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The Pythagorean Theorem

TEACHING TIPS

COMBINING LIKE TERMS – MISCONCEPTION ALERT
Teaching Tip 1
Preview/Warmup
Since the proof of the Pythagorean theorem in this lesson requires some symbol
manipulation, practice is provided in the warmup. Be prepared for student mistakes when
combining like terms .

If students falter by stating that ab + ab = a2b2 , remind them of previous experiences with
variables. For example, the cost of two slices of pepperoni pizza can be represented as p
+ p = 2p. Therefore, a + a =2a might represent the cost of two apples. It follows that
might represent the cost of two halves of apples, which is the cost of one
apple, or simply a.

Later in the lesson, polygon cutouts link a visual model to this symbol manipulation .
Emphasize that two triangles combined from the second square are congruent to one
rectangle in the first square. Since the area of each triangle is , and the area of each
rectangle is ab, it follows that

EXPLORING THE PYTHAGOREAN THEOREM AND ITS CONVERSE
Teaching Tip 2
Practice
In general, if we have a theorem that says, “If A is true, then B is true”, then the converse
of that statement is, “If B is true, then A is true.”

Therefore,

Pythagorean Theorem: If a triangle is a right triangle, then the sum of the squares
of the two shorter legs is equal to the square of the hypotenuse.

Converse of the Pythagorean theorem: If the sum of the squares of the two
shorter legs is equal to the square of the hypotenuse, then the triangle is a right
triangle. Replace “legs” with “sides” and “hypotenuse” with “longest side” – the
converse is starting with a mystery triangle so avoid right triangle names.

Students can use the Pythagorean theorem together with its converse to verify numerically
whether a triangle is right. Draw one triangle with sides of length 5, 11, and 12 units, and
another triangle with sides of length 5, 12, and 13 units. Do not label right angles on either
triangle. Ask students to determine which of the triangles are right (if any) by using only
their side lengths. Since 52 + 112 is not equal to 122, they can deduce that the 5-11-12
triangle is NOT right by the Pythagorean theorem. On the other hand, since 52 +122 = 132 ,
the converse to the Pythagorean theorem says that the 5-12-13 triangle is in fact a right
triangle.
 

PREVIEW / WARMUP
Whole Class

SP1, OH1
Ready, Set, Go

Teaching Tip 1

• Introduce the goals and standards for the lesson. Underline important
vocabulary.

• Students find the areas of the figures given and simplify the given
expressions . Discuss answers and possible misconceptions as needed.

INTRODUCE 1
Whole Class

SP2, OH2
Two Right
Triangles

• Discuss basic properties of a right triangle (three sides, two acute
angles, one right angle) and vocabulary (legs, hypotenuse) associated
with the naming of the sides of a right triangle.

Which side of a right triangle must be the longest? Why? The
hypotenuse must be the longest side because it is opposite the largest (right) angle.

• Focus attention on the small triangle on the left. Demonstrate how to
draw squares on the legs.

What is the area of the square on the shorter leg? 9 square units on
the longer leg? 16 square units

How do the side lengths relate to the areas of these squares? Side
length squared is equal to area, and square root of area is equal to side length.

• Demonstrate how to draw a square on the hypotenuse using the given
vertices as guides. Then encase the square on the hypotenuse with a
larger square as shown.

What steps can be taken to find the area of
the square on the hypotenuse? Find the area
of the outside square and subtract the area of four
triangles.

Once you know the area of the square on
the hypotenuse, how can you find the
length of the hypotenuse? Take square root of
the area of the square. This will give the length of the
side of the square.

EXPLORE 1
Pairs/ Individual

SP2
Two Right Triangles

• Students find the area of the square on the hypotenuse for the small
triangle and then find the length of the hypotenuse by taking its square
root. Ask questions to guide computation as needed.

What is the area of the large square? 49 sq units The area of each
triangle? 6 sq units The four triangles? 24 sq units The square on the
hypotenuse? 25 sq units

How are the areas of the squares on the legs related to the area of
the square on the hypotenuse? 9 + 16 = 25.

• Students find lengths of sides and areas of squares on the sides for the
larger triangle. Note that the hypotenuse of the larger triangle is a nonintegral
square root. This length may be left in square root form, or it
may be estimated.

SUMMARIZE 1
Whole Class

SP2, OH2
Two Right
Triangles

• Invite students to explain their work and calculations on the overhead or
board, leading them to conjecture the Pythagorean theorem based on
two numerical illustrations .

What appears to be a relationship between the area of the square
on the hypotenuse and the areas of the squares on the two legs? In
a right triangle, the area of the square on the hypotenuse appears to be equal to the
sum of the squares on the two legs.

• Explain to students that this conjecture is among the most well known
mathematical theorems. It was discovered thousands of years ago, and
it is called the Pythagorean theorem. The Pythagorean Theorem was
understood to varying degrees in many ancient civilizations. Ancient
Babylonians recorded many Pythagorean triples on stone tablets
between 1900 and 1600 BCE. Although the theorem as we know it
today is commonly attributed to the Greek philosopher and
mathematician Pythagoras (who lived during the 6th century BCE), the
earliest known formal proofs of the Pythagorean theorem and its
converse are recorded in Euclid’s Elements. (References: 1. Boyer and
Merzbach, A History of Mathematics, Wiley 1991, 2. Wikipedia)

INTRODUCE 2
Whole Class

 SP3-4
Pythagorean
Theorem (Part 1)

R1
Pythagorean
Theorem Cut Ups

• Lead students through a cut-up proof of the Pythagorean theorem.
Show how each of the squares was constructed using side lengths
from the right triangle. Label some right angles and lengths.

What does it mean to say that two shapes are congruent? They have
the exact same shape and size.

Are the two large shapes congruent? Yes. How do you know? They
are both squares with a side length of a + b.

How do their areas compare? They are same.

• Write the area inside each triangle, rectangle, and square.

• Cut out both squares, and cut them into the smaller polygons.

• Arrange two triangles to form a rectangle.

What does this mean geometrically? The area of two triangles is the same
as the area of one rectangle. algebraically?

• Separate the cut up pieces into two piles, keeping dissected pieces from
each square together. Ask students to state an equation that shows
that the sum of the area of the shaded pieces is equal to that of the
unshaded pieces. Record the equation on the board.

What equation is illustrated?
 

• Have students rearrange pieces so that the pieces with equal area are
clearly identifiable.

What simplified equation does the picture suggest?
 

EXPLORE 2
Individual/Pairs

SP3-4
Pythagorean
Theorem (Part 2)

• Students answer questions that lead them to record for themselves this
common proof of the Pythagorean theorem. Circulate as students work,
giving reminders and hints only if needed.
SUMMARIZE 2
Whole Class

SP5, OH
Pythagorean
Theorem (Part 2)

Math Background 1

• Ask individuals or pairs to come to the overhead to explain the different
parts of the problem.

• Congratulate students for proving the Pythagorean theorem.

What is the Pythagorean theorem? For a right triangle, the sum of the
squares of the lengths of the legs is equal to the square of the length of the
hypotenuse.

• Using one triangle and the three squares, arrange the pieces to show
that the sum of the squares on the legs of the right triangle is equal to
the square on the hypotenuse. Remind students that this was their
earlier conjecture and it is the Pythagorean theorem.

PRACTICE
Individuals

SP6-7
Pythagorean
Theorem Practice

Teaching Tip 2

• This group of problems uses both the Pythagorean theorem and its
converse. Use for additional practice or homework.
EXTEND
Whole Class
Math Background 2
• Share the Garfield proof of the Pythagorean theorem with students if
desired.
CLOSURE
Whole Class

 SP1, OH1
Ready, Set

• Review the goals and standards for the lesson.

SOLUTIONS

SP1-Ready Set Go
1. 48 sq. units
2. xy sq. units
3. 24 sq. units
4. 1/2xy sq. units.
5. a+a=2a
6. ab+ab=2ab
7. (1/2)a+(1/2)a = a
8. (1/2)ab+(1/2)ab=ab
 

SP2—Two Right Triangles
1. 3, 4
2. 4, 7
3. 9, 16
4. 16, 49
5. 25, 65
6.
7. In a right triangle, the area of the square on the hypotenuse is equal to the sum
of the areas of the squares on the two legs.

SP5—Right Triangle ABC

3. In a right triangle, the square of the length of the hypotenuse is equal to the
sum of the squares of the lengths of the legs.
4. Pythagorean theorem

SP6-7—Pythagorean Theorem Practice
1. 9 + 4 = 13; 32 + 22 = ( 13)2
2. 4 squared + 5 squared does not equal 9 squared. 16+25 does not equal
81.
3. 13
4. no
5. 3-4-5, 6-8-10

6. only the 6-8-10 triangle is a right triangle, by the converse to the Pythagorean
Theorem. The 4-6-8 triangle is not a right triangle, by the Pythagorean theorem.

7. This cannot be right because of the converse to the Pythagorean Theorem. The
given triangle is not a right triangle. Note: The Pythagorean Theorem itself does not
justify Tommy’s answer. The converse to the Pythagorean Theorem goes beyond
that, saying that Tommy’s answer also cannot be a lucky guess

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