Inductive and Deductive Reasoning

Problem Solving 101

Using 3R2S

Problems with Solving: Algebra Edition

• Lots of Condensed Information

• Looking how to connect numbers (wrong)

• belief that all equations look the exact same

• there can only be one variable.

The solution to Solving: Algebra Edition

• Read for understanding

• Look for relationships

• Recognizing quantities and the simple relationship between them.

• Putting pieces of a puzzle together.

Writing Equations For Word Problems

• Look for key words that will help you write the equation. Highlight the key words and write an equation to match the problem.

• Check Solution to see if answer makes sense.

• 
  

KEY WORDS

These keywords give you an idea of operations that will be used in a problem or an idea of what your final answer would look like.

Example #1

Linda was selling tickets for the school play. She sold 10 more adult tickets than children tickets and she sold twice as many senior tickets as children tickets.

Write an expression to represent the number of tickets sold.

Multiple Choice

think of the relationship between all qualtities mentioned.

Which is the best to build an equation to solve this problem?

• tickets

• adult tickets

• children tickets

• senior tickets

Step 1: Define your variable

reminder to start by saying

x = ..........

Step 2: Build around the variable

Now that we have starting points, let's write a word equation to guide my mathematics.

Write an expression to represent the number of tickets sold.

• (# of children tix) + (# of adult tix) + ( # of senior tix )

• by using words, at the end you will ensure you have accounted for each part of the problem in your expression/equation

Fill in the Blanks

Write an expression to represent the number of adult tickets sold.

Fill in the Blanks

Write an expression to represent the number of senior tickets sold.

Fill in the Blanks

Linda was selling tickets for the school play. She sold 10 more adult tickets than children tickets and she sold twice as many senior tickets as children tickets.

Write an expression to represent the number of tickets sold.

Open Ended

Adult tickets cost $5, children's tickets cost $2, and senior tickets cost $3.

Write an expression to represent the total amount that is made from the ticket sales.

Open Ended

Linda made $700.

How many children's tickets were sold for the play? How many adult tickets were sold? How many senior tickets were sold?

Open Ended

You have $70 and your sister has $115. You are saving $8 per week and your sister is saving $5 per week.

1) write an equation to represent this scenario.

Open Ended

You have $60 and your sister has $120. You are saving $7 per week and your sister is saving $5 per week. How long will it be before you and your sister have the same amount of money?

MINI-LESSON

Types of Reasoning

Some text here about the topic of discussion

Deductive reasoning begins with general statements accepted to be true (the premises) and ends up with a specific factual statement.

LET'S PRACTICE!!

Deductive Reasoning

Some text here about the topic of discussion

Multiple Choice

Using the statements: If it is Friday, then I will get paid. It is Friday.

Choose the deduction.

• It is not Friday

• It is Friday

• I will not get paid

• I will get paid

Multiple Choice

What can you deduce from the given true statements?

If an animal is a dog, then they like biscuits.

Sammy is a dog.

• Sammy is a Great Dane

• Sammy likes biscuits.

• Sammy is really a cat.

• We can't conclude anything.

Multiple Choice

What does deductive reasoning use to form its arguments?

• patterns

• observed examples

• specific cases

• facts and laws of logic

Multiple Choice

What can you deduce from the given true statements?

No human can fly.

I am a human.

• I can fly.

• I cannot fly.

• I am not a human.

• We can't conclude anything.

Multiple Choice

What can you deduce from the given true statements?

Every child needs love and attention.

Jose is a child.

• Jose does not need love and attention.

• Jose needs love and attention.

• Jose is not a child anymore.

• We can't conclude anything.

Multiple Choice

What can you deduce from the given true statements?

If you live in Orlando, then you live in Florida.

You live in Orlando.

• You don't live in Florida.

• You hate snow.

• You live in Florida.

• We can't conclude anything.

Inductive Reasoning

Using specific observations to arrive at general conclusions.

Abductive Reasoning is similar, but it's when you have incomplete information and knowledge of patterns to offer the simplest, most likely explanation. (BEST GUESS!)

I hope this doctor is using inductive reasoning, not abductive.

Multiple Choice

The type of reasoning where a person makes  conclusions based on observations and patterns is called...

• Inductive reasoning

• Deductive reasoning

• Conjecture

• Experiments

Multiple Choice

How would you describe this pattern's rule?
65, 62, 59, 56, 53, 50

• Subtract 2

• Subtract 3

• Subtract 4

• Add 3

Multiple Choice

What comes next in the pattern?

• up, down, right

• down, up, left

• up, right, down

• down, left, right

LET'S PRACTICE!!

DEDUCTIVE or INDUCTIVE??

It might be challenging, but don't give up!

These are tough concepts, but I believe in you!!

Some text here about the topic of discussion

Multiple Choice

The first marker I pulled out the box is blue. The second one I pulled out is blue, too. So, all the markers in here are probably blue.

• Deductive

• Inductive

Multiple Select

I heard lots of barking last night. The neighbor’s dog must’ve been pretty upset about something, since he rarely barks

• Deductive

• Inductive

Multiple Select

Jack is taller than Jill. Jill is taller than Joey. Therefore, Jack is taller than Joey.

• Deductive

• Inductive

Multiple Select

It was the pizza that made my stomach churn. What else could it be? I was fine until I ate it.

• Deductive

• Inductive

Poll

How did you do in your FIRST reasoning lesson??

• I really struggled. This is difficult for me.

• Not bad, but I need more practice!

• Pretty good! I'm learning it!

• GREAT! I could teach this to others!

​When a detective solves a case, is the detective more likely to use inductive or deductive reasoning? Explain

When might you want to make a conjecture about a set of numbers?​

​Is one counterexample enough to prove that a conjecture is false? Explain

​Suppose you use deductive reasoning to show that an angle is not acute. Can you conclude that the angle is obtuse? Explain

Section 2.2: Deductive and Inductive Reasoning

​

Inductive Reasoning

Using patterns to make conjectures

​The following picture shows a pattern.

Multiple Choice

What is the next figure in the pattern?

​The following shows a number pattern.

Fill in the Blanks

Fill in the next number in the pattern

0, 2, 6, 12, 20, ...

Open Ended

What is the next letter in the sequence?

J, F, M, A, M, J, J, A,

COUNTEREXAMPLE

Proves a conjecture wrong

Conjecture: All dogs are hairy.

COUNTEREXAMPLE: The Argentine Pila Dog is hairless.

Open Ended

What is a counterexample for the conjecture:

All months have 31 days

Open Ended

Give a counterexample for the statement:

The sum of 2 odd numbers is always odd.

DEDUCTIVE REASONING

There are 2 types of Deductive Reasoning

• Law of Detachment

• Law of Syllogism

• https://www.youtube.com/watch?v=YBNrkXEqCiI

Deductive Reasoning

uses laws, facts, definitions, rules, etc.

Consider the sentence:
If I don't understand, then I will ask questions.

The BLUE "if" section is what we will call part "A".

The RED "then" section is what we will call part "B".

Multiple Choice

Which is the "A" section in this statement?

If it is cold, then I wear glove.

• if it is cold

• then I wear gloves

• I don't like to be cold

• it was cold yesterday

Multiple Choice

Which is the "B" section in this statement?

If it is cold, then I wear glove.

• if it is cold

• then I wear gloves

• I don't like to be cold

• it was cold yesterday

Both DETACHMENT and SYLLOGISM will start with a sentence in this form.

The sentence that follows determines if you can use detachment, syllogism, or neither.

To use detachment, the next sentence has to restate the "A" section.

Note: You don't have to restate the "A" section word for word, just the general idea.

You would then conclude the "B" section.

Multiple Choice

What can you conclude using detachment?

If I have a dog, then I have to take it out for walks. I have a Poodle (type of dog).

• Conclusion: Poodles are cute.

• Conclusion: I like taking my dog for walks.

• Conclusion: I have to take my dog out for walks.

Open Ended

What can you conclude using detachment?

If it is after 9pm, then I am in the bed. It is 9:30 pm.

Both DETACHMENT and SYLLOGISM will start with a sentence in this form.

The sentence that follows determines if you can use detachment, syllogism, or neither.

To use syllogism, the next sentence has to start with the "B" section and also has to give a new "then" section, which we will call part "C".

The green is the new then ("C") section.

You would then conclude the "A" and the "C" section in a new if-then sentence.

The "B" sections that are both the end and beginning of a sentence are canceled.

Multiple Choice

What can you conclude using syllogism?

If I exercise, then I will be healthy. If I am healthy, then I will be happy.

• Exercising makes you happy.

• Conclusion: If I exercise, then I will be happy.

• If i am happy, then I exercised.

If these patterns are not followed, you cannot make a valid conclusion.

Multiple Choice

Is the conclusion is based on inductive or deductive reasoning?

Students at Olivia’s high school must have a B average in order to participate in sports. Olivia has a B average, so she concludes that she can participate in sports at school.

• inductive

• deductive

Multiple Choice

Is the conclusion is based on inductive or deductive reasoning?

Holly notices that every Saturday, her neighbor mows his lawn. Today is Saturday. Holly concludes her neighbor will mow his lawn.

• inducutive

• deductive