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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, 21β¦ β¦ππ, ππ, ππ Fibonacci Sequence B 0,1,3,6,10,15,21,28β¦ β¦ππ, ππ, ππ Triangular Numbers C 0, 1, 4, 9, 16, 25, 36β¦ β¦ππ, ππ, ππ Square Numbers
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Concept: Defining Conjecture
Conjecture: an educated guess based on known information
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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. Γ2 Γ3 Γ4 Γ5 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 β 240 or 1440. Answer: 1440
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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. +6 +9
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Example 1B: Patterns and Conjectures
Conjecture: Notice that 6 is 3 Γ 2 and 9 is 3 Γ 3. The next figure will increase by 3 Γ 4 or 12 segments. So, the next figure will have or 30 segments. Answer: 30 segments Check Draw the next figure to check your conjecture.
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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. π, π π , π π , π ππ , π ππ ,β¦ π π π , π π π , π π π , π π π , π π π ,β¦ Answer: π ππ
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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
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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
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Example 2A: Reasoning WEATHER Determine whether the conclusion is based on inductive or deductive reasoning. In Miguelβs 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: Miguelβs conclusion is based on a pattern of observation, so he is using inductive reasoning.
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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.
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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 Macyβs 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
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Activity: Building Conjectures
Step 1: Extend Lines Step 2: Measure Angles ππΒ° ππΒ° ππΒ° ππΒ° Step 3: Sum the Angles ππΒ° ππΒ° ππΒ° +ππΒ° +ππΒ° +ππΒ° +ππΒ° +ππΒ° =?Β°
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Concept: Counterexamples
A counterexample is an example used to show that a given statement is not always true (or false)
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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.
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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.
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Example 5B: Finding Counter Examples
For any value for π₯, π₯ 2 β₯π₯ π π π π π π π π π π π π π π π ππ π π π π π True True True True True False False Only takes ONE counterexample to prove a conjecture false
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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 itβs Easter, then itβs 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 πππΒ°+πΒ°=πππΒ° and πππΒ° is obtuse
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End of Lesson Observe the pattern. Make a conjecture about the next figure in the sequence Make a conjecture based on the data shown
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