Step 2: Inquiry-based Lesson Design in Mathematics
Title of Lesson: Pythagorean Theorem
UFTeach students’ Names: Hang Nguyen and Emily Nelson
Teaching Date and Time: April 14, 2011 at 9:30
Length of Lesson: 50 Minutes
Grade / Topic: 6th grade/Pythagorean Theorem
Source of the Lesson: Gizmo GeoBoard lesson, TI-nspire,
http://www.geom.uiuc.edu/~demo5337/Group3/hist.html
Appropriateness for Middle School Students:
This will be a technological-based lesson, which will give students a chance to work with
computers. A change of scenery will hopefully get the kids excited about our lesson. We will begin
by simply asking about a real-world situation in which you need to know the length of the side of a
triangle. This quick scenario is meant to stimulate critical problem solving. We will also show a short
clip, which will engage visual learners, and help all the students make some connection. The activity
on the computer will be done individually, but we will circulate and help those who may be having
trouble. There are a few handouts for the students to record data and fill in as we go over the
explanation of the lesson. The students will not be expected to write out everything we go over, but
rather fill in smaller things to avoid detracting from learning by excessive note-taking
Concepts
One of the most fundamental theorems on which mathematics is based is the Pythagorean
Theorem, a2 + b2 = c2. It is applicable to most any level of mathematics from geometry to calculus.
To really understand the theorem in the context of geometry requires a familiarity with triangles and
their construction, specifically the right triangle. The length of the hypotenuse, which is the longest
side of a triangle or the opposite side from the 90 degree angle, is represented by the variable c.
The lengths of the other two shorter sides, forming the 90 degree angle, are represented by
variables a and b. The simplest application of this theorem is that of solving for the length of one
side of a right triangle given the lengths of the remaining two sides. That is the equation can be
manipulated to the forms of c2 - b2 = a2, c2 – a2 = b2. In this way, a, b, or c can be found by taking the
square root of its equivalent.
Because the Euclidean plane operates on a grid-like system, the Pythagorean Theorem can
be transposed into the distance formula. Given the points (1, 2) and (5, 5), we can construct side “a”
to be the difference between the values of the x-coordinates from our two points, 1 & 5. In the same
way, we can construct side “b” to be the difference between the values of the y-coordinates, 2 & 5.
Plugging it into the Pythagorean Theorem, the total distance between the two points is represented
by “c.” At its root, the Pythagorean Theorem is built on a strong understanding of angles and on a
higher level, the study of trigonometry is most applicable.
http://www.geom.uiuc.edu/~demo5337/Group3/hist.html
Step 2: Inquiry-based Lesson Design in Mathematics
Performance Objectives: Students will be able to…
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Identify right triangles.
Explain how two sides of right triangle relate to the third side.
Compute the length of a side given the lengths of two other sides in a right triangle.
Florida State Standards:
Benchmark Number:
MA.6.G.4.3
Benchmark Description:
Determine a missing dimension of a plane figure or prism given its area or volume and some of the dimensions, or determine
the area or volume given the dimensions.
Materials List and Student Handouts
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Guided Exploration Handout (21)
Pre-tests (21)
Post-tests (21)
Triangle handout to record findings (21)
Definitions handout (21)
Advance Preparations
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Pre-test will be given in advance by teacher.
Have packets ready for each student.
Set up Gizmo site for each computer desk
Powerpoint will be uploaded and ready for the start of the lesson.
Safety

There are no significant safety concerns.
Page 2 of 8
Step 2: Inquiry-based Lesson Design in Mathematics
ENGAGEMENT
What the Teacher Will Do
Time: 5 minutes
Probing Questions
Student Responses and
Potential Misconceptions
Good morning class! Today we
are going to be talking about
triangles!
Justin Beiber wants to build this
awesome treehouse to impress
Selena Gomez. (Show
transposition of triangle on
treehouse pediment).
Unfortunately, he only knows
the length of the base of the
triangle.
Show water proof of
Pythagorean theorem.
“Can he figure out the
lengths of the other two
sides without physically
measuring this model?”
What if he had the height
of the triangle as well?
Think about these
questions as we watch
this short clip. (watch clip)
Page 3 of 8
Yes, because he can just
measure them, he can’t
because there isn’t enough
information, [if he knew the
altitude of the triangle he might
be able to], he can figure out all
the sides by adding them up…
Step 2: Inquiry-based Lesson Design in Mathematics
EXPLORATION
What the Teacher Will Do
Time: 15 minutes
Probing/Eliciting
Questions
Student Responses and
Misconceptions
Each of you has been signed
onto the Gizmo site that looks
like this (on board). (We will get
go to the computer lab early
and setup. To save time, we
will we log onto the Gizmo site
on each of the computers.)
Handout Pythagorean Theorem
Gizmo packet.
Work through the “Using the
Interactive Geoboard” with the
students.
1. Click and drag one of
the dots located in the
top left corner of the
geoboard. Drop the
endpoint wherever you
like. Then drag the
other endpoint of the
segment and place it in
any new location.
2. Click near the middle of
the line segment and
drag away from the
segment. Notice that
you have made a third
vertex.
What type of figure do
you have?
A line segment
How many endpoints
does this figure have?
2
What is a vertex?
The corners of a shape/polygon
[The corners where two lines
meet]I don’t know…
What type of figure do
you have?
A triangle
How many vertices does
it have?
3
3. Make a quadrilateral
using the methods you
just learned. Click on
show side lengths and
show right angles.
What is a quadrilateral?
A shape with 4 sides.
4. Turn the quadrilateral
you made into a square.
Be sure to use the right
angles and side lengths
as a guide.
What are the lengths of
the side of the square
that you made?
1,2,3,…answers will vary
depending on how large the
student made their squares.
How do you find the area
of a square?
Square the side. 12, 22. 32,…
Multiply two sides. 1X1, 2x2,…
Page 4 of 8
Step 2: Inquiry-based Lesson Design in Mathematics
(The students will be working
individually through the packet
to derive the formula of the
Pythagorean theorem.) There
are questions throughout the
packet that the students need
to fill out.
Now that you are familiar
with how to use the
Geoboard. Click clear
and work through the
“Area and the
Pythagorean Theorem”
section.
Walk around the room to see if
the students have any
questions or problems working
the packet.
Go over the questions
throughout the packet.
Build a 3, 4, 5 right triangle.
Which side are the legs of
the right triangle?
3,4 [5]
What is the hypotenuse?
The side opposite the right angle
[The longest side]
5
Which side is the
hypotenuse?
Build squares on each side of
the triangle using the show
sides and right angles as guide.
Now test this for the 5, 12, 13
right triangle.
What are the areas of
each square?
32=9
42=16
52=25
How do the areas relate
to one another? Think
back to the water proof
video.
The sums of the smaller areas (3
and 4) equal the area of the
hypotenuse.
What is the equation for
the 3, 4, 5 right triangle?
32+42=52
If we label “a” and “b” as
our sides and “c” as our
hypotenuse, what is the
formula now?
a2+b2=c2
Does the Pythagorean
Theorem work for this
triangle?
Make another right triangle with
different lengths then we
already used and test the
Pythagorean Theorem.
Page 5 of 8
Yes. [No, maybe…]
Step 2: Inquiry-based Lesson Design in Mathematics
EXPLANATION
What the Teacher Will Do
Pythagoras was a Greek
philosopher who travelled the
ancient world extensively
studying mathematics - a
subject he held in high esteem.
He, along with his cultish
followers, considered numbers
divine, and swore to keep their
discoveries secret.
Now, that we’ve done that
activity, let’s consider some
important definitions. Please fill
in your blanks on the sheet as
we go through each one.
(If students are still displaying
trouble, will replay the waterproof clip from earlier.)
Time: 10 min
Probing/Eliciting
Questions
What is the Pythagorean
Theorem?
Student Responses and
Misconceptions
2
2
a + b =c2
Who can tell me what the
area of a square is?
Area = length x width, [A = one
side squared], add up all the
sides, half base x height…
How are squares related
to the Pythagorean
theorem?
[The areas of the squares of two
sides are related to the area of
the square of the other side], all
the squares add up to the whole
area, the squares inside the
triangle determine the sides…
Who can tell me what
taking the square root
means?
It is one number times itself, like
the opposite of an exponent,
undoing a square, it’s like the
one side of a square…
What are some
characteristics of a right
triangle?
It has three sides, one side is
longer than the other, it has one
90 degree angle, it is related to
squares, it has sides all of the
same length, it has angles of 30,
60, and 90 degrees, it has
angles of equal measure….
Who can tell me what the
hypotenuse is?
The longest side of a right
triangle, the greatest area, the
length of all of the sides…
(Students should be able to
recognize and understand the
use of the Pythagorean
theorem.)
Page 6 of 8
Step 2: Inquiry-based Lesson Design in Mathematics
ELABORATION
What the Teacher Will Do
Now let’s think back to the
question we posed at the
beginning of class. Remember,
Hunter was trying to build this
tree house. (We will show slide
with picture of the tree house
and a triangle transposed over
the pediment.)
(We will again transpose the
height onto the triangle and
show its measurement.)
We will also split the base, to
illustrate the two right triangles
more clearly.
We will repeat this same
process for problems where the
hypotenuse and one side are
given.
Time: 15 minutes
Probing/Eliciting
Questions
Is it possible to find the
lengths of the side of this
tree house when you
have the length of the
base only?
Student Responses and
Misconceptions
Yes, because you could still
measure it, no because you
need at least two dimensions to
find a third…
Can anyone think of a
way we might apply the
Pythagorean theorem to
solve this problem?
[If you knew the height or
altitude, you could split the large
triangle into two smaller right
triangles, then each side we are
trying to find would be the
hypotenuse], you could divide
the base by 2…
What other piece of
information that we don’t
have might be critical to
using this theorem?
The length of the sides, half of
the base, the height of the
triangle…
Is there a way we could
split this large triangle
into two right triangles?
Yes, no, maybe, I’m not sure…
So, if we know the length
of the base, and the
length of the height, how
could we find the
hypotenuse of each
triangle?
(If the kids still are not getting it,
we will move onto simpler, stepby-step questions)
What is the length of the
base now for each right
triangle?
Half of the base, the total
base….
Which side would we
denote a? Which side
would be b? Does it really
matter?
No, it doesn’t really matter.
Who can tell me the
answer?
82 + 62 = √100, so the
hypotenuse, or the length of the
sides is 10 feet long.
Page 7 of 8
Step 2: Inquiry-based Lesson Design in Mathematics
EVALUATION
What the Teacher Will Do
Time: 5 minutes
Probing/Eliciting
Questions
Hand out post-assessment test
to give students a chance to
show their understanding.
Page 8 of 8
Student Responses and
Misconceptions