21… …𝟑𝟒, 𝟓𝟓, 𝟖𝟗 Fibonacci Sequence B 0,1,3,6,10,15,21,28… …𝟑𝟔, 𝟒𝟓, 𝟓𝟓 Triangular Numbers C 0, 1, 4, 9, 16, 25, 36… …𝟒𝟗, 𝟔𝟒, 𝟖𝟏 Square Numbers

Concept: Defining Conjecture Conjecture: an educated guess based on known information

Example 1A: Patterns and Conjectures Write a conjecture that describes the pattern 2, 4, 12, 48, 240. Then use your conjecture to find the next item in the sequence. Step 1 : Look for a pattern. 2 4 12 48 240 ×2 ×3 ×4 ×5 Step 2: Make a conjecture The numbers are multiplied by 2, 3, 4, and 5. The next number will be multiplied by 6. So, it will be 6 ● 240 or 1440. Answer: 1440

Example 1B: Patterns and Conjectures Write a conjecture that describes the pattern shown. Then use your conjecture to find the next item in the sequence. 3 9 18 +6 +9

Example 1B: Patterns and Conjectures Conjecture: Notice that 6 is 3 × 2 and 9 is 3 × 3. The next figure will increase by 3 × 4 or 12 segments. So, the next figure will have 18 + 12 or 30 segments. Answer: 30 segments Check Draw the next figure to check your conjecture.

Student Led Example 1A: Patterns and Conjecture Write a conjecture that describes the pattern in the sequence. Then use your conjecture to find the next item in the sequence. 𝟏, 𝟏 𝟒 , 𝟏 𝟗 , 𝟏 𝟏𝟔 , 𝟏 𝟐𝟓 ,… 𝟏 𝟏 𝟐 , 𝟏 𝟐 𝟐 , 𝟏 𝟑 𝟐 , 𝟏 𝟒 𝟐 , 𝟏 𝟓 𝟐 ,… Answer: 𝟏 𝟑𝟔

Student Led Example 1B: Patterns and Conjecture Write a conjecture that describes the pattern in the sequence. Then use your conjecture to find the next item in the sequence. 1 3 6 10 Answer: 15

Concept: Inductive vs Deductive Reasoning Inductive Reasoning: Reasoning that uses a number of specific examples to arrive at a plausible predication. Deductive Reasoning: A system of reasoning that use facts, rules, definition, properties to reach a logical conclusion. Uses examples to predict Make conclusions based on given evidence

Example 2A: Reasoning WEATHER Determine whether the conclusion is based on inductive or deductive reasoning. In Miguel’s town, the month of April has had the most rain for the past 5 years. He thinks that April will have the most rain this year. Answer: Miguel’s conclusion is based on a pattern of observation, so he is using inductive reasoning.

Example 2B: Reasoning WEATHER Determine whether the conclusion is based on inductive or deductive reasoning. Sandra learned that if it is cloudy at night it will not be as cold in the morning than if there are no clouds at night. Sandra knows it will be cloudy tonight, so she believes it will not be cold tomorrow morning. Answer: Sandra is using facts that she has learned about clouds and temperature, so she is using deductive reasoning.

Student Led Example 2: Reasoning Determine whether the conclusion is based on inductive or deductive reasoning. The library charges $0.25 per day for overdue books. Kyle returns a book that is 3 days overdue. Kyle concludes that he will be charged a $0.75 fine. Deductive Macy’s mother orders pizza for dinner every Thursday. Today is Thursday. Macy concludes that she will have pizza for dinner tonight. Inductive Every time Lauren has a healthy meal before her run, she is able to run farther than her sister who does not like healthy foods. Lauren has eaten a healthy meal, so she concluded that she will run farther than her sister Inductive

Activity: Building Conjectures Step 1: Extend Lines Step 2: Measure Angles 𝟔𝟎° 𝟔𝟎° 𝟔𝟎° 𝟔𝟎° Step 3: Sum the Angles 𝟔𝟎° 𝟔𝟎° 𝟔𝟎° +𝟔𝟎° +𝟔𝟎° +𝟔𝟎° +𝟔𝟎° +𝟔𝟎° =?°

Concept: Counterexamples A counterexample is an example used to show that a given statement is not always true (or false)

Example 5A: Finding Counterexamples Answer: Perry has a population of 9,652, and it has a higher rate of unemployment than Butler, which has a population of 203,709. UNEMPLOYMENT Based on the table showing unemployment rates for various counties in Alabama, find a counterexample for the following statement. The unemployment rate is highest in the counties with the most people.

Student Led Example 5A: Finding Counterexamples Answer: Alabama and West Virginia serve as counterexamples to the statement DRIVING This table shows selected states, the 2000 population of each state, and the number of people per 1000 residents who are licensed drivers in each state. Based on the table, which two states could be used as a counterexample for the following statement? The greater the population of a state, the lower the number of drivers per 1000 residents.

Example 5B: Finding Counter Examples For any value for 𝑥, 𝑥 2 ≥𝑥 𝒙 𝒙 𝟐 𝟏 𝟐 𝟑 𝟒 𝟎 𝟏 𝟐 𝟐 𝟑 𝟏 𝟒 𝟗 𝟏𝟔 𝟎 𝟏 𝟒 𝟒 𝟗 True True True True True False False Only takes ONE counterexample to prove a conjecture false

Student Led Example 5B: Finding Counterexamples Determine if the following are true or false. If false, provide a counterexample If you live in Chicago, then you live in Illinois If it’s Easter, then it’s April Easter occurs in March as well All even numbers are divisible by 10 2 is not divisible by 10 The only shape with four equal sides is a square Rhombus The product of two odd integers is an odd integer All prime numbers are odd 2 is even and a prime number If two angles are supplementary, then they are both acute 𝟏𝟕𝟗°+𝟏°=𝟏𝟖𝟎° and 𝟏𝟕𝟗° is obtuse

End of Lesson Observe the pattern. Make a conjecture about the next figure in the sequence Make a conjecture based on the data shown