Middle Grade Mathematics Task

Proving the Pythagorean Theorem The Pythagorean Theorem is one of the most well known and important theorems in all of mathematics. It was named after a Greek mathematician and philosopher. However, there is evidence to suggest that the Babylonians knew about this relationship some 1000 years earlier. The Chou-pei, an ancient Chinese text containing a table of Pythagorean triples also shows that the Chinese were aware of the relationship between the sides of a right triangle long before Pythagoras. It was, however Pythagoras, or one of his colleagues in the Pythagorean society, that proved the theorem and so it was named after him. Since the time of Pythagoras, there have been many proofs for this theorem. Here we will examine several. Pythagorean Theorem: A triangle is a right triangle if and only if the square of the hypotenuse is equal to the sum of the squares of the shorter two sides. 1) Stockbroker’s Proof In 1830, Henry Perigal, a London stockbroker, discovered a way to indicate the truth of the Pythagorean Theorem using paper and scissors. • • • • • • Using a straight edge, draw a right triangle. Label the longest side of your right triangle c and the other two a and b. Draw a square off of side a. Draw a square off of side b. Using your protractor (to ensure right angles), draw a square off of side c. What is the area of each of these squares? _________ _________ _________ (Label the areas) • • Find the center of 𝑏 2 , mark this point w. Using your protractor, draw a line segment through point w that is perpendicular to side c of your triangle. Draw a line segment that passes through point w that is perpendicular to the line segment you just drew. (Square 𝑏 2 should now be broken into 4 pieces) • • • • Cut out all three of your squares (𝑎2 , 𝑏 2 , 𝑎𝑛𝑑 𝑐 2 ) Cut square 𝑏 2 into 4 pieces using the line segments that you have drawn. Using squares 𝑎2 and the 4 pieces from square 𝑏 2 , construct a square congruent to square 𝑐 2 . (Hint: you may want to lay the pieces on top of square 𝑐 2 .) 2) Pythagoras’s Proof Pythagoras lived in the fifth of sixth century B.C. and founded the Pythagorean School in Crotona where students studied philosophy, mathematics, and natural science. Pythagoras and his colleagues are credited with many contributions to mathematics. • Draw two squares of the same size. • On Square 1, draw a vertical line segment so that you split the top and bottom sides of the square into two unequal pieces. Label the shorter piece a and the longer piece b. Next, draw a horizontal line segment so that you split the right and left sides of the square into two pieces of size a and b. • • ➢ What is the length of one side of the square? _______________________________________________________________ ➢ What is the area of the square (express using an exponent)? _______________________________________________________________ ➢ What are the areas of each of the four pieces that your square has been cut up into? ______________ ______________ ________________ _______________ ➢ Find the sum of these four areas. ________________________________________________________________ ➢ Write an equation representing the relationship between the area of the large square and the area of its parts. ________________________________________________________________ • • • On Square 2, split each of the four sides into segments of lengths a and b and mark this point. Be sure to alternate between a and b so that no two a’s will be consecutive and no two b’s will be consecutive as you move around the square. Connect the top point with the point on the left side of the square. Label this segment c. Connect the left point with the bottom point. Connect the bottom point with the point on the right side of the square. Connect the point on the right side of the square to the top point. ➢ What is the length of one side of the square? _______________________________________________________________ ➢ What is the area of the square (express using an exponent)? _______________________________________________________________ ➢ What are the areas of each of the five pieces that your square has been cut up into? ___________ ___________ ____________ ___________ ____________ ➢ Find the sum of these five areas. ________________________________________________________________ ➢ Write an equation representing the relationship between the area of the large square and the area of its parts. ________________________________________________________ • Using substitution, set the equation found for Square 1 equal to the equation found for Square 2 and simplify. 3) President Garfield’s Proof President Garfield discovered this proof 5 years before becoming president of the United States. He came up with this proof while having a mathematics discussion with some of the members of congress. • • Using a straight edge, draw a right triangle. Label the longest side of your right triangle c and the other two a and b. • Extend side a by b units (in the direction that is not by the right angle). You should now have a segment that is (a+b) units long. Construct a new right triangle, with the same dimensions as the first, using your newly drawn segment b. (Be sure that the segments labeled a do not touch.) Connect the two vertices of the two triangles that are currently not connected (not the right angles). Note that you have now created a third right triangle. Use your protractor to verify. • • • What shape have you created? ________________________________ 1 ➢ So, knowing that the area of a trapezoid is 2 (𝑏𝑎𝑠𝑒1 + 𝑏𝑎𝑠𝑒2 )𝑥 ℎ𝑒𝑖𝑔ℎ𝑡, find the area of the trapezoid you have drawn. (Remember the height of the trapezoid is the distance between the two parallel bases.) _______________________________________________________________ ➢ What are the areas of each of the three triangles that your trapezoid has been cut up into? __________________ __________________ ___________________ ➢ Find the sum of these three areas. ________________________________________________________________ ➢ Write an equation representing the relationship between the area of the trapezoid and the area of its parts. ➢ ________________________________________________________________ 4) Proof of the Pythagorean Theorem using Similar Triangles • Start with circle O having diameter GH and radius of length c. Point F lies of the circle so FO is of length c. Drop a perpendicular from F to point K on the diameter GH. Denote the length of FK as a and the length of KO as b. Joining points F, K, and O forms a ____________ triangle. For ∆FKO, the length of the hypotenuse is ___________ and the lengths of the legs are _________ and __________. F c a G • c b O H K Construct segments FG and FH. Since FK is perpendicular to GH (by construction), we note that both ∆FKH and ∆GKF are _______________ triangles. F c a G • c O b K H Note that angle HFG is inscribed in a semicircle, and as such, is a right angle. Therefore, the measure of angle HFG = ___________. Angles HFG and KFG share a common side FK and lie in a right angle; therefore, angle HFK is complementary to angle KFG. But, angle KFG is also complementary to angle FGK since the two non-right angles in a right traignle must have measure that sum to 90 degrees. Therefore, angle HFK must be congruent to angle FGK. It follows then that right triangles FKH and GKF are similar triangles. That is ∆FKG ~∆GKF. • • In ∆FKG, why is the length of leg HK equal to c-b? __________________________________________ __________________________________________________________________________________ In ∆GKF, why is the length of leg GK equal to c+b? __________________________________________ __________________________________________________________________________________ F a c-b G • c+b K H Since ∆FKG ~∆GKF, corresponding parts are proportional to one another. The triangles are shown below with the same orientation so that it is easier to see the proportional relationships. H F c-b F a K a G c+b • • K Write the proportional relationship between the two legs of the triangles above. Simplify algebraically. Does the result look familiar? p Principles and Standards for School Mathematics advocates an experimentation approach to middle-grades geometry study (NCTM 2000). Students are asked to explore and examine a variety of geometric shapes and discover their characteristics and properties using hands-on materials. They also create inductive arguments about the Pythagorean relationship. This empirical approach to the Pythagorean theorem, for example, will lay the foundation for analytical proofs. Incorporating experimentation as part of middle school geometry is also consistent with research on how students learn best. According to van Hiele, students need opportunities to develop their geometric thinking through five levels: (1) visualization, (2) analysis, (3) informal deduction, (4) deduction, and (5) rigor (Fuys, Geddes, and Tischler 1988; van Hiele 1986). Levels 2 and 3, analysis and informal deduction, are especially relevant in the middle grades. At the analysis level, students can go beyond their perception of shapes and analyze the components, parts, and properties of shapes. Students use descriptions instead of definitions. They also discover and prove properties or rules by folding, measuring, or using a grid or a diagram. At the informal deduction level, students learn the role of definitions, analyze the relationships between figures, order figures hierarchically according to their characteristics, and deduce facts logically from previously accepted facts using informal arguments. The activities presented here will help students make the transition from level 2, analysis, to level 3, informal deduction. In activities 1 and 2, middle-grades students explore the Pythagorean theorem by using jelly beans to estimate the areas of squares and semicircles. In activity 3, students use the result they obtained for squares and apply deductive reasoning to establish the result for semicircles on the Jelly Beans Jeong Oak Yun and Alfinio Flores 202 Mathematics Teaching in the Middle School ● Vol. 14, No. 4, November 2008 Copyright © 2008 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. iStockphoto.com The Pythagorean Theorem with sides of the right triangle. In activity 4, students use their discovery about the areas of semicircles and deductive reasoning to establish further relations among figure areas. THE JELLY BEAN STRUCTURE Activity 1 uses a cardboard mat that is constructed with sides to hold jelly beans. Figure 1 shows a right triangle and its corresponding squares. Figure 2 illustrates the mat with cardboard fencing to hold one layer of jelly beans. To construct the mat, draw the shapes by hand or with computer software and paste the colored figures on cardboard that is about the size of a shoe box. To make the fences, cut strips of corrugated cardboard that are about 2 cm wide. (We recommend cutting perpendicularly to the ridges Jeong Oak Yun, joyun@asu.edu, is a graduate student at Arizona State University and a teacher of public secondary school in Seoul, Korea. She likes to use dynamic models to help students understand mathematical concepts. Alfinio Flores, alfinio@udel.edu, is a professor of mathematics education at the University of Delaware in Newark, where he teaches mathematics and mathematics education courses. He likes to make mathematical concepts more tangible for students and help teachers develop a profound understanding of school mathematics. Fig. 2 The jelly bean structure (a) The mat’s outside fence is glued in place. (b) The triangular fence is added but not glued. Photographs by Jeong Oak Yun; all rights reserved Fig. 1 The mat (c) A layer of jelly beans covers the legs. Vol. 14, No. 4, November 2008 ● Mathematics Teaching in the Middle School 203 given different mats based on different shapes, such as isosceles right triangles (see fig. 2c) and scalene right triangles (fig. 3), and then compare results. Students could move from one station to the next, where the layout of each mat is different, and conduct a similar experiment. Photographs by Jeong Oak Yun; all rights reserved Fig. 3 A scalene right triangle ➞ Photographs by Jeong Oak Yun; all rights reserved Fig. 4 Jelly beans on the move (a) Slide the jelly beans from the squares on the two legs to the square on the hypotenuse. (b) Jelly beans cover the square on the hypotenuse. so that the strips can be bent easily.) Paste the strips on the mat along the outside of the figures with instant glue. Make the fence for the inner right triangle separately and do not glue it down, since it will be removed during the experiment. Estimating the area with jelly beans works well as long as students abstain from eating the candy because (1) jelly beans are easy to layer; (2) empty space, or gaps, around the jelly beans will be kept to a minimum; and (3) students like the bright colors. Although jelly beans are threedimensional objects, we are using one layer, which is two dimensional. In this activity, students will not have to count the number of jelly beans to compare areas. The activities are designed to allow students to move jelly beans from one section of the mat to another to see whether or not all the jelly beans fit into a new shape. There are several ways in which the activities could be conducted. The teacher can demonstrate the first activity and discuss the relationship between the areas of the squares. Students can also work with the mat in groups. Different groups could be 204 Mathematics Teaching in the Middle School ● Vol. 14, No. 4, November 2008 Activity 1: The Pythagorean relationship Students will use a mat that consists of a right triangle with squares on each of the three sides. With the cardboard fence glued around the three squares, students insert the triangular fence around the triangle (see fig. 2b). They should notice that a square is formed on each side of the right triangle. Students fill the two squares on the legs of the right triangle with jelly beans, ensuring that jelly beans completely fill the two squares without overlapping, forming just one layer (see fig. 2c). Next, students remove the cardboard triangle that is inside the frame and push all the jelly beans into the square on the hypotenuse (see fig. 4a). Finally, they re-insert the triangular fence and flatten the layer of jelly beans. Students verify whether the jelly beans completely cover the square on the hypotenuse (see fig. 4b). The teacher guides students to explicitly express in their own words what they found about the sum of the areas of the squares on the legs of the right triangle compared with the area of the square on the hypotenuse. Then students label the two legs of the right triangle as a and b, and the hypotenuse as c, and express the relationship using an algebraic equation: a 2 + b 2 = c 2. Activity 2: EXTENDING the Pythagorean theorem Students can also use jelly beans to explore the relationships between areas when other similar shapes are constructed on the sides of a right triangle. Students use a mat with Fig. 6 Extending the Pythagorean theorem for semicircles Photographs by Jeong Oak Yun; all rights reserved Fig. 5 Precise semicircles ➞ Photographs by Jeong Oak Yun; all rights reserved (a) Three semicircles surround a right triangle. (b) Three fences form three semicircles. three semicircles adjacent to each side of a right triangle (see fig. 5a). The diameters of the semicircles are the sides of the right triangle. With the cardboard fence around the three semicircles glued to the mat, students insert the triangular fence around the right triangle. They need to notice that the two fences together form three semicircles on the sides of the right triangle (see fig. 5b). When making the structure for students, be sure that the arcs are precise semicircles. In the same way as in the first activity, students pour one layer of jelly beans in the two semicircles on the legs of the right triangle, completely covering the two semicircles (see fig. 6). Then they remove the cardboard triangle that is inside the frame and slide the jelly beans into the semicircle on the hypotenuse. Next, they re-insert the triangular fence, flatten the layer of jelly beans, and verify whether the jelly beans completely cover the semicircle on the hypotenuse. Students can describe in their own words the relationship they see between the sum of the areas of the semicircles on the legs of the right triangle and the area of the semicircle on the hypotenuse. The teacher will also guide students to see that in each of the activities, the three shapes on the sides of the right triangle are similar to each other, that is, squares are similar to other squares, and semicircles are similar to other semicircles. These two activities correspond to van Hiele’s level 2, because the verifications are done empirically. Other Extensions Students can also experiment with different shapes constructed on the sides of the right triangle, as long as all three shapes are similar and their corresponding sides are placed on the sides of the right triangle (see figs. 7a and 7b). Of course, these extensions to the Pythagorean theorem in terms of similar shapes on the sides of a right triangle are not Vol. 14, No. 4, November 2008 ● Fig. 7 Other extensions of the Pythagorean theorem (a) An equilateral triangle is along each side. (b) A quarter circle is along each side. new (Flores 1992; Heath 1956; Pólya 1948), but students are usually surprised that the Pythagorean cc2 2 relationship holds true for shapes C C other than squares. aa2 2 Mathematics Teaching in the Middle School 205 a1 c1 c1 a1 t t Activity 3: Formulas for the areas of the semicircles and connections with algebra Students can derive the extension of the Pythagorean theorem with semicircles by using some algebraic skills, such as factoring and simplifying algebraic expressions. They can do this activity working in pairs or small groups. Students can denote the two legs of the right triangle as a and b, and the hypotenuse as c (see fig. 5a) and express the relationship among the areas of the squares on the sides of the right triangle as a 2 + b 2 = c2. They need also remember that the diameter of each semicircle is congruent to the corresponding side of the triangle and that the sum of the areas of the semicircles on the legs of the right triangle is equal to the area of the semicircle on the hypotenuse. The teacher can guide students to write algebraic expressions for the areas of the semicircles and for their relationships. Students can write an algebraic expression for each radius of the semicircles in terms of the length of each side of the triangle and write an expression for the area of each semicircle. They can use algebraic notation to state that the sum of the areas of the semicircles on legs a and b of the right triangle is equal to the area of the semicircle on xxxxxx the hypotenuse, c: 2 2 2 1  a 1  b 1 c × π+ ×  π = ×  π 2  2  2  2 2  2 Students then factor and simplify both sides of the equation to show that it is equivalent to the equation a 2 + b 2 = c 2. To verify, they can reverse all the steps; starting with a 2 + b 2 = c 2, they can show how to obtain from this equation the relation among semicircles. This activity corresponds to van Hiele’s level 3 because students make deductive arguments based on previously accepted facts. 206 Fig. 8 Crescents formed by semicircles c2 C a2 a1 c1 t A B Activity 4: Area of the crescents of Hippocrates Hippocrates of Chios, a Greek mathematician from the fifth century BC, discovered that two figures bounded by arcs of circles have the same area as a figure bounded by straight lines. In figure 8, semicircles are cons