Lesson Plans

Picking Pythagoras

Prep for Teachers

Triangle Activity Packets
1. Make one copy of the “Triangle Strips Organizer” for each group of 2-3 students in your class. Each “Triangle Strips Organizer” will become a Triangle Activity Packet.
2. Measure the strips and make sure they are 3 inches, 4, 5, 6, 8, and 10 inches.
3. Cut the strips from the page, making sure their measurements are fairly exact as it's important for the “Exploring Triangles Activity.”
4. Bind each set with a paper clip.

Make a class set of right triangles on grid paper. This is available at Nova's Pythagorean Puzzle Web site at http://www.pbs.org/wgbh/nova/proof/puzzle/papertriangle.html.

Make sure all organizers are copied and Web sites are bookmarked.

When using media, provide students with a FOCUS FOR MEDIA INTERACTION, a specific task to complete and/or information to identify during or after viewing of video segments, Web sites, or other multimedia elements.

Review the definition and classification of triangles with the class. Triangles are closed three-sided polygons. Triangles can be classified by side (scalene, isosceles, equilateral) or by angle (acute, right, obtuse).

Tell the class that they will now do an activity that explores the importance of the sides of a triangle in determining its angle class. Break the class up into groups of two or three each. Distribute Triangle Activity Packets and the “Exploring Triangles Activity” sheet to each group. Explain that they will follow the instructions on the sheet and record data on the organizer table. Students will select three strips at a time from the packet and try to make triangles. Not all combinations of three strips will create triangles. The organizer contains a table that records the measurements of each combination, if a triangle can be formed using these three strips, and if so the type of the triangle's largest angle.

After they complete the activity, have groups report their findings. Elicit what logic or pattern they used in selecting the three strips. (Some groups may have tried all possibilities with two sides being constant, e.g. 3-4-5, 3-4-6. 3-4-8. and so on.)

Ask students to compare the sides that formed triangles with the sides that could not form triangles. Does any pattern or conclusion emerge? (For those sides that formed triangles, the sum of the two smaller sides always is greater than the longest side.)

Can they use this fact to predict whether any three sides will form a triangle? (Yes.)

Elicit three sides not on the table that will form a triangle and three that will not. (Answers will vary.)

Ask students to compare the sides that created obtuse angles with those that created acute angles. Does any pattern or conclusion emerge? (Generally, the sum of the two smaller sides was closer to the longest for obtuse triangles than acute triangles.)

Can they use this to predict whether three sides will form an obtuse or acute triangle? (No. From the table, a 4-5-6 triangle is acute but a 5-6-8 is obtuse. In both cases, the sum of the two smaller sides is three more than the longest.)

Compare the sides that formed right triangles. Does any pattern or conclusion emerge? (No.)

Side measurement greatly influences the angle class of a triangle, but how can students use them to predict triangle classification?

Step 1:

Students will learn the Pythagorean Theorem and how to use it to predict the angle class of a triangle. Review squares and squaring with the class. A square is a closed four-sided polygon with right angles and equal sides. Squaring is a process of multiplying a number by itself to produce a multiple. Squaring integers produces multiples called perfect squares.

The ancient world knew something about right triangles. The Egyptians, Babylonians, and Greeks knew that if you drew a square on the short side of a right triangle, a square on the medium side, and another square on the longest side, the two smaller squares added together would equal the third. To the Greeks, drawing a square meant squaring a number. Using our definition of squaring, we can write what the ancients knew in the following sentence:

short side² + medium side² = long side²

Elicit from the class the sides of a triangle from the “Exploring Triangles Activity” that produced a right triangle. (3-4-5)

Tell students that they can write a mathematical sentence that shows the relationship of the sides according to ancient wisdom:

32 + 42 = 52 or 9 + 16 = 25

Note that the sum of the smaller squares equals the larger square.

Although this fact was well known, no one proved it until Pythagoras, a Greek mathematician and philosopher who lived around 560 BC.

Break the class into groups of two or three each. Distribute the Nova: Demonstrate the Pythagorean Theorem worksheet containing a right triangle on a grid. Also distribute the class set of scissors and rulers. Have the class log on to Nova: Demonstrate the Pythagorean Theorem Web site at http://www.pbs.org/wgbh/nova/proof/puzzle/theoremsans.html. Provide a FOCUS FOR MEDIA INTERACTION, telling the class to follow the directions on the Web page starting with Step 3. They do not have to cut out the triangle on the Nova handout sheet.

When groups are finished, have them report their findings. (Students will draw squares using the side measures of a 3-4-5 triangle. They will cut them out and fit the smaller squares on the larger square, thereby proving the Pythagorean Theorem.)

Tell the class that because Pythagoras showed the ancient world how the sides of a right triangle relate to each other, the ancient fact became known as the Pythagorean Theorem. If we use A to represent the shortest side, B to represent the medium side, and C to represent the longest side, we can write the theorem in modern form like this:

A² + B² = C²

Does this work for all right triangles? Have the class log on to International Education Software Pythagorean Theorem #1 at http://www.ies.co.jp/math/Java/geo/pythasvn/pythasvn.html. Provide a FOCUS FOR MEDIA INTERACTION, telling them to determine if the Pythagorean Theorem works for three different right triangles of their choice. Have students note that the site is Japanese and they use A for the longest side instead of C. (Students will create a right triangle by moving a red dot. Squares on two of the triangle's shorter sides have been cut into polygons. The polygons inside the squares change as the shape of the right triangle changes. When students define a right triangle they want to work with, they try to fit the polygons into the largest square by moving them. In each case it works, thereby demonstrating that the two smaller squares added together equal the largest.)

Elicit from the class how they would use the Pythagorean Theorem to predict if three sides can form a right triangle. (If the sum of the two smaller squares equals the largest square, the triangle is a right triangle.)

Elicit from the class another right triangle from the “Exploring Triangle Activity.” Ask them to test the sides using the Pythagorean Theorem.

Step 2:

Students will learn that squaring the sides of a triangle will also predict obtuse and acute triangles.

Elicit from the class the side measurements of one triangle that is obtuse and another that is acute. (Answers will vary. For our purposes, use 4-5-6 for acute and 5-6-8 for obtuse.)

If students square the sides, would they expect the sum of the two smaller ones to equal the largest? (No, they are not right triangles.)

Ask students to try squaring the sides of each triangle and then comparing the sum of the two smaller squares to the largest square. (42 + 52 > 62 or 16 + 25 > 36, and 52 + 62 < 82 or 25 + 36 < 64)

What theory can students make from squaring the sides of obtuse and acute triangles? (If the sum of the two smaller squares is less than the largest, the triangle is obtuse; if greater, then it is acute.)

Before students can confirm this conclusion, have the class test it on other obtuse triangles from the “Exploring Triangle Activity.” For other acute triangles, use an equilateral triangle with sides that measure 5.

Elicit from the class which obtuse triangles they tested and if squaring accurately predicted their classification. (Students should conclude yes.)

Ask students to complete the sentence, "To determine a triangle's classification by angle, we..." (...square the sides, add the two smaller squares and compare the sum to the largest square. If equal, then it is a right triangle; less, then it is obtuse; greater, then it is acute.)

Distribute the “Triangle Classification Organizer” and have students work either individually or in groups. When finished, CHECK answers. (Students will be given four activities: first they will test side measurement to determine if a triangle can be made; second they will use side measurement to determine triangle classification by angle; third, they will combine both skills; and last, they will be given a triangle's classification, the measures of the two shorter sides, and asked to determine possible third sides.)

Step 3:

Students will use the Pythagorean Theorem to find the measure of missing sides of a right triangle. Review square root and the parts of a right triangle. Square root is one of two equal factors of a number. Some square roots are irrational and should be rounded off. Right triangles have two legs and a hypotenuse. The hypotenuse is the longest side and can always be found opposite the right angle. In the western world, we use C to represent the hypotenuse in the Pythagorean Theorem.

Tell students that in the “Triangles Classification Activity,” they were asked to find the largest side of a right triangle given the two shorter sides. The Pythagorean Theorem can be used to find any missing side by using it and solving it like an equation. If the shorter sides are 5 and 12 and they were asked to find the largest side (hypotenuse) they would set the problem up as follows:

1. Substitute	A² + B² = C² or 52 + 122 = C²
2. Calculate	25 + 144 = C²
3. Combine	169 = C²
4. Square root	¯169 = ¯C²
5. Answer	13 = C

Notice that square root undoes squaring. They are opposite processes.

Distribute the class set of calculators and demonstrate the square root key.

Have students find the hypotenuse of a right triangle that has legs of 7 and 24. (25)

Sometimes you are asked to find a missing leg rather than the hypotenuse. If one of the legs is 6 and the hypotenuse is 10, students can find the missing leg using the Pythagorean Theorem as follows:

1. Substitute	A² + B² = C² or 62 + B² = 102
2. Calculate	36 + B² = 100
3. Balance	36 - 36 + B² = 100 - 36
4. Subtract	0 + B² = 64
5. Square root	¯B2 = ¯64
6. Answer	B = 8

Have students find the leg of a right triangle that has one leg of 8 and a hypotenuse of 17. (15)

Sometimes square roots are not integers. When this happens, round off. The problem will usually tell you what place to round off. If the two legs of a right triangle are 10 and 12, we can find the hypotenuse to the nearest tenth as follows:

1. Substitute	A² + B² = C² or 102 + 122 = C²
2. Calculate	100 + 144 = C²
3. Combine	244 = C²
4. Square root	¯244 = ¯C²
5. Answer not rounded	15.6204... = C
6. Answer rounded	15.6 = C

Have students practice finding the missing side. Include figures. Here are some examples:

Find the missing side. Round answers off to the nearest tenth.
1. A = 10, B= 24, C= ?
2. A = 7, B = ?, C = 20
3.

Step 4:

Students will learn the process of triangulation to solve problems. By cutting polygons into right triangles, the Pythagorean Theorem can be used to find missing information.

Break students up into groups of two or three each. Have them log on to the Geometry Center of the University of Minnesota at http://www.geom.umn.edu/~demo5337/Group3/bball.html. Provide a FOCUS FOR MEDIA INTERACTION by telling them to solve the problems about baseball using the Pythagorean Theorem to determine the distance a player has to throw to reach certain points on the field. When groups are finished, have them report their answers. (The page will ask them how far will a catcher have to throw to second base and how far will a third basemen have to throw to first base. If they divide a baseball diamond in half, they get two right triangles with legs of 90 and 90. Rounded to the nearest tenth, the distance both players will have to throw is 127.3 feet.)

Draw the baseball diamond on the chalkboard and ask how groups cut the diamond to get right triangles. Tell them such a process is called triangulation and is frequently used to find missing information.

Students will use triangulation to find the perimeter of polygons on the coordinate plane. Review with the students how to graph points on the coordinate plane. Also, define perimeter (the sum of the linear measures of a polygons sides).

Break the class up into groups of two or three each. Distribute the Triangulation Sampler Organizer, graph paper, rulers, and class set of calculators. Tell students they will calculate the perimeter of the polygon drawn on the coordinate axes.

On the organizer, have students write the coordinates of points A, B, C, D, and E. [(1,4), (3,8), (5,9), (8,7), (7,4)]

What type of polygon is ABCDE? (Pentagon.)

Elicit the length of side AE. (6)

Ask if the measure of AB is 4. (The line is slanted and is greater than 4.)

Point out the creation of Right Triangle I. How can we use the right triangle to find the length of AB? (Pythagorean Theorem.)

What are the lengths of the legs of Right Triangle I? (2 and 4)

Calculate the length of AB to the nearest tenth. (4.5)

Point out the right triangles (II, III, and IV) that were created to find the length of the other sides. In all cases, the Pythagorean Theorem can be used to find the length of the missing side.

Ask the class to find the length of BC, CD, and DE. (2.2, 3.6, 3.2)

What is the perimeter of ABCDE? (6 + 4.5 + 2.2 + 3.6 + 3.2 = 19.5)

On the first sheet of graph paper, ask the class to graph and draw the polygon with the following vertices: A (1,6), B (4,7), C (7,2), D (2,-3), and E (-3,1).

Have the class use triangulation to find the perimeter of a polygon with coordinates AB = 3.2, BC = 5.8, CD = 7.1, DE = 6.4, EA = 6.4, perimeter = 28.9)

Have each group write five coordinates representing the vertices of a polygon and give it to another group. Groups will graph and draw the polygon on the second sheet of graph paper, then find the perimeter using triangulation as before.

SOCIAL STUDIES
Have students research the life of Pythagoras and write a short biography.

ART AND MATHEMATICS
Have students create a Pythagoras tree. Students should log on to the International Education Software Web site at http://www.ies.co.jp/math/Java/geo/pytree/pytree.html and follow the instructions for building a tree.

MUSIC
Research Pythagoras' influence on musical scales.

SOCIAL STUDIES AND ART
Have students create their own pyramid using scaled down measurements of Egyptian pyramids. At Nova Online Adventure http://www.pbs.org/wgbh/nova/pyramid/geometry/model.html they can assemble a scale paper model of the great pyramids using an already prepared diagram available as a link on the page. Once students understand the process, the page gives two more pyramids which they must scale down and find missing measures before assembling.

Have a master carpenter speak to the class on how carpenters use the Pythagorean Theorem in building.
Have an architect speak to the class on how the Pythagorean Theorem is used to build effective ramps for highways and buildings.