Square Roots and Pythagorean Theorem Lesson Plan

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					Dominique Davis, Molly Duffy, Julie Hernandez, Kelly Holton, Brandi Zars
CI 473
Pre-Reading Lesson Plan

Square Roots and Pythagorean Theorem Lesson Plan
Grade Level: 9th grade Algebra I 334
Demographics:

20 students (10AA, 7 Caucasian, 2 Latino/a, 1 Asian)
(15 freshman, 4 sophomores, 1 junior)

Estimated Time: 1 class period (about 50 min)

Concept:
Students will first thoroughly reexamine the concepts of perfect squares and square roots.
The main focus of this lesson will demonstrate to students how to use square roots and
the Pythagorean Theorem to find distance. They will also become familiar with real
world applications. By the end of the lesson students will have a clear understanding of
the importance of the Pythagorean Theorem and how to apply it.

Objectives:
Students will review the meaning of a perfect square
Student will be practice how to find the square root of a number
Students will understand real world applications of the Pythagorean Theorem
Students will use visual representation to measure and compute distance
Students will use the Pythagorean Theorem to compute distances

Materials:
 - 11” x 17” paper
- ruler
- protractor
- pencil
- calculator with square root function
- principles of square roots and Pythagorean Theorem worksheet (homework)

Procedure:
 1) Good Morning Class! While I am taking attendance, I want you to write all the
numbers I have written on the board in your notebook and circle the ones that are perfect
squares. You should all be familiar with the concept of perfect squares, but if you cannot
 remember what a perfect square is – ask your partner what they are, then circle them on
your own!

2, 4, 9, 12, 16, 25, 36, 44, 55, 64, 81, 100
(After attendance is complete) Alright, could someone please raise their hand and tell me
which of these numbers are perfect squares? (Call on student)
The perfect squares are 4, 9, 16, 25, 36, 64, 81, 100

When we have a number multiplied by itself, it gives us a perfect square
Let’s take a look at these numbers that were written on the board. Ask students what
number multiplied by itself gives us these numbers.
2x2 = 4
3x3 = 9
4x4 = 25
6x6 = 36
8x8 = 64
9x9 = 100


When we take the square root of any perfect square, we get an integer. The symbol for a
square root is called a radical. To remind you all, this is what a radical looks like



So could someone raise their hand and tell me what the answer to √9 is?
                                                     What about √25?

Can anyone find √8? If you are having difficult that is okay. This problem is more
difficult to solve because the number under the square root, 8, is not a perfect square.
When we do not have perfect square under the radical it becomes more complicated.
For today we are going to use the square root button on our calculators to find the
approximate decimal answers of square roots such as these. (Show class where and how
to use square root button on calculator. Later on in this unit we will take a look at how to
rewrite √8.

Alright, now that we have re-familiarized ourselves with how to do square roots, let us
understand where in the world we are ever going to have to use them!

Have all the students look up at the board. Draw a Rectangle on the board and label one
side 3 and the other side 4. Ask them if they can see clearly the lengths of all four sides
of this rectangle? Now draw a diagonal through the rectangle and erase everything above
the diagonal. Now ask the class if they know how to find the length of this side (the
diagonal). We’ve never had to figure out something like this before, but today we will
learn using something called the Pythagorean Theorem!

I bet some of you are still wondering, “What is the importance of right triangles and the
Pythagorean Theorem?”
Quickly explain the following to the class
Electricians: use right triangle to make off-sets when bending conduit
Carpenters: apply the Pythagorean Theorem to check the diagonal foundation or framer
wall to determine if it is a square
Plumbers: have fittings that allow them to make perfect 90° angles
Surveyors: use sophisticated instruments

All of these professions need this kind of math!

Before we go into the details of the Pythagorean Theorem, let us just review a few terms
most of us should remember from before which have to do with the Pythagorean
Theorem (draw pictures to explain):
Altitude: The perpendicular distance from the vertex to the base
Angle: The union of two rays with a common endpoint
Base: The bottom of a plane figure or three-dimensional figure
Diagonal: The line segment connecting two nonadjacent vertices in a polygon
Right angle: An angle whose measure in 90°
Hypotenuse: The side opposite the right angle in a right triangle
Square Root (√): The square root of x is the number that, when multiplied by itself,
gives the number x.
It is especially important that we remember what a right triangle is, because the
Pythagorean Theorem will ONLY work with right triangles. Recall the following two
triangles on the board:




                     30°-60°-90°                                    45°, 45°, 90°

Alright, I think we are all caught up on everything we are going to need to know for the
Pythagorean Theorem:

Demonstration for 3-4-5 triangle:
Now what I want everyone to do is take out your 11x17 paper that I handed out.
(Demonstrate on board while students follow on their paper)
1. First measure the corner of your paper with your protractor to discover the degree of
the angle of the corner (90°) – which is what we need to use the Pythagorean Theorem.
2. With your ruler, measure 3” up on the end of the paper and mark.
3. Measure from that same corner in the other direction out 4” and mark.
4. Draw a line from Mark to Mark
5. Measure the length of the line you drew from mark to mark (5”)
Ask the class if everyone got a number for the hypotenuse approximately equal to 5”.
The teacher should walk around the room to see if it looks like everyone has gotten
approximately 5”.

Great Job Class. I think that everyone is ready to learn the formula to find the
Hypotenuse without a ruler!
If we take a right triangle and label the altitude a, the base b, and the hypotenuse c




                  The Pythagorean Theorem is the following formula: a² + b² = c²
                  So using the rectangle that we just drew on our sheet of paper, if we
only knew the lengths 3 and 4, how would we figure out the hypotenuse?

We would plug our 3 and 4 in as our a and b.
So we would get 3² + 4² = c²
                9 + 16 = c²
                    25 = c²
                     5=c

The answer is 5, which matches what we measured on our paper!

Now I will draw another triangle on the board with altitude 5, and base 12. To find the
hypotenuse of this how would we use the equation? We would plug 5 and 12 in as our a
and b, so we would get: 5² +12² = c²
                      25 + 144 = c²
                            169 = c²
                             13 = c²

We were lucky on these last two problems. These were special right triangles that gave
us perfect squares. Sometimes a² + b² will not always give us a perfect square:
What if a=2 and b=5, then what does c=? It is okay to use your calculator on this one.

 2² + 5² = c²
4 + 25 = c²
     29 = c²
   5.39 = c

I want you to do this next one on your own. When you have the answer put your pencil
down and I will come around the room to see if you got the right answer. Here is the
problem: (write on board) Given a and c, find b
A = 3, c = 7 then b = ?

a² + b² = c²
3² + b² = 7²
9 + b² = 49
   b² = 49-9
   b² = 40
   b = 6.32

Now, on the other corner of your paper, on your own pick any two different lengths for
sides of your triangle and make your marks. Then draw and measure the hypotenuse
with your ruler and see the length. After you have done this, use the Pythagorean
Theorem to see that c is the same length as your hypotenuse. It should match!
(Go around the room and see if students successfully matched their hypotenuse measure
with their c calculation)

Great Job today! Tonight’s homework will be principles of square roots and Pythagorean
Theorem worksheet. It will be more problems like what we were doing in class today,
Solving for a, b, or c. This will be due tomorrow after you ask questions at the start of
class.

Assessment: Assessment is done throughout class by observing students and walking
around the room to check answers and see if students are understanding. At the end of
class students will receive a worksheet which goes over the Pythagorean Theorem.
Homework is collected the following class period and graded.

Modifications: If we run out of time, students can measure and do Pythagorean Theorem
calculations at home. If we have extra time, students can get started on tonight’s
homework.

Rationale for strategy use:
The lesson is begun with perfect squares and square roots to refresh students’ memory on
how to do these. This should be very elementary for most students, and is intended for
students who have struggled with these concepts in the past. The section of the lesson
dedicated to the actual profession is intended to target those who have limited interest in
math as well as other students who enjoy it and understand its applications. This helps
students realize where they may apply these concepts in the real world. The “triangle
terms” are revisited and defined again to refresh students memory with an appropriate
background of what is needed for the Pythagorean Theorem, before the teacher begins
using terms they have forgotten. Finally the lesson slowly moves into the main idea of
the lesson, the Pythagorean Theorem, using hands on materials and basic methods on the
board.

Standards:
Illinois Learning Standards:
9.D.3 Compute distances, lengths, and measures of angles using proportions, the
Pythagorean theorem and its converse

NCTM Standards:
Create and critique inductive and deductive arguments concerning geometric ideas and
relationships, such as congruence, similarity, and the Pythagorean
Dominique Davis, Molly Duffy, Julie Hernandez, Kelly Holton, Brandi Zars
CI 473
During-Reading Lesson Plan

Pythagorean Theorem Lesson Plan
Grade Level: 9th grade Algebra I 334

Demographics:
20 students (10 AA, 7 Caucasian, 2 Latino/a, 1 Asian
(15 freshman, 4 sophomores, 1 junior)

Estimated Time: 1 class period (about 50 minutes)

Concept:
Students will have a deeper understanding of the Pythagorean Theorem and how it is
used. After this lesson is completed, students will be able to find the length of the
hypotenuse, the length of a side of a right triangle, understand the concept of a
Pythagorean Triple, and they will be able to figure out if a triangle is a right triangle
when given its side lengths.

Objectives:
Students will review and fully understand the Pythagorean Theorem.
Students will understand real world applications of the Pythagorean Theorem.
Students will use the Pythagorean Theorem to compute distances.
Students will know the concept of a Pythagorean Triple.
Students will use the Pythagorean Theorem to see if a triangle is a right triangle.

Materials:
Notes
Loose Leaf Paper
Pencil
Transparency
Overhead
Homework Handout

Procedure:
1) Good morning class! Who can tell me what we learned about yesterday? Hopefully
at this time a student will raise their hand and tell me that they learned about perfect
squares, radicals, and the Pythagorean Theorem. If not, I will remind them quickly about
what we went over and what we learned. Thank you, that’s great! Yesterday we
learned about the Pythagorean Theorem and how that applies to right triangles. Before
we start today, I would like all of you to take out a sheet of paper and fold it into fourths.
This is going to by your summary sheet. I would like you to label the first section
Finding the Length of the Hypotenuse. Label the second section Finding the Length of a
Leg, label the third section Pythagorean Triples, and label the fourth section Checking for
Right Triangles. As we go through our lesson today, try to fill out these sections. For the
first section, I want you to think about how you would find the hypotenuse for a right
triangle –what do we need to know about the triangle in order to find the hypotenuse, and
what do we have to do once we have that information? Do the same for the second
section except we’re trying to find the length of one of the legs of a right triangle instead
of the hypotenuse – so what do we need to know, and what do we need to do once we
have that information? For the third section, just tell me what a Pythagorean Triple is –
describe it to me. And for the fourth section, explain how you would go about checking
to see if a triangle is a right triangle – how do you know when a triangle is a right triangle
and when it isn’t? Any questions about what is expected of you? Great, what I need for
you to do now is take out your notes and work on the problem on the board while I am
taking attendance. Problem on the board: The roller coaster “Superman: Ride of Steel”
in Agawam, Massachusetts, is one of the world’s tallest roller coasters at 208 feet. It
also boasts one of the world’s steepest drops, measured at 78 degrees, and it reaches a
maximum speed of 77 miles per hour. If at the highest point, the rollercoaster is 208 feet
and the base is approximately 44 feet, what is the length of the first hill?



                                      208 feet       c


                                           44 feet

After I am done taking attendance: Was anyone able to find the answer? That’s great!
Let’s take a look at how you came up with that. Can anyone first remind me of the
formula for the Pythagorean Theorem? Thank you, it’s a2 + b2 = c2 where a and b are
lengths of the legs of the right triangle and c is the length of the hypotenuse. Now, if we
plug in for a and b we can easily solve for c.

                                       2082 + 442 = c2
                                     43,264 + 1936 = c2
                                         45,200 = c2
                                        45,200 = c 2
                                       212.6 feet ≈ c

So, not only can electricians, carpenters, plumbers, and surveyors use the Pythagorean
Theorem like we learned yesterday, but it’s applicable in many other facets of life as well
– such as finding the length of a drop on a rollercoaster.

2) In this last problem, what was our unknown? Yes, it was the hypotenuse. What
would happen if one of our legs of the right triangle was our unknown instead of the
hypotenuse? Would we be able to use the Pythagorean Theorem to solve for that side?
What do you think? We looked at something like this yesterday. Here, I will let them
brainstorm for a while and explain to me how we could find the length of the missing side
using the Pythagorean Theorem to make sure they fully understand what it is we learned
yesterday. Now let’s look at an example of this and work through it together. In a right
triangle, let c = 25 and b = 10. What is the length of a? Here we will work through the
problem on the board:

                                      a2 + 102 = 252
                                      a2 + 100 = 625
                                      a2 = 625 – 100
                                         a2 = 525
                                         a 2 = 525
                                         a ≈ 22.91

That was great! Now we know how to find the length of any side of a right triangle,
given that only one side is missing.

3) We had a few examples of triangles yesterday where each side of the right triangle
was a whole number. When this happens, they are called Pythagorean Triples. The
examples we saw yesterday included a 3-4-5 triangle and a 5-12-13 triangle. Can you
think of any other examples of Pythagorean Triples? If you think you have one, use the
Pythagorean Theorem to make sure it’s a right triangle. If you’re having trouble, see if
you can come up with a Pythagorean Triple based off of the ones we already know.
Were any of you able to find some? That’s great! I don’t expect you to memorize or
remember how to do what I am about to show you. I just want you to be exposed to it in
case it comes up in one of your future math classes. By seeing it now, hopefully when it
does appear, you’ll be able to think back to this and it will be easier for you to retrieve.
Okay, so there is a simple formula that gives you all of the Pythagorean Triples. Suppose
m and n are two positive integers with m < n, then a = n2 - m2, b = 2mn, and c = n2 + m2.
Look at this table: (We will make this into a transparency and put it on the overhead)

      m=1                 2              3                4              5               6

n=
2      [3, 4, 5]

3      [8, 6, 10]      [5, 12, 13]

4      [15, 8, 17]     [12, 16, 20]   [7, 24, 25]

5      [24, 10, 26]    [21, 20, 29]   [16, 30, 34]     [9, 40, 41]

6      [35, 12, 37]    [32, 24, 40]   [27, 36, 45]     [20, 48, 52]   [11, 60, 61]

7       [48, 14, 50]   [45, 28, 53]   [40, 42, 58]     [33, 56, 65]   [24, 70, 74]   [13,
84, 85]

As you can see in the 3-4-5 right triangle, when m = 1 and n = 2, a = 4 – 1 = 3, b =
2(2)(1) = 4, and c = 4 + 1 = 5. The table can go on forever, and I thought it was very
interesting to see that you can find EVERY Pythagorean Triple. Anyways, this is just for
your own benefit, so take it as you wish.

4) Next, let’s think about what we were doing when we were figuring out if our
Pythagorean Triples really were Pythagorean Triples. We were taking our a’s, b’s, and
c’s and seeing if they worked in the Pythagorean Theorem. By doing this, we were
determining whether the side measures of a triangle form a right triangle. Therefore, if a2
+ b2 = c2, then the triangle is a right triangle. However, if a2 + b2 ≠ c2, then the triangle
is not a right triangle. Let’s see if these next triangles are actually right triangles. First
let’s examine the triangle with sides 20, 21, and 29. Of these three sides, which one is the
hypotenuse? Great, and why is the longest side the hypotenuse? What angle is always
opposite the hypotenuse? Why? Excellent, now let’s plug in these values.

                                     202 + 212 =(?) 292
                                     400 + 441 =(?) 841
                                         841 = 841

As we can see, a triangle with side lengths of 20, 21, and 29 form a right triangle. What
about a triangle with the sides 8, 10, 12? Let’s try this one too.

                                      82 + 102 =(?) 122
                                      64 + 100 =(?) 144
                                          164 ≠ 144

As we can see, this triangle is not a right triangle.

5) So, who can tell me what we did today? What did you fill out in your four sections of
your paper? As you explain it to me, I will write it out on the board so everyone can see.
I will let the students take their time in summarizing the main topics of today and then we
will have a class discussion about it.

Modifications:
If we run out of time, we will continue our summary on the board the following day. If
we have extra time, they will be able to start on their homework.

Assessment:
Assessment is done throughout class by observing students and walking around the room
to make sure they are staying on task. At the end of class, students will receive a handout
with the following questions on it:
1) Draw a right triangle and label each side and angle. Be sure to indicate the right
angle.
2) Explain how you can determine which angle is the right angle of a right triangle if you
are given the lengths of the three sides.
3) Write an equation you could use to find the length of the diagonal d of a square with
side length s.
4) What was your favorite thing you learned today and why? Explain it to me.
5) Can you think of where else you might use the Pythagorean Theorem in your life?
They will have to answers these on a separate sheet of paper and hand them in the
following class period.

Rational:
We began our lesson by discussing what was learned the previous day in hopes of getting
the students thinking and getting them engaged with the text at the very beginning of
class. We also let the students brainstorm by having them tell us how to find the length
of a side of a right triangle instead of just giving them the answer. We did this because it
will help them synthesize and organize the information they are learning and have
already learned. Furthermore, we constantly questioned them throughout the lesson
which guided their understanding into what we wanted them to know. By answering
these questions, not only were we guiding their learning, but they were reflecting upon
the information that was being presented to them. This was also accomplished by having
them fill out the summary sheet with the four sections throughout the class. We gave
them explicit directions and told them exactly what we wanted from them and what we
were looking for in each box. By doing this they were able to reflect up on the
information we went over in class, and they were able to put it in their own words. At the
end of class, we had a discussion about what we learned. We had the students describe in
their own words what they wrote in their boxes. After that, we wrote this information out
on the board, in sentences, so everyone could see. We did this because group summaries
help the students review and remember the information that was presented to them during
class.

Standards:

Illinois Learning Standards:
9.D.3 Compute distances, lengths, and measures of angles using proportions, the
Pythagorean Theorem and its converse

NCTM Standards:
Create and critique inductive and deductive arguments concerning geometric ideas and
relationships, such as congruence, similarity, and the Pythagorean Theorem.

Dominique Davis, Molly Duffy, Julie Hernandez, Kelly Holton, Brandi Zars
CI 473
Writing in Mathematics Lesson Plan

Pythagorean Theorem Lesson Plan
Grade Level: 9th grade Algebra I 334

Demographics:
20 students (10 AA, 7 Caucasian, 2 Latino/a, 1 Asian
(15 freshman, 4 sophomores, 1 junior)
Estimated Time: 1 class period (about 50 minutes)

Concept:
Students will have a deeper understanding of the Pythagorean Theorem and why it
actually “works”. Prior to this lesson, students applied the theorem to find lengths of
missing sides of right triangles and to identify right triangles given dimensions of
arbitrary triangles. After this lesson, they will understand why they were able to apply
the formula a2 + b2 = c2 to solve right triangles. Students will experiment with specific
online manipulatives that demonstrate visual proofs of the theorem. After experimenting
with these visual proofs, students will be challenged to write in their own words an
explanation of the proof.

Objectives:
Students will review and fully understand the Pythagorean Theorem.
Students will understand visual proofs of the Pythagorean Theorem.
Students will be describe the principles of the Pythagorean Theorem using their own
words.

Materials:
Notes
Loose Leaf Paper/Journal
Overhead/transparencies
Pencil
Computer Lab

Procedure:
1) Good morning class! Yesterday we worked with the Pythagorean Theorem, and we
used it to discover the missing side lengths and hypotenuses of given right triangles. We
also used the theorem to determine if an arbitrary triangle was indeed a right triangle,
given the dimensions of that triangle. So yesterday we actually applied the formula a2 +
b2 = c2 to solve right triangles. Today, however, we are going to focus on your
understanding of the theorem. We’re going to discover why we can apply the
Pythagorean Theorem to triangles. Today we will ask questions like, why does this
theorem work? Why can we simply use the formula a2 + b2 = c2 to solve right triangles?
 2). So today, I’m going to provide you with a visual proof of the theorem, then you will
individually work with online manipulatives to understand why the proof works. After
we return from the computer lab, you will all write in your own words, a brief
explanation of the proof, and why it works. We have written proofs in this class already,
so you should be familiar with the concept of a proof. But for a quick review, a proof is a
logical argument that consists of a sequence of steps, statements, or demonstrations that
leads to a valid conclusion. In other words, it’s a series of logical statements of facts that
“prove” something to be true. Today we will discuss why the Pythagorean Theorem
holds true for any given right triangle and focus specifically on a visual proof of the
theorem.
3). The Pythagorean Theorem is a statement about triangles containing a right angle. The
Pythagorean Theorem actually states that: "The area of the square built upon the
hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the
remaining sides." We have simplified that into an easily memorized formula a2 + b2 = c2,
but we need to discover why we can simply use this formula. Here we have a right
triangle ABC. We say that the side AB=a, side BC=b and side CA= c.

                     A



                 a             c



                  B       b        C

Now, if we construct squares upon the sides and hypotenuse of the triangle we get an




object that looks like this:



Now our theorem states that the area of the square built upon the hypotenuse is equal to
the sum of the areas of the squares upon the remaining sides. Let’s look at the picture
below to get an idea of what this statement means.
According to the Pythagorean Theorem then, the sum of the areas of the two red squares
A and B, is equal to the area of the blue square, square C. We find the areas of the
squares as follows: Area of Square A = a2. Area of Square B = b2 . And finally, Area of
Square C = c2 . This is how we arrive at our algebraic formula, a2 + b2 = c2 . So our
formula is simply the algebraic application of the Pythagorean Theorem for a right
triangle with sides of lengths a, b, and c, where c is the length of the hypotenuse.

4). So, we have simply broken down the statement of the Pythagorean Theorem, and
discovered what it means visually. Now we’ll go to the computer lab and discover a
visual proof of the theorem.

(Once at the lab, have students go to http://www.usna.edu/MathDept/mdm/pyth.html,
also have overheads used in class available to see in the lab)

So class, we have a figure similar to the ones on our overheads in class. Here we have a
triangle ABC, with squares built upon each side of the triangle. So similarly in this
animated proof, we start with a right triangle and squares on each side. The middle size
square is cut into four congruent quadrilaterals. In our overhead, this would be square A,
cut into four congruent quadrilaterals. That means that each one is equal in dimension
and area. The square is separated into parts such that the cuts are made through the
center of the square, and run parallel to the sides of our square C. The animation then
moves the quadrilaterals. They are hinged and rotated and shifted to the big square, or
square C. Finally the smallest square, in our case square B, is translated to cover the
remaining middle part of the biggest square C. So it just so happens that it is a perfect fit.
Thus we have just seen visually, that the sum of the squares on the smaller two sides
equals the square on the biggest side. In other words, the sum of the areas of squares A
and B, is equal to the area of square C. So indeed a2 + b2 = c2.
5). Now I want you to try a few online manipulatives to further discover what we have
been learning today. Please go to the websites listed on the board:

1) http://www.dynamicgeometry.com/javasketchpad/pythagoras.php

2) http://nlvm.usu.edu/en/nav/frames_asid_164_g_4_t_3.html?open=instructions

3) http://cinderella.de/files/HTMLDemos/1G01_Pythagoras.html

The first and second websites are visual applications of the Pythagorean theorem. I want
you to play around with each manipulative and make your own discoveries about the
theorem. The first website uses a similar object to the one in our overhead and the visual
proof website. It is a geometer’s sketchpad interactive applet that allows you to move the
pieces by dragging and dropping the red points on the polygons. It shows that the area of
square BCDE = the area of square BFGA + the area of square AHIC.

Our second website is a little bit different. In this you are asked to solve two puzzles that
illustrate the proof of the Pythagorean theorem. In this manipulative, we have a right
triangle with its sides labeled. You can move any triangle or square by clicking the piece
and dragging it around. You rotate the pieces by clicking on the corner of the piece and
rotate your mouse in a circular motion. At the bottom, there are ways to reset your pieces
to start from scratch. You can also go onto the next puzzle. As you solve the puzzles, I
want you to draw a picture of what the shape should look like. You will turn this in along
with an explanation in your own words of what you learned today.

Our third website is optional, and it is simply another visual proof of the Pythagorean
Theorem. You may go to this site if you need further clarification on the subject matter.
(Allow students 20 minutes to work in the lab)

6). Alright class, now that we have returned from the lab, I want you to take out a sheet
of paper. Please redraw your sketches of your solutions to the two different puzzles from
our second website. There should be a total of four sketches, since each puzzle had two
different objects. When you have done that, I would then like for you to tell me IN
YOUR OWN WORDS, what the Pythagorean Theorem states, and why we are able to
use the formula a2 + b2 = c2 to solve right triangles. You are allowed to use any
explanation that you see fit, whether or not you want to describe what we did in class, or
what you learned from the online manipulatives, it is up to you. Please write at least four
to five sentences about what you learned from today. You will be turning these in at the
end of class as an evaluation of the effort you put into today’s activities. (Allow students
at least 10 minutes to summarize their thoughts. Have a class discussion in which
individual students share their writings, if time permits.)


Modifications:
If we run out of time, we will have a short review of the activity, and students will
summarize and put into their own words their explanations of the Pythagorean theorem
on the following day. If we have extra time, we will have a class discussion in which
individual students are to share their writings with the class.

Assessment:
Assessment is done throughout class by observing students and walking around the
computer lab to make sure they are staying on task. Also, the in-class writing activity
includes their solutions to the online puzzles, which should keep students focused during
lab time. Each individual student’s written submissions should also give the instructor
feedback of student progress and comprehension.

Rationale:
We began the lesson by reviewing what was accomplished in previous lessons to help the
students understand the material for the day, to get them thinking about previous material
and to help them become engaged with the day’s lesson. We want our students to have a
deeper understanding of the Pythagorean theorem and also see why they are able to apply
a simple formula in order to solve right triangles. We prefaced the lesson by explaining
that in previous lessons we simply applied the formula a2 + b2 = c2 , but that this alone,
will not tell us why or how the Pythagorean theorem actually works. We gave them clear
objectives for the day by telling them we were going to see visual proofs of the
Pythagorean theorem and that by the end of the period, they were to be able to write in
their own words an explanation of the proofs. In order to introduce the material, we
wanted to first break down the theorem and give the students a visual representation of
the theorem. This would provide students with further clarification of the theorem,
allowing them to synthesize and organize the information with a visual aid. Our activity
in the computer lab was intended to engage students with the material and force them to
think critically about what we previously learned in class. We also gave them explicit
directions about what would be expected of them at the completion of the task. The
writing activity was intended to the students summarize their thoughts. It was a method
for checking each student’s individual comprehension and ability to synthesize the
material. The written submission and the class discussion are intended to effectively
summarize and review the information from the day.

Standards:
Illinois Learning Standards:
9.C.4,5 Select and apply technology tools for research, information analysis, problem-
solving, and decision-making in content learning.
NCTM Standards:
Create and critique inductive and deductive arguments concerning geometric ideas and
relationships, such as congruence, similarity, and the Pythagorean Theorem.