Lesson Plan: Pythagorean Theorem and Special Right Triangles

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10 Οκτ 2013 (πριν από 3 χρόνια και 8 μήνες)

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Project AMP Dr. Antonio R. Quesada

Director, Project AMP


Lesson Plan: Pythagorean Theorem and Special Right Triangles


Lesson Summary

This lesson will provide students with the opportunity to explore concepts including
the Pythagorean Theorem, rational and irrational numbers, and the relationship of
side lengt
hs of special right triangles.


Key Words



Hypotenuse



Irrational Numbers



Legs



Pythagorean Theorem



Rational Numbers



Right Triangles



Simplest Radical Form


Background Knowledge

We are assuming the students are proficient at calculating square roots and squar
es,
solving multi
-
step equations as well as basic quadratic equations. We are also
assuming they have a strong background in basic geometry concepts including
finding area and perimeter and classifying angles. Lastly, we assume students have
basic graphin
g calculator proficiency in finding squares, finding square roots, and
using the table function.


NCTM Standards Addressed



Geometry and Spatial Sense Standard:

Students identify, classify, compare
and analyze characteristics, properties and relationship
s of one
-
, two
-

and
three
-
dimensional geometric figures and objects. Students use spatial
reasoning, properties of geometric objects and transformations to analyze
mathematical situations and solve problems.

o

Grade 7
:
Characteristics and Properties



Indicat
or #3
: Use and demonstrate understanding of the
properties of triangles.


For example:

a.

Use Pythagorean Theorem to solve problems involving right
triangles.

b.

Use triangle angle sum relationships to solve problems.

o

Grade 8
:
Characteri
stics and Properties

Project AMP Dr. Antonio R. Quesada

Director, Project AMP



Indicator #1
: Make and test conjectures about characteristics and
properties (e.g., sides, angles, symmetry) of two
-
dimensional figures
and three
-
dimensional objects.



Number, Number Sense and Operations Standard
: Students demonstrate
number sense, including an understanding of number systems and operations
and how they relate to one another. Students compute fluently and make
reasonable estimates using paper and pencil, technology
-
supported and mental
methods.

o

Grade 7
:
Number and Numb
er Systems



Indicator #3
: Describe differences between rational and irrational
numbers; e.g., use technology to show that some numbers
(rational) can be expressed as terminating or repeating decimals
and others (irrational) as non
-
terminating and non
-
repea
ting
decimals.


Learning Objectives



Understand Pythagorean Theorem



Appl
y

Pythagorean Theorem to real
-
world
scenario



Understand the difference between rational versus irrational numbers



Recognize the relationships of side lengths in special right triangle
s



Apply knowledge of special right triangles to real
-
world scenarios


Materials



Ziploc bags containing colored straws of different lengths
. Use lengths so that
not all combinations will result in a triangle.



Calculators



Activity and extension activity shee
ts



Pencils


Suggested Procedure



The

time frame for this lesson is 2

days.



Day 1

o


Group students into heterogeneous groups of 3.

o

Attention Getter
: Pass out Ziploc bags with straws to students and pose
the question “Can you get a right triangle from any 3 s
ide lengths?” Ask
students to work individually on this question and then discuss their
findings with their group members. Facilitate class discussion after 2
minutes.

o

Pass out Pythagorean Theorem activity sheet and have students work on
this activity sh
eet within their groups. Walk around and observe.
Project AMP Dr. Antonio R. Quesada

Director, Project AMP

Remember to let students determine conjectures and solutions


do not
give answers! After completion of this activity sheet, facilitate a debrief
discussion of the activity.

o

Ask students to summarize ind
ividually in their math journals what they
learned today.




Day 2

o

Pass out Extension Sheet
s

on 30°, 60°
, right triangles and
45°, 45°,
right

triangles.


o

Have stu
dents work in same groups on the
s
e

activit
ies
.

o

At conclusion of activities
, debrief as a whole
class.

o

Ask students to summarize individually in their math journals what they
learned today.



Assessments



Observation



Class Discussion



Written solutions and explanations



Math Journals



Future quiz and/or test





















The Pythagorean Theorem an
d Special Right Triangles



Project AMP Dr. Antonio R. Quesada

Director, Project AMP


Group Members

____________________________


______________________________







____________________________


______________________________



Before We Begin


Can a right triangle be formed using any three lengths of sides?

Select three straws from the bag. Use the corner of a piec
e of notebook paper for a

right angle. Use two of your straws

placed along the edges of your notebook paper as the
legs

of the right triangle. The third straw will form the
hypotenuse

of the right
triangle. Do your
straws form a right triangle? Compare your results with those of your members. How many
of you were able to form a right triangle?


Lesson 1


We recall the area formula for
a
square is A
=
s
2
.
Using

the dimensions given, complete
the tab
le
by finding the area of t
he squares in the diagram below
.


1.)



















2.) What relationship do you notice with
the areas of the three squares
?



a

b

c

a
2

b
2

c
2

3

4

5




7

10





6

8

10




5


14




Project AMP Dr. Antonio R. Quesada

Director, Project AMP



3.) Based on the patterns observed in the table, what conjecture can you make?






4.) Notice two of the dimensions in the table are expressed as rad
icals, why do you think

they are expressed as radicals and not in decimal form?





5.) When writing the decimal form, is it possible to give an exact answer? Why or why not?





Definition:

Irrational number


An irrational number is a number that cann
ot be expressed as
a
fraction

for any
integers

and
. Irrational numbers have

decimal expansions

that neither
terminate nor
repeat.

For the remainder of this lesson, you may want to express irrational answers as decimal
approximations (f
or real world applications) or in simplest radical form (to express a more
precise value).


6
.) Now, create your own dimensions. Does your conjecture still hold?





7
.) Now, use that relationship to find the missing length in
each of
the
right
triangl
es below.



a.

)

b.)

c.)



8.) Summarize, in your own words, what you have learned today.




60 ¾ in

57 in

b

c

164.01 cm

a

100 cm

119 ft

57 ft

Project AMP Dr. Antonio R. Quesada

Director, Project AMP















Real World Application:
Play Ball!


















The distance from e
ach consecutive base is 90 feet and it can be necessary to
determine how far the catcher will have to throw to get
the ball from home plate to
2
nd

plate or the distance of the throw from the 3
rd

baseman to 1
st

base. We will use
the Pythagorean Theorem to find answers to these questions.


1.

What other shape is the baseball diamond?


2.

Explain how you might use the Pythago
rean Theorem to determine a value for x.

1
st

Base


2
nd

Base

3
rd

Base

Home
Plate

90 ft.

90 ft.

90 ft.

90 ft.

X

Project AMP Dr. Antonio R. Quesada

Director, Project AMP



3.

Where is a right triangle located?



4.

Determine what side of the right triangle is the hypotenuse.



5.

Determine the legs of the triangle.



6.

Now, set up the problem so that you can use the Pythagorean Theorem to find

out how far the
catcher will have to throw the ball from home plate to 2
nd

base to the nearest foot.





7.


How far will the third baseman have to throw to get to 2
nd

base. How can you determine this
answer from the work that you have already done. Explain
.















Extension 1


1.) Complete the table for the special right triangle below.

Express irrational values in
simplest radical form
.


Project AMP Dr. Antonio R. Quesada

Director, Project AMP








2.) Do you notice a relationship between the lengths of the sides of this special right
triangle?





3.) Express the above relationship in terms of
x
.







a

b

c

a
2

b
2

c
2

1






3


6





5

10









144

c

b

a

30º

60º

Project AMP Dr. Antonio R. Quesada

Director, Project AMP

Using the TI
-
Nspire


From the
Home Menu
, select
Lists & Spreadsheets
.






























Label the first
column


sl” (for short leg),
the second column
“ll” (for long leg), and
the
third column
“hyp” (for hypotenuse
)
.















































Project AMP Dr. Antonio R. Quesada

Director, Project AMP

Type in the formula “=a(
)” in the formula line of column b.



















Enter any values you wish into column
a

to test your conjecture.












Was your conjecture accurate? Explain.






4
.) What conclusion
s

can you make about 30
-
60 right triangles?














Project AMP Dr. Antonio R. Quesada

Director, Project AMP

Real world application



Locked Out!


José has

locked himself out his house. Fortunately, he did leave an
upstairs window open and does have access to an extension ladder. The
ladder, when fully extended safely, will extend to 24 feet. For optimal
safe
ty reasons, he wants to maintain a 60
-
degree angle with the ground.
If he extends the ladder to 18 feet, (a)

how far will the base have to be
away from the wall and (b)how high up on the wall will the ladder reach?







































Project AMP Dr. Antonio R. Quesada

Director, Project AMP

Extension 2


1.) Complete the table for the special right triangle below.

Express irrational values in
simplest radical form.















2.) Do you notice a relationshi
p between the lengths of the sides of this special right
triangle?





3.) What conclusion can you make about 45
-
45 right triangles?















a

b

C

a
2

b
2

c
2

5

5





3


3







49







1


45º

45º

a

b

c

Project AMP Dr. Antonio R. Quesada

Director, Project AMP


Real World Application


































6 feet


An Isosceles Right Triangle is the sha
pe of the opening of a tent. What is the largest
sized object that can be put through the opening without touching the sides of the
opening?