Logic
Inductive & Deductive Reasoning
The Pythagorean Converse
By
Holly Young
Math 7-12 Trainer
NW RPDP
861-1237
Essential Understandings:
Logic:
1) Can you define hypothesis, conclusion and identify them in a
conditional statement?
2) Can you represent logical relationships using conditional
statements and determine reasonable conclusions?
3) Can you provide or identify counterexamples for a conditional
statement?
Inductive & Deductive Reasoning:
1) Can you distinguish between inductive and deductive reasoning?
2) Can you analyze and complete a Venn Diagram given multiple
facts/rules?
3) Can you determine which counterexample is appropriate to use to
disprove a logical argument?
4) Can you examine a series of conditional statements and use
deductive reasoning to come to a logical conclusion?
5) Can you examine a pattern and use inductive reasoning to make a
conjecture/formula/or rule to extend the pattern?
Pythagorean Theorem and Converse:
1) Can you show how the Pythagorean Theorem works using a variety
of different methods?
2) Can you determine the measure of the missing side of a right
triangle?
3) Can you use the Pythagorean Theorem in a real-life situation?
4) Can you use the Pythagorean Theorem Converse to determine if a
triangle is a right triangle?
Logic:
1) Can you define hypothesis, conclusion and identify them in a conditional statement?
2) Can you represent logical relationships using conditional statements and determine
reasonable conclusions?
3) Can you provide or identify counterexamples for a conditional statement?
A) Define vocabulary using graphic organizer. Have students define the words with
teacher help, then place an example in each of the spaces below the definitions. It
sometimes helps logic learners to start with a real world example first, then move
to a math example. “If you leave your luggage unattended at the airport, then it
will be confiscated.” “If a closed figure has 4 sides, then it is a quadrilateral.”
B) Read any of Laura Numeroff’s books, “If you …., “ and have the students find the
original hypothesis, the final conclusion, and two conditional statements that happen
in the book on their vocabulary graphic organizer. When you are done reading the
book, students can determine a counterexample for each of their two statements
that they found. I am using If You Take a Mouse to the Movies.
a. Additional assignment – write a mini-geometry book using conditional
statements to get from one hypothesis to an unlikely conclusion; however each
step involved must make logical sense. See example on student instruction
sheet.
C) Play “I have – who has?” Activity. Divide the class in half. The shaded cards should be
copied once and passed out to both sides of the class. Everyone should have a non-shaded
question card. Have one person read a question card, if it is a conditional statement, then
a person who has the counterexample shaded card has to stand up before the person that
has the same card on the other side of the class stands. When a winner has been decided
have them read the card and decide if it is true, if it isn’t, then the other team’s winner
gets a chance to read their card. To discourage random standing, you can take points for
someone being wrong. Play resumes to reading the next non-shaded card in line. If the
card is a hypothesis only, then the logical conclusion shaded card holder must stand. Some
hypothesis cards may have more than one logical conclusion worked.
Logic
Conditional Statement Hypothesis:
Example:
Example:
Conclusion:
Example:
Counterexample:
If you take a Mouse to the Movies
The initial hypothesis of the
book is:
Final Conclusion given in the book:
One Conditional statement that is given in
the book
Counterexample:
Another Conditional statement that is given in
the book
Counterexample:
Mini-Geometry Books
If you…
If
If
If
If
If
If
If
If
Then
Then
Then
Then
Then
Then
Then
Then
Drawing:
Drawing:
Drawing:
Drawing:
Drawing:
Drawing:
Drawing:
Drawing:
Original hypothesis + final conclusion: _______________________________________________________________
Hypothesis starters:
If you have 2 points
If you have 2 lines
If you have a triangle
If you have a point
If you have a segment
If you have 3 points
If you have 3 lines
If you have a quadrilateral
If you have a ray
If you have an angle
Conclusions:
Then you have a square
Then you have a rhombus
Then you have a parallelogram Then you have parallel lines
Then you have a kite
Then you have congruent triangles
Then you have a trapezoid
Then you have a circle
Then you have a rectangle
YOUR CHOICE!!
Then you have an equilateral triangle
Logic
Conditional Statement Hypothesis:
Example:
Example:
Conclusion:
Example:
Counterexample:
If you take a Mouse to the Movies
The initial hypothesis of the
book is:
Final Conclusion given in the book:
One Conditional statement that is given in
the book
Counterexample:
Another Conditional statement that is given in
the book
Counterexample:
I Have – Who Has Activity?
Counterexample: A trapezoid has one set of
If opposite sides are parallel, then the
figure is a parallelogram.
parallel sides and is not a parallelogram.
If a quadrilateral has 4 congruent sides,
Counterexample: A rhombus has 4
then it is a square.
congruent sides.
If a figure has one set of opposite sides
Counterexample: A parallelogram has one
parallel, then it is a trapezoid.
set of opposite sides parallel, it just
happens to have the other sides be parallel
too.
If a figure has all angles congruent, then it
Counterexample: An equilateral triangle has
all angles congruent.
is a square.
If a quadrilateral’s diagonals bisect each
Counterexample: A square has diagonals
that bisect each other.
other, then it is a rhombus.
If a quadrilateral’s diagonals are
Counterexample: A rhombus has
perpendicular, then it is a square.
perpendicular diagonals.
If a quadrilateral has both sets of opposite Counterexample: A parallelogram has both
sides parallel, then it is a rectangle.
sets of opposite sides parallel.
If a quadrilateral’s adjacent angles are
Counterexample: A rhombus has adjacent
supplementary, then it is a rectangle.
angles that are supplementary.
If a quadrilateral’s adjacent angles are
Counterexample: A rectangle has adjacent
congruent, then it is a square.
angles that are congruent.
If a quadrilateral has all 4 angles congruent, Counterexample: A rectangle has all 4
then it is a square.
angles congruent.
If base angles in a figure are congruent,
Counterexample: An isosceles triangle has
base angles congruent.
then the figure is an isosceles trapezoid.
If opposite angles are congruent in a
Counterexample: A parallelogram has
opposite angles congruent.
quadrilateral, then it is a rectangle.
If opposite sides in a quadrilateral are
Counterexample: A rectangle has opposite
sides congruent.
congruent, then it is a square.
If a quadrilateral has perpendicular
Counterexample: A kite has perpendicular
diagonals, then it is a rhombus.
diagonals.
If the interior angles of a quadrilateral sum Counterexample: A trapezoid interior
angles sum to 360 degrees.
up to 360 degrees, then it is a
parallelogram.
If a both sets of opposite sides of a
Then the quadrilateral is a parallelogram.
quadrilateral are parallel,
If base angles of a trapezoid are congruent, Then it is an isosceles trapezoid.
If the diagonals of a parallelogram are
Then the quadrilateral is a rhombus.
perpendicular,
If the diagonals of a quadrilateral are
perpendicular and no sides are parallel,
If both sets of opposite sides of a
parallelogram are congruent,
If both sets of opposite angles in a
quadrilateral are congruent and equal to 90
degrees,
If the diagonals of a parallelogram bisect
each other and intersect at a right angle,
If a set of parallel lines are cut by a
transversal,
If you add up the interior angles of a
quadrilateral,
If you add up the interior angles of a
triangle,
If you add up the interior angles of a
pentagon,
If a quadrilateral has a horizontal and
vertical line of symmetry,
If a quadrilateral has 4 congruent sides,
If a figure’s area can be found by taking
half the base times the height,
If a figure’s area can be found by taking the
base times the height,
Then the quadrilateral is a kite.
Then the quadrilateral is a rectangle.
Then the quadrilateral is a rectangle.
Then the quadrilateral is a square.
Then alternate interior angles are
congruent.
Then you get 360 degrees.
Then you get 180 degrees.
Then you get 540 degrees.
Then the quadrilateral is a rectangle.
Then the quadrilateral is a rhombus.
Then the figure is a triangle.
Then the figure is a parallelogram.
Inductive & Deductive Reasoning:
1) Can you distinguish between inductive and deductive reasoning?
2) Can you analyze and complete a Venn Diagram given multiple facts/rules?
3) Can you determine which counterexample is appropriate to use to disprove a logical
argument?
4) Can you examine a series of conditional statements and use deductive reasoning to
come to a logical conclusion?
5) Can you examine a pattern and use inductive reasoning to make a
conjecture/formula/or rule to extend the pattern?
Lesson Notes:
A) Deductive Reasoning notes on note-maker – Using facts, definitions, accepted
properties, and the laws of logic to make a logical argument. (The truth of the
premises guarantees the truth of the conclusion) i.e. if a quadrilateral has both
sets of opposite sides parallel and all its angles equal 90 degrees, then it is a
rectangle.
i. Smith owns only blue pants and brown pants. Smith is wearing a pair of
his pants today. So Smith is wearing either blue or brown pants today.
ii. Example – Neighborhood watch puzzle.
B) Faulty Deductive Reasoning - show movie clip of A Few Good Men – the courtroom
scene.
a. Have students note as many deductive reasoning conditional statements that
they hear or are implied.
b. Pass out the breakdown of the argument sheet to see how many they found.
1-5 leads to one conclusion and 6-10 leads to the opposite conclusion which
using deductive reasoning means that the COL is lying.
c. Work on some venn plexors with faulty arguments. Make sure that they draw
the venn diagram. These are from the book Venn Perplexors Level C by Evelyn
Christensen.
C) Inductive Reasoning – a process that looks for patterns and makes conjectures.
(The premises make the conclusion likely, but they don’t guarantee that the
conclusion is true – how Law & Order works) i.e. look at the following pattern, what
will the 20th number be?
i. January has always been cold here in Siberia. Today is January 14, so
it is going to be another cold day in Siberia.
ii. Have students complete the tiered “continue the number patterns and
devise a formula” worksheet. There are three levels to this worksheet.
The triangle worksheet is the easiest and has the answers: x+4, x-5, 3x, 3x+2, just intersecting. The rhombus worksheet is the medium
difficulty one with answers: 2x-2, 3x-1, -2x+2, -4x-7, just
intersecting. The infinity worksheet is very difficult with the answers:
-4x-7, x2+1, 2x2, x2+x, the lines are parallel.
D) Show movie clip of Young Sherlock Holmes to illustrate inductive reasoning.
a. Have students note as many examples of inductive reasoning that they hear
and which ones lead or could lead to a faulty conclusion.
E) Reasoning Tic-Tac-Toe
a. The answers to the CRT column are Alg-B, Geo – B, and Practical – it shouldn’t
be the PRODUCT of the two numbers times 2, but the SUM.
b. The answers to the Venn Diagrams are given below. They came from the book
Venn Perplexors – Level D by Evelyn Christensen
My teacher’s definition of Deductive
Reasoning:
My teacher’s example:
My example (which is much better):
CRT Questions
Faulty Logic with Venn
Diagrams
Faulty Logic from A Few
Good Men
Summary: Deductive reasoning is
Why is there a picture with an upside-down triangle on this page?
My teacher’s definition of Inductive Reasoning:
This TV Show illustrates inductive reasoning by:
CRT Questions
Examples from Young
Sherlock Holmes
Summary: Inductive reasoning is
Why is there a picture with a triangle on this page?
Pattern Expanding:
HERE ARE THE CLUES – PREDICT A FORMULA Δ
Table of Values
What is the pattern?
1
5
2
6
3
7
4
8
5
9
-3
-8
-2
-7
-1
-6
0
-5
1
-4
What is the formula?
What are the next
two numbers in the
sequence?
-7 -6 -5 -4 -3
21 18 15 12 9
-1
-1
0
2
1
5
2
8
3
11
Inductive reasoning question: If you have the line y= -3x-4, what can you conclude about the
line y= 3x+2? Is parallel, perpendicular, or just intersect the original line? Explain what
clues help you come to that conclusion.
You meet a person for the first time. The person is wearing a tie. What job do you think
this person has and are they male or female? Explain your reasons for your answer.
HERE ARE THE CLUES – PREDICT A FORMULA ◊
Table of Values
2
2
What is the pattern?
-1
-4
0
-2
1
0
-4
-3
-2
-1
0
-13
-10
-7
-4
-1
-2
6
0
2
2
-2
4
-6
6
3
-2
0
5
-4
-19
1
-7
-27
9
What is the formula?
What are the next
two numbers in the
sequence?
3
4
-10
Inductive reasoning question: If you have the line y= -3x-4, what can you conclude about the
line y= 3x+2? Is parallel, perpendicular, or just intersect the original line? Explain what
clues help you come to that conclusion.
You meet a person for the first time. The person is wearing a tie. What job do you think
this person has and are they male or female? Explain your reasons for your answer.
HERE ARE THE CLUES – PREDICT A FORMULA ∞
Table of Values
What is the pattern?
3
-2
0
5
-4
-19
1
-7
-27
9
-1
2
0
1
1
2
-3 -2
10 5
What is the formula?
What are the next
two numbers in the
sequence?
-7 -6 -5 -4 -3
98 72 50 32 18
-1
0
0
0
1
2
2
6
3
12
Inductive reasoning question: If you have the line y= -3x-4, what can you conclude about the
line that passes through the points (1,1) and (1,-2)? Is parallel, perpendicular, or just
intersect the original line? Explain what clues help you come to that conclusion.
You meet a person for the first time. The person is wearing a tie. What job do you think
this person has and are they male or female? Explain your reasons for your answer.
Reasoning Boxes
Essential Understandings:
1) Can you distinguish between inductive and deductive reasoning?
2) Can you analyze and complete a Venn Diagram given multiple facts/rules?
3) Can you determine which counterexample is appropriate to use to disprove a logical argument?
4) Can you examine a series of conditional statements and use deductive reasoning to come to a logical conclusion?
5) Can you examine a pattern and use inductive reasoning to make a conjecture/formula/or rule to extend the pattern?
Instructions: You need to pick 2 boxes from each row, in addition to all the CRT boxes. You may not pick the same combination
of boxes for every row.
Logical conclusions
Venn Diagrams
Counterexamples
CRT problems
Algebra
If a number pattern given
is 3, 7, 11, 15, 19, explain
the number pattern, write
it as an equation using x,
and write what you would
get if x were 15 and if x
were 22.
Pg. 19 – attached
Using the table of
values:
-1 0
1
2
3
-2 0
2
5
6
A person claims that the
equation to describe it is
y=2x. Give a
counterexample to
prove that isn’t true.
Given the table:
1
2
3
4
5
5
8
10 14 18
An equation to describe
it is given to be y=3x+2.
a) It is a reasonable
equation because it
fits all the points.
b) It is a reasonable
equation because it
fits most points.
c) It is not a reasonable
equation because it
fits no points.
d) It is not a reasonable
equation because
you can’t fit
equations to a table
of points.
Logical conclusions
Geometry Given the following
description about a
geometric shape, tell
which shape it is, draw the
shape, and give 2 reasons
why it couldn’t be any
other shape. The shape is
a polygon with 4 sides, 4
90 degree angles, and the
4 sides are not all the
same length.
Now, make up your own
example like the one
above and answer your
own problem.
Practical
Finish the logical
conclusion:
If it is Tuesday, then we
eat pizza. If we eat pizza,
then we need to exercise.
If we exercise, then we
need to go to the gym. If
we go to the gym, then we
need to have a water
bottle. So, if it is Tuesday,
then…
Write 3 more examples
like the one above and
state the logical
conclusion. They must
each have 5 statements.
Go back and look at all the boxes above.
Venn Diagrams
Take the following properties and
put them in a tri-venn diagram for
rhombus, rectangle, and square:
Diagonals bisect each other,
diagonals are perpendicular,
opposite angles congruent, all
angles = 90 degrees, all 4 sides
congruent, adjacent angles are
supplementary, both sets of
opposite sides are parallel,
diagonals are congruent, is a
quadrilateral, and is a polygon.
Counterexamples
Decide if the following
argument is valid or
invalid. If it is invalid,
explain the faulty
reasoning. If it is valid,
draw the venn diagram
to show its validity. All
4-sided figures with 4
right angles are
rectangles. A square
has 4 equal sides and 4
right angles. Therefore
a square is a rectangle.
CRT problems
If a quadrilateral has 4
equal side lengths, then
you can only conclude:
a) It is a square.
b) It is a rhombus.
c) It is a rectangle.
d) It is a non-special
quadrilateral.
Pg. 7 – attached
Decide if the following
argument is valid or
invalid. If it is invalid,
explain the faulty
reasoning. If it is valid,
draw the venn diagram
to show its validity. All
mammals are warmblooded vertebrates. A
dolphin is a vertebrate
with warm blood.
Therefore a dolphin is a
mammal.
Elise was calculating
the perimeter of her
rectangular room. She
measured along one
wall and then measured
the adjacent wall. In
order to find her total
perimeter, she
multiplied the product of
the two measured walls
by 2. Explain why her
reasoning was correct
or incorrect.
Put a star in all the boxes that are using deductive reasoning!
Pythagorean Theorem & the Converse
Essential Understandings:
1) Can you show how the Pythagorean Theorem works using a variety of different
methods?
2) Can you determine the measure of the missing side of a right triangle?
3) Can you use the Pythagorean Theorem in a real-life situation?
4) Can you use the Pythagorean Theorem Converse to determine if a triangle is a right
triangle?
Pre-Assessment and note-maker – Place your bets:
A) Have students fill out the place your bets activity first. When you have gone over the
answers to check to see where they are at, send them to the appropriate work stations
depending on the area that they did not understand, or do well. Depending on how the
students do, you can assign more than one station to the students or give them one station
(or more) to complete and then have the class complete a scavenger hunt on all the
completed stations. See the grid below for where to assign students a station – remember
they need to go to a station if their CONFIDENCE was low, not if they missed it or got it
right. The answers are (going across rows): F(c is the longest side),T,F(only in a right
triangle),T,T,F(you can use Pythagorean converse),T,F(doesn’t imply that) ,T,F(should be
4),T,T,T,T,F(should be 3 root 2),F(should be 10),T,T,F(should be 4 root 2),T
Station:
Station:
Station:
Station:
1
2
2
1
Station: 1
Station: 2
Station: 4
Station: 3
Station 1: Can you
determine the
measure of the
missing side of a
right triangle?
B)
Station: 1
Station: 2
Station: 2
Station: 1
Station 2: Can you
use the Pythagorean
Theorem Converse to
determine if a
triangle is a right
triangle?
Station: 4
Station: 2
Station: 3
Station: 3
Station 3: Can you
use the Pythagorean
Theorem in a reallife situation?
Station: 3
Station: 1
Station: 4
Station: 3
Station 4: Can you
show how the
Pythagorean
Theorem works using
a variety of
different methods?
At each station, students will be working in groups of 4. If there are
more than 4 students, then make more than one group working at a station.
Each group of 4 needs to create their own poster and therefore needs
their own supplies. Once each station has finished their poster, every
C)
student fills out a “scavenger hunt/note-maker” by going to a poster from
each station and filling in examples and answering questions.
Using their note-maker, have students fill out Place Your Bets – PostAssessment. This activity does not use True-False answers, but is still
tracking confidence. Teachers can assess to see if students have a better
confidence on individual tasks. If there is an area that a large portion of
students are lacking, then a large class discussion/re-teach can be done.
The answers are 1) a-leg of a right triangle, b-leg of a right triangle, and
c is the hypotenuse of a right triangle, 2) b 2  c 2  a 2 3) longest side in a
right triangle, 4) distance is a 2  b 2  c 2 solved for c, 5) d=s√2, 6) see if
a 2  b 2  c 2 , 7) a 2  b 2  c 2 , 8) many answers possible
3 2  4 2  5 2 but 33  4 3  53 , 9) 5 2  12 2  132 which is 25 + 144 = 169, 10) 4,
11) 5 2  7 2  9 2 so it isn’t right, 12) 5, 13) 30 2  40 2  50 2 , 14) by the
diagonal, 15) 3√2, 16) 10, 17) 14√2, 18) 5√5, 19) 4√2, 20) √13
Place Your Bets
Circle your answer to each question as true or false. Once you have given an answer it is
your chance to decide how confident you are in your answer by placing your bet. You need
to circle a number from 1-5, 1 being “not very sure” and 5 being “totally sure.”
The Pythagorean
Theorem is
a 2  b 2  c 2 where
a and b represent
the shortest and
longest side of the
triangle and c is the
medium side.
True/False
Bet: $1 2 3 4 5
The only way to tell
if a triangle is a
right triangle is to
measure the angle
and see if it equals
ninety degrees.
The hypotenuse is
the name for the
longest side in any
triangle.
The Pythagorean
Theorem and the
Distance Formula
are the exact same
formula.
The diagonal of any
square can be
found by using the
True/False
Bet: $1 2 3 4 5
True/False
Bet: $1 2 3 4 5
True/False
Bet: $1 2 3 4 5
If a set of three
numbers fits the
pattern:
If a set of three
numbers fits the
pattern:
5,13, and 12 are
considered a
Pythagorean triple.
a 2  b 2  c 2 then
a 2  b 2  c 2 then
they are called a
Pythagorean triple.
they will also fit the
pattern:
True/False
Bet: $1 2 3 4 5
If one side of a
right triangle has
length 3, and the
hypotenuse is 5,
the other side is
A different way to
write the
Pythagorean
Theorem is
a2  c2  b2 .
formula d  s 2 .
5 3 .
a3  b3  c3.
True/False
Bet: $1 2 3 4 5
True/False
Bet: $1 2 3 4 5
The sides of a
triangle are 5, 7,
and 9; therefore it
is not a right
triangle.
A point on a grid is
at (0,4) and
another point is
located at (3,0).
The shortest
distance between
them has a length
of 5.
True/False
Bet: $1 2 3 4 5
c
6
True/False
Bet: $1 2 3 4 5
A square has a side
length of 14, the
length of its
diagonal is 14 2 .
8
True/False
Bet: $1 2 3 4 5
A triangle with side
lengths of 30, 40,
and 50 is a right
triangle.
True/False
Bet: $1 2 3 4 5
b
10
15
The length of
b= 5 5
The length of
c= 48
True/False
Bet: $1 2 3 4 5
True/False
Bet: $1 2 3 4 5
True/False
Bet: $1 2 3 4 5
True/False
Bet: $1 2 3 4 5
Television sets
that are given to
have a 32 inch or
36 inch screen size
are measured by
the length of the
diagonal of the
rectangle that they
make.
True/False
Bet: $1 2 3 4 5
A 6-ft ladder is
placed against a
wall with its base
2-ft from the wall.
The top of the
ladder is 2 2 ft
above the ground.
True/False
Bet: $1 2 3 4 5
True/False
Bet: $1 2 3 4 5
A point M is
located at (1,1) and
a point N is located
at (-2,-2). The
distance between
M and N is 3 3.
True/False
Bet: $1 2 3 4 5
If one side of a
deck is 2 feet and
the intersecting
side is 3 feet, the
diagonal connecting
them must be
13 for the sides
to make a right
angle.
True/False
Bet: $1 2 3 4 5
1)Right/Wrong
1)Right/Wrong
1)Right/Wrong
4)Right/Wrong
3)Right/Wrong
Confidence: ___
2)Right/Wrong
Confidence: ___
2)Right/Wrong
Confidence: ___
2)Right/Wrong
Confidence: ___
2)Right/Wrong
Confidence: ___
1)Right/Wrong
Confidence: ___
2)Right/Wrong
Confidence: ___
4)Right/Wrong
Confidence: ___
2)Right/Wrong
Confidence: ___
3)Right/Wrong
Confidence: ___
4)Right/Wrong
Confidence: ___
1)Right/Wrong
Confidence: ___
3)Right/Wrong
Confidence: ___
1)Right/Wrong
Confidence: ___
3)Right/Wrong
Confidence: ___
3)Right/Wrong
Confidence: ___
Confidence: ___
Confidence: ___
Confidence: ___
Confidence: ___
Record your confidence scores in the boxes below for all 1’s, 2’s, 3’s, and 4’s. Then find
the average in each box. The lowest average(s) is the station(s) that you will be working at
during station time.
Station 1: Can you
determine the
measure of the
missing side of a
right triangle?
Confidence scores:
Average: _______
Station 2: Can you
use the Pythagorean
Theorem Converse to
determine if a
triangle is a right
triangle?
Confidence scores:
Station 3: Can you
use the Pythagorean
Theorem in a reallife situation?
Average: _______
Average: _______
Confidence scores:
Station 4: Can you
show how the
Pythagorean
Theorem works using
a variety of
different methods?
Confidence scores:
Average: _______
Place Your Bets – Post-Assessment
Give an answer to each question. Once you have given an answer, it is your chance to
decide how confident you are in your answer by placing your bet. You need to circle a
number from 1-5, 1 being “not very sure” and 5 being “totally sure.”
The Pythagorean
Theorem is
a 2  b 2  c 2 where
Solve the
Pythagorean
Theorem for b2.
The hypotenuse is
the name for what?
The Pythagorean
Theorem and the
Distance Formula
are the exact same
formula because:
The diagonal of any
square can be
found by using
what formula?
Bet: $1 2 3 4 5
Bet: $1 2 3 4 5
Bet: $1 2 3 4 5
Bet: $1 2 3 4 5
A Pythagorean
triple is a set of
three numbers that
follows what
pattern?
Provide a
counterexample to
show that
Show that 5,13,
and 12 are a
Pythagorean triple.
If one side of a
right triangle has
length 3, and the
hypotenuse is 5,
then the other side
is what?
Bet: $1 2 3 4 5
Bet: $1 2 3 4 5
Television sets
that are given to
have a 32 inch or
36 inch screen size
are measured how?
A point M is
located at (1,1) and
a point N is located
at (-2,-2). The
distance between
M and N is what?
The length of b=
Bet: $1 2 3 4 5
A 6-ft ladder is
placed against a
wall with its base
2-ft from the wall.
How far is the
ladder up the wall?
Bet: $1 2 3 4 5
If one side of a
deck is 2 feet and
the intersecting
side is 3 feet, the
diagonal connecting
them must be how
long for the sides
to make a right
angle?
Bet: $1 2 3 4 5
Bet: $1 2 3 4 5
Bet: $1 2 3 4 5
a ,b, and c
represent what?
Bet: $1 2 3 4 5
One way to tell if a
triangle is a right
triangle without
measuring the angle
is to check what?
if a  b  c then
the same numbers
will also fit the
pattern:
2
2
2
a3  b3  c3.
Bet: $1 2 3 4 5
Bet: $1 2 3 4 5
The sides of a
triangle are 5, 7,
and 9; show that
these are or aren’t
the sides of a right
triangle.
A point on a grid is
at (0,4) and
another point is
located at (3,0).
The shortest
distance between
them is how long?
Bet: $1 2 3 4 5
Bet: $1 2 3 4 5
c
6
8
A square has a side
length of 14, the
length of its
diagonal is how
long?
Bet: $1 2 3 4 5
Show that a triangle
with side lengths of
30, 40, and 50 is a
right triangle.
Bet: $1 2 3 4 5
b
10
15
The length of c=
Bet: $1 2 3 4 5
Bet: $1 2 3 4 5
1)Right/Wrong
1)Right/Wrong
1)Right/Wrong
4)Right/Wrong
3)Right/Wrong
Confidence: ___
2)Right/Wrong
Confidence: ___
2)Right/Wrong
Confidence: ___
2)Right/Wrong
Confidence: ___
2)Right/Wrong
Confidence: ___
1)Right/Wrong
Confidence: ___
2)Right/Wrong
Confidence: ___
4)Right/Wrong
Confidence: ___
2)Right/Wrong
Confidence: ___
3)Right/Wrong
Confidence: ___
4)Right/Wrong
Confidence: ___
1)Right/Wrong
Confidence: ___
3)Right/Wrong
Confidence: ___
1)Right/Wrong
Confidence: ___
3)Right/Wrong
Confidence: ___
3)Right/Wrong
Confidence: ___
Confidence: ___
Confidence: ___
Confidence: ___
Confidence: ___
Record your confidence scores in the boxes below for all 1’s, 2’s, 3’s, and 4’s. Then find
the average in each box. The lowest average(s) is the station(s) that you will be working at
during station time.
Station 1: Can you
determine the
measure of the
missing side of a
right triangle?
Confidence scores:
Average: _______
Station 2: Can you
use the Pythagorean
Theorem Converse to
determine if a
triangle is a right
triangle?
Confidence scores:
Station 3: Can you
use the Pythagorean
Theorem in a reallife situation?
Average: _______
Average: _______
Confidence scores:
Station 4: Can you
show how the
Pythagorean
Theorem works using
a variety of
different methods?
Confidence scores:
Average: _______
PYTHAGOREAN THEOREM SCAVENGER HUNT/NOTE-MAKER
You need to travel to a poster from all 4 stations and fill in the boxes below as completely as possible.
Write an example of solving
Write 2 examples of
Write the Pythagorean
Station 1
for
the
hypotenuse
(
c
)
using
solving for a side other
Theorem
out
and
label
what
Can you
the Pythagorean Theorem:
than the hypotenuse:
each part means (include all
determine
important vocabulary):
the
measure of
the missing
side of a
right
triangle?
Station 2
Can you use
the
Pythagorean
Theorem
Converse to
determine
if a triangle
is a right
triangle?
Station 3
Can you use
the
Pythagorean
Theorem in
a real-life
situation?
What is the Pythagorean
Theorem Converse? Give an
example:
What are Pythagorean triples
and give examples of some
common ones:
How can you find out if a
triangle has a right
angle? Give an example:
Write down how to find the
length of a diagonal of a
square:
Write down an example of a
“ladder” problem:
Write down an example
of how to find the
diagonal of a rectangle:
Explain using an example or in
words how to use the
Pythagorean Theorem instead
of the distance formula:
Here is an example of
finding the distance
between 2 points using
Pythagorean Theorem:
What is the distance formula
Station 4
and give an example of how to
Can you
use it:
show how
the
Pythagorean
Theorem
works using
a variety of
different
methods?
Station 1 Goal: Can you determine the measure of the missing side of a right triangle?
Supplies: scissors, graph paper, glue sticks, chart paper, and Algebra book with a section
on the Pythagorean theorem tagged.
Part 1 - Vocabulary building: a, b, c, legs, right angle, hypotenuse, sides, Pythagorean
Theorem.
On part of the poster, students will need to define each of the terms above with an
explanation in words of their own and a picture (all except the Pythagorean Theorem).
Part 2 - Directions: On graph paper students need to cut out 3- 3x3 squares, 3- 4x4’s, 3 5x5’s, and 1 each of the following: 6x6, 7x7, 8x8, 9x9, 10x10, 11x11, 12x12, and 13x13.
Once their squares are cut out, they need to write the area of each square on the inside.
Before gluing their final results to a poster for presentation, they need to make some
practice triangles. Give them the copy of a right angle already drawn for them. They need
to place squares along the two given edges and see if they can find a square to complete
the hypotenuse of the triangle. For each triangle that they construct, they need to
record it in the table. They must fill out the entire table with as many possibilities as they
can find. Once they are finished with the table, they need to answer the questions under
the table as completely as possible. When this is completed, they are to make (by gluing
their squares on the poster) 3 right triangles and show with numbers or diagrams next to
the triangles how the Pythagorean Theorem works for those particular numbers. They
must also make a non-right triangle and show with numbers how the Pythagorean Theorem
doesn’t work.
Part 3 – Directions: 1) Students will do some practice problems and transfer some (their
choice, but at least one of each kind) to the poster. The poster must have the
Pythagorean Theorem solved for a2 and b2 on it above those types of problems. It is
important that students read through the examples of the two kinds of problems before
solving problems on their own.
2) Students need to make up their own problem and show its solution worked out step by
step.
Part 4 – Summary – Students need to write a small paragraph summary about what is
contained on their poster. Their goal is to answer, “so what does it mean”?
Station 1 - Investigation
a2
a
b2
b
c2
c
Right
Triangle?
Questions:
1) If the Pythagorean Theorem states that “if a triangle is a right triangle then,
a 2  b 2  c 2 ” explain how your table helps prove this fact.
2) If you know for sure that you have a right triangle and you know the lengths of two
of the sides, explain how you can find the length of the missing side?
Student Instructions
Station 1 Goal: Can you determine the measure of the missing side of a right triangle?
Part 1 - Vocabulary building: a, b, c, legs, right angle, hypotenuse, sides, Pythagorean
Theorem.
On part of the poster, your group will need to define each of the terms above with
an explanation in words of your own and a picture (all except the Pythagorean Theorem
needs a picture).
Part 2 - Directions: On graph paper you need to cut out 3- 3x3 squares, 3- 4x4’s, 3 5x5’s, and 1 each of the following: 6x6, 7x7, 8x8, 9x9, 10x10, 11x11, 12x12, and 13x13.
Once your squares are cut out, you need to write the area of each square on the inside.
Given to you is a copy of a right angle already drawn with a table below. You need to place
squares along the two given edges and see if you can find an appropriate square to
complete the hypotenuse of the triangle. For each triangle that you construct, you need to
record it in the table. You must fill out the entire table with as many possibilities as you
can find. Not all the triangles that you try to make will be a right triangle! Once you are
finished with the table, you need to answer the questions under the table as completely as
possible. When this is completed, you are to make (by gluing your squares on the poster) 3
right triangles and show with numbers or diagrams next to the triangles how the
Pythagorean Theorem works for those particular numbers. You must also make a non-right
triangle and show with numbers how the Pythagorean Theorem doesn’t work.
Part 3 – Directions: It is important that you read through the examples of the two kinds
of problems before solving problems on your own.
1) You need to do some practice problems from the practice worksheet. How many do you
need to complete? As many as it takes until you can do them without making any errors!
Now, you will need to transfer some problems of your choice to the poster. You need to
have a problem on the poster where you solve for a, one for b, and one for c. The poster
must have the Pythagorean Theorem solved for a2 and b2 (normally, it is solved for c2) on it
above those types of problems depending on which problem you are showing. Each problem
on the poster needs to have the solution worked out step by step.
2) You need to make up your own problem and show its solution worked out step by step.
Part 4 – Summary – You need to write a small paragraph summary about what is contained
on your poster. Your goal is to answer, “How can you determine the measure of the missing
side of a right triangle?”
Station 2 – Goal - Can you use the Pythagorean Theorem Converse to determine if a
triangle is a right triangle?
Supplies: scissors, glue stick, ruler, protractor, markers, poster paper
Part 1 – Introduction and explanation of the Pythagorean Theorem Converse. Students will
read the following statement that if three sides of a triangle follow the pattern – the
length of the legs squared added together equals the length of the hypotenuse squared,
then the triangle is a right triangle ( if a 2  b 2  c 2 then the triangle is a right triangle). If
the pattern holds for a set of three numbers, then the three numbers are referred to as a
Pythagorean triple. Students are to show that the converse is true on their poster by
drawing an example where it works and makes a right triangle (measure with a protractor
to check) and one where the numbers don’t make a triple and the angle measured is not a
right angle (write the angle measure on the poster). Students must write a summary in
their own words and put it on the poster to explain how the converse works and what it
says.
Part 2 – Examination of Pythagorean triples and patterns that emerge. Students are to
look at some popular Pythagorean triples and see if they can make any generalizations from
them.
1) Verify which of the following are Pythagorean Triples:
a. 3-4-5, 1-2-3, 6-8-10, 3-6-9, 9-12-15, 5-12-13, 8-10-12, 7-24-25, 30-40-50
2) By looking at which groups of three numbers are Pythagorean triples, can you
find any patterns that hold true. For example, if you know one or two basic
triples, can you generate a whole bunch of other ones that work too. Show and
explain any patterns that you found on your poster.
3) If a 2  b 2  c 2 holds true for a set of three numbers, will a 3  b 3  c 3 work as well?
Explain on your poster.
Part 3 – Application – Students need to draw a picture and solve the problem and explain all
the steps on their poster.
When laying out a deck, a carpenter measures along one side a distance of 6 feet
and along the adjacent side a distance of 7 feet. What must the measure of the
hypotenuse be in order to see if the corners of the deck form a right angle?
STUDENT INSTRUCTIONS – STATION 2
Goal - Can you use the Pythagorean Theorem Converse to determine if a triangle is a right
triangle?
Part 1 – Introduction and explanation of the Pythagorean Theorem Converse. Your group
needs to examine the following statement: if three sides of a triangle follow the pattern –
the length of the legs squared added together equals the length of the hypotenuse
squared, then the triangle is a right triangle ( if a 2  b 2  c 2 then the triangle is a right
triangle). If the pattern holds for a set of three numbers, then the three numbers are
referred to as a Pythagorean triple. You are to show that the converse is true on your
poster by drawing an example where it works and makes a right triangle (measure with a
protractor to check) and one where the numbers don’t make a triple and the angle
measured is not a right angle (write the angle measure on the poster). You must write a
summary in your own words and put it on the poster to explain how the converse works and
what it says.
Part 2 – Examination of Pythagorean triples and patterns that emerge. You are going to
look at some popular Pythagorean triples and see if you can make any generalizations from
them.
1) Verify which of the following are Pythagorean Triples, show the ones that work
on your poster:
a. 3-4-5, 1-2-3, 6-8-10, 3-6-9, 9-12-15, 5-12-13, 8-10-12, 7-24-25, 30-40-50
2) By looking at the groups of three numbers that are Pythagorean triples, can you
find any patterns between the numbers that hold true? For example, if you know
one or two basic triples, can you generate a whole bunch of other ones by
doubling numbers or multiplying all of the numbers by the same factor? Show and
explain any patterns that you find on your poster.
3) If a 2  b 2  c 2 holds true for a set of three numbers, will a 3  b 3  c 3 work as well?
What about a+b=c? Explain on your poster.
Part 3 – Application – You need to draw a picture, solve the problem, and explain all the
steps on your poster.
When laying out a deck, a carpenter measures along one side a distance of 6 feet
and along the adjacent side a distance of 7 feet. What must the measure of the
hypotenuse be in order to see if the corners of the deck form a right angle?
Station 3 Goal - Can you use the Pythagorean Theorem in a real-life situation?
Supplies: scissors, glue sticks, poster paper, & markers
Part 1 – Examining the diagonals of squares. Students will need to fill out a table of values
for computing the length of the diagonal of a square. Make sure that they are breaking
down the square root to lowest terms. After completing the table, they will need to
generate a formula in their own words for finding this value. Their formula needs to be put
on their poster with an explanation and example.
Part 2 – Studying Real Life Examples explained. Students need to read over the examples
to see how they work and are explained.
Part 3 – Students need to cut apart the problem squares. They will need to keep them in
their respective piles of “picture,” “problem,” “solution worked out step-by-step,” and
“answer.” They are trying to match 4 squares together (1 from each pile), work out the
problem to make sure that it matches the solution, then glue them in a group of 4 to their
poster. The solution worked out should explain the answer.
Part 4 – Summary – On the poster students need to provide a summary for anyone reading
the poster. What are some important steps to complete in order to use the Pythagorean
Theorem in a real-life situation?
The Diagonal of a Square – Investigation
You will need to fill out the table, make sure to keep track of all of your work. It is very
important to break-down all square roots to their lowest form. For example, you should not
write 8 but 2 2 .
Length of
the side of
a square
Draw the square,
label the sides,
and draw the
diagonal
Find the length of the diagonal using the
Pythagorean theorem (the diagonal cuts the
square into 2 equal right triangles)
a2  b2  c2 .
Length of
the diagonal
3
4
5
6
7
Now it is time to see if you can write a formula that would work for all squares. The length
of a diagonal (d) is:
Picture
Problem
A builder is laying out
the foundation for a
house. The measure
along one side is 12 ft,
the adjacent side is 16
ft, and hypotenuse is 21
ft. Determine whether
the corner is a right
angle.
A square has a diagonal
15.5
19.5
Solution worked out step-by-step
Answer
24.91,
or 25
inches
8 ft
length of 17 2 , what is
the length of the side?
19.5
15.5
5
3
21
12
A rectangular television
set measures
approximately 15.5 in
high and 19.5 in. wide.
What size should it be
advertised?
5.83 mi
A 5 foot ladder is placed
against a wall with its
base 3 ft from the wall.
How high above the
ground is the top of the
ladder?
24.91 ft
Jackson is 5 miles from
Lazy R Resort. Ontario
is 3 miles south of
Jackson. If you build a
shortcut road to connect
Ontario and Lazy R, find
the length of the new
road.
A farmer needs to put a
fence across the diagonal
of his square pen with a
No,
400<441
16
25
7
17√2
x
side length of 3 2 . Ft.
How long is this fence?
17
19
x
3√2
x
5
3
A wire from the top of a
19.5 ft flagpole is
attached to a point 15.5
feet from the base of
the flagpole. Find the
length of the wire.
A 25-foot ladder is
placed against a vertical
wall of a building with
the bottom of the ladder
standing on concrete 7
feet from the base of
the building. If the top
of the ladder slips down
4 feet, then the bottom
of the ladder will slide
out how many feet?
A television screen is
advertised as being 19
inches. Give a possible
dimension for the height
of the side of the TV?
4 ft
13 in
6 ft
Station 3 – Student Instructions
Goal: Can you use the Pythagorean Theorem in a real-life situation?
Part 1 – Examining the diagonals of squares using the Pythagorean Theorem. You will need
to fill out a table of values for computing the length of the diagonal of a square. Make sure
that you are breaking down the square root to lowest terms. After completing the table,
you will need to generate a formula in your own words for finding this value. Your formula
needs to be put on your poster with an explanation and example.
Part 2 – You need to read over the real-life examples to see how they work and are
explained. Discuss them as a group: What do all the problems have in common? What helps
to solve the problems?
Part 3 – You need to cut apart the problem squares. You will need to keep them in their
respective piles of “picture,” “problem,” “solution worked out step-by-step,” and “answer.”
The “solution worked out step-by-step” squares are empty and need to filled out as a group.
You are trying to match 4 squares together (1 from each pile), work out the problem to
make sure that it matches the solution, then glue them in a group of 4 to their poster. The
solution worked out should explain the answer. You also might to add more parts to the
picture if it helps solve the problem.
Part 4 – Summary – On the poster you need to provide a summary for anyone reading the
poster. What are some important steps to complete in order to use the Pythagorean
Theorem in a real-life situation?
Station 4 – Goal - Can you show how the Pythagorean Theorem works using a variety of
different methods?
Supplies: scissors, glue sticks, graph paper, poster paper, markers
Part 1 – Vocabulary – distance formula, Pythagorean Theorem, x-coordinate, ycoordinate, hypotenuse, and legs.
On the poster, students will need to define the vocabulary words with an example or
picture. If the vocabulary word is a formula, then show what parts of the formula
stand for.
Part 2 – The students’ job is to show with pictures and examples that the distance
formula and the Pythagorean Theorem are really the same formula in a slightly
different form. They are going to use problems normally geared toward using the
distance formula and solving them using the Pythagorean Theorem instead. Students
plot the point (0,4) and the point (3,0) on a piece of graph paper. Calculate the distance
between the two points using the distance formula: d  ( x2  x1 ) 2  ( y 2  y1 ) 2 Now, on a
different graph, students will need to graph (0,4) and (3,0) again, draw the distance
between the two points, draw a right triangle using the origin as the corner where the
right angle is located. Label the legs of the right triangle. Show by using the
Pythagorean Theorem that the length of the distance between the two points is the
same that was obtained using the distance formula. Both problems with solutions
should be put on the group poster. As a group, students need to consider the following
questions: How is the distance formula like the Pythagorean Theorem? How can you
use the Pythagorean Theorem instead of the distance formula to find the distance
between two points on a coordinate plane? Does it matter how you draw the right
triangle? What happens when some or all of the coordinates are negative? When the
group feels that they can confidently answer each of the questions, then the
explanations should be presented in a full summary on the poster (It can be done with a
step by step instruction on how to use the Pythagorean theorem to find the distance
between two points on a graph). Students will show how to find the distance between
the points (3,5) and (6,4) by drawing a picture and using the Pythagorean Theorem.
They will then show how they get the same answer by using the distance formula. The
work for both parts needs to be displayed on the poster. Students will show how to
find the distance between the points (1,2) and (-3,7) by drawing a picture and using the
Pythagorean Theorem. They will then show how they get the same answer by using the
distance formula. The work for both parts needs to be displayed on the poster.
Part 3 – Challenge Problem – Find the value of a if G(4,7) and H(a,3) are 5 units apart.
Show your work explaining each step on your poster. This requires knowledge of solving
quadratics, so it can be a great challenge for advanced students. It can also be solved
using guess and check and would be a good challenge for those students who like to
move a little faster than others. The answer is either 7 or 1.
STUDENT INSTRUCTIONS FOR STATION 4
Goal - Can you show how the Pythagorean Theorem works using a variety of different
methods?
Part 1 – Vocabulary – distance formula, Pythagorean Theorem, x-coordinate, ycoordinate, hypotenuse, and legs.
On your poster, your group will need to define the vocabulary words with an example
or picture. If the vocabulary word is a formula, then show what each part of the
formula stands for.
Part 2 – Your group’s job is to show with pictures and examples that the distance
formula and the Pythagorean Theorem are really the same formula in a slightly
different form. You are going to use problems normally geared toward using the
distance formula and solve them using the Pythagorean Theorem instead. On graph
paper, plot the point (0,4) and the point (3,0). Calculate the distance between the two
points using the distance formula: d  ( x2  x1 ) 2  ( y 2  y1 ) 2
Now, on a different graph,
you will need to graph (0,4) and (3,0) again, draw the distance between the two points,
and draw a right triangle using the origin as the corner where the right angle is located.
Label the lengths of the legs of the right triangle. Show by using the Pythagorean
Theorem that the length of the distance between the two points is the same that was
obtained using the distance formula. Both problems with solutions should be put on the
group poster. As a group, you need to consider the following questions as you do the
beginning problem and the two additional examples: How is the distance formula like
the Pythagorean Theorem? How can you use the Pythagorean Theorem instead of the
distance formula to find the distance between two points on a coordinate plane? Does
it matter how you draw the right triangle? What happens when some or all of the
coordinates are negative?
When the group feels that they can confidently answer each of the questions, then the
explanations should be presented in a full summary on the poster (It can be done with
step-by-step instructions on how to use the Pythagorean Theorem to find the distance
between two points on a graph).
Examples: You will show how to find the distance between the points (3,5) and (6,4) by
drawing a picture and using the Pythagorean Theorem. You will then show how you get
the same answer by using the distance formula. The work for both parts needs to be
displayed on the poster. You will show how to find the distance between the points
(1,2) and (-3,7) by drawing a picture and using the Pythagorean Theorem. You will then
show how you get the same answer by using the distance formula. The work for both
parts needs to be displayed on the poster.
Part 3 – Challenge Problem – Find the value of a if G(4,7) and H(a,3) are 5 units apart.
Show your work explaining each step on your poster.