MATHEMATICS 20-2
INDUCTIVE AND DEDUCTIVE REASONING
High School collaborative venture with
Edmonton Christian, Institutional Services, Jasper Place,
Millwoods Christian, Queen Elizabeth and Victoria Schools
Edmonton Christian: Jenn Johnson
Institutional Services: Eric Hanson
Jasper Place: Jessica Noselski
Millwoods Christian: Ken Scharf
Queen Elizabeth: David Hernandez-Rivera
Victoria: Gina MacKechnie
Facilitator: John Scammell (Consulting Services)
Editor: Jim Reed (Contracted)
2010 - 2011
Mathematics 20-2
Inductive and Deductive Reasoning
Page 2 of 46
TABLE OF CONTENTS
STAGE 1
DESIRED RESULTS
PAGE
Big Idea
4
Enduring Understandings
4
Essential Questions
4
Knowledge
5
Skills
6
STAGE 2
ASSESSMENT EVIDENCE
Transfer Task
Detective Challenge
Teacher Notes for Transfer Task
Transfer Task
Glossary and Rubric
Possible Solution
7
8
14
16
STAGE 3 LEARNING PLANS
Lesson #1
Identifying Patterns and Writing Conjectures
18
Lesson #2
Inductive Reasoning & Counterexamples
25
Lesson #3
Deductive Reasoning
30
Lesson #4
Proofs by Induction
37
Lesson #5 Proofs by Deduction
42
Mathematics 20-2
Inductive and Deductive Reasoning
Page 3 of 46
Mathematics 20-2
Inductive and Deductive Reasoning
STAGE 1
Desired Results
Big Idea:
In the world around us it is important to see all types of problems from different
perspectives. We can use inductive and deductive reasoning to make general
assumptions and to validate an argument.
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to
it often.
Enduring Understandings:
Students will understand …





Patterns exist in many forms and can be identified and described.
Inductive reasoning is based on patterns and assumptions that can leave an
argument open to flaws.
Deductive reasoning attempts to show that a conclusion follows from a set of
premises.
Inductive and deductive reasoning can be applied to contextual problems.
Number sense and logic can be used to validate/disprove arguments.
Essential Questions:




When is it appropriate to use deductive and/or inductive reasoning?
What are some real-life applications of deductive and inductive reasoning?
What is the role of a counterexample?
Is there a unique proof for each argument?
Implementation note:
Ask students to consider one of the
essential questions every lesson or two.
Has their thinking changed or evolved?
Mathematics 20-2
Inductive and Deductive Reasoning
Page 4 of 46
Knowledge:
Enduring
Understanding
Students will understand…




Inductive reasoning is
based on patterns and
assumptions that can
leave your argument
open to flaws.
Deductive reasoning
attempts to show that
a conclusion follows
from a set of premises.
Inductive and
deductive reasoning
can be applied to
contextual problems.
Number sense and
logic can be used to
validate/disprove
arguments.
8888
I*N =
Specific
Outcomes
Description of
Knowledge
Students will know …
*N1, N2









how to identify a pattern
what deductive reasoning is
what inductive reasoning is
when to use inductive reasoning
when to use deductive reasoning
what makes a counter example valid
if an argument is valid
how to identify an error
that if an error occurs, two outcomes can arise:
o the entire case is disproved
o an exception is revealed
Number
Mathematics 20-2
Inductive and Deductive Reasoning
Page 5 of 46
Skills:
Enduring
Understanding
Specific
Outcomes
Students will understand…
Description of
Skills
Students will be able to…
*N1, N2




Inductive reasoning is
based on patterns and
assumptions that can
leave your argument
open to flaws.
Deductive reasoning
attempts to show that
a conclusion follows
from a set of premises.
Inductive and
deductive reasoning
can be applied to
contextual problems.
Number sense and
logic can be used to
validate/disprove
arguments.







use appropriate tools to demonstrate their
arguments
demonstrate their arguments
use appropriate tools to prove/disprove an
argument
prove using algebraic methods
recognize false conjectures
state a conjecture by observing patterns
use
o divisibility rules
o number properties
o algebra and number relations in order to
justify an argument
*N = Number
Implementation note:
Teachers need to continually ask
themselves; if their students are
acquiring the knowledge and skills
needed for the unit.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 6 of 46
STAGE 2
1
Assessment Evidence
Desired Results Desired Results
Detective Challenge
Teacher Notes
There is one transfer task to evaluate student understanding of the concepts relating
to inductive and deductive reasoning. A photocopy-ready version of the transfer task is
included in this section.
Implementation note:
Students must be given the transfer task & rubric at
the beginning of the unit. They need to know how
they will be assessed and what they are working
toward.
Notes are included in the sample solution following the transfer task.
Implementation note:
Teachers need to consider what performances and
products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Mathematics 20-2
Inductive and Deductive Reasoning
Page 7 of 46
Detective Challenge - Student Assessment Task
Stanley Skeezeball has recently been a victim of a theft crime. He is accusing
Johnny B. Good, his former business partner, of stealing company property and
money from their office on the 11th floor of The Big Tall Towers Office Building.
The lead detective has obtained information. Inspect the Crime Report and use
your knowledge of logic and reasoning to help the Edmonton Police
Department. Answer the following questions and present your findings in the
Intern Report.
o List the information given.
o Summarize how the crime was committed?
o Use inductive reasoning to prove who committed the crime
o Use inductive reasoning to prove if Johnny B. Good was involved.
o Is he guilty?
o Provide a counterexample to support your conclusions.
o Explain how your inductive reasoning may lead to a false conclusion.
Crime/Incident Report
C
R
I
M
E
W
I
T
N
E
S
S
Type of Incident:
Year
Day of Week
Time
Break and Enter, Theft, Vandalism
2010
Wednesday
8:30 pm
Location of Incident (or address)
City
Private Office of Stan Skeezeball and Johnny B. Good
Edmonton
Big Tall Towers Office Building – 11th floor
Witness Name:
Beefy "Bulldog" Jones (Security Guard)
Witness Statement:
The following witness statement is an account of the events according to Beefy ‘Bulldog’ Jones (Security Guard) from his kiosk in the
front lobby.

At 8:30 pm a key card was used to gain entry to the front lobby of The Big Tall Towers Office Building

Male 1 approximately 6’3” with short brown hair entered the mail lobby. As Mr. Jones returned from a nightly walk he
witnessed the male enter the stairwell on the North side of the building at 8:31 pm. He only saw the male from behind

At 8:35 pm a different key card was used to gain entry to the front lobby of The Big Tall Towers Office Building.

Mr. Jones witnessed a second male enter the main lobby and proceed to the elevator at 8:36 pm. Male 2 was approximately
5’10” and wore a ball cap.

At 8:37 pm the security camera caught male 1 walking the corridors on the 11th floor.

At 8:38 pm male 2 exited the elevator on the 10th floor (security camera).

At 8:42 pm male 2 was seen walking down the corridor on the 11th floor (security camera).

At 8:45 pm Mr. Jones departed the kiosk to complete a nightly walk around the perimeter of the building.

Male 2 entered the elevator on the 10th floor at 8:52 pm.

Male 2 entered lobby and departed from the building at 8:54 pm.

Mr. Jones witnessed male 1 departing from the back entry of the building while on his nightly walk around the building.
Male 1 departed at 8:58 pm.
Witness Name:
Mr. Sid Clean (Custodian)
Witness Statement:
Mr. Sid Clean (Custodian) provided the following witness statement.

Mr. Clean began his shift at 6:00 pm; he was assigned to clean floors 10, 11 and 12.

Mr. Clean witnessed a male (6’3” with short brown hair) on the 12th floor just before his break.

Mr. Clean stated he normally takes his break at 8:45 pm. The break room is on the 11th floor.

During his break Mr. Clean heard banging from down the hall.

Mr. Clean received a text message from his girlfriend at 8:51 pm right before he went to the bathroom on the 10th floor.

On his way to the bathroom Mr. Clean stopped right outside the bathroom door and saw a male with a ball cap on walking
towards the elevator.
M
O
I
N
F
O
R
M
A
T
I
O
N
P
R
O
P
E
R
T
Y
S
U
S
P
E
C
T
S
N
A
R
R
A
T
I
V
E
Total # of witnesses at crime: 2
Place
of attack:
Business
Vehicle
Street/Alley
Lot/Park/Yard
Vessels
Other
Description of Surrounding area:
Residential Business
Industrial/Mftg.
Recreational
Institutional
Open Space
School
Marine/Water
Force Tool Weapon
Specify:
How Used:
Hammer
Broke Lock on door
Type of Structure
N/A
Security Used
Suspect Actions
High Rise Office Building
Security guard on duty.
Point of Entry
Key card used to enter lobby of
office building
Item
Stolen
Miscellaneous Description
No. Article Name
Received
1
Laptop
stolen
MacBook Pro - neon pink, sticker on cover "I love Butterflies"
Includes personal information from clients and banking records and codes
2
Money
stolen
Taken from safe, $50 bills bundled in groups of 25, 2 bundles
Arrested
Suspect #1
Sex
Age
DOB
Height
Weight
Build
Johnny B. Good
Male
38
February 29, 1972
5'10''
195 lbs
Average
Y
N
Additional information/Further suspect description (i.e., glasses, tattoos, teeth, birthmarks, jewellery, scars, etc.
None
Suspect’s clothing
Red New York Yankees Baseball Cap, Dark Grey Jacket, Black Pants
Hair Length/Type
Hair Style
Facial Hair
Complexion
Short, brown hair, slightly balding
Crew Cut
None
Normal
Evidence:
No fingerprints found
Hammer found near office door
Officer’s statement/investigation
Upon further investigation it was discovered that the security tapes were damaged beyond repair.
Other
Value
$1200
$2500
Hair color
brown
Eye color
brown
General Appearance
N/A
Intern Report
List the information given:
Summarize the crime that was committed…
Write a proof statement about who committed the crime (male 1 or male 2?):
Statement
Reason
Write a proof statement about who Johnny B. Good is (male 1 or male 2?):
Statement
Reason
Johnny B. Good is _________________.
(guilty/not guilty)
Counterexample (How could you prove that the other person is guilty/not guilty?):
False Conclusion (how could your inductive reasoning be incorrect? Where are
the gaps in your logic?)
Glossary
conjecture - A generalization made through inductive reasoning
deductive reasoning – A type of reasoning by which generalizations are drawn from
specific patterns in observed data
inductive reasoning – A type of reasoning by which generalizations are drawn from
general patterns in observed data
pattern – A repeated sequence or arrangement about which predictions can be made
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Assessment
Mathematics 20-2
Inductive and Deductive Reasoning
Rubric
Level
4
3
2
1
Excellent
Proficient
Adequate
Limited *
Criteria
Shows thorough
understanding;
explanations are
effective and thorough.
Shows
understanding;
explanations are
appropriate.
Explains Choice
Shows a solution for
the problem; provides
a logical and
insightful explanation.
Solves and explains
the solution using
appropriate
terminology
conjecture, deductive
inductive reasoning).
The student(s)
has/have a good
understanding of
terminology and are
able to correctly solve
and explain the
solution.
Shows a solution for
the problem;
provides an
adequate
explanation.
For the most part
student(s) has/have
an understanding of
terminology and can
solve and explain the
solution.
Communicates
findings in a
thoughtful and
convincing manner.
Student(s) show(s)
clear understanding
of inductive and
deductive reasoning
Communicates
findings in a
thoughtful and
convincing manner,
using specific
mathematical
vocabulary.
Explanations of proof
Communicates
findings in an
interesting and
coherent manner,
using appropriate
mathematical
vocabulary.
Shows partial
understanding;
explanations are
often incomplete or
somewhat confusing.
Shows a solution for the
problem; provides
explanations that are
complete but vague.
Shows very limited
understanding;
explanations
are omitted or
inappropriate.
Shows a solution for
the problem; provides
explanations that are
incomplete or
confusing.
Student(s) has/have a
partial understanding of
terminology and have
errors in explaining the
solution.
Student(s) has/have
limited understanding
of terminology and
cannot correctly
explain or solve the
question.
Communicates findings
in a predictable and
simplistic manner using
inconsistent
mathematical
vocabulary.
Communicates
findings in a
confusing and vague
manner using
inappropriate
mathematical
vocabulary.
When work is judged to be limited or insufficient, the teacher makes decisions
about appropriate intervention to help the student improve.
Possible Solution to Detective Challenge
Mathematics 20-2
Inductive and Deductive Reasoning
Page 15 of 46
Mathematics 20-2
Inductive and Deductive Reasoning
Page 16 of 46
Mathematics 20-2
Inductive and Deductive Reasoning
Page 17 of 46
STAGE 3
Learning Plans
Lesson 1
Identifying Patterns and Writing Conjectures
STAGE 1
BIG IDEA:
In the world around us it is important to see all types of problems from different perspectives. We can
use inductive and deductive reasoning to make general assumptions and to validate an argument.
ENDURING UNDERSTANDINGS:
Students will understand …

Patterns exist in many forms and can be
identified and described.
ESSENTIAL QUESTIONS:
 What are some real-life applications of
deductive and inductive reasoning?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 how to identify a pattern
 state a conjecture by observing patterns
Implementation note:
Each lesson is a conceptual unit and is not intended to
be taught on a one lesson per block basis. Each
represents a concept to be covered and can take
anywhere from part of a class to several classes to
complete.
Lesson Summary
Students will be able to identify patterns pictorially and numerically and write a
conjecture regarding the observed properties.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 18 of 46
Lesson Plan
Initial Activity:
From the information provided, find the names and positions of the first eight to finish
the marathon. Sean finishes the marathon in fourth place; he finishes after John, but
before Sandra. Sandra finishes before Robert but after Liam. John finishes after Rick
but before Alex. Anne finishes two places after Alex. Liam is sixth to finish the race.
Answer:
1st – Rick
2nd – John
3rd – Alex
4th – Sean
5th – Anne
6th – Liam
7th – Sandra
8th – Robert
http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.PATT&I
D2=AB.MATH.JR.PATT.PATT&lesson=html/object_interactives/patterns/use_it.html
Mathematics 20-2
Inductive and Deductive Reasoning
Page 19 of 46
Ask Students:
What is a pattern and how do we know something has a pattern? Give examples of
patterns in real life?
Ask Students:
What is one looking for when they ask someone to write a conjecture?
Show examples of patterns and have students write conjectures of what they expect
to happen if the patterns continue. Here are some examples you could use:
Fractal Patterns in Nature
Fractals are unpredictable in specific details yet deterministic when viewed as a total
pattern. In many ways this reflects what we observe in the small details and total
pattern of life in all its physical and mental varieties.
Ferns
Mountains in Tibet
Source: http://www.miqel.com/fractals_math_patterns/visual-math-natural-fractals.html
Sunflower and Fibonacci
Source: http://brittgow.globalteacher.org.au/files/2010/03/fibonacci-sunflower.jpg
Mathematics 20-2
Inductive and Deductive Reasoning
Page 20 of 46
Draw Your Own Fractal
Source: http://math.youngzones.org/Fractal webpages/fractal_intro.html
Other Patterns
1. 2, 8, 27, 64, …
2. 34, 7, 29, 11, 23, 16, 16, 22, 8 …
3. 1, 2, 3, 7, 22, …
4. F, S, T, F, F, S, . . .
5. 7, -5, 2, 1, -3, 7, -8, 13, ?, 19
Mathematics 20-2
Inductive and Deductive Reasoning
Page 21 of 46
Going Beyond
The teacher can bring in various music pieces and talk about the relationship of music
with patterns.
Show TED talk regarding fractals in nature:
http://www.ted.com/talks/ron_eglash_on_african_fractals.html
Mathematics 20-2
Inductive and Deductive Reasoning
Page 22 of 46
Another way students can extend their learning is through a game called SET.
Select "Daily Puzzle" from: http://www.setgame.com/
November 23, 2010 sample:
Resources
Principle of Mathematics - Nelson
Section 1.2-1.3 (pages 16-23)
Provide examples of number patterns and/or patterns that can be seen in nature.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 23 of 46
Supporting
http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.PATT&I
D2=AB.MATH.JR.PATT.PATT&lesson=html/object_interactives/patterns/use_it.html
Video of fractals in nature: http://www.youtube.com/watch?v=kkGeOWYOFoA
Assessment
Students need to demonstrate an ability to recognize patterns and write a conjecture
describing the pattern and predicating what will occur if the pattern continues.
Glossary
conjecture - A generalization made through inductive reasoning
pattern – A repeated sequence or arrangement about which predictions can be made
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Other
Mathematics 20-2
Inductive and Deductive Reasoning
Page 24 of 46
Lesson 2
Inductive Reasoning & Counterexamples
STAGE 1
BIG IDEA:
In the world around us it is important to see all types of problems from different perspectives. We can
use inductive and deductive reasoning to make general assumptions and to validate an argument.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …

Inductive reasoning is based on patterns and
assumptions that can leave your argument
open to flaws.
 When is it appropriate to use deductive and/or
inductive reasoning?
 What are some real-life applications of
deductive and/or inductive reasoning?
 What is the role of a counterexample?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 what inductive reasoning is
 when to use inductive reasoning
 what makes a counter example valid
 demonstrate their arguments
 recognize false conjectures
 use appropriate tools to prove/disprove an
argument
Lesson Summary
1. Students will be able to recognize examples of inductive reasoning, by
describing if the reasoning provided is weak or strong.
2. Through counterexamples students will disprove a given conjecture.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 25 of 46
Lesson Plan
Initial Activity:
Bank A
Bank B
7.00%
7.00%
6.00%
6.00%
5.00%
15 Year
5.00%
4.00%
30 Year
4.00%
5 Year
3.00%
3.00%
2.00%
2.00%
1.00%
1.00%
0.00%
Feb
15 Year
30 Year
5 Year
0.00%
Mar
Feb
Mar
The mortgage rates for March decreased from February
Ask Students: What conclusions can you make from this graph about mortgages?
What is Inductive Reasoning?
Inductive reasoning is reasoning based on detailed facts and general principles, which
are eventually used to reach a specific conclusion. There are two types of conjectures,
strong and weak. The strong inductive reasoning is specific and does not lead to a
false conjecture. A weak inductive reasoning leaves it self open to false conjecture.
Example of Strong Inductive Reasoning:
All the tigers observed in a particular region have yellow stripes, therefore all the tigers
native to this region have yellow stripes.
Examples of Weak Inductive Reasoning:
I always run the red light; therefore everybody runs the red light.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 26 of 46
More Examples of Inductive Reasoning:
Students should be able to:


Classify each statement of the following as strong or weak inductive reasoning
examples.
Explain why each example is strong or weak.
1. "Every time you eat shrimp, you get cramps. Therefore I get cramps because I ate
shrimp."
2. "Mikhail is from Russia and Russians are tall. Therefore Mikhail is tall."
3. "When chimpanzees are exposed to rage, they tend to become violent. Humans
are similar to chimpanzees. Therefore they tend to get violent when exposed to
rage."
4. "All men are mortal. Socrates is a man. Therefore Socrates is mortal."
5. "The woman in the neighboring apartment has a shrill voice. I can hear a shrill
voice from outside. Therefore the woman in the neighboring apartment is
shouting."
Counter Examples:
Introduce the idea that not all inductive/deductive reasoning is true. Only ONE counter
example is needed to disprove a conjecture.
For each inductive reasoning example, have the students:

Write two examples that confirm the conjecture.

Write one counter example.
1. In Edmonton we had a white Christmas in 2008, 2009, and 2010.
2. A number that is not positive is negative.
3. The square of a number is always greater than that number.
4. A larger number divided by a smaller number is always greater than 1.
5. A number raised to a whole number exponent will always produce a number
greater than or equal to the base.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 27 of 46
Conclusion:
Bank A
Bank B
7.00%
7 .0 0 %
6.00%
6 .0 0 %
5.00%
4.00%
3.00%
15 Year
5 .0 0 %
30 Year
4 .0 0 %
5 Year
3 .0 0 %
2.00%
15 Y e a r
30 Year
5 Year
2 .0 0 %
1.00%
1.0 0 %
0.00%
Feb
Mar
0 .0 0 %
F eb
M ar
Ask Students:
Is this graph a strong or weak inductive reasoning example?
What can be done to the graph to make it a strong inductive reasoning example?
Going Beyond
Student Activity: Make your own strong and weak inductive reasoning example with
solutions.
http://www.learnalberta.ca/content/me20tle/TLE11/patterns/patterns_desc.html
Another way students can extend their learning is through a game called Set.
Select "Daily Puzzle" from: http://www.setgame.com/November 23, 2010 sample
Mathematics 20-2
Inductive and Deductive Reasoning
Page 28 of 46
Resources
Principle of Mathematics - Nelson
Section 1.3 (pages 18-23)
http://www.buzzle.com/articles/inductive-reasoning-examples.html
Supporting
Video of fractals in nature: http://www.youtube.com/watch?v=kkGeOWYOFoA
Assessment
Written formative assessments
Glossary
inductive reasoning – A type of reasoning by which generalizations are drawn from
patterns in observed data
Other
Mathematics 20-2
Inductive and Deductive Reasoning
Page 29 of 46
Lesson 3
Deductive Reasoning
STAGE 1
BIG IDEA:
In the world around us it is important to see all types of problems from different perspectives. We can
use inductive and deductive reasoning to make general assumptions and to validate an argument.
ENDURING UNDERSTANDINGS:
Students will understand …


Deductive reasoning attempts to show that a
conclusion follows from a set of premises.
Inductive and deductive reasoning can be
applied to contextual problems.
ESSENTIAL QUESTIONS:
 When is it appropriate to use deductive and/or
inductive reasoning?
 What are some real-life applications of
deductive and inductive reasoning?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 what deductive reasoning is
 when to use deductive reasoning
 use appropriate tools to demonstrate their
arguments
 demonstrate their arguments.
 use appropriate tools to prove/disprove an
argument
Lesson Summary


Students will be able to describe and recognize examples of deductive
reasoning.
Through counterexamples students will disprove a given conjecture.
Mathematics 20-2
Inductive and Deductive Reasoning
Page 30 of 46
Lesson Plan
Define Deductive Reasoning and provide a few examples of deductive reasoning:
Deductive reasoning – A type of reasoning by which generalizations are drawn from
patterns in observed data
or
- Making a conclusion based on statements we believe to be
true.
Deductive Reasoning Examples
1. All oranges are fruits. All fruits grow on trees. Therefore, all oranges grow on trees.
2. All bachelors are single. Johnny is single. Hence, Johnny is a bachelor.
Source: http://www.buzzle.com/articles/deductive-reasoning-examples.html
Given: Triangle ABC and line CD are parallel
Prove: ∠1 + ∠2 + ∠3 = 180o
Proof: ∠1 = ∠4
∠2 = ∠5
∠4 + ∠3 + ∠5 = 180o
∠1 + ∠3 + ∠2 = 180o
Mathematics 20-2
Inductive and Deductive Reasoning
Page 31 of 46
Deductive Reasoning Puzzles
Have students work through a deductive puzzle
1. The Hospital Staff
The staff at the hospital consists of 16 doctors and nurses, including me. The
following fact applies to the staff members; whether you include me or not does
not make a difference. The staff consists of:




more nurses than doctors
more male doctors than male nurses.
more male nurses than female nurses.
at least one female doctor.
What is the sex and occupation of the speaker? (Answer: woman nurse)
Source: Test Your Logic (by George J Summers, Dover Publications, 1972)
(Pan American and International Copyright Conventions)
2. The Woman Freeman will Marry:
Freeman knows five women: Ada, Bea, Cyd, Deb, and Eve.

The women are in two age brackets: three women are under 30 and two
women are over 30.

Two women are teachers and the other three women are secretaries.

Ada and Cyd are in the same age bracket.

Deb and Eve are in different age brackets.

Bea and Eve have the same occupation.

Cyd and Deb have different occupations.

Of the five women, Freeman will marry the teacher over 30.
Whom will Freeman marry? (Answer: Deb)
Source: Test Your Logic (by George J Summers, Dover Publications, 1972)
(Pan American and International Copyright Conventions
3. Given 34, 7, 29, 11, 23, 16, 16, 22, 8, _, _, _, what are the next three numbers?
4. Given 7, -5, 2, 1, -3, 7, -8, 13, ?, 19 What is the value of the missing term?
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5. A snail is at the bottom of a well and wants to get out. He manages to crawl up the
wall 3 feet each day, but at night slips back down 2 feet when he rests. If the well is
30 feet deep, how long will it take him to get out?
6. An Old friend of the family left $666,666 to 2 fathers and 2 sons to be split equally.
After careful consideration they each happily received $222,222. Why and how
was this possible?
7. If a snail crawls halfway around a circle then turns around and crawls halfway
back, is it now back where it started?
8. What day of the year has 25 hours in it?
Source for 3-8: http://www.justriddlesandmore.com/logicquestions.html
9. Triangles Puzzle 1: Look at the two top triangles, and then work out what number
should replace the question mark in the bottom triangle.
Source:
http://www.justriddlesandmore.com/logicquestions.html
http://www.buzzle.com/articles/deductive-reasoning-examples.html
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Inductive Reasoning vs. Deductive Reasoning
Inductive reasoning makes conclusions based on several examples showing that
same result.
Deductive reasoning makes conclusions based on statements we believe to be true.
Deductive reasoning is one of the two basic forms of valid reasoning, the other one
being inductive reasoning. The main difference between these two types of reasoning
is that, inductive reasoning argues from a specific to a general base whereas
deductive reasoning goes from a general to a specific instance. Thus, deductive
reasoning is the method by which conclusions are drawn using proofs.
Going Beyond
Activity:
Students make their own strong and weak inductive reasoning with solutions
Another way students can extend their learning is through a game called SET.
Select "Daily Puzzle" from: http://www.setgame.com/
November 23, 2010 sample:
Mathematics 20-2
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See "Reasoning and Analyzing Conjectures" on the following site for additional
information (you may have to install the MathReader):
http://www.learnalberta.ca/content/me20tle/TLE11/patterns/patterns_desc.html
Resources
Principle of Mathematics - Nelson
Section 1.4 (pages 27-33)
http://www.buzzle.com/articles/deductive-reasoning-examples.html
Supporting
Set game: Select "Daily Puzzle" from: http://www.setgame.com/
http://www.learnalberta.ca/content/me20tle/TLE11/patterns/patterns_desc.html
Video of fractals in nature: http://www.youtube.com/watch?v=kkGeOWYOFoA
Assessment
Students need to demonstrate an ability to recognize examples of inductive reasoning
and provide true or false conjectures.
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Glossary
deductive reasoning – A type of reasoning by which generalizations are drawn from
patterns in observed data
inductive reasoning – A type of reasoning by which generalizations are drawn from
patterns in observed data
Other
Mathematics 20-2
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Lesson 4
Proofs by Induction
STAGE 1
BIG IDEA:
In the world around us it is important to see all types of problems from different perspectives. We can
use inductive and deductive reasoning to make general assumptions and to validate an argument.
ENDURING UNDERSTANDINGS:
Students will understand …


Inductive and deductive reasoning can be
applied to contextual problems.
Inductive reasoning is based on patterns and
assumptions that can leave your argument
open to flaws.
ESSENTIAL QUESTIONS:
 Is there a unique proof for each argument?
 When is it appropriate to use deductive and/or
inductive reasoning?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 what inductive reasoning is
 when to use inductive reasoning
 if an argument is valid
 use appropriate tools to prove/disprove an
argument
Mathematics 20-2
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Lesson Summary

Students will be able to support inductive and deductive reasoning based on
patterns and assumptions when give a contextual problem.
Lesson Plan
Review what inductive reasoning is. Students can provide examples for inductive
reasoning.
1. What is the relationship between the diameter and the circumference of a circle?
Solution:
C
=π
d
2. What is the pattern when adding two odd numbers together?
a. Add 3 + 5 =
b. Add 7 +13 =
c. Add 9 + 21 =
Solution:
The sum of two odd numbers is an even number. (Answers will vary for the
following tasks).
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
Try other examples to prove or disprove adding two odd numbers is an even
number.

Show the proof for this example.

Does a pattern exist when 3 odd numbers are added together?

Does a pattern exist when 4 or 5 odd numbers are added together?

Are there any patterns that become apparent when adding odd numbers
together?
3. What are the sums of the angles of a quadrilateral?
Solution: 360o
4. Compare the diagonal length to side length ratio of a square.
Solution: diagonal =
ratio:
Mathematics 20-2
s 2 + s 2 = 2s 2 = 2s
2s
= 2
s
Inductive and Deductive Reasoning
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5. Does this diagonal length to side length ratio exist for other quadrilaterals?
Solution: No. The diagonal =
l2 + w2 .
Going Beyond
Resources
Principle of Mathematics - Nelson
Section 1.1 (pages 6-14)
Supporting
Assessment
Written formative assessments
Mathematics 20-2
Inductive and Deductive Reasoning
Page 40 of 46
Glossary
deductive reasoning – A type of reasoning by which generalizations are drawn from
specific patterns in observed data
inductive reasoning – A type of reasoning by which generalizations are drawn from
general patterns in observed data
Other
Mathematics 20-2
Inductive and Deductive Reasoning
Page 41 of 46
Lesson 5
Proofs by Deduction
STAGE 1
BIG IDEA:
In the world around us it is important to see all types of problems from different perspectives. We can
use inductive and deductive reasoning to make general assumptions and to validate an argument.
ENDURING UNDERSTANDINGS:
Students will understand …



Deductive reasoning attempts to show that a
conclusion follows from a set of premises.
Inductive and deductive reasoning can be
applied to contextual problems.
Number sense and logic can be used to
validate/disprove arguments.
ESSENTIAL QUESTIONS:
 When is it appropriate to use deductive and/or
inductive reasoning?
 What are some real-life applications of
deductive and inductive reasoning?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 what inductive reasoning is
 when to use inductive reasoning
 if an argument is valid
 use appropriate tools to demonstrate their
arguments
 prove using algebraic methods
 state a conjecture by observing patterns
 use
 divisibility rules
 number properties
Lesson Summary

Students are to support their answers to contextual problems using deductive
reasoning.
Mathematics 20-2
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Lesson Plan
Introduction Activity A:
Example:
1. Pick any number
2. Multiply that number by 3
3. Add 45
4. Double your number
5. Divide your number by 6
6. Subtract by the number you initially picked
7. The number you have left is 15
Task: Prove 15 is the final answer.
Solution:
1. Pick any number
n
2. Multiply that number by 3
3n
3. Add 45
3n + 45
4. Double your number
6n + 90
5. Divide your number by 6
n + 15
6. Subtract by the number you initially picked
n + 15 – n
7. The number you have left is 15
15
Introduction Activity B:
Example:
1. Pick any number
2. Subtract by 1
3. Multiply by 3
4. Add 12
5. Divide by 3
6. Add 5
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7. Subtract by the number you initially picked
8. The answer is 8
Task: Prove 15 is the final answer.
Solution:
1. Pick any number
n
2. Subtract by 1
n–1
3. Multiply by 3
3n – 3
4. Add 12
3n + 9
5. Divide by 3
n+3
6. Add 5
n+8
7. Subtract by the number you initially picked n + 8 – n
8. The answer is 8
9
Task: Have students write their own number magic trick.
Solution: Will vary.
Prerequisite:
Deduction:


Reasoning based on statements, which are known to be true.
Like theorems
Lesson:
1. Prove deductively that when two odd numbers are added, the result is an even
number.
Solution:
2n + 1 = first odd number
2m + 1 = second odd number
(2n + 1) + (2m + 1) = 2n + 2m + 2 = 2(n + m + 1)
Since 2 is a factor, the result must be an even number.
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2. Prove deductively that the difference between an odd number and an even number
is always odd.
Solution:
(2n + 1) – (2m) = 2n – 2m + 1 = 2(n – m) + 1
Whenever 1 is added to an even number, the result is odd.
3. ABCD is a square and ΔABE is an equilateral triangle. Explain why
ΔAED is isosceles.
Solution:
AB = AD
Properities of a square
AB = AE
Properties of an equalteral triangle
AD = AE
Transitive property
Therefore ΔADE is Isosceles (Triangle with 2 equal sides)
4. [extension problem]:
The difference of the squares of two odd numbers is divisible by 4.
Solution:
First odd number = 2n – 1
Second odd number = (2n – 1) + 2 = 2n + 1
(2n + 1)2 - ( 2n - 1) 2 = [(2n + 1) + ( 2n - 1)] [(2n + 1) - ( 2n - 1)]
= 4n (2)
= 8n
Going Beyond
Prove by Deduction
1. Pick a 3-digit number with all three digits being the same. Ex. 333, 999, etc.
2. Add the three digits together
3. Divide your three digit number by the sum you found
4. Your answer is 37
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Proof:
Let x be the repeating digit
100x + 10x + x = your number
x + x + x = 3x
(100x + 10x + x) / 3x
= 100x/ 3x + 10x/ 3x + x/ 3x
= 33 1/3 + 3 1/3 + 1/3
= 37
Resources
Principle of Mathematics - Nelson
Section 1.4 (pages 27-33)
Supporting
Assessment
Written formative assessments
Glossary
deductive reasoning – A type of reasoning by which generalizations are drawn from
specific patterns in observed data
inductive reasoning – A type of reasoning by which generalizations are drawn from
general patterns in observed data
Other
Mathematics 20-2
Inductive and Deductive Reasoning
Page 46 of 46