gsp teaching math.pdf

8/17/2019 GSP Teaching Math.pdf

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Teaching NotesSample Activities

Windows ® /Macintosh ®

Dynamic Geometry ® Softwarefor Exploring Mathematics

Teaching Mathematics with

Key College PublishingKey Curriculum Press


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Teaching Mathematics with The Geometer’s Sketchpad

Limited Reproduction Permission© 2002 Key Curriculum Press. All rights reserved. Key Curriculum Press grants the teacherwho purchases Teaching Mathematics with The Geometer’s Sketchpad the right to reproduceactivities and example sketches for use with his or her own students. Unauthorized copying

of Teaching Mathematics with The Geometer’s Sketchpad is a violation of federal law.

® The Geometer’s 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


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© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • iii

Contents

Teaching Notes ..................................................................................................................................... 1The Geometer’s Sketchpad and Changes in Mathematics Teaching ....................................... 1Where Sketchpad Came From ........................................................................................................ 2

Using Sketchpad in the Classroom ................................................................................................ 3A Guided Investigation: Napoleon’s Theorem ............................................................................ 4An Open -Ended Exploration: Constructing Rhombuses .......................................................... 5A Demonstration: A Visual Demonstration of the Pythagorean Theorem ............................ 6Using 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 ................................................................................................................................ 8Using Sketchpad as a Presentation Tool ....................................................................................... 8Using Sketchpad as a Productivity Tool ....................................................................................... 9The Geometer’s Sketchpad and Your Textbook ........................................................................ 10

Sample Activities ............................................................................................................................... 11Introduction ...................................................................................................................................... 11Angles ................................................................................................................................................ 12Constructing a Sketchpad Kaleidoscope .................................................................................... 13Properties of Reflection .................................................................................................................. 16Tessellations Using Only Translations ........................................................................................ 18The Euler Segment .......................................................................................................................... 20

Napoleon’s 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 ................................................................................................... 35

Parabolas in Vertex Form .............................................................................................................. 38Reflection in Geometry and Algebra ........................................................................................... 41Walking Rex: An Introduction to Vectors ................................................................................. 44Leonardo da Vinci’s Proof ............................................................................................................. 46The Folded Circle Construction .................................................................................................... 49


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iv • Teaching Mathematics with The Geometer’s 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


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© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 1

Teaching NotesIf you’ve read the Learning Guide, you’ve learnedhow to use The Geometer’s Sketchpad ® andyou’ve probably discovered that the range of things you can do with the software is greaterthan you first imagined. For all its potential

uses 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 sampleactivities —touching on a range of schoolmathematics topics —follow these teaching notes.By exploring the sample documents that areinstalled with the software, you’ll find even moreideas. Try them with your students for a sense of how Sketchpad can serve your classroom best.

The Geometer’s Sketchpad and Changes inMathematics Teaching

The way we teach mathematics —geometry in particular —has 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 mathematics by making it easy to create simple geometric figures and make conjectures about theirproperties. Learning geometry could become a series of open -ended explorations of relationships in geometric figures —a process of discovery that motivates proof, ratherthan a rehashing of proofs of theorems that students take for granted or don’t 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 from

time -consuming, mundane tasks and giving them the means to see and explore interestingrelationships.By publishing the first edition of Michael Serra’s 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 aren’t expected to prove their discoveries until after they’ve mastered

1+Φ = 1/Φ

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.


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2 • Teaching Mathematics with The Geometer’s 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 Geometer’s 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 students

to 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 Geometer’s 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 Geometer’s Sketchpad established the current generation of educational softwarethat has accelerated the change begun by The Geometric Supposers and that was spurred on by publications like Discovering Geometry and the NCTM Standards. Sketchpad’s 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 algebraiccapabilities —as well as the unbounded extensibility offered by its custom tools —Sketchpad broadens the scope of what it’s 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 widelyembraced — by mathematics and educational researchers, by teachers across the curriculum,and by millions of students —that 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 Geometer’s Sketchpad was developed as part of the Visual Geometry Project, a NationalScience Foundation –funded 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 software

tested 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.


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© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 3

In Sketchpad’s 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 the

program’s analytic and graphing capabilities. By 1999, the Teaching, Learning, and Computingnational teacher survey conducted by the University of California, Irvine, found that thenation’s 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 program’s usefulness in algebra, pre -calculus, and calculus classes, while increasing boththe ease of use in earlier grades and the software’s curriculum development authoring tools.Classroom research continues to form the basis for further development of the software andaccompanying materials.

Using Sketchpad in the Classroom

The Geometer’s 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 Sketchpad’s powerful transformationcapabilities and to custom tools allowing students to explore non -Euclidean geometries. Evenartists and mechanical drawing professionals have been enthralled by Sketchpad’s powerand elegance. It’s 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. You’ll find tools forconstructing regular polygons, defining mathematical symbols, exploring non -Euclideangeometries, composing and combining functions, and much more.) In this section, we’llconcentrate 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, you’ll want students to collaborate andcommunicate their findings. Sketchpad’s annotation features encourage students to articulatemathematical ideas. Whatever approach you take to using Sketchpad, it can serve as aspringboard for discussion and communication. We’ll 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 Geometer’sSketchpad, © 1999 by Key Curriculum Press.


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4 • Teaching Mathematics with The Geometer’s Sketchpad © 2002 Key Curriculum Press

A Guided Investigation: Napoleon’s 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 this —and, in fact, any —Sketchpadinvestigation 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 discussed

their findings in pairs orsmall groups, it’s importantto discuss them as a largegroup. Ask students to shareany special cases they’vediscovered, 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 shape

triangle 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.

Napoleon ’ s Theorem Name(s):French emperor Napoleon Bonaparte fancied himself as something of anamateur geometer and liked to hang out with mathematicians. Thetheorem you ’ ll 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 you ’ releft with a figure like the one shown at right.

4. Make a custom tool for this construction.Next, you ’ ll 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 Napoleon ’ s theorem might be.

Explore More

1. Construct segments connecting each vertex of your original trianglewith 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 chooseCreate New Tool

from the CustomTools menu

in the Toolbox(the bottom tool).

A C

B

Be sure to attacheach equilateral

triangle to a pair oftriangle ABC ’s

vertices. If yourequilateral triangle

goes the wrong way(overlaps the interior

of ∆ ABC ) or is notattached properly,

undo and tryattaching it again.


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© 2002 Key Curriculum Press Teaching Mathematics with The Geometer ’s Sketchpad • 5

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 students

used. Offering students anopen -ended constructionproblem also gives you theopportunity to emphasizethe important distinction between 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 that ’snot 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


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6 • Teaching Mathematics with The Geometer ’s Sketchpad © 2002 Key Curriculum Press

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 ” of the theorem. The sketch usedin the lesson is a pre -madesketch of some complexity.Students aren ’t expected tocreate this constructionthemselves to discover the

Pythagorean 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 they ’ve beendoing other investigationsrelated to the Pythagoreantheorem.

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


1. Open the sketch Shear Pythagoras.gsp .You’ll see a right triangle with squareson the sides.

2. Measure the areas of the squares.

3. Drag point A onto the line that’s

perpendicular to the hypotenuse.Note that as the square becomes aparallelogram its area doesn’t 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 doesn’t change either. It should appearcongruent to the shape you made with the two smallerparallelograms.

b

a c

A

C

B

Step 3

b

a c

B

A

C

Step 4

b

a 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?)

b

a c

C

A

B

All sketchesreferred to in this

bookletcan be found in

Sketchpad |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.


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Using Sketchpad in Different Classroom Settings

Schools 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 you ’re in a classroom with one computer, one

computer 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 of students 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 a

demonstration 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 Geometer ’s 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 which

you 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 they ’re expected to do.• That students have some kind of written explanation of the investigation or problem

they ’re to work on. It ’s 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 of sketches 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.


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• 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 of

teachers using The Geometric Supposers) suggests that even if enough computers areavailable for students to work individually, it ’s perhaps best to have students work in pairs.Students learn best when they communicate about what they ’re 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 they ’re doingand to compare their findings with those of their nearest neighbor —they 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 Tool

You ’ll find that Sketchpad ’s features —especially its text capabilities, multi - page documentstructure, and action buttons —make 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 they ’ve done in a sketch.Teachers and students can use action

buttons 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 a button. 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 isn ’t 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

π = 0.760 in.

m AB = 1.347 in.

Area ABA'B'= 1.815 in 2Area CC' = 1.815 in 2Radius 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


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The Geometer ’s Sketchpad and Your Textbook

The variety of ways Sketchpad can be used makes it the ideal tool for exploring schoolmathematics, regardless of the text you ’re 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 you

require 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 Serra ’s 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 can be done using Sketchpad.The Discovering Geometry student text includes ten Geometer ’s Sketchpad Projects and

numerous Investigations and Take Another Look suggestions for using Sketchpad. Morethan 60 lessons best - suited for exploration with Sketchpad were adapted and collected as blackline masters in the ancillary book Discovering Geometry with The Geometer’s 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 Teacher’s Resource Book comeswith demonstration sketches corresponding to Discovering Geometrylessons.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 Press ’s Discovering Geometry combined with The Geometer ’s Sketchpad.If you ’re using a text other than Discovering Geometry, ask the publisher whether Sketchpadancillaries are available.

Exploring Geometry with The Geometer’s 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 year ’s worth of activities to cover nearly allthe content of a typical high school geometry course using The Geometer ’s Sketchpad. Andother activity books could occupy a large part of the year in other mathematics courses, too.We don ’t, however, advocate that you abandon other teaching methods in favor of usingthe computer. It ’s 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, it ’s important that students don ’t get the mistaken impression thatmathematics exists only in their books and on their computer screens.


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From Geometry Activities for Middle School Students with The Geometer ’s Sketchpad


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 Sketchpad ’scalculator 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 Sketchpad ’s 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) interior

angles.Write any conjectures based on your exploration.


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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 space

to deselect objects. b. Select three or four points in

clockwise 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 .


18/100

Constructing a Sketchpad Kaleidoscope (continued)



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 if you 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 image by 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


19/100

Constructing a Sketchpad Kaleidoscope (continued)



c. Using the Compass tool, construct another circle with its center at thecenter of your kaleidoscope, but this time let the radius be about half the 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


20/100

From Exploring Geometry with The Geometer ’s Sketchpad


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, you ’ll investigate theproperties of reflections that make a reflection the “mirror image ” of the

original.

Sketch and Investigate: Mirror Writing

1. Construct verticalline AB.

2. Construct point Cto the right of the line.

3. Mark d AB 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 Figures

8. Turn off Trace Points forpoints C and C´.

9. In the Display menu, chooseErase Traces .

10. Construct j CDE.

11. Reflect j CDE (sides andvertices) over d AB.

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 Display

menu, you ’ll seeTrace Points

checked. Choose itto uncheck it.

D'

E'

C'A

C

DEB

Select the entirefigure; then, in the

Transform menu,choose Reflect .


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Properties of Reflection (continued)



13. Measure the lengths of the sides of triangles CDE and C´DÉ´.

14. Measure one angle in j CDE and measure the corresponding anglein j C´DÉ´.

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 j CDE, are the verticesoriented in a clockwise or counter - clockwise direction? In whatdirection (clockwise or counter -clockwise) are vertices C´, D´, and Eóriented 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 relationships between the dashed segments andthe mirror line.

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

Explore More

1. Suppose Sketchpad didn ’t have a Transform menu. How could youconstruct a given point ’s 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 yourmiddle selection.

Then, in theMeasure 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

D

EB

You may wish toconstruct points of

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


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

Sketch

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

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.We ’ll 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 C

and 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 still

have vector AC marked.)10. Repeat this process until you have

a column of tiles all the way upyour sketch. Change the color onevery other tile to create a pattern.

C

A B

C'

Steps 1 –3

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 8 –10

Select the vertices inconsecutive order;

then, in theConstruct menu,

choose PolygonInterior .


23/100

Tessellations Using Only Translations (continued)



11. Mark vector AB. Thenselect all the polygoninteriors in your columnof tiles and translate them by 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 More

1. Animate your tessellation. To do this, select the original polygon (orany 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, polygoninteriors —anything 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


24/100



The Euler Segment Name(s):In this investigation, you ’ll look for a relationship among four points of concurrency: the incenter, the circumcenter, the orthocenter, and thecentroid. You ’ll 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 triangle ’svertices 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 angle bisectors 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 circumcenter

a label that identifies it.7. Use the Orthocenter tool on the

same 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 Ce Ci

I


25/100

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 More

1. Construct a circle centered at the midpoint of the Euler segment andpassing 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 them

have to do with the altitudes and the orthocenter.)2. Once you ’ve constructed the nine -point circle, drag your triangle

around 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.)


26/100



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


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 you ’releft with a figure like the one shown at right.

4. Make a custom tool for this construction.

Next, you ’ll use your custom tool to construct equilateral triangles on the

sides of an arbitrary triangle.5. Open a new sketch.

6. Construct j ABC.7. Use the custom tool to construct

equilateral triangles on each sideof j 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 j ABC.

Q1 State what you think Napoleon ’s theorem might be.

Explore More

1. Construct segments connecting each vertex of your original trianglewith 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 ABC ’s

vertices. If yourequilateral triangle

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

properly, undoand try attaching

it again.


27/100


28/100



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


1. 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 of the 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 paper

if 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


29/100



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? You ’ll discover the answer in this investigation. Or, if you have a hunch already, this investigation will help confirm your hunch

and give you more insight into it.


1. Construct s AB.

2. Construct s AC on s AB.3. Construct lines

perpendicular to s ABthrough points A and C.

4. Construct circle CB.5. Construct point D where

this circle intersects theperpendicular line.

6. Construct a line through point D, parallel to s AB.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 s AB, point A,and point C . Then,

in the Constructmenu, choose

PerpendicularLine .

Be sure to releasethe mouse —or click

the second time —with the pointer

over point B .

Select the vertices ofthe rectangle in

consecutive order.Then, in the Construct

menu, choose

QuadrilateralInterior .

Select point A andpoint C . Then, in the

Measure menu,choose Distance .

Repeat tomeasure AE .


30/100

A Rectangle with Maximum Area (continued)



In Steps 12 –14, you ’ll explore this relationship graphically.

12. Plot the measurements for the length of s AC 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 cm

m AC = 3.69 cm

Perimeter ACDE = 9.41 cmArea ACDE = 3.74 cm 2

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 More

1. Investigate area/perimeter relationships in other polygons. Make aconjecture about what kinds of polygons yield the greatest area for agiven perimeter.

2. What ’s the equation for the graph you made? Let AC be x and let AB be (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,m s AC and Area

ACDE . Then choosePlot As (x, y) from

the Graph menu.If you can ’t 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.


31/100


32/100



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, they ’re used often in architecture,especially the classical architecture of ancientGreece. In this activity, you ’ll construct a goldenrectangle and find an approximation to the goldenratio. Then you ’ll see how smaller golden rectangles are found withina golden rectangle. Finally, you ’ll construct a golden spiral.


1. Use a custom tool to construct a square ABCD. Then construct thesquare ’s 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 s AD.4. Construct circle EC.

E

CB

A D

G

FE

CB

A D

G

F

CB

A D

Steps 1 – 4 Steps 5 – 8 Steps 9 – 11

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

6. Construct point F where f AD intersects the circle.

7. Construct a line perpendicular to f AD through point F.

8. Construct point G where this perpendicular intersects f BC. Rectangle AFGB is a golden rectangle.

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

10. Hide s AD, s DC, and s BC.

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 choosethe Ray tool.

Select the objects;then, in the Display

menu, choose HideObjects .

a

b

a + b

b =

b

a


33/100

The Golden Rectangle (continued)



11. Construct s BG, s GF, and s FA.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 ( ø ) is often used to represent the golden ratio.Write an approximation for ø .

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 More

1. Let the short side of a golden rectangle havelength 1 and the long side have length ø .Write a proportion, cross -multiply, anduse the quadratic formula to calculate an exact value for ø .

2. Calculate ø2 and 1/ ø . How are these numbers related to ø ?Use algebra to demonstrate why these relationships hold.

Select, in order,s AF and s AB ; then, inthe Measure menu,

choose Ratio .

Choose Calculatefrom the Measure

menu to openthe Calculator.Click once on a

measurement toenter it into a

calculation.

G

F

CB

A D

Select , in order, thecircle and points B

and D . Then chooseArc On Circle

from theConstruct menu.

Select the entirefigure; then choose

Create New Toolfrom the Custom

Tools menuin the Toolbox

(the bottom tool).

If your rectanglegoes the wrong way

when you use thecustom tool, undo

and try applying it in

the opposite order.

B

A


34/100



A Sine Wave Tracer Name(s):In this exploration, you ’ll 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 s AB.

4. Construct a line perpendicular to s AB through point D.5. Construct point E on the circle.

6. Construct a line parallel to s AB through point E.7. Construct point F, the point of intersection of the vertical line through

point 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, you ’ll create an animation in your sketch that combinesthese two motions. But first try to guess what the path of point F will be 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 D and s AB ; then, in

the Constructmenu, choose

PerpendicularLine .

Don ’t worry, this isn ’ta trick question!


35/100

A Sine Wave Tracer (continued)



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

9. Move point D so that it ’s 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 actual

path 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 s AB is the value of thesine of ∠EAD.

5 10

F

A

BC

D

E

Q5 If the circle has a radius of 1 grid unit, what is its circumference ingrid units? (Calculate this yourself; don ’t 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 What ’s 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.


36/100

From Exploring Algebra with The Geometer ’s Sketchpad


Adding Integers Name(s):They say that a picture is worth a thousand words . In the next twoactivities, you ’ll 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 that ’smodeled: 8 + 5 = 13. Thendrag the two “drag ”circles 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 referred

toin this booklet

can be found inSketchpad |

Samples | Teach -ing Mathematics



drag

drag

1-1-2-3-4-50

+ -3

-6


37/100

Adding Integers (continued)



Q4 Model the problem 5 + –5.According to your sketch,what is the sum of 5 and –5?

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?

Q6 Model the problem 4 + –7.According to your sketch,what is the sum of 4 and –7?

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


38/100

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 andif the negative number is larger. ( “Larger ” here means fartherfrom zero.)

Explore More

1. The Commutative Property of Addition says that for any two numbersa and b, a + b = b + a. In other words, order doesn ’t matter in addition!Model two addition problems on your sketch ’s 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 commute means to travelback and forth.

The Commutative Property of Addition

basically says that

addends cancommute across anaddition sign without

affecting the sum.


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Points “Lining Up ” in the Plane Name(s):If you ’ve seen marching bands perform at football games, you ’veprobably 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, you ’ll start toanswer this question by exploring simple patterns of dots in the x- y plane.



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 its

“ positive value. ”The absolutevalue of both5 and – 5 is 5.


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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


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Points “Lining Up ” in the Plane (continued)



Backward Thinking

In Q1, you were given descriptions and asked to apply them to points.Here, we ’ll reverse the process and let you play detective.

6. Open the sketch Line Up.gsp from the folder 2_Lines .You’ll 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 More

1. Each of the “descriptions ” in this activity can be written as anequation. 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.

2. Add your own action buttons to those in Line Up.gsp , then see if your classmates can come up with descriptions or equations for yourpatterns. Instructions on how to do this are on page 2 of the sketch.

All sketches referredto in this bookletcan be found in

Sketchpad |

Samples | Teach -ing Mathematics




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Parabolas in Vertex Form Name(s):Things with bilateral symmetry—such as the human body —have parts onthe sides that come in pairs (such as ears and feet) and parts down themiddle there ’s just one of (such as the nose and bellybutton). Parabolas arethe same way. Points on one side have corresponding points on the other.But one point is unique: the vertex. It ’s right in the middle, and —like your

nose —there ’s just one of it. Not surprisingly, there ’s a common equationform for parabolas that relates to this unique point.


1. Open the sketch Vertex Form.gsp from the folder 3_Quads .You’ll see an equation in the form y = a(x – h)2 + k, with a, h, and k filled in,and sliders for a, h, and k. Adjust the sliders (by dragging the points at theirtips) and watch the equation change accordingly. There’s no graph yetbecause we wanted you to practice using Sketchpad’s graphing features.

2. Choose Plot New Functionfrom the Graph menu.The New Function dialogbox appears. If necessary,move it so that you can seea, h, and k’s measurements.

3. Enter a*(x–h)̂ 2+ k andclick OK.

-5

-2

2y = 1.4(x – (0.9))2 – 1.6

f x( ) = a ⋅ x-h( )2+k

xP = 2.5

k = -1.6

h = 0.9

a = 1.4P

Sketchpad plots the function for the current values of a, h, and k.You ’ll now plot the point on the parabola whose x-coordinate is the sameas point P’s.4. Calculate f (xP), the value of the function f evaluated at xP.

You’ll see an equation for f(xP), the value of the function f evaluated at xP.

5. Select, in order, xP and f (xP); then choose Plot as (x, y) from theGraph menu. A point is plotted on the parabola.

Q1 Using paper and pencil or a calculator, show that the coordinates of the new point satisfy the parabola ’s equation. Write your calculation

below. If the numbers are a little off, explain why this might be.

All sketches referredto in this bookletcan be found in

Sketchpad |Samples | Teach -ing Mathematics



To enter a, h, and k,click on their

measurements in thesketch. To enter x,

click on the x inthe dialog box.

Choose Calculatefrom the Measure

menu. Click on thefunction equation

from step 3.Then click onxP to enter it.

Now type a closeparenthesis —“ )”—

and click OK.


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Parabolas in Vertex Form (continued)



Exploring Families of Parabolas

By dragging point P, you ’re exploring how the variables x and y varyalong one particular parabola with particular values for a, h, and k . Forthe rest of this activity, you ’ll change the values of a, h, and k, whichwill change the parabola itself, allowing you to explore whole familiesof parabolas.

Q2 Adjust a’s slider and observe the effect on the parabola.Summarize a’s role in the equation y = a(x – h)2 + k. Be sure to discussa’s sign (whether it ’s positive or negative), its magnitude (how big orsmall it is), and anything else that seems important.

Q3 Dragging a appears to change all the points on the parabola but one:the vertex. Change the values of h and k; then adjust a again, focusingon where the vertex appears to be.How does the location of the vertex relate to the values of h and k ?

Q4 Adjust the sliders for h and k . Describe how the parabola transformsas h changes. How does that compare to the transformation thatoccurs as k changes?

Here ’s how the Plot as (x, y) command in the Graph menu works:Select two measurements and choose the command. Sketchpad plots apoint whose x-coordinate is the first selected measurement and whose y-coordinate is the second selected measurement.

6. Use Plot as (x, y) to plot the vertex of your parabola.


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Parabolas in Vertex Form (continued)



Q5 Write the equation in vertex form y = a(x – h)2 + k for each paraboladescribed. As a check, adjust the sliders so that the parabola is drawnon the screen.

a. vertex at (1, −1); y- intercept at (0, 4)

b. vertex at ( −4, −3); contains the point ( −2, −1)

c. vertex at (5, 2); contains the point (1, −6)

d. same vertex as the parabola

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