teaching math

Teaching NotesSample Activities

Windows/Macintosh

Dynamic Geometry Softwarefor Exploring Mathematics

Teaching Mathematics with

Key College PublishingKey Curriculum Press

Teaching Mathematics with The Geometers Sketchpad

Limited Reproduction Permission 2002 Key Curriculum Press. All rights reserved. Key Curriculum Press grants the teacherwho purchases Teaching Mathematics with The Geometers Sketchpad the right to reproduceactivities and example sketches for use with his or her own students. Unauthorized copyingof Teaching Mathematics with The Geometers Sketchpad is a violation of federal law.

The Geometers Sketchpad, Dynamic Geometry, and Key Curriculum Press areregistered trademarks of Key Curriculum Press. Sketchpad is a trademark of KeyCurriculum Press. All other brand names and product names are trademarks or registeredtrademarks of their respective holders.

Key Curriculum Press1150 65th StreetEmeryville, California [email protected]://www.keypress.com

10 9 8 7 6 5 4 3 2 05 04 03 02 ISBN 1-55953-582-2

2002 Key Curriculum Press Teaching Mathematics with The Geometers Sketchpad i i i

ContentsTeaching Notes ..................................................................................................................................... 1

The Geometers Sketchpad and Changes in Mathematics Teaching ....................................... 1Where Sketchpad Came From ........................................................................................................ 2Using Sketchpad in the Classroom ................................................................................................ 3A Guided Investigation: Napoleons Theorem ............................................................................ 4An Open-Ended Exploration: Constructing Rhombuses .......................................................... 5A Demonstration: A Visual Demonstration of the Pythagorean Theorem ............................ 6

Using Sketchpad in Different Classroom Settings ...................................................................... 7A Classroom with One Computer ................................................................................................. 7One Computer and a Projection Device ........................................................................................ 7A Classroom with a Handful of Computers ................................................................................ 7A Computer Lab ................................................................................................................................ 8

Using Sketchpad as a Presentation Tool ....................................................................................... 8Using Sketchpad as a Productivity Tool ....................................................................................... 9The Geometers Sketchpad and Your Textbook ........................................................................ 10

Sample Activities ............................................................................................................................... 11Introduction ...................................................................................................................................... 11Angles ................................................................................................................................................ 12Constructing a Sketchpad Kaleidoscope .................................................................................... 13Properties of Reflection .................................................................................................................. 16Tessellations Using Only Translations ........................................................................................ 18The Euler Segment .......................................................................................................................... 20Napoleons Theorem ...................................................................................................................... 22Constructing Rhombuses ............................................................................................................... 23Midpoint Quadrilaterals ................................................................................................................ 24A Rectangle with Maximum Area ............................................................................................... 25Visual Demonstration of the Pythagorean Theorem ................................................................ 27The Golden Rectangle .................................................................................................................... 28A Sine Wave Tracer ......................................................................................................................... 30Adding Integers .............................................................................................................................. 32Points Lining Up in the Plane ................................................................................................... 35Parabolas in Vertex Form .............................................................................................................. 38Reflection in Geometry and Algebra ........................................................................................... 41Walking Rex: An Introduction to Vectors ................................................................................. 44Leonardo da Vincis Proof ............................................................................................................. 46The Folded Circle Construction .................................................................................................... 49

i v Teaching Mathematics with The Geometers Sketchpad 2002 Key Curriculum Press

The Expanding Circle Construction ............................................................................................ 53Distances in an Equilateral Triangle ............................................................................................ 56Varignon Area ................................................................................................................................. 60Visualizing Change: Velocity ........................................................................................................ 64Going Off on a Tangent ................................................................................................................. 68Accumulating Area ......................................................................................................................... 71

Activity Notes for Sample Activities ................................................................................. 75

2002 Key Curriculum Press Teaching Mathematics with The Geometers Sketchpad 1

Teaching NotesIf youve read the Learning Guide, youve learnedhow to use The Geometers Sketchpad andyouve probably discovered that the range ofthings you can do with the software is greaterthan you first imagined. For all its potentialuses though, Sketchpad was designed primarilyas a teaching and learning tool. In this section,we establish a context for Sketchpad ingeometry teaching and offer suggestions forusing Sketchpad in different ways in differentclassroom settings. More than 20 sampleactivitiestouching on a range of schoolmathematics topicsfollow these teaching notes.By exploring the sample documents that areinstalled with the software, youll find even moreideas. Try them with your students for a sense ofhow Sketchpad can serve your classroom best.

The Geometers Sketchpad and Changes inMathematics TeachingThe way we teach mathematicsgeometry in particularhas changed, thanks to a fewimportant developments in recent years. Alternatives to a strictly deductive approach areavailable after more than a century of failing to reach a majority of students. (The NationalAssessment of Educational Progress found in 1982 that doing proofs was the least likedmathematics topic of 17-year-olds, and less then 50% of them rated the topic as important.)First, in 1985, Judah Schwartz and Michal Yerushalmy of the Education Development Centerdeveloped a landmark piece of instructional software that enabled teachers and students touse computers as teaching and learning tools rather than just as drillmasters. The GeometricSupposers, for Apple II computers, encouraged students to invent their own mathematicsby making it easy to create simple geometric figures and make conjectures about theirproperties. Learning geometry could become a series of open-ended explorations ofrelationships in geometric figuresa process of discovery that motivates proof, ratherthan a rehashing of proofs of theorems that students take for granted or dont understand.

In 1989, the National Council of Teachers of Mathematics (NCTM) published Curriculum andEvaluation Standards for School Mathematics (the Standards) which called for significant changesin the way mathematics is taught. In the teaching of geometry, the Standards called fordecreased emphasis on the presentation of geometry as a complete deductive system anda decreased emphasis on two-column proofs. Across the curriculum, the Standards calledfor an increase in open exploration and conjecturing and increased attention to topics intransformational geometry. In their call for change, the Standards recognized the impactthat technology tools have on the way mathematics is taught, by freeing students fromtime-consuming, mundane tasks and giving them the means to see and explore interestingrelationships.

By publishing the first edition of Michael Serras Discovering Geometry: An Inductive Approachin 1989, Key Curriculum Press joined the forces of change. Discovering Geometry, a highschool geometry textbook, takes much the same approach that the creators of The GeometricSupposers espoused: Students should create their own geometric constructions andthemselves formulate the mathematics to describe relationships they discover. WithDiscovering Geometry, students working in cooperative groups do investigations using toolsof geometry to discover properties. Students look for patterns and use inductive reasoning tomake conjectures. They arent expected to prove their discoveries until after theyve mastered

1+ = 1/

Although it remains a matter of dispute, some architects andmathematicians believe the Parthenon was designed to utilize theGolden Mean. This sketch shows how the Parthenon roughly fits intoa Golden Rectangle.

2 Teaching Mathematics with The Geometers Sketchpad 2002 Key Curriculum Press

geometry concepts and can appreciate the significance of proof. Now in its second edition,Discovering Geometry lets students take advantage of a broader range of tools, including pattypapers and The Geometers Sketchpad.

This approach is consistent with research done by the Dutch mathematics educators Pierrevan Hiele and Dina van Hiele-Geldof. From classroom observations, the van Hieles learnedthat students pass through a series of levels of geometric thinking: Visualization, Analysis,Informal Deduction, Formal Deduction, and Rigor. Standard geometry texts expect studentsto employ formal deduction from the beginning. Little is done to enable students to visualizeor to encourage them to make conjectures. A main goal of The Supposers, DiscoveringGeometry, and, now, The Geometers Sketchpad is to bring students through the first threelevels, encouraging a process of discovery that more closely reflects how mathematics isusually invented: A mathematician first visualizes and analyzes a problem, makingconjectures before attempting a proof.

The Geometers Sketchpad established the current generation of educational softwarethat has accelerated the change begun by The Geometric Supposers and that was spurred onby publications like Discovering Geometry and the NCTM Standards. Sketchpads uniqueDynamic Geometry enables students to explore relationships interactively so that theycan see change in mathematical diagrams as they manipulate them. With this breakthrough,along with the completeness of its construction, transformation, analytic, and algebraiccapabilitiesas well as the unbounded extensibility offered by its custom toolsSketchpadbroadens the scope of what its possible to do with mathematics software to an extentnever seen before. In the ten years of its existence, teachers have taken Sketchpad outsidethe geometry classroom and into algebra, calculus, trigonometry, and middle-schoolmathematics courses; and ongoing development of the software has refined it for thesewider uses. The Dynamic Geometry paradigm pioneered by Sketchpad has been so widelyembracedby mathematics and educational researchers, by teachers across the curriculum,and by millions of studentsthat the 2000 edition of the Standards now call for DynamicGeometry by name. Concurrent development of Macintosh, Windows, Java, and handheldversions of Sketchpad in a number of different languages ensures the most powerful andup-to-date geometry tool is always available to a wide variety of school computingenvironments throughout the world.

Where Sketchpad Came FromThe Geometers Sketchpad was developed as part of the Visual Geometry Project, a NationalScience Foundationfunded project under the direction of Dr. Eugene Klotz at SwarthmoreCollege and Dr. Doris Schattschneider at Moravian College in Pennsylvania. In addition toSketchpad, the Visual Geometry Project (VGP) has produced The Stella Octangula and ThePlatonic Solids: videos, activity books, and manipulative materials also published by KeyCurriculum Press. Sketchpad creator and programmer Nicholas Jackiw joined the VGP in thesummer of 1987. He began serious programming work a year later. Sketchpad for Macintoshwas developed in an open, academic environment in which many teachers and other usersexperimented with early versions of the program and provided input to its design. Nicholascame to work for Key Curriculum Press in 1990 to produce the beta version of the softwaretested in classrooms. A core of 30 schools soon grew to a group of more than 50 sites as wordspread and more people heard of Sketchpad or saw it demonstrated at conferences. Theopenness with which Sketchpad was developed generated an incredible tide of feedbackand enthusiasm for the program. By the time of its release in the spring of 1991, it had beenused by hundreds of teachers, students, and other geometry lovers and was already the mosttalked about and awaited piece of school mathematics software in recent memory.


In Sketchpads first year, Key Curriculum Press began to study how the program was beingused effectively in schools. Funded in part by a grant for small businesses from theNational Science Foundation, this research is reflected in these teaching notes, in curriculummaterials, and in new versions of Sketchpad. Version 2 of the program, released in April1992, introduced improved transformation and presentation capabilities and brought toolsfor the graphical exploration of recursion and iteration into the hands of Sketchpad users.Version 3 for Macintosh and Windows, a major upgrade released in April 1995, expanded theprograms analytic and graphing capabilities. By 1999, the Teaching, Learning, and Computingnational teacher survey conducted by the University of California, Irvine, found that thenations mathematics teachers rated Sketchpad the most valuable software for students bya large margin. Version 4 of the software, introduced in the fall of 2001, dramatically expandsthe programs usefulness in algebra, pre-calculus, and calculus classes, while increasing boththe ease of use in earlier grades and the softwares curriculum development authoring tools.Classroom research continues to form the basis for further development of the software andaccompanying materials.

Using Sketchpad in the ClassroomThe Geometers Sketchpad was designed initially primarily for use in high schoolgeometry classes. Testing has shown, though, that its ease of use makes it possible foryounger students to use Sketchpad successfully, and the power of its features has made itattractive to instructors of college-level mathematics and teacher pre-service and inservicecourses. College instructors are drawn particularly to Sketchpads powerful transformationcapabilities and to custom tools allowing students to explore non-Euclidean geometries. Evenartists and mechanical drawing professionals have been enthralled by Sketchpads powerand elegance. Its a testament to the versatility of the software that the same toolcan be used by six-year-olds and college professors to explore new mathematical concepts.(Be sure to browse the sample documents that come installed with Sketchpad for additionaltools that help particularize the program to your classroom needs. Youll find tools forconstructing regular polygons, defining mathematical symbols, exploring non-Euclideangeometries, composing and combining functions, and much more.) In this section, wellconcentrate on ways Sketchpad might be used in a high school geometry class.

As a high school geometry teacher, you may want to guide your students towarddiscovering a specific property or small set of properties, or you may want to pose anopen-ended question or problem and ask students to try to discover as much as they canabout it. Alternatively, you may want to prepare for students an interactive demonstrationthat models a particular concept. In any case, youll want students to collaborate andcommunicate their findings. Sketchpads annotation features encourage students to articulatemathematical ideas. Whatever approach you take to using Sketchpad, it can serve as aspringboard for discussion and communication. Well look at examples of three approachesto using Sketchpad in the classroom: a guided investigation, an open-ended exploration, anda demonstration. These three examples come from Exploring Geometry with The GeometersSketchpad, 1999 by Key Curriculum Press.


A Guided Investigation: Napoleons TheoremThe purpose of this investigation is to guide students to some specific conjectures. They aregiven instructions to construct a figure with certain specifically defined relationships: in thiscase, a triangle with equilateral triangles constructed on its sides. Students manipulate theirconstruction to see what relationships they find that can be generalized for all triangles. Afterthis experimentation, students are asked to write conjectures.

An important aspect of thisand, in fact, anySketchpadinvestigation is that bymanipulating a single figurea student can potentially seeevery possible case of thatfigure. Here they have visualproof that the Napoleontriangle of an arbitrary triangleis always equilateral, even asthe original triangle changesfrom acute to right to obtuse,from scalene to isosceles toequilateral.

Suggestions are made forfurther, open-ended investi-gation for students who finishfirst. In this Explore Moresuggestion, students candiscover that the segments inquestion are congruent, areconcurrent, and intersect toform 60 angles.

After students have discussedtheir findings in pairs orsmall groups, its importantto discuss them as a largegroup. Ask students to shareany special cases theyvediscovered, and use yourquestions to emphasizewhich relationships can begeneralized for all triangles:Was the Napoleon trianglealways equilateral even asyou changed your original triangle from being acute to being obtuse? Were the threesegments you constructed in Explore More congruent and concurrent no matter what shapetriangle you had? In this wrap-up you can introduce vocabulary or special names forproperties students discover (for example, the point of concurrency they discover in ExploreMore is called the Fermat point) and agree as a class on wording for students conjectures as away of checking for understanding.

Napoleons Theorem Name(s): French emperor Napoleon Bonaparte fancied himself as something of anamateur geometer and liked to hang out with mathematicians. Thetheorem youll investigate in this activity is attributed to him.

Sketch and Investigate

1. Construct an equilateral triangle. You can use a pre-made custom toolor construct the triangle from scratch.

2. Construct the center of the triangle.

3. Hide anything extra you may have constructed toconstruct the triangle and its center so that youreleft with a figure like the one shown at right.

4. Make a custom tool for this construction.

Next, youll use your custom tool to construct equilateral triangles on thesides of an arbitrary triangle.

5. Open a new sketch.

6. Construct ABC.

7. Use the custom tool to constructequilateral triangles on each sideof ABC.

8. Drag to make sure each equilateraltriangle is stuck to a side.

9. Construct segments connecting thecenters of the equilateral triangles.

10. Drag the vertices of the original triangleand observe the triangle formed by thecenters of the equilateral triangles. Thistriangle is called the outer Napoleon triangle of ABC.

Q1 State what you think Napoleons theorem might be.

Explore More1. Construct segments connecting each vertex of your original triangle

with the most remote vertex of the equilateral triangle on the oppositeside. What can you say about these three segments?

One way toconstruct the center

is to construct twomedians and their

point of intersection.

Select the entirefigure; then choose

Create New Toolfrom the Custom

Tools menuin the Toolbox

(the bottom tool).

A C

B

Be sure to attacheach equilateral

triangle to a pair oftriangle ABCs

vertices. If yourequilateral triangle

goes the wrong way(overlaps the interior

of ABC) or is notattached properly,

undo and tryattaching it again.


An Open-Ended Exploration: Constructing RhombusesIn an open-ended exploration there is not a specific set of properties that students areexpected to discover as outcomes of the lesson. A question or problem is posed with a fewsuggestions about how to use Sketchpad to explore the problem. Different students willdiscover or use different relationships in their constructions and write their findings in theirown words.

In this example, studentsare asked to come up withas many ways as they canto construct a rhombus.

Again, various constructionmethods should be discussedin small groups, then withthe whole class. To bringclosure to the lesson youmight want to compile onthe chalkboard a list of allthe properties your studentsused. Offering students anopen-ended constructionproblem also gives you theopportunity to emphasizethe important distinctionbetween a drawing and aconstruction. For example,if students have actuallyused defining propertiesof a rhombus in theirconstructions, it should bepossible to manipulate theirfigure into any size or shaperhombus and it should beimpossible to distort thefigure into anything thatsnot a rhombus.

Constructing Rhombuses Name(s): How many ways can you come up with toconstruct a rhombus? Try methods that usethe Construct menu, the Transform menu, orcombinations of both. Consider how you mightuse diagonals. Write a brief description of eachconstruction method along with the propertiesof rhombuses that make that method work.

Method 1:

Properties:

Method 2:

Properties:

Method 3:

Properties:

Method 4:

Properties:

D

A

C

B


A Demonstration: A Visual Demonstration of the Pythagorean TheoremA teacher (or for that matter, a student) can use Sketchpad to prepare a demonstration forothers to use. Sometimes a complex construction can nicely show a property, but it might beimpractical to have all students do the construction themselves. In that case, teachers mightuse a demonstration sketch accompanied by an activity sheet.

Before using this demonstration,students can actually discoverthe Pythagorean theoremthemselves in a guided investi-gation. The purpose of thislesson, though, is as a demon-stration of a visual proof ofthe theorem. The sketch usedin the lesson is a pre-madesketch of some complexity.Students arent expected tocreate this constructionthemselves to discover thePythagorean theorem, butthey have a chance with thisdemonstration to look at it ina new and interesting way.

This demonstration might bedone most efficiently as awhole-class demonstrationwith you or a student workingat an overhead projector.Alternatively, you couldreproduce the activity masterfor students to use on theirown time or at the end of a labperiod in which theyve beendoing other investigationsrelated to the Pythagoreantheorem.

Visual Demonstration of thePythagorean Theorem Name(s): In this activity, youll do a visual demonstration of the Pythagoreantheorem based on Euclids proof. By shearing the squares on the sides of aright triangle, youll create congruent shapes without changing the areasof your original squares.

Sketch and Investigate1. Open the sketch Shear Pythagoras.gsp.

Youll see a right triangle with squareson the sides.

2. Measure the areas of the squares.

3. Drag point A onto the line thatsperpendicular to the hypotenuse.Note that as the square becomes aparallelogram its area doesnt change.

4. Drag point B onto the line. It shouldoverlap point A so that the twoparallelograms form a singleirregular shape.

5. Drag point C so that the large square deforms to fill in the triangle.The area of this shape doesnt change either. It should appearcongruent to the shape you made with the two smallerparallelograms.

ba c

A

C

B

Step 3

ba c

BA

C

Step 4

ba c

B

C

A

Step 5

Q1 How do these congruent shapes demonstrate the Pythagoreantheorem? (Hint: If the shapes are congruent, what do you know abouttheir areas?)

ba c

C

A

B

All sketchesreferred to in this

bookletcan be found inSketchpad |

Samples | Teach-ing Mathematics

(Sketchpad isthe folder that

contains theapplication itself.)

Click on a polygoninterior to select it.

Then, in theMeasure menu,

choose Area.

To confirm that thisshape is congruent,

you can copy andpaste it. Drag thepasted copy ontothe shape on the

legs to see thatit fits perfectly.

To confirm that thisworks for any right

triangle, changethe shape of the

triangle and try theexperiment again.


Using Sketchpad in Different Classroom SettingsSchools use computers in a variety of classroom settings. Sketchpad was designed with thisin mind, and its display features can be optimized for these different settings. Teachingstrategies also need to be adapted to available resources. What follows are some suggestionsfor using and teaching with Sketchpad if youre in a classroom with one computer, onecomputer and an overhead display device, a handful of computers, or a computer lab.

A Classroom with One ComputerPerhaps the best use of a single computer without a projector is to have small groups ofstudents take turns using the computer. Each group can investigate or confirm conjecturesmade working at their desks or tables using standard geometry tools such as a compass andstraightedge. In that case, each group would have an opportunity during a class period to usethe computer for a short time. Alternatively, you can give each group a day on which to doan investigation on the computer while other groups are doing the same or differentinvestigations at their desks. A single computer without a projection device or large-screenmonitor has limited use as a demonstration tool. Although preferences can be set inSketchpad for any size or style of type, a large class will have difficulty following ademonstration on a small computer screen.

One Computer and a Projection DeviceA variety of devices are available that plug into computers so that the display can be outputto a projector, a large-screen monitor, an LCD device used with an overhead projector, or alarge-format touch panel. The Geometers Sketchpad was designed to work well with theseprojection devices, increasing your options considerably for classroom uses. You or a studentcan act as a sort of emcee to an investigation, asking the class as a whole things like, Whatshould we try next? Where should I construct a segment? Which objects should I reflect?What do you notice as I move this point? With a projection device, you and your studentscan prepare demonstrations, or students can make presentations of findings that they madeusing the computer or other means. Sketchpad becomes a dynamic chalkboard on whichyou or your students can draw more precise, more complex figures that, best of all, can bedistorted and transformed in an infinite variety of ways without having to erase and redraw.Many teachers with access to larger labs also find that giving one or two introductorydemonstrations on Sketchpad in front of the whole class prepares their students to use it in alab with a minimum of lab-time lost to training. For demonstrations, we recommend usinglarge display text in a bold style and formatting illustrations with thick lines to make text andfigures clearly visible from all corners of a classroom.

A Classroom with a Handful of ComputersIf you can divide your class into groups of three or four students so that each group hasaccess to a computer, you can plan whole lessons around doing investigations with thecomputers. Make sure of the following: That you introduce the whole class to what it is theyre expected to do. That students have some kind of written explanation of the investigation or problem

theyre to work on. Its often useful for that explanation to be on a piece of paper onwhich students have room to record some of their findings; but for some open-endedexplorations the problem or question could simply be written on the chalkboard ortyped into the sketch itself. Likewise, students written work could be in the form ofsketches with captions and comments.

That students work so that everybody in a group has an opportunity to actually operatethe computer.

That students in a group who are not actually operating the computer are expected tocontribute to the group discussion and give input to the student operating the computer.


That you move among groups posing questions, giving help if needed, and keepingstudents on task.

That students findings are summarized in a whole-class discussion to bring closure tothe lesson.

A Computer LabThe experience of teachers in using Sketchpad in the classroom (as well as the experience ofteachers using The Geometric Supposers) suggests that even if enough computers areavailable for students to work individually, its perhaps best to have students work in pairs.Students learn best when they communicate about what theyre learning, and studentsworking together can better stimulate ideas and lend help to one another. If you do havestudents working at their own computers, encourage them to talk about what theyre doingand to compare their findings with those of their nearest neighborthey should peek overeach others shoulders. The suggestions above for students working in small groups apply tostudents working in pairs as well.

If your laboratory setting has both Macintosh computers and computers running Windows,your students can read sketches created on one type of machine with the other. Use PC-formatted disks (Macintoshes can read them, but Windows PCs cannot read Mac-formatteddisks) or a network to exchange documents between platforms.

Using Sketchpad as a Presentation ToolYoull find that Sketchpads featuresespecially its text capabilities, multi-page documentstructure, and action buttonsmake it ideally suited for teacher and student presentations.Sketchpad provides a powerful medium for mathematical communication.

With the Text tool, students and teachers can annotate their sketches with captions thatdescribe salient features of a construction. Captions can highlight properties that aconstruction demonstrates, or they can provide instructions for manipulating a construction,including what to look for as the construction changes. In this way, students and teachers cancommunicate about what theyve done in a sketch.

Teachers and students can use actionbuttons to simplify complex sketches.Buttons can be used to show and hidegeometric objects and text or to initiateanimations. Buttons can also besequenced so that proceduresand explanations of a constructioncan be played with the click of abutton. In other words, action buttonsturn sketches into presentations.

Text and action buttons make possiblepresentations without presenters: Asufficiently annotated sketch couldspeak for itself when opened byanother user at a time when the sketchcreator isnt around to explain it. Apresentation, in this context, is notnecessarily designed for a groupaudience looking at an overheaddisplay. The audience for an annotatedsketch might be a fellow student or ateacher. Teachers who ask students tohand in assignments in the form of sketches can ask students to create presentations usingaction buttons and to explain their work in captions.

A Captioned Sketch

m AB( )2 = 0.760 in.

m AB = 1.347 in.

Area ABA'B' = 1.815 in2Area CC' = 1.815 in2Radius CC' = 0.760 in.

the figure.Press the action buttons to transform

circle.that quantity and constructed thetranslated the center of the square byof a circle with the same area. Finally, Ilength AB. Then I calculated the radiusFirst I constructed a square with sidea square and a circle with equal areas.Given a segment AB, I've constructed

D. Bennett 7.6.01

The Circle Squared

circle me!square me!

C'C

A B


Sketchpads web integration facilities allows you to draw on the full resourcesof the Internet. Action buttons allow you to link to web resources to provide additionalexplorations, survey real-world applications, or establish the historical context of a particularmathematics exploration. In addition, if youre interested in publishing web pages of yourown, Sketchpad allows you to export your activities to the web, where you can integratethem with the full set of multimedia components and hyperlinked resources available toweb page authors, and share them over the net with users across the world. Users who visityour web page will be able to interact with your pages Dynamic Geometry illustrationswhether they have Sketchpad or not!

By browsing through the sample documents that come with Sketchpad you can get ideas fordifferent ways sketch captions can be used to communicate mathematically.

Using Sketchpad as a Productivity ToolThe Reference Manual describes how to use the Edit menu to cut, copy, and paste Sketchpadobjects into other applications, such as graphics or word processing programs. These featuresmake Sketchpad an extremely useful productivity tool for anyone, including teachers andstudents, who wants to easily create and store geometric figures. Teachers, for example,can create figures in Sketchpad and paste them into a test or worksheet created in a wordprocessing program. All of the graphics in the sample activities and most of the graphicsin the documentation were created in Sketchpad and pasted into Microsoft Word.

Sketchpad stores objects in the clipboard both as Sketchpad objects, which behave as suchwhen pasted back into a sketch, and graphic images, which are recognized by virtually anyprogram that deals with graphics. Sketchpad graphics will act exactly like images producedin most other graphics programs and will give excellent results when printed. If yourewriting a book or article that will be printed professionally, Sketchpad graphics can even beoutput on a typesetting machine with very high quality results. Lines and rays are truncatedwhen pasted into other programs, just as they are when printed in Sketchpad.

10 cm

yx 5 in.y

x

Solve for x and y:

y

8 cmx

You can save Sketchpad sketches as libraries of figures that you usein tests and worksheets. Then you can easily change figures if youneed variations. You can edit labels and type in measurements ofangles and lengths. Even figures that you might find easier to drawby hand have the advantage, when done with Sketchpad, that theycan be saved, easily modified, and used again and again.

Excircles of a Triangle

1 0 Teaching Mathematics with The Geometers Sketchpad 2002 Key Curriculum Press

The Geometers Sketchpad and Your TextbookThe variety of ways Sketchpad can be used makes it the ideal tool for exploring schoolmathematics, regardless of the text youre using. Use Sketchpad to demonstrate conceptspresented in the text. Or have students use Sketchpad to explore problems given as exercises.If your text presents theorems and proves them (or asks students to prove them) along theway, give your students an opportunity to explore the concepts with Sketchpad before yourequire them to do a proof. Working out constructions using Sketchpad and interacting withdiagrams dynamically will deepen students understanding of concepts and, in formalcontexts, will make proof more relevant.

Sketchpad is ideally suited for use with books that take a discovery approach to teachingand learning geometry. In Michael Serras Discovering Geometry, for example, studentsworking in small groups do investigations and discover geometry concepts for themselves,before they attempt proof. Many of these investigations call for constructions that could bedone with Sketchpad. Many other investigations involving transformations, measurements,calculations, or graphs can also be done effectively and efficiently with Sketchpad. In fact,most investigations in Discovering Geometry or any other book with a similar approach canbe done using Sketchpad.

The Discovering Geometry student text includes ten Geometers Sketchpad Projects andnumerous Investigations and Take Another Look suggestions for using Sketchpad. Morethan 60 lessons best-suited for exploration with Sketchpad were adapted and collected asblackline masters in the ancillary book Discovering Geometry with The Geometers Sketchpad.These Sketchpad lessons have the same titles and guide students to the same conjecturesas the corresponding Discovering Geometry lessons. A collection of Sketchpad documentsaccompany this book on CD-ROM. The Discovering Geometry Teachers Resource Book comeswith demonstration sketches corresponding to Discovering Geometry lessons.

Ancillary Sketchpad materials are also available for some secondary texts from otherpublishers, though for a geometry course, none provide as complete a technology packageas Key Curriculum Presss Discovering Geometry combined with The Geometers Sketchpad.If youre using a text other than Discovering Geometry, ask the publisher whether Sketchpadancillaries are available.

Exploring Geometry with The Geometers Sketchpad, available from Key Curriculum Press,contains more than 100 reproducible activities that can be used with any text. A CD-ROMwith activities for Macintosh and Windows computers accompany the activities. Many othertopic-specific volumes of activities are also available from Key Curriculum Press. Sampleactivities from some of these books are included in this booklet. These books are listed anddescribed on the back cover of this booklet.

Exploring Geometry could supply a teacher with a years worth of activities to cover nearly allthe content of a typical high school geometry course using The Geometers Sketchpad. Andother activity books could occupy a large part of the year in other mathematics courses, too.We dont, however, advocate that you abandon other teaching methods in favor of usingthe computer. Its our belief that students learn best from a variety of learning experiences.Students need experience with hands-on manipulatives, model building, function plotting,compass and straightedge constructions, drawing, paper and pencil work, and mostimportantly, group discussion. Students need to apply mathematics to real-life situationsand see where it is used in art and architecture and where it can be found in nature. ThoughSketchpad can serve as a medium for many of these experiences, its potential will be reachedonly when students can apply what they learn with it to different situations. As engaging asusing Sketchpad can be, its important that students dont get the mistaken impression thatmathematics exists only in their books and on their computer screens.

2002 Key Curriculum Press Teaching Mathematics with The Geometers Sketchpad 1 1

Sample ActivitiesIntroductionThese sample classroom activity masters will give you an idea of some of the types of learning experiencesthat are possible using The Geometers Sketchpad. In the Teaching Notes, you saw three different types oflessons from Exploring Geometry with The Geometers Sketchpad: an investigation, an exploration, and ademonstration. Here youll find more activities from Exploring Geometry along with samples from other KeyCurriculum Press publications. This collection is neither a complete curriculum nor a comprehensive set ofactivities to keep you and your students occupied for a school year.

The topics of the activities range from creating geometric art to calculus. Their difficulty ranges from beingappropriate for middle school students to presenting challenges to college undergraduate math majors. Thereare 25 activities here, and youre obviously not going to be able to use them all with the same class. While wecertainly hope that teachers will find some of the activities in this sample useful in their classes, the collectionhere is designed to show you a range of possibilities.

Exploring Geometry contains over 100 activities. That volume does represent a nearly complete curriculum,though we would caution teachers from overusing it. (See Teaching Notes, page 10.)

The list below shows the names of activities sampled here and the titles of the books theyre from.

From Geometry Activities for Middle SchoolStudents with The Geometers Sketchpad

Angles

Constructing a Sketchpad Kaleidoscope

From Exploring Geometry with The GeometersSketchpad

Properties of Reflection

Tessellations Using Only Translations

The Euler Segment

Napoleons Theorem

Constructing Rhombuses

Midpoint Quadrilaterals

A Rectangle with Maximum Area

Visual Demonstration of the PythagoreanTheorem

The Golden Rectangle

A Sine Wave Tracer

From Exploring Algebra with The GeometersSketchpad

Adding Integers

Points Lining Up in the Plane

Parabolas in Vertex Form

Reflection in Geometry and Algebra

Walking Rex: An Introduction to Vectors

From Pythagoras Plugged InLeonardo da Vincis Proof

From Exploring Conic Sections with TheGeometers Sketchpad

The Folded Circle Construction

The Expanding Circle Construction

From Rethinking Proof with The GeometersSketchpad

Distances in an Equilateral Triangle

Varignon Area

From Exploring Calculus with The GeometersSketchpad

Visualizing Change: Velocity

Going Off on a Tangent

Accumulating Area

Try some or all of these activities yourself and with your students to explore Sketchpads potential and learnhow you can use it in the classroom. (You may reproduce these sheets for use with your classes.) Then join usin creating the most comprehensive teacher support materials ever to accompany new classroom softwarematerials that reflect what teachers and students can accomplish with state-of-the-art teaching and learningtools.

If youre interested in contributing worksheets, sample sketches, or custom tools for possible inclusion infuture teacher materials and sample disks, contact the Editorial Department at Key Curriculum Press.

From Geometry Activities for Middle School Students with The Geometers Sketchpad1 2 Teaching Mathematics with The Geometers Sketchpad 2002 Key Curriculum Press

Angles Name(s):


2. Construct a triangle.

3. Extend one side by constructing a ray using two vertices.

A

C

B

D

4. Measure each of the interior angles.

5. Go to the Measure menu and choose Calculate. Use Sketchpadscalculator to determine the sum of the three interior angles.

Q1 Drag any vertex of the triangle and observe the measures of theinterior angles and their sum.Write any conjectures based on your exploration.

6. Click somewhere on the ray outside the triangle to construct a point.Measure the exterior angle.

7. Use Sketchpads calculator to determine the sum of the two interiorangles that are not adjacent to the exterior angle.

Q2 Drag any vertex of the triangle and compare the measure of theexterior angle to the sum of the two remote (nonadjacent) interiorangles.Write any conjectures based on your exploration.

From Geometry Activities for Middle School Students with The Geometers Sketchpad 2002 Key Curriculum Press Teaching Mathematics with The Geometers Sketchpad 1 3

Constructing a Sketchpad Kaleidoscope Name(s): Follow the directions below to construct a Sketchpad kaleidoscope. Thenumbered steps tell you in general what you need to do, and the letteredsteps give you more detailed instructions. Make sure you did each step correctlybefore you go on to the next step.

1. Open a new sketch and construct a many-sided polygon.

a. Go to the File menu and choose New Sketch.

b. Use the Segment tool to construct a polygon with manysides (make it long and somewhat slender).

2. Construct several polygon interiors within your polygon.Shade them different colors.

a. Click on the Selection Arrowtool. Click in any blank spaceto deselect objects.

b. Select three or four points inclockwise or counter-clockwiseorder.

c. Go to the Construct menu andchoose Triangle Interior orQuadrilateral Interior.

Step b Step c Step e

d. While the polygon interior is still selected, go to the Display menu andchoose a color for your polygon interior.

e. Click in any blank space to deselect objects. Repeat steps b, c, and d until youhave constructed several polygon interiors with different colors or shades.

3. Mark the bottom vertex point of your polygon as the center. Hide the pointsand rotate the polygon by an angle of 60.

a. Click in any blank space to deselect objects.

b. Select the bottom vertex point. Go to the Transform menu and chooseMark Center.

c. Click on the Point tool. Go to the Edit menu and choose Select All Points.Go to the Display menu and choose Hide Points.

Constructing a Sketchpad Kaleidoscope (continued)

From Geometry Activities for Middle School Students with The Geometers Sketchpad1 4 Teaching Mathematics with The Geometers Sketchpad 2002 Key Curriculum Press

d. Click on the Selection Arrow tool.Use a selection marquee to selectyour polygon. Go to the Transformmenu and choose Rotate.

e. Enter 6 0 and then click Rotate.(Pick a different factor of 360 ifyou wish.)

Rotate Dialog Box (Mac)

4. Continue to rotate the new rotated imagesuntil you have completed your kaleidoscope.

a. While the new rotated image is still selected, goto the Transform menu and rotate this imageby an angle of 60. Remember to click Rotate.

b. When the newer rotated image appears, andwhile it is still selected, go to the Transform menu and rotate this image by anangle of 60. Remember to click Rotate.

c. Repeat this process until you have constructed your complete kaleidoscope.

d. Go to the Display menu and choose ShowAll Hidden. You should see the pointson the original arm reappear.

5. Construct circles with their centers at thecenter of your kaleidoscope.

a. Click in any blank space to deselect allobjects.

b. Click on the Compass tool. Press on thecenter point of your kaleidoscope and draga circle with a radius a little larger than theoutside edge of your kaleidoscope.

selection marquee after 60 rotation

control point

control point

control point

Constructing a Sketchpad Kaleidoscope (continued)

From Geometry Activities for Middle School Students with The Geometers Sketchpad 2002 Key Curriculum Press Teaching Mathematics with The Geometers Sketchpad 1 5

c. Using the Compass tool, construct another circle with its center at thecenter of your kaleidoscope, but this time let the radius be about halfthe radius of your kaleidoscope. Repeat for a circle with a radius aboutone-third the radius of your kaleidoscope.

Note: Make sure you release your mouse in a blank space between two arms of yourkaleidoscope. You do not want the outside control points of your circles to beconstructed on any part of your kaleidoscope.

6. Merge points of your kaleidoscope onto the three circles.

a. Click on the Selection Arrow tool. Click inany blank space to deselect objects.

b. Select one point on the original polygonnear the outside circle and select the outsidecircle (do not click on one of the controlpoints of the circle). Go to the Edit menuand choose Merge Point To Circle.

c. Click in any blank space to deselectall objects. Repeat step b. for the middlecircle and a point near the middle circle.Do this one more time for the smallestcircle and a point near the smallest circle.

7. Animate points of your kaleidoscope on thethree circles.

a. Click in any blank space to deselect all objects.

b. Select the three points you merged ontocircles in the previous step.

c. Go to the Edit menu, chooseAction Button, and drag to theright and choose Animation. Clickon OK in the Animate dialog box.

d. When the Animate Points button appears,click on it to start the animation. Watchyour kaleidoscope turn!

e. To hide all the points, click on thePoint tool. Go to the Edit menu andchoose Select All Points. Go to theDisplay menu and choose Hide Points.Click on the Compass tool, select all the circles, and hide them.

merged points

Animate Points

From Exploring Geometry with The Geometers Sketchpad1 6 Teaching Mathematics with The Geometers Sketchpad 2002 Key Curriculum Press

Properties of Reflection Name(s): When you look at yourself in a mirror, how far away does your image inthe mirror appear to be? Why is it that your reflection looks just like you,but backward? Reflections in geometry have some of the same propertiesof reflections you observe in a mirror. In this activity, youll investigate theproperties of reflections that make a reflection the mirror image of theoriginal.

Sketch and Investigate: Mirror Writing1. Construct vertical

line AB.

2. Construct point Cto the right ofthe line.

3. Mark dAB as amirror.

4. Reflect point C toconstruct point C.

5. Turn on Trace Points for points C and C.

6. Drag point C so that it traces out your name.

Q1 What does point C trace?

7. For a real challenge, try dragging point C so that point C traces outyour name.

Sketch and Investigate: Reflecting Geometric Figures8. Turn off Trace Points for

points C and C.

9. In the Display menu, chooseErase Traces.

10. Construct jCDE.

11. Reflect jCDE (sides andvertices) over dAB.

12. Drag different parts of eithertriangle and observe how thetriangles are related. Also dragthe mirror line.

C'

B

C

A

Double-click onthe line.

Select the twopoints; then, in the

Display menu,choose Trace

Points. A checkmark indicates that

the command isturned on. Choose

Erase Traceswhen you wish to

erase your traces.

Select points C andC. In the Displaymenu, youll seeTrace Points

checked. Choose itto uncheck it.

D'

E'

C'A

C

DEB

Select the entirefigure; then, in theTransform menu,choose Reflect.

Properties of Reflection (continued)

From Exploring Geometry with The Geometers Sketchpad 2002 Key Curriculum Press Teaching Mathematics with The Geometers Sketchpad 1 7

13. Measure the lengths of the sides of triangles CDE and CDE.

14. Measure one angle in jCDE and measure the corresponding anglein jCDE.

Q2 What effect does reflection have on lengths and angle measures?

Q3 Are a figure and its mirror image always congruent? State youranswer as a conjecture.

Q4 Going alphabetically from C to D to E in jCDE, are the verticesoriented in a clockwise or counter-clockwise direction? In whatdirection (clockwise or counter-clockwise) are vertices C, D, and Eoriented in the reflected triangle?

15. Construct segments connectingeach point and its image: C to C,D to D, and E to E. Make thesesegments dashed.

16. Drag different parts of the sketcharound and observe relationshipsbetween the dashed segments andthe mirror line.

Q5 How is the mirror line related to asegment connecting a point and itsreflected image?

Explore More1. Suppose Sketchpad didnt have a Transform menu. How could you

construct a given points mirror image over a given line? Try it. Startwith a point and a line. Come up with a construction for the reflectionof the point over the line using just the tools and the Construct menu.Describe your method.

2. Use a reflection to construct an isosceles triangle. Explain whatyou did.

Select three pointsthat name the angle,with the vertex your

middle selection.Then, in the

Measure menu,choose Angle.

Your answer to Q4demonstrates that a

reflection reversesthe orientation

of a figure.

Line Width is inthe Display menu.

D'

E'

C'A

C

DE

B

You may wish toconstruct points of

intersection andmeasure distancesto look for relation-ships between themirror line and thedashed segments.


Tessellations UsingOnly Translations Name(s): In this activity, youll learn how to construct an irregularly shaped tilebased on a parallelogram. Then youll use translations to tessellate yourscreen with this tile.

Sketch

1. Construct sAB in the lower left corner of yoursketch, then construct point C just above sAB.

2. Mark the vector from point A to point B andtranslate point C by this vector.

3. Construct the remaining sides of yourparallelogram.

C

A B

C' C

A B

C' C

A B

C' C

A B

C'

Step 4 Step 5 Step 6 Step 7

4. Construct two or three connected segments from point A to point C.Well call this irregular edge AC.

5. Select all the segments and points of irregular edge AC and translatethem by the marked vector. (Vector AB should still be marked.)

6. Make an irregular edge from A to B.

7. Mark the vector from point A to point Cand translate all the parts of irregularedge AB by the marked vector.

8. Construct the polygon interior of theirregular figure. This is the tile youwill translate.

9. Translate the polygon interior by themarked vector. (You probably stillhave vector AC marked.)

10. Repeat this process until you havea column of tiles all the way upyour sketch. Change the color onevery other tile to create a pattern.

C

A B

C'

Steps 13

Select, in order,point A and point B;

then, in theTransform menu,

choose MarkVector. Select

point C; then, in theTransform menu,

choose Translate.

C

A B

C'

Steps 810

Select the vertices inconsecutive order;

then, in theConstruct menu,

choose PolygonInterior.

Tessellations Using Only Translations (continued)


11. Mark vector AB. Thenselect all the polygoninteriors in your columnof tiles and translate themby this marked vector.

12. Continue translatingcolumns of tiles untilyou fill your screen.Change shades andcolors of alternatingtiles so you can seeyour tessellation.

13. Drag vertices of your original tile until you get a shape that you likeor that is recognizable as some interesting form.

Explore More1. Animate your tessellation. To do this, select the original polygon (or

any combination of its vertex points) and choose Animate from theDisplay menu. You can also have your points move along paths youconstruct. To do this, construct the paths (segments, circles, polygoninteriorsanything you can construct a point on) and then mergevertices to paths. (To merge a point to a path, select both and chooseMerge Point to Path from the Edit menu.) Select the points you wishto animate and, in the Edit menu, choose Action Buttons | Animation.Press the Animate button. Adjust the paths so that the animationworks in a way you like, then hide the paths.

2. Use Sketchpad to make a translation tessellation that starts with aregular hexagon as the basic shape instead of a parallelogram.(Hint: The process is very similar; it just involves a third pair of sides.)

C

A B

C'

Steps 11 and 12


The Euler Segment Name(s): In this investigation, youll look for a relationship among four points ofconcurrency: the incenter, the circumcenter, the orthocenter, and thecentroid. Youll use custom tools to construct these triangle centers, eitherthose you made in previous investigations or pre-made tools.

Sketch and Investigate1. Open a sketch (or sketches) of yours that

contains tools for the triangle centers:incenter, circumcenter, orthocenter, andcentroid. Or, open Triangle Centers.gsp.

2. Construct a triangle.

3. Use the Incenter tool on the trianglesvertices to construct its incenter.

4. If necessary, give the incenter a label that identifies it, such as I forincenter.

5. You need only the triangle and the incenter for now, so hide anythingextra that your custom tool may have constructed (such as anglebisectors or the incircle).

6. Use the Circumcenter tool on thesame triangle. Hide any extrasso that you have just the triangle,its incenter, and its circumcenter.If necessary, give the circumcentera label that identifies it.

7. Use the Orthocenter tool on thesame triangle, hide any extras,and label the orthocenter.

8. Use the Centroid tool on the same triangle, hide extras, and labelthe centroid. You should now have a triangle and the four trianglecenters.

Q1 Drag your triangle around and observe how the points behave.Three of the four points are always collinear. Which three?

9. Construct a segment that contains the three collinear points. This iscalled the Euler segment.

I

TriangleCenters.gsp

can be foundin Sketchpad |

Samples |Custom Tools.



O CeCi

I

The Euler Segment (continued)


Q2 Drag the triangle again and look for interesting relationships on theEuler segment. Be sure to check special triangles, such as isosceles andright triangles. Describe any special triangles in which the trianglecenters are related in interesting ways or located in interesting places.

Q3 Which of the three points are always endpoints of the Euler segmentand which point is always between them?

10. Measure the distances along the two parts of the Euler segment.

Q4 Drag the triangle and look for a relationship between these lengths.How are the lengths of the two parts of the Euler segment related?Test your conjecture using the Calculator.

Explore More1. Construct a circle centered at the midpoint of the Euler segment and

passing through the midpoint of one of the sides of the triangle. Thiscircle is called the nine-point circle. The midpoint it passes through isone of the nine points. What are the other eight? (Hint: Six of themhave to do with the altitudes and the orthocenter.)

2. Once youve constructed the nine-point circle, drag your trianglearound and investigate special triangles. Describe any triangles inwhich some of the nine points coincide.

To measure thedistance between

two points, select thetwo points. Then, inthe Measure menu,choose Distance.

(Measuring thedistance betweenpoints is an easyway to measure

the length of partof a segment.)


Napoleons Theorem Name(s): French emperor Napoleon Bonaparte fancied himself as something of anamateur geometer and liked to hang out with mathematicians. Thetheorem youll investigate in this activity is attributed to him.

Sketch and Investigate1. Construct an equilateral triangle. You can use a pre-made custom tool

or construct the triangle from scratch.

2. Construct the center of the triangle.

3. Hide anything extra you may have constructed toconstruct the triangle and its center so that youreleft with a figure like the one shown at right.

4. Make a custom tool for this construction.

Next, youll use your custom tool to construct equilateral triangles on thesides of an arbitrary triangle.


6. Construct jABC.

7. Use the custom tool to constructequilateral triangles on each sideof jABC.

8. Drag to make sure each equilateraltriangle is stuck to a side.

9. Construct segments connecting thecenters of the equilateral triangles.

10. Drag the vertices of the original triangleand observe the triangle formed by thecenters of the equilateral triangles. Thistriangle is called the outer Napoleon triangle of jABC.

Q1 State what you think Napoleons theorem might be.

Explore More1. Construct segments connecting each vertex of your original triangle

with the most remote vertex of the equilateral triangle on the oppositeside. What can you say about these three segments?

One way toconstruct the center

is to construct twomedians and their

point of intersection.




(the bottom tool).

A C

B

Be sure to attacheach equilateral

triangle to a pair oftriangle ABCs

vertices. If yourequilateral triangle

goes the wrongway (overlaps theinterior of jABC)or is not attached

properly, undoand try attaching

it again.


Constructing Rhombuses Name(s): How many ways can you come up with toconstruct a rhombus? Try methods that usethe Construct menu, the Transform menu, orcombinations of both. Consider how you mightuse diagonals. Write a brief description of eachconstruction method along with the propertiesof rhombuses that make that method work.

Method 1:

Properties:

Method 2:

Properties:

Method 3:

Properties:

Method 4:

Properties:

D

A

C

B


Midpoint Quadrilaterals Name(s): In this investigation, youll discover something surprising about the quad-rilateral formed by connecting the midpoints of another quadrilateral.

Sketch and Investigate1. Construct quadrilateral ABCD.

2. Construct the midpoints of the sides.

3. Connect the midpoints to constructanother quadrilateral, EFGH.

4. Drag vertices of your originalquadrilateral and observe themidpoint quadrilateral.

5. Measure the four side lengths of thismidpoint quadrilateral.

Q1 Measure the slopes of the four sides of the midpoint quadrilateral.What kind of quadrilateral does the midpoint quadrilateral appearto be? How do the measurements support that conjecture?

6. Construct a diagonal.

7. Measure the length and slope ofthe diagonal.

8. Drag vertices of the originalquadrilateral and observe how thelength and slope of the diagonal arerelated to the lengths and slopes of thesides of the midpoint quadrilateral.

Q2 The diagonal divides the original quadrilateral into two triangles.Each triangle has as a midsegment one of the sides of the midpointquadrilateral. Use this fact and what you know about the slope andlength of the diagonal to write a paragraph explaining why theconjecture you made in Q1 is true. Use a separate sheet of paperif necessary.

H

E

F

G

A

B

C

D

If you select all foursides, you can

construct all fourmidpoints at once.

H

E

F

G

A

B C

D


A Rectangle with Maximum Area Name(s): Suppose you had a certain amount of fence and you wanted to use it toenclose the biggest possible rectangular field. What rectangle shape wouldyou choose? In other words, what type of rectangle has the most area for agiven perimeter? Youll discover the answer in this investigation. Or, ifyou have a hunch already, this investigation will help confirm your hunchand give you more insight into it.

Sketch and Investigate

1. Construct sAB.

2. Construct sAC on sAB.

3. Construct linesperpendicular to sABthrough points A and C.

4. Construct circle CB.

5. Construct point D wherethis circle intersects theperpendicular line.

6. Construct a line through point D, parallel to sAB.

7. Construct point E, the fourth vertex of rectangle ACDE.

8. Construct polygon interior ACDE.

9. Measure the area and perimeter of this polygon.

10. Drag point C back and forth and observe how this affects the area andperimeter of the rectangle.

11. Measure AC and AE.

Q1 Without measuring, state how AB is related to the perimeter of therectangle. Explain why this rectangle has a fixed perimeter.

Q2 As you drag point C, observe what rectangular shape gives thegreatest area. What shape do you think that is?

E D

A BC

Select sAB, point A,and point C. Then,

in the Constructmenu, choose

PerpendicularLine.

Be sure to releasethe mouseor click

the second timewith the pointer

over point B.

Select the vertices ofthe rectangle in

consecutive order.Then, in the Construct

menu, chooseQuadrilateral

Interior.

Select point A andpoint C. Then, in the

Measure menu,choose Distance.

Repeat tomeasure AE.

A Rectangle with Maximum Area (continued)


In Steps 1214, youll explore this relationship graphically.

12. Plot the measurements for the length of sAC and the area of ACDEas (x, y). You should get axes and a plotted point H, as shown below.

13. Drag point C to see the plotted point move to correspond to differentside lengths and areas.

-5

2

-10

m AE = 1.01 cmm AC = 3.69 cm

Perimeter ACDE = 9.41 cmArea ACDE = 3.74 cm2

H

E D

A B

F

C

G

14. To see a graph of all possible areas for this rectangle, construct thelocus of plotted point H as defined by point C. It should now be easyto position point C so that point H is at a maximum value for the areaof the rectangle.

Q3 Explain what the coordinates of the high point on the graph are andhow they are related to the side lengths and area of the rectangle.

15. Drag point C so that point H moves back and forth between the twolow points on the graph.

Q4 Explain what the coordinates of the two low points on the graph areand how they are related to the side lengths and area of the rectangle.

Explore More1. Investigate area/perimeter relationships in other polygons. Make a

conjecture about what kinds of polygons yield the greatest area for agiven perimeter.

2. Whats the equation for the graph you made? Let AC be x and let ABbe (1/2)P, where P stands for perimeter (a constant). Write anequation for area, A, in terms of x and P. What value for x (in termsof P) gives a maximum value for A?

Select, in order,msAC and Area

ACDE. Then choosePlot As (x, y) from

the Graph menu.If you cant see

the plotted point,drag the unit point

at (1, 0) to scalethe axes.

Select point H andpoint C; then, in the

Construct menu,choose Locus.

You may wish toselect point H and

measure itscoordinates.


Visual Demonstration of thePythagorean Theorem Name(s): In this activity, youll do a visual demonstration of the Pythagoreantheorem based on Euclids proof. By shearing the squares on the sides of aright triangle, youll create congruent shapes without changing the areasof your original squares.

Sketch and Investigate1. Open the sketch Shear Pythagoras.gsp.

Youll see a right triangle with squareson the sides.

2. Measure the areas of the squares.

3. Drag point A onto the line thatsperpendicular to the hypotenuse.Note that as the square becomes aparallelogram its area doesnt change.

4. Drag point B onto the line. It shouldoverlap point A so that the twoparallelograms form a singleirregular shape.

5. Drag point C so that the large square deforms to fill in the triangle.The area of this shape doesnt change either. It should appearcongruent to the shape you made with the two smallerparallelograms.

ba c

A

C

B

Step 3

ba c

BA

C

Step 4

ba c

B

C

A

Step 5

Q1 How do these congruent shapes demonstrate the Pythagoreantheorem? (Hint: If the shapes are congruent, what do you know abouttheir areas?)

ba c

C

A

B

All sketches referredto in this bookletcan be found inSketchpad |




Click on an interiorto select it. Then, inthe Measure menu,

choose Area.

To confirm that thisshape is congruent,

you can copy andpaste it. Drag thepasted copy ontothe shape on the

legs to see thatit fits perfectly.

To confirm that thisworks for any right

triangle, changethe shape of the

triangle and try theexperiment again.


The Golden Rectangle Name(s): The golden ratio appears often in nature: in the proportions of a nautilusshell, for example, and in some proportions in our bodies and faces. Arectangle whose sides have the golden ratio is called a golden rectangle.In a golden rectangle, the ratio of the sum of thesides to the long side is equal to the ratio of thelong side to the short side. Golden rectangles aresomehow pleasing to the eye, perhaps becausethey approximate the shape of our field of vision.For this reason, theyre used often in architecture,especially the classical architecture of ancientGreece. In this activity, youll construct a goldenrectangle and find an approximation to the goldenratio. Then youll see how smaller golden rectangles are found withina golden rectangle. Finally, youll construct a golden spiral.

Sketch and Investigate1. Use a custom tool to construct a square ABCD. Then construct the

squares interior.

2. Orient the square so that the control points are on the left side, oneabove the other (points A and B in the figure).

3. Construct the midpoint E of sAD.

4. Construct circle EC.

E

CB

A D

G

FE

CB

A D

G

F

CB

A D

Steps 14 Steps 58 Steps 911

5. Extend sides AD and BC with rays, as shown.

6. Construct point F where fAD intersects the circle.

7. Construct a line perpendicular to fAD through point F.

8. Construct point G where this perpendicular intersects fBC. RectangleAFGB is a golden rectangle.

9. Hide the lines, the rays, the circle, and point E.

10. Hide sAD, sDC, and sBC.

You can usethe tool 4/Square

(By Edge) fromthe sketch

Polygons.gspthat comes with

the program.

Hold down themouse button on the

Segment tool toshow the Straight

Objects palette.Drag right to choose

the Ray tool.

Select the objects;then, in the Display

menu, choose HideObjects.

a

b

a + bb =

ba

The Golden Rectangle (continued)


11. Construct sBG, sGF, and sFA.

12. Measure AB and AF.

13. Measure the ratio of AF to AB.

14. Calculate (AB + AF)/AF.

15. Drag point A or point B to confirm that your rectangle is alwaysgolden.

Q1 The Greek letter phi () is often used to represent the golden ratio.Write an approximation for .

Continue sketching to investigate the rectangle further and to construct agolden spiral.

16. Construct circle CB.

17. Construct an arc on the circlefrom point B to point D, thenhide the circle.

18. Make a custom tool for thisconstruction.

19. Make the rectangle as big as you can,then use the custom tool on points F and D. You should find thatthe rectangle constructed by your custom tool fits perfectly in theregion DFGC.

Q2 Make a conjecture about region DFGC.

20. Continue using the custom tool within yourgolden rectangle to create a golden spiral.Hide unnecessary points.

Explore More1. Let the short side of a golden rectangle have

length 1 and the long side have length .Write a proportion, cross-multiply, anduse the quadratic formula to calculate an exact value for .

2. Calculate 2 and 1/. How are these numbers related to ?Use algebra to demonstrate why these relationships hold.

Select, in order,sAF and sAB; then, inthe Measure menu,

choose Ratio.

Choose Calculatefrom the Measure

menu to openthe Calculator.

Click once on ameasurement to

enter it into acalculation.

G

F

CB

A D

Select, in order, thecircle and points B

and D. Then chooseArc On Circle

from theConstruct menu.




(the bottom tool).

If your rectanglegoes the wrong way

when you use thecustom tool, undo

and try applying it inthe opposite order.

B

A


A Sine Wave Tracer Name(s): In this exploration, youll construct an animation engine that traces outa special curve called a sine wave. Variations of sine curves are the graphsof functions called periodic functions, functions that repeat themselves. Themotion of a pendulum and ocean tides are examples of periodic functions.

Sketch and Investigate1. Construct a horizontal segment AB.

F

A B

CD

E

2. Construct a circle with center A and radius endpoint C.

3. Construct point D on sAB.

4. Construct a line perpendicular to sAB through point D.

5. Construct point E on the circle.

6. Construct a line parallel to sAB through point E.

7. Construct point F, the point of intersection of the vertical line throughpoint D and the horizontal line through point E.

Q1 Drag point D and describe what happens to point F.

Q2 Drag point E around the circle and describe what point F does.

Q3 In a minute, youll create an animation in your sketch that combinesthese two motions. But first try to guess what the path of point F willbe when point D moves to the right along the segment at the sametime as point E is moving around the circle. Sketch the path youimagine below.

Select point Dand sAB; then, in

the Constructmenu, choose

PerpendicularLine.

Dont worry, this isnta trick question!

A Sine Wave Tracer (continued)


8. Make an action button that animates point D forward on sAB andpoint E forward on the circle.

9. Move point D so that its just to the right of the circle.

10. Select point F; then, in the Display menu, choose Trace Point.

11. Press the Animation button.

Q4 In the space below, sketch the path traced by point F. Does the actualpath resemble your guess in Q3? How is it different?

12. Select the circle; then, in the Graph menu, choose Define Unit Circle.You should get a graph with the origin at point A. Point B should lieon the x-axis. The y-coordinate of point F above sAB is the value of thesine of EAD.

5 10

F

AB

CD

E

Q5 If the circle has a radius of 1 grid unit, what is its circumference ingrid units? (Calculate this yourself; dont use Sketchpad to measureit because Sketchpad will measure in inches or centimeters, not gridunits.)

13. Measure the coordinates of point B.

14. Adjust the segment and the circle until you can make the curvetrace back on itself instead of drawing a new curve every time.(Keep point B on the x-axis.)

Q6 Whats the relationship between the x-coordinate of point B and thecircumference of the circle (in grid units)? Explain why you think thisis so.

Select points D andE and choose Edit |Action Buttons |

Animation.Choose forward in

the Directionpop-up menu for

point D.

From Exploring Algebra with The Geometers Sketchpad3 2 Teaching Mathematics with The Geometers Sketchpad 2002 Key Curriculum Press

Adding Integers Name(s): They say that a picture is worth a thousand words. In the next twoactivities, youll explore integer addition and subtraction using a visualSketchpad model. Keeping this model in mind can help you visualizewhat these operations do and how they work.

Sketch and Investigate1. Open the sketch Add

Integers.gsp from thefolder 1_Fundamentals.

2. Study the problem thatsmodeled: 8 + 5 = 13. Thendrag the two dragcircles to model otheraddition problems. Noticehow the two upper arrowsrelate to the two lower arrows.

Q1 Model the problem 6 + 3.According to your sketch,what is the sum of 6 and 3?

3. Model three more problems in which you add two negative numbers.Write your equations (2 + 2 = 4, for example) below.

Q2 How is adding two negative numbers similar to adding two positivenumbers? How is it different?

Q3 Is it possible to add two negative numbers and get a positive sum?Explain.

Definition:Integers are positiveand negative wholenumbers, includingzero. On a number

line, tick marksusually represent the

integers.

drag

drag

1-1 2 3 4 5 100

+ 5

8All sketches referredtoin this booklet

can be found inSketchpad |




drag

drag

1-1-2-3-4-5 0

+-3

-6

Adding Integers (continued)

From Exploring Algebra with The Geometers Sketchpad 2002 Key Curriculum Press Teaching Mathematics with The Geometers Sketchpad 3 3


4. Model four more problems in which the sum is zero. Have the firstnumber be positive in two problems and negative in two problems.Write your equations below.

Q5 What must be true about two numbers if their sum is zero?


5. Model six more problems in which you add one positive and onenegative number. Have the first number be positive in three problemsand negative in three. Also, make sure that some problems havepositive answers and others have negative answers. Write yourequations below.

Q7 When adding a positive number and a negative number, how can youtell if the answer will be positive or negative?

drag

drag

1-1 2-2 3-3 4-4 5-5 0

+-5

5

drag

drag

1-1 2-2 3-3 4-4 5-5 0

+-7

4

Adding Integers (continued)


Q8 A classmate says, Adding a positive and a negative number seemsmore like subtracting. Explain what he means.

Q9 Fill in the blanks:

a. The sum of a positive number and a positive number is always a number.

b. The sum of a negative number and a negative number is always a number.

c. The sum of any number and is always zero.

d. The sum of a negative number and a positive number is if the positive number is larger and if the negative number is larger. (Larger here means fartherfrom zero.)

Explore More1. The Commutative Property of Addition says that for any two numbers

a and b, a + b = b + a. In other words, order doesnt matter in addition!Model two addition problems on your sketchs number line thatdemonstrate this property.

a. Given the way addition is represented in this activity, why does theCommutative Property of Addition make sense?

b. Does the Commutative Property of Addition work if one or bothaddends are negative? Give examples to support your answer.

To commutemeans to travelback and forth.

The CommutativeProperty of Addition

basically says thataddends can

commute across anaddition sign without

affecting the sum.


Points Lining Up in the Plane Name(s): If youve seen marching bands perform at football games, youveprobably seen the following: The band members, wandering in seeminglyrandom directions, suddenly spell a word or form a cool picture. Canthese patterns be described mathematically? In this activity, youll start toanswer this question by exploring simple patterns of dots in the x-y plane.

Sketch and Investigate1. Open a new sketch.

2. Choose the Point tool from the Toolbox. Then, while holding downthe Shift key, click five times in different locations (other than on theaxes) to construct five new points.

3. Measure the coordinates of thefive selected points.A coordinate system appears andthe coordinates of the five pointsare displayed.

4. Hide the points at (0, 0) andat (1, 0).

5. Choose Snap Points from theGraph menu.From now on, the points will only landon locations with integer coordinates.

Q1 For each problem, drag the five points to different locations thatsatisfy the given conditions. Then copy your solutions onto the gridson the next page.

For each point,

a. the y-coordinate equals the x-coordinate.

b. the y-coordinate is one greater than the x-coordinate.

c. the y-coordinate is twice the x-coordinate.

d. the y-coordinate is one greater than twice the x-coordinate.

e. the y-coordinate is the opposite of the x-coordinate.

f. the sum of the x- and y-coordinates is five.

g. the y-coordinate is the absolute value of the x-coordinate.

h. the y-coordinate is the square of the x-coordinate.

Holding down theShift key keeps all

five points selected.

To measurethe coordinates

of selectedpoints, choose

Coordinates fromthe Measure menu.

2

-2

E: (1.00, 2.00)D: (3.00, -1.00)C: (2.00, -2.00)B: (-1.00, -1.00)A: (3.00, 3.00)

A

B

C

D

E

To hide objects,select them and

choose Hide fromthe Display menu.

The absolute valueof a number is itspositive value.

The absolutevalue of both5 and 5 is 5.

Points Lining Up in the Plane (continued)


a. b.

-6

-3

6

3

-10 -5 105

-6

-3

6

3

-10 -5 105

c. d.

-6

-3

6

3

-10 -5 105

-6

-3

6

3

-10 -5 105

e. f.

-6

-3

6

3

-10 -5 105

-6

-3

6

3

-10 -5 105

g. h.

-6

-3

6

3

-10 -5 105

-6

-3

6

3

-10 -5 105

Points Lining Up in the Plane (continued)


Backward ThinkingIn Q1, you were given descriptions and asked to apply them to points.Here, well reverse the process and let you play detective.

6. Open the sketch Line Up.gsp from the folder 2_Lines.Youll see a coordinate system with eight points (A through H), theircoordinate measurements, and eight action buttons.

Q2 For each letter, press the corresponding button in the sketch. Like themembers of a marching band, the points will wander until theyform a pattern. Study the coordinates of the points in each pattern,then write a description (like the ones in Q1) for each one.

a.

b.

c.

d.

e.

f.

g.

h.

Explore More1. Each of the descriptions in this activity can be written as an

equation. For example, part b of Q1 (the y-coordinate is one greaterthan the x-coordinate) can be written as y = x + 1. Write an equationfor each description in Q1 and Q2.

teaching math

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