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 6^th 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