Lampshade Mathematics

A new series timed for today’s changing needs!• Delivers rich, rigorous content to ALL students • Incorporates the College Board Standards for College Success • Integrates the American Diploma Project Mathematics Benchmarks • Applies state-of-the-art technology, including lessons using TI-nspire™ handheld 1-800-334-7344 glencoe.com • glencoe.com/catalog com arwin’s adaptation of a well-known Chinese proverb to mathematics may parallel the feelings of futility that many students may have regarding mathematics. Matthews and Greg Gross A mathematician is a blind man in a dark room looking for a black cat which isn’t there. we take several geometric concepts—similarity. the line segments BD and AF are the radii of the bases. Inc. The solution to this problem is complicated by the size and shape of the material from which the lamp shade is cut. We simplify the problem by restricting 332 Mathematics Teacher | Vol. Each customer specifies the height and the radii of the two circular bases of his or her lamp shade. One response to these feelings is for mathematics teachers to show students the beauty of mathematics through realistic and intriguing projects. and arcs and sectors of circles—and integrate them in a context that encourages students to apply the mathematics they have learned. This project explores how to create. right triangles. Using a lamp shade. 5 • December 2008/January 2009 Copyright © 2008 The National Council of Teachers of Mathematics. AF. from a rectangular piece of fabric. —Charles Darwin Illuminating the Mathematics of D iStockphoto.Lamp Shades Creating lamp shades to specific design parameters allows rich explorations in the mathematics of circles and triangles. The challenge is to discover how to cut material into a lamp shade pattern given the lengths BD. 102. and AB. a lamp shade according to specific design parameters. Michael E.nctm. . www. All rights reserved. No. In figure 1. We introduce this problem by having students imagine a specialty design company that creates lamps. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.org. and AB is the height of the shade. E Making the connection between what the F customer gives the designer (BD. the class C F GB R GB +h A determines that the net a portion of = rh = h →appropriate R i GB = r i is GB + rh → GB r R R – r a sector of a circle.G C B D G C B D Photograph by Michael Matthews. label the lamp G shade as in figure 1. Note that students do not draw figure 3 with the labels C or C′ and F or F′ until after they have discovered the appropriate locations G for points C and F (see fig. As Fig. 1 Lamp shade with relevant quantities labeled Y Making a truncated cone to visualize point G’s location C F A E D E Y X Z Z X Fig. something like that shown in figure 3. G F' we usually review the relationship between arc C C ' as length and the corresponding central angle. we 2 ask them to try to describe which quantities in the  rh  2 F net would theA shape of the lamp shade. wrap these nets up into a truncated cone (lamp shade). all the students use labeled sample nets to solve these problems. and D m∠CGC′) is a challenging spatial reasoning problem. No. These tasks challenge even those students who were able to find point G easily. 4). thus narrowing the options of possible lamp shades but not excessively so. Recognizing that point G is above the lamp shade is critical (see fig. Also note that points F' F' r we give students is to ask them The first task GC . 3) would G occur on the actual lamp shade (see fig. GF. Eventually. 102. illustrated in figure 2 and the equation below: measure of arc YZ G measure of angle ZXY = circumference 360 B C students describe and label the quantities that affect the shape of the lamp shade (the measures of GC. we use a “prototype” net F Don the front board to facilitate the discussion. Although some C Y C' students are able to visualize point G’s location. C ' At this point. 1). and the F formula for the circumference of a circle. 5). and ∠CGC′). We also provide string-and-pencil compasses to help students create the larger circles needed during the project. Once the students discover the net. 5 • December 2008/January 2009 | Mathematics Teacher 333 CF = h2 + R – r 2 → GF =  rh  2 + r 2 + h2 + R – r 2 B . F r nets produce a shape similar to that of a party hat) are discovered and rejected. Next we challenge students to locate points A–F on a lamp shade’s net. the concept C C' of AA similarity. Inevitably. 2 The ratio of the measure of arc YZ to the circle’s circumference is the same as the ratio of the measure of angle YXZ to 360°. the concept that F corresponding angles formed when parallel lines F' are cut by a transversal are congruent. others eventually make sample nets. A E GF. 3 The net of the lamp shade R GC = affect + r   R − r  G C B D F' F C B C' A D ( ) ( ) E G F A E Vol. To proceed successfully. to discover which basic net (two-dimensional shape) produces a lamp shade. AF. all rights reserved C F B A D E F A E F C B A B D E Fig. we ask students to shed some light on F where point G on the net (from fig. and AB) and what is needed to draw theC net (GC. the concept that corresponding lengths of similar triangles are proportional. C Rectangles GF . and finally unwrap the lamp Z shade to see where these labels go when the shade is X flattened back into net form. Further. the size of the paper to 18-by-12-inch construction paper. We allow time for h B and trapezoids (whose exploration. students need to be familiar with the following mathematical ideas: the Pythagorean theorem. of circle GCC 360 ′ 36 6 0 2 π GC Y   rh as “What can you say about the measures of angles measure e of arc CC ′ m 2π r 2 ∠CGC ′ GC =  + r2  →  rh=  R − r   2o 2 GCB and GFA?” to guide students to see the AA G2 → circum. using the Pythagorean theorem again.25 lengths.   GC. circumference 360 F C D GC A. R the length AF . 6). 334 Mathematics Teacher | Vol. A and B are not really located on the net. We eventually ask leading questions. such following equation): 2 circum. and a circle’s central angle G C' to this project. Some students do not readily think 20 of this relationship. The students G 11. of GF circle ′ CF = h2 + R –r = GCC + r + h + R –r 36 6 0 2 π GC   R– r  similarity. of circle GCC 360 ′ 36 6 0 2π GC known quantities. realization that the radii needed to make the net = GB 360 GB + h circumference =  → R i GB = r i stuGB + rh → GB (GC and GF) are not the same as the radii of the Then. F Z the problem becomes finding R − r   2 generalized formulas for GC. G Fig. Thus. 2 20 2 2 Z  rh  2 2 Using these similar triangles. students then come CF = h2 + R – r → GF = +r + h + R –m r ∠CGC ′ measure e of arc CC ′  m ∠ CGC 2 π r ′  X  up with a general formula for finding GB from the =R – r  o → = → 20 circum. we usually re20 label some quantities. they could then use the Pythagorean students 2 discover with little 2 guidance that  rh  F Usually. as shown here: rh  solution GF . 5 Students are challenged to locate points on the net and relate them to the lamp shade.  R– r  2 GB GB + h rh measure of GB. 2 r R R –r We generally encourage students to find GC first. F A E E Z Fig. For instance. 3 and 4). 5 • December 2008/January 2009 . r R 2 top and bottom circles of the lamp shade (compare dents geta general for GC. C B D GF . 25 circum. 2+ rh → GB = = → R i rh GBCF = r i2 G B = h + R – r for →R GF =  + r2 theorem to calculate the length of GC. 2  rh  D 2 2 2 CF = h + R – r → GF = + r + h + R–r They usually notice that if they could discover the The students’ next challenge is  to find GF.GC. 7 A portion of the net may help in generalizing relationships. circumferences.25 20 F E ( ) ( ) ( ) ( ) ( ( ) ( ( ) ( ) ) ( ) ( ) ( ( ) ( ) ( ( )) ( ) ( ) ) ( ) C B D G E C r h B C A F B E A D 11. measure of arc YZ measure of angle ZXY = circumference 360 GF . the m∠CGC ′ measure e−of arc m∠ CGC 2π r ′ ′ 7 and C = = → C' → o and GAF. One way can find a generalized solution CF r to Rthey – r(and GC = +r F' R– r     r find GB comes from the similarity of triangles GBC thus GF )R once GC CC is known (see fig. 102. Sometimes we need to collapse figure 4 to help students see the relationship (see fig. R − r   = → R i GB = r i GB + rh → GB = GF . a CC general measure 20 = formula → = of ∠CGC → m∠CGC ′ = o  11 . GF.25 F F E C C B D F C' G Fig. we might let r The students’ subsequent task is to come up measure ewith of arc m∠ CGC ′for the 2π r m∠CGC ′ ′ ′ represent the lengths BD and BC. 4 Realizing that point G exists helps students visualize F R A the problem.= A GC +E r2 E figs. at times we do a quick side 20 activity to help them rediscover it. and m∠CGC′ from  rh  X + r2 GB GB + h GC =  rh  the given customer-specified lengths. 20 the proportional relationship involving arc apply F' 11. ofon circle 360must 36 6 0 2π GCStudents and h the G length AB: the GCC basis′ of the known values. 6 A collapse of the image in figure 4 D F C F C G E D C' C' C r B F' F' F h R A Fig. For convenience. No. The next GB GB + h rh = → R i GB = r i GB + rh → GB = big “aha!” moment comes with the nonintuitive Y r YZ R R–r measure of arc measure of angle ZXY GC. we occasionally use leading questions  R − r  and physical models to shine light on some of the 20 On the basis of these general formulas. Using this example. and h) given in table 1 produce lamp shades exciting. an important skill approximately 7 3/4 inches and 4 1/2 inches. h 6. and rewarding. and challenging nature of this project. the corners too dark to see clearly. We believe that projects 161°. all rights reserved .500 8. decrease the amount of wasted construction paper by GREG GROSS teaches geometry and precalculus placing point G off the construction paper. For instance. The lamp shade project has several benefits: GF = 320 inches + 511. disengaged students or her compass at the center of the 18-inch side of become enthusiastic about mathematics. quickly of arc CC r R R–r Then. MATTHEWS teaches preservice different courses.5 4. However.25 GB GB + h rh = see that → they R i need GB =the r i measure GB + rh → GB = ′. 2 20 +r  As their guides. create circles with radii of improve their spatial reasoning. variations of the solution Solving.25 project. shades …”—show that they enjoy the hands-on.125 1.5 2.500 0. and students the construction paper.. if r = 2 inches.5 and 2 inches.000 0.0 4. GC = 320 inches.75 2. that can be made from 18-by-12-inch sheets.750 7.r 0. and m∠CGC′ ≈ Students see the blending of mathematical topics 161°. 102.000 1. and for successful nonstandard problem solving (Presmeg then cut out a sector along an angle measuring and Balderas-Cañas 2001). not + r   R − r   yet in the repertoire of geometry students.625 0. and exploratory. No.measure of arc YZ measure of angle ZXY = Table 1 circumference 360 Dimensions (in inches) of lamp shades made from 18-by-12-inch construction paper ure of an ngle ZXY GC . Norma C.5 10. minimizing 36 60 2π GC  rh  2 better approached by using the tools of calculus. 4 (2001): 289–313. sometimes students teachers at the University of Nebraska at Omaha. of circle GCC ′ 360 classes.50 3. while others are practically cylinders. That is.25 inches.125 1. students typically want to try a few yeah. Informal comments design company can now create any lamp shade to from students—“Wow. R = 3. rh r i GB top + rh → GB =  circle 2 GC = R –+ rr  they realize r  R −that  the measure of arc CC′ in the net equals the circumference of the smaller circle in the lamp shade (or 2pr). worthwhile. circles with radii 3. Studying the no.0 ( ) ( ) ( ) ( ) ( ) 2 2  rh  2 360 r measure of arc CC 2π r m∠CGC ′ ′ – r m∠CGC ′ + re + h2 + R   = → = → m∠CGC ′ =  R– r o  waste is a question 2 circum. R. Balderas-Cañas. then GB = 4 inches. ∞ spectrum of variations of this problem brings out mathematical nuances that may be appropriate for MICHAEL E. all rights reserved 2 F= ( ) ( ) () A variety of student results ( ) () Vol. h = 3 inches. With this realization. The dimensions that they learn to see mathematical investigations as (r. For example. Moreover. some flare out wide (like mega­ Presmeg. 11. “Visualization and Affect in Nonroutine Problem As with all rich problems. trigonometry and so may be appropriate only for some Photographs by Colina Matthews and Jill M. a student could place his typically taught separately. I never knew that so many a customer’s specifications.0 1. students engage in some 2 360 36 60o 2π GC  rh  intricate mathematics as they develop their solutions. they generally discover that CC′ wraps around and becomes the 2 rh (or smaller) of the lamp shade. Both are discovering point G’s ideal placement requires interested in novel problem solving and spatial visualization. given time to think about it. These produce some distinct variations in the shapes of REFERENCE lamp shades. this is just like the time when we cut those examples to verify that the formulas really work. This construction would create a net for a like this one can help students see mathematics as lamp shade that is 3 inches high with openings for more than just a futile search in the dark—indeed. at Blair High School in Blair.” Mathematical Thinking and Learning 3.25 3.0 1. phones). 2 the students substitute appropriate 2  known rh  values 2 CF the = proportion h + R – r to→ GFup = with + r 2 + h2 + R – r into come the  r  R–  following general formula for the measure of ∠CGC′: 5. and the problem itself are plentiful. 5 • December 2008/January 2009 | Mathematics Teacher 335 Photograph by Michael Matthews. Nebraska.0 2.5 3. 360 r ∠CGC ′ 2π r m∠CGC ′ → = → m∠CGC ′ = In the lamp shade project.5 inches.50  360 R 4. Gross.50 GF . At this point in the different topics could go together like this” and “Oh. and Patricia E.000 1.