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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 teacher who purchases Teaching Mathematics with The Geometer’s Sketchpad the right to reproduce activities 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 are registered trademarks of Key Curriculum Press. Sketchpad is a trademark of Key Curriculum Press. All other brand names and product names are trademarks or registered trademarks of their respective holders. Key Curriculum Press 1150 65th Street Emeryville, California 94608 USA 510-595-7000 http://www.keypress.com/ Spectrum Educational Supplies Limited 125 Mary Street Aurora, Ontario L4G 1G3 1-800-668-0600 [email protected] http://www.spectrumed.com/index.html
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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 teacher who purchases Teaching Mathematics with The Geometer’s Sketchpad the right to reproduce activities 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 are registered trademarks of Key Curriculum Press. Sketchpad is a trademark of Key Curriculum Press. All other brand names and product names are trademarks or registered trademarks of their respective holders. Key Curriculum Press 1150 65th Street Emeryville, California 94608 USA 510-595-7000 http://www.keypress.com/

Spectrum Educational Supplies Limited 125 Mary Street

Aurora, Ontario L4G 1G3 1-800-668-0600

[email protected] http://www.spectrumed.com/index.html

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

Contents Teaching Notes ............................................................................................................1

The Geometer’s Sketchpad and Changes in Mathematics Teaching.................................................. 1 Where Sketchpad Came From.................................................................................................................. 2 Using Sketchpad in the Classroom.......................................................................................................... 2 Using Sketchpad in Different Classroom Settings ................................................................................ 7 Using Sketchpad as a Presentation Tool................................................................................................. 8 Using Sketchpad as a Productivity Tool................................................................................................. 9 The Geometer’s Sketchpad and Your Textbook .................................................................................... 9

Sample Activities ......................................................................................................11 Angles ........................................................................................................................................................ 12 Constructing a Sketchpad Kaleidoscope .............................................................................................. 13 Properties of Reflection ........................................................................................................................... 16 Tessellations Using Only Translations.................................................................................................. 18 The Euler Segment ................................................................................................................................... 20 Napoleon’s Theorem ............................................................................................................................... 22 Constructing Rhombuses........................................................................................................................ 23 Midpoint Quadrilaterals ......................................................................................................................... 24 A Rectangle with Maximum Area ......................................................................................................... 25 Visual Demonstration of the Pythagorean Theorem .......................................................................... 27 The Golden Rectangle ............................................................................................................................. 28 A Sine Wave Tracer ................................................................................................................................. 30 Adding Integers........................................................................................................................................ 32 Points “Lining Up” in the Plane............................................................................................................. 35 Parabolas in Vertex Form........................................................................................................................ 38 Reflection in Geometry and Algebra..................................................................................................... 41 Walking Rex: An Introduction to Vectors ............................................................................................ 44 Leonardo da Vinci’s Proof ...................................................................................................................... 46 The Folded Circle Construction ............................................................................................................. 48 The Expanding Circle Construction ...................................................................................................... 52 Distances in an Equilateral Triangle...................................................................................................... 55 Varignon Area .......................................................................................................................................... 59 Visualizing Change: Velocity ................................................................................................................. 63 Going Off on a Tangent........................................................................................................................... 67 Accumulating Area.................................................................................................................................. 70

Also Available from Spectrum...............................................................................74

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

Teaching Notes If you’ve read the Learning Guide, you’ve learned how to use The Geometer’s Sketchpad and you’ve probably discovered that the range of things you can do with the software is greater than you first imagined. For all its potential uses though, Sketchpad was designed primarily as a teaching and learning tool. In this section, we establish a context for Sketchpad in geometry teaching and offer suggestions for using Sketchpad in different ways in different classroom settings. More than 20 sample activities—touching on a range of school mathematics topics—follow these teaching notes. By exploring the sample documents that are installed with the software, you’ll find even more ideas. Try them with your students for a sense of how Sketchpad can serve your classroom best.

The Geometer’s Sketchpad and Changes in Mathematics Teaching The way we teach mathematics—geometry in particular—has changed, thanks to a few important developments in recent years. Alternatives to a strictly deductive approach are available after more than a century of failing to reach a majority of students. (The National Assessment of Educational Progress found in 1982 that doing proofs was the least liked mathematics 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 Center developed a landmark piece of instructional software that enabled teachers and students to use computers as teaching and learning tools rather than just as drillmasters. The Geometric Supposers, for Apple II computers, encouraged students to invent their own mathematics by making it easy to create simple geometric figures and make conjectures about their properties. Learning geometry could become a series of open-ended explorations of relationships in geometric figures—a process of discovery that motivates proof, rather than 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 and Evaluation Standards for School Mathematics (the Standards) which called for significant changes in the way mathematics is taught. In the teaching of geometry, the Standards called for decreased emphasis on the presentation of geometry as a complete deductive system and a decreased emphasis on two-column proofs. Across the curriculum, the Standards called for an increase in open exploration and conjecturing and increased attention to topics in transformational geometry. In their call for change, the Standards recognized the impact that 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 interesting relationships. By publishing the first edition of Michael Serra’s Discovering Geometry: An Inductive Approach in 1989, Key Curriculum Press joined the forces of change. Discovering Geometry, a high school geometry textbook, takes much the same approach that the creators of The Geometric Supposers espoused: Students should create their own geometric constructions and themselves formulate the mathematics to describe relationships they discover. With Discovering Geometry, students working in cooperative groups do investigations using tools of geometry to discover properties. Students look for patterns and use inductive reasoning to make conjectures. They aren’t expected to prove their discoveries until after they’ve mastered 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 patty papers and The Geometer’s Sketchpad. This approach is consistent with research done by the Dutch mathematics educators Pierre van Hiele and Dina van Hiele-Geldof. From classroom observations, the van Hieles learned that 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 visualize or to encourage them to make conjectures. A main goal of The Supposers, Discovering Geometry, and, now, The Geometer’s Sketchpad is to bring students through the first three levels,

1+Φ = 1/Φ

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

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

encouraging a process of discovery that more closely reflects how mathematics is usually invented: A mathematician first visualizes and analyzes a problem, making conjectures before attempting a proof. The Geometer’s Sketchpad established the current generation of educational software that 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 unique Dynamic Geometry enables students to explore relationships interactively so that they can see change in mathematical diagrams as they manipulate them. With this breakthrough, along with the completeness of its construction, transformation, analytic, and algebraic capabilities—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 extent never seen before. In the ten years of its existence, teachers have taken Sketchpad outside the geometry classroom and into algebra, calculus, trigonometry, and middle-school mathematics courses; and ongoing development of the software has refined it for these wider uses. The Dynamic Geometry paradigm pioneered by Sketchpad has been so widely embraced—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 Dynamic Geometry by name. Concurrent development of Macintosh, Windows, Java, and handheld versions of Sketchpad in a number of different languages ensures the most powerful and up-to-date geometry tool is always available to a wide variety of school computing environments throughout the world.

Where Sketchpad Came From The Geometer’s Sketchpad was developed as part of the Visual Geometry Project, a National Science Foundation–funded project under the direction of Dr. Eugene Klotz at Swarthmore College and Dr. Doris Schattschneider at Moravian College in Pennsylvania. In addition to Sketchpad, the Visual Geometry Project (VGP) has produced The Stella Octangula and The Platonic Solids: videos, activity books, and manipulative materials also published by Key Curriculum Press. Sketchpad creator and programmer Nicholas Jackiw joined the VGP in the summer of 1987. He began serious programming work a year later. Sketchpad for Macintosh was developed in an open, academic environment in which many teachers and other users experimented with early versions of the program and provided input to its design. Nicholas came 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 word spread and more people heard of Sketchpad or saw it demonstrated at conferences. The openness with which Sketchpad was developed generated an incredible tide of feedback and enthusiasm for the program. By the time of its release in the spring of 1991, it had been used by hundreds of teachers, students, and other geometry lovers and was already the most talked about and awaited piece of school mathematics software in recent memory. In Sketchpad’s first year, Key Curriculum Press began to study how the program was being used effectively in schools. Funded in part by a grant for small businesses from the National Science Foundation, this research is reflected in these teaching notes, in curriculum materials, and in new versions of Sketchpad. Version 2 of the program, released in April 1992, introduced improved transformation and presentation capabilities and brought tools for 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 Computing national teacher survey conducted by the University of California, Irvine, found that the nation’s mathematics teachers rated Sketchpad the “most valuable software for students” by a large margin. Version 4 of the software, introduced in the fall of 2001, dramatically expands the program’s usefulness in algebra, pre-calculus, and calculus classes, while increasing both the 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 and accompanying materials.

Using Sketchpad in the Classroom The Geometer’s Sketchpad was designed initially primarily for use in high school geometry classes. Testing has shown, though, that its ease of use makes it possible for younger students to use Sketchpad successfully, and the power of its features has made it attractive to instructors of college-level mathematics and teacher pre-service and inservice courses. College instructors are drawn particularly to Sketchpad’s powerful transformation capabilities and to custom tools allowing students to explore non-Euclidean geometries. Even artists and mechanical drawing professionals have been enthralled by Sketchpad’s power and elegance. It’s a

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

testament to the versatility of the software that the same tool can 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 additional tools that help particularize the program to your classroom needs. You’ll find tools for constructing regular polygons, defining mathematical symbols, exploring non-Euclidean geometries, composing and combining functions, and much more.) In this section, we’ll concentrate 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 toward discovering a specific property or small set of properties, or you may want to pose an open-ended question or problem and ask students to try to discover as much as they can about it. Alternatively, you may want to prepare for students an interactive demonstration that models a particular concept. In any case, you’ll want students to collaborate and communicate their findings. Sketchpad’s annotation features encourage students to articulate mathematical ideas. Whatever approach you take to using Sketchpad, it can serve as a springboard for discussion and communication. We’ll look at examples of three approaches to using Sketchpad in the classroom: a guided investigation, an open-ended exploration, and a demonstration. These three examples come from Exploring Geometry with The Geometer’s Sketchpad, © 1999 by Key Curriculum Press. (This publication is available in Canada from Spectrum Educational Supplies.)

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

A Guided Investigation: Napoleon’s Theorem The purpose of this investigation is to guide students to some specific conjectures. They are given instructions to construct a figure with certain specifically defined relationships: in this case, a triangle with equilateral triangles constructed on its sides. Students manipulate their construction to see what relationships they find that can be generalized for all triangles. After this experimentation, students are asked to write conjectures. An important aspect of this— and, in fact, any—Sketchpad investigation is that by manipulating a single figure a student can potentially see every possible case of that figure. Here they have visual proof that the Napoleon triangle of an arbitrary triangle is always equilateral, even as the original triangle changes from acute to right to obtuse, from scalene to isosceles to equilateral. Suggestions are made for further, open-ended investi-gation for students who finish first. In this Explore More suggestion, students can discover that the segments in question are congruent, are concurrent, and intersect to form 60° angles. After students have discussed their findings in pairs or small groups, it’s important to discuss them as a large group. Ask students to share any special cases they’ve discovered, and use your questions to emphasize which relationships can be generalized for all triangles: “Was the Napoleon triangle always equilateral even as you changed your original triangle from being acute to being obtuse? Were the three segments 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 for properties students discover (for example, the point of concurrency they discover in Explore More is called the Fermat point) and agree as a class on wording for students’ conjectures as a way 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 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 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 Rhombuses In an open-ended exploration there is not a specific set of properties that students are expected to discover as outcomes of the lesson. A question or problem is posed with a few suggestions about how to use Sketchpad to explore the problem. Different students will discover or use different relationships in their constructions and write their findings in their own words. In this example, students are asked to come up with as many ways as they can to construct a rhombus. Again, various construction methods should be discussed in small groups, then with the whole class. To bring closure to the lesson you might want to compile on the chalkboard a list of all the properties your students used. Offering students an open-ended construction problem also gives you the opportunity to emphasize the important distinction between a drawing and a construction. For example, if students have actually used defining properties of a rhombus in their constructions, it should be possible to manipulate their figure into any size or shape rhombus and it should be impossible to distort the figure into anything that’s not 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 Theorem A teacher (or for that matter, a student) can use Sketchpad to prepare a demonstration for others to use. Sometimes a complex construction can nicely show a property, but it might be impractical to have all students do the construction themselves. In that case, teachers might use a demonstration sketch accompanied by an activity sheet. Before using this demonstration, students can actually discover the Pythagorean theorem themselves in a guided investi-gation. The purpose of this lesson, though, is as a demon-stration of a visual “proof” of the theorem. The sketch used in the lesson is a pre-made sketch of some complexity. Students aren’t expected to create this construction themselves to discover the Pythagorean theorem, but they have a chance with this demonstration to look at it in a new and interesting way. This demonstration might be done most efficiently as a whole-class demonstration with you or a student working at an overhead projector. Alternatively, you could reproduce the activity master for students to use on their own time or at the end of a lab period in which they’ve been doing other investigations related to the Pythagorean theorem.

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.

Sketch and Investigate

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’sperpendicular 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 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.

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© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 7

Using Sketchpad in Different Classroom Settings Schools use computers in a variety of classroom settings. Sketchpad was designed with this in mind, and its display features can be optimized for these different settings. Teaching strategies also need to be adapted to available resources. What follows are some suggestions for 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 Computer Perhaps 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 conjectures made working at their desks or tables using standard geometry tools such as a compass and straightedge. In that case, each group would have an opportunity during a class period to use the computer for a short time. Alternatively, you can give each group a day on which to do an investigation on the computer while other groups are doing the same or different investigations at their desks. A single computer without a projection device or large-screen monitor has limited use as a demonstration tool. Although preferences can be set in Sketchpad 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 Device A variety of devices are available that plug into computers so that the display can be output to a projector, a large-screen monitor, an LCD device used with an overhead projector, or a large-format touch panel. The Geometer’s Sketchpad was designed to work well with these projection devices, increasing your options considerably for classroom uses. You or a student can act as a sort of emcee to an investigation, asking the class as a whole things like, “What should 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 students can prepare demonstrations, or students can make presentations of findings that they made using 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 be distorted 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 introductory demonstrations on Sketchpad in front of the whole class prepares their students to use it in a lab with a minimum of lab-time lost to training. For demonstrations, we recommend using large display text in a bold style and formatting illustrations with thick lines to make text and figures clearly visible from all corners of a classroom.

A Classroom with a Handful of Computers If you can divide your class into groups of three or four students so that each group has access to a computer, you can plan whole lessons around doing investigations with the computers. 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 on which students have room to record some of their findings; but for some open-ended explorations the problem or question could simply be written on the chalkboard or typed 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 operate the computer. • That students in a group who are not actually operating the computer are expected to contribute 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 keeping students on task. • That students’ findings are summarized in a whole-class discussion to bring closure to the lesson.

A Computer Lab The 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 are available 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 students working together can better stimulate ideas and lend help to one

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

another. If you do have students working at their own computers, encourage them to talk about what they’re doing and to compare their findings with those of their nearest neighbor—they should peek over each others’ shoulders. The suggestions above for students working in small groups apply to students 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-formatted disks) 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 document structure, 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 that describe salient features of a construction. Captions can highlight properties that a construction 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 can communicate 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 hide geometric objects and text or to initiate animations. Buttons can also be sequenced so that procedures and explanations of a construction can be “played” with the click of a button. In other words, action buttons turn sketches into presentations. Text and action buttons make possible presentations without presenters: A sufficiently annotated sketch could speak for itself when opened by another user at a time when the sketch creator isn’t around to explain it. A presentation, in this context, is not necessarily designed for a group audience looking at an overhead display. The audience for an annotated sketch might be a fellow student or a teacher. Teachers who ask students to hand in assignments in the form of sketches can ask students to create presentations using action buttons and to explain their work in captions. Sketchpad’s web integration facilities allows you to draw on the full resources of the Internet. Action buttons allow you to link to web resources to provide additional explorations, survey real-world applications, or establish the historical context of a particular mathematics exploration. In addition, if you’re interested in publishing web pages of your own, Sketchpad allows you to export your activities to the web, where you can integrate them with the full set of multimedia components and hyperlinked resources available to web page authors, and share them over the net with users across the world. Users who visit your web page will be able to interact with your page’s Dynamic Geometry illustrations whether they have Sketchpad or not! By browsing through the sample documents that come with Sketchpad you can get ideas for different ways sketch captions can be used to communicate mathematically.

A Captioned Sketch

m AB( )2

š = 0.760 in.

m AB = 1.347 in.

Area ABA'B' = 1.815 in2

Area CC' = 1.815 in2

Radius 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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© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 9

Using Sketchpad as a Productivity Tool The Reference Manual describes how to use the Edit menu to cut, copy, and paste Sketchpad objects into other applications, such as graphics or word processing programs. These features make Sketchpad an extremely useful productivity tool for anyone, including teachers and students, 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 word processing program. All of the graphics in the sample activities and most of the graphics in the documentation were created in Sketchpad and pasted into Microsoft Word. Sketchpad stores objects in the clipboard both as Sketchpad objects, which behave as such when pasted back into a sketch, and graphic images, which are recognized by virtually any program that deals with graphics. Sketchpad graphics will act exactly like images produced in most other graphics programs and will give excellent results when printed. If you’re writing a book or article that will be printed professionally, Sketchpad graphics can even be output on a typesetting machine with very high quality results. Lines and rays are truncated when 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 use in tests and worksheets. Then you can easily change figures if you need variations. You can edit labels and type in measurements of angles and lengths. Even figures that you might find easier to draw by hand have the advantage, when done with Sketchpad, that they can be saved, easily modified, and used again and again.

Excircles of a Triangle

The Geometer’s Sketchpad and Your Textbook The variety of ways Sketchpad can be used makes it the ideal tool for exploring school mathematics, regardless of the text you’re using. Use Sketchpad to demonstrate concepts presented 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 the way, 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 with diagrams dynamically will deepen students’ understanding of concepts and, in formal contexts, will make proof more relevant. Sketchpad is ideally suited for use with books that take a discovery approach to teaching and learning geometry. In Michael Serra’s Discovering Geometry, for example, students working in small groups do investigations and discover geometry concepts for themselves, before they attempt proof. Many of these investigations call for constructions that could be done 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. More than 60 lessons best-suited for exploration with Sketchpad were adapted and collected as blackline masters in the ancillary book Discovering

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

Geometry with The Geometer’s Sketchpad. These Sketchpad lessons have the same titles and guide students to the same conjectures as the corresponding Discovering Geometry lessons. A collection of Sketchpad documents accompany this book on CD-ROM. The Discovering Geometry Teacher’s Resource Book comes with demonstration sketches corresponding to Discovering Geometry lessons. Ancillary Sketchpad materials are also available for some secondary texts from other publishers, though for a geometry course, none provide as complete a technology package as 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 Sketchpad ancillaries are available. Exploring Geometry with The Geometer’s Sketchpad, available in Canada from Spectrum Educational Supplies, contains more than 100 reproducible activities that can be used with any text. A CD-ROM with activities for Macintosh and Windows computers accompany the activities. Many other topic-specific volumes of activities are also available from Spectrum Educational Supplies. Sample activities from some of these books are included in this booklet. These books are listed and described on the back cover of this booklet. Exploring Geometry could supply a teacher with a year’s worth of activities to cover nearly all the content of a typical high school geometry course using The Geometer’s Sketchpad. And other 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 using the 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 most importantly, group discussion. Students need to apply mathematics to real-life situations and see where it is used in art and architecture and where it can be found in nature. Though Sketchpad can serve as a medium for many of these experiences, its potential will be reached only when students can apply what they learn with it to different situations. As engaging as using Sketchpad can be, it’s important that students don’t get the mistaken impression that mathematics exists only in their books and on their computer screens.

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© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 11

Sample Activities These sample classroom activity masters will give you an idea of some of the types of learning experiences that are possible using The Geometer’s Sketchpad. In the Teaching Notes, you saw three different types of lessons from Exploring Geometry with The Geometer’s Sketchpad: an investigation, an exploration, and a demonstration. Here you’ll find more activities from Exploring Geometry along with samples from other Key Curriculum Press publications. This collection is neither a complete curriculum nor a comprehensive set of activities 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 being appropriate for middle school students to presenting challenges to college undergraduate math majors. There are 25 activities here, and you’re obviously not going to be able to use them all with the same class. While we certainly hope that teachers will find some of the activities in this sample useful in their classes, the collection here 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 they’re from. (The various books are available in Canada from Spectrum Educational Supplies.)

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

Angles Constructing a Sketchpad Kaleidoscope

From Exploring Geometry with The Geometer’s Sketchpad

Properties of Reflection Tessellations Using Only Translations The Euler Segment Napoleon’s Theorem Constructing Rhombuses Midpoint Quadrilaterals A Rectangle with Maximum Area Visual Demonstration of the Pythagorean Theorem The Golden Rectangle A Sine Wave Tracer

From Exploring Algebra with The Geometer’s Sketchpad

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 In Leonardo da Vinci’s Proof

From Exploring Conic Sections with The Geometer’s Sketchpad

The Folded Circle Construction The Expanding Circle Construction

From Rethinking Proof with The Geometer’s Sketchpad

Distances in an Equilateral Triangle Varignon Area

From Exploring Calculus with The Geometer’s Sketchpad

Visualizing Change: Velocity Going Off on a Tangent Accumulating Area

Try some or all of these activities yourself and with your students to explore Sketchpad’s potential and learn how you can use it in the classroom. (You may reproduce these sheets for use with your classes.) Then join us in creating the most comprehensive teacher support materials ever to accompany new classroom software—materials that reflect what teachers and students can accomplish with state-of-the-art teaching and learning tools. If you’re interested in contributing worksheets, sample sketches, or custom tools for possible inclusion in future teacher materials and sample disks, contact the Editorial Department at Key Curriculum Press.

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

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

Angles

1. Open a new sketch.

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’s calculator to determine the sum of the three interior angles.

Q1 Drag any vertex of the triangle and observe the measures of the interior 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 interior angles that are not adjacent to the exterior angle.

Q2 Drag any vertex of the triangle and compare the measure of the exterior angle to the sum of the two remote (nonadjacent) interior angles.

Write any conjectures based on your exploration.

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

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 13

Constructing a Sketchpad Kaleidoscope

Follow the directions below to construct a Sketchpad kaleidoscope. The numbered steps tell you in general what you need to do, and the lettered steps give you more detailed instructions. Make sure you did each step correctly before 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 many sides (make it long and somewhat slender).

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

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

b. Select three or four points in clockwise or counterclockwise order.

c. Go to the Construct menu and choose Triangle Interior or Quadrilateral Interior.

Step b Step c Step e

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

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

3. Mark the bottom vertex point of your polygon as the center. Hide the points and 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 choose Mark 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.

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Constructing a Sketchpad Kaleidoscope (continued)

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

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

d. Click on the Selection Arrow tool. Use a selection marquee to select your polygon. Go to the Transform menu and choose Rotate.

e. Enter 60 and then click Rotate. (Pick a different factor of 360 if you wish.)

Rotate Dialog Box (Mac)

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

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

b. When the newer rotated image appears, and while it is still selected, go to the Transform menu and rotate this image by an angle 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 Show All Hidden. You should see the points on the original arm reappear.

5. Construct circles with their centers at the center of your kaleidoscope.

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

b. Click on the Compass tool. Press on the center point of your kaleidoscope and drag a circle with a radius a little larger than the outside edge of your kaleidoscope.

selection marquee after 60˚ rotation

control point

control point

control point

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Constructing a Sketchpad Kaleidoscope (continued)

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

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 15

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

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

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

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

b. Select one point on the original polygon near the outside circle and select the outside circle (do not click on one of the control points of the circle). Go to the Edit menu and choose Merge Point To Circle.

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

7. Animate points of your kaleidoscope on the three circles.

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

b. Select the three points you merged onto circles in the previous step.

c. Go to the Edit menu, choose Action Button, and drag to the right and choose Animation. Click on OK in the Animate dialog box.

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

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

merged points

Animate Points

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From Exploring Geometry with The Geometer’s Sketchpad

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

Properties of Reflection

When you look at yourself in a mirror, how far away does your image in the mirror appear to be? Why is it that your reflection looks just like you, but backward? Reflections in geometry have some of the same properties of reflections you observe in a mirror. In this activity, you’ll investigate the properties of reflections that make a reflection the “mirror image” of the original.

Sketch and Investigate: Mirror Writing

1. Construct vertical line AB.

2. Construct point C to the right of the line.

3. Mark dAB as a mirror.

4. Reflect point C to construct 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 out your name.

Sketch and Investigate: Reflecting Geometric Figures

8. Turn off Trace Points for points C and C´.

9. In the Display menu, choose Erase Traces.

10. Construct jCDE. 11. Reflect jCDE (sides and

vertices) over dAB.

12. Drag different parts of either triangle and observe how the triangles are related. Also drag

C'

B

C

A

Double-click on the line.

Select the two points; then,

in the Display menu, choose

Trace Points. A check mark

indicates that the command is

turned on. Choose Erase

Traces to erase your traces.

Select points C and C´. In the Display menu,

you’ll see Trace Points

checked. Choose it to uncheck

it.

D'

E'

C'A

C

DEB

Select the entire figure;

then, in the Transform menu, choose Reflect.

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Properties of Reflection (continued)

From Exploring Geometry with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 17

the mirror line.

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

Q2 Measure one angle in jCDE and measure the corresponding angle in jC´D´E´.

What effect does reflection have on lengths and angle measures?

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

Q4 Going alphabetically from C to D to E in jCDE, are the vertices oriented in a clockwise or counterclockwise direction? In what direction (clockwise or counterclockwise) are vertices C´, D´, and E´ oriented in the reflected triangle?

14. Construct segments connecting each point and its image: C to C´, D to D´, and E to E´. Make these segments dashed.

Q5 Drag different parts of the sketch around and observe relationships between the dashed segments and the mirror line.

How is the mirror line related to a segment connecting a point and its reflected image?

Explore More

1. Suppose Sketchpad didn’t have a Transform menu. How could you construct a given point’s mirror image over a given line? Try it. Start with a point and a line. Come up with a construction for the reflection of 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 what you did.

Select three points that

name the angle, with the vertex

your middle selection.

Then, in the Measure menu, choose Angle.

Your answer to Q4 demonstrates

that a reflection

reverses the orientation

of a figure.

Line Weight is in the Display

menu.

D'

E'

C'A

C

DE

B

You may wish to construct points of

intersection and measure

distances to look for

relation-ships between the

mirror line and the dashed segments.

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From Exploring Geometry with The Geometer’s Sketchpad

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

Tessellations Using Only Translations

In this activity, you’ll learn how to construct an irregularly shaped tile based on a parallelogram. Then you’ll use translations to tessellate your screen with this tile.

Sketch

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

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

3. Construct the remaining sides of your parallelogram.

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 translate them 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 irregular edge AB by the marked vector.

8. Construct the polygon interior of the irregular figure. This is the tile you will translate.

9. Translate the polygon interior by the marked vector. (You probably still have vector AC marked.)

10. Repeat this process until you have a column of tiles all the way up your sketch. Change the shading or color on every other tile to create a pattern.

C

A B

C'

Steps 1–3

Select, in order, point A

and point B; then, in the

Transform menu, choose Mark

Vector. Select point C; then,

in the Transform menu,

choose Translate.

C

A B

C'

Steps 8–10

Select the

vertices in consecutive

order; then, in the Construct menu, choose

Polygon Interior.

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Tessellations Using Only Translations (continued)

From Exploring Geometry with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 19

11. Mark vector AB. Then select all the polygon interiors in your column of tiles and translate them by this marked vector.

12. Continue translating columns of tiles until you fill your screen. Change shades and colors of alternating tiles so you can see your tessellation.

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

Explore More

1. Animate your tessellation. To do this, select the original polygon (or any combination of its vertex points) and choose Animate from the Display menu. You can also have your points move along paths you construct. To do this, construct the paths (segments, circles, polygon interiors—anything you can construct a point on) and then merge vertices to paths. (To merge a point to a path, select both and choose Merge Point to Path from the Edit menu.) Select the points you wish to animate and, in the Edit menu, choose Action Buttons | Animation. Press the Animate button. Adjust the paths so that the animation works in a way you like, then hide the paths.

2. Use Sketchpad to make a translation tessellation that starts with a regular 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

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From Exploring Geometry with The Geometer’s Sketchpad

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

The Euler Segment

In this investigation, you’ll look for a relationship among four points of concurrency: the incenter, the circumcenter, the orthocenter, and the centroid. You’ll use custom tools to construct these triangle centers, either those you made in previous investigations or pre-made tools.

Sketch and Investigate

1. Open a sketch of yours that contains tools for the triangle centers: incenter, circumcenter, orthocenter, and centroid. Or, open Triangle Centers.gsp.

2. Construct a triangle.

3. Use the Incenter tool on the triangle’s vertices to construct its incenter.

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

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

6. Use the Circumcenter tool on the same triangle. Hide any “extras” so 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 label the centroid. You should now have a triangle and the four triangle centers.

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 is called the Euler segment.

I

Triangle Centers.gsp

can be found in Sketchpad |

Samples | Custom Tools. (Sketchpad is

the folder that contains the application

itself.)

O CeCi

I

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The Euler Segment (continued)

From Exploring Geometry with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 21

Q2 Drag the triangle again and look for interesting relationships on the Euler segment. Be sure to check special triangles, such as isosceles and right triangles.

Describe any special triangles in which the triangle centers are related in interesting ways or located in interesting places.

Q3 Which of the three points are always endpoints of the Euler segment and 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 and passing through the midpoint of one of the sides of the triangle. This circle is called the nine-point circle. The midpoint it passes through is one 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, as described above, drag your triangle around and investigate special triangles. Describe any triangles in which some of the nine points coincide.

To measure the distance

between two points, select

the two points. Then, in the

Measure menu, choose

Distance. (Measuring the

distance between points is an easy way

to measure the length of

part of a segment.)

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From Exploring Geometry with The Geometer’s Sketchpad

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

Napoleon’s Theorem

French emperor Napoleon Bonaparte fancied himself as something of an amateur geometer and liked to hang out with mathematicians. The theorem 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 tool or construct the triangle from scratch.

2. Construct the center of the triangle.

3. Hide anything extra you may have constructed to construct the triangle and its center so that you’re left 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 jABC.

7. Use the custom tool to construct equilateral triangles on each side of jABC.

8. Drag to make sure each equilateral triangle is stuck to a side.

9. Construct segments connecting the centers of the equilateral triangles.

10. Drag the vertices of the original triangle and observe the triangle formed by the centers of the equilateral triangles. This triangle is called the outer Napoleon triangle of jABC.

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

Explore More

1. Construct segments connecting each vertex of your original triangle with the most remote vertex of the equilateral triangle on the opposite side. What can you say about these three segments?

One way to construct the center is to

construct two medians and

their point of intersection.

Select the entire figure;

then choose Create New Tool from the Custom

Tools menu in the Toolbox

(the bottom tool).

A C

B

Be sure to attach each equilateral

triangle to a pair of

triangle ABC’s vertices. If

your equilateral

triangle goes the wrong

way (overlaps the interior of

jABC) or is not attached

properly, undo and try

attaching it again.

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From Exploring Geometry with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 23

Constructing Rhombuses

How many ways can you come up with to construct a rhombus? Try methods that use the Construct menu, the Transform menu, or combinations of both. Consider how you might use diagonals. Write a brief description of each construction method along with the properties of 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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From Exploring Geometry with The Geometer’s Sketchpad

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

Midpoint Quadrilaterals

In this investigation, you’ll discover something surprising about the quadrilateral formed by connecting the midpoints of another quadrilateral.

Sketch and Investigate

1. Construct quadrilateral ABCD.

2. Construct the midpoints of the sides.

3. Connect the midpoints to construct another quadrilateral, EFGH.

4. Drag vertices of your original quadrilateral and observe the midpoint quadrilateral.

5. Measure the four side lengths of this midpoint quadrilateral.

Q1 Measure the slopes of the four sides of the midpoint quadrilateral. What kind of quadrilateral does the midpoint quadrilateral appear to 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 original quadrilateral and observe how the length and slope of the diagonal are related to the lengths and slopes of the sides 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 midpoint quadrilateral. Use this fact and what you know about the slope and length of the diagonal to write a paragraph explaining why the conjecture 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 four sides,

you can construct all

four midpoints at once.

H

E

F

G

A

B C

D

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From Exploring Geometry with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 25

A Rectangle with Maximum Area

Suppose you had a certain amount of fence and you wanted to use it to enclose the biggest possible rectangular field. What rectangle shape would you choose? In other words, what type of rectangle has the most area for a given 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.

Sketch and Investigate

1. Construct sAB.

2. Construct sAC on sAB.

3. Construct lines perpendicular to sAB through points A and C.

4. Construct circle CB.

5. Construct point D where this circle intersects the perpendicular 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 and perimeter of the rectangle.

11. Measure AC and AE.

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

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

E D

A BC

Select sAB, point A, and

point C. Then, in the

Construct menu, choose

Perpendicular Line. Be sure to

release the mouse—or click

the second time—

with the pointer

over point B.

Select the vertices of the

rectangle in consecutive

order. Then, in the Construct menu, choose

Quadrilateral Interior.

Select point A and point C. Then, in the

Measure menu, choose

Distance. Repeat to

measure AE.

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A Rectangle with Maximum Area (continued)

From Exploring Geometry with The Geometer’s Sketchpad

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

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

12. Plot the measurements for the length of sAC and the area of ACDE as (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 different side 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 the locus of plotted point H as defined by point C. It should now be easy to position point C so that point H is at a maximum value for the area of the rectangle.

Q3 Explain what the coordinates of the high point on the graph are and how 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 two low points on the graph.

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

Explore More

1. Investigate area/perimeter relationships in other polygons. Make a conjecture about what kinds of polygons yield the greatest area for a given 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 an equation for area, A, in terms of x and P. What value for x (in terms of P) gives a maximum value for A?

Select, in order, msAC and Area ACDE. Then

choose Plot As (x, y) from the Graph

menu. If you can’t

see the plotted

point, drag the unit point

at (1, 0) to scale

the axes.

Select point H and point C; then, in the

Construct menu, choose Locus.

You may wish to select point H and measure its

coordinates.

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From Exploring Geometry with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 27

Visual Demonstration of the Pythagorean Theorem

In this activity, you’ll do a visual demonstration of the Pythagorean theorem based on Euclid’s proof. By shearing the squares on the sides of a right triangle, you’ll create congruent shapes without changing the areas of your original squares.

Sketch and Investigate

1. Open the sketch Shear Pythagoras.gsp. You’ll see a right triangle with squares on 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 a parallelogram its area doesn’t change.

4. Drag point B onto the line. It should overlap point A so that the two parallelograms form a single irregular 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 appear congruent to the shape you made with the two smaller parallelograms.

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 Pythagorean theorem? (Hint: If the shapes are congruent, what do you know about their areas?)

ba c

C

A

B

All sketches referred to in

this booklet can be found in

Sketchpad | Samples | Teach-ing Mathematics

(Sketchpad is the folder that

contains the application

itself.) Click on a

polygon interior to select it.

Then, in the Measure menu, choose Area.

To confirm that this shape is

congruent, you can copy and

paste it. Drag the pasted copy onto the shape on the legs to

see that it fits

perfectly.

To confirm that this works for

any right triangle,

change the shape of the triangle and try the experiment

again.

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From Exploring Geometry with The Geometer’s Sketchpad

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

The Golden Rectangle

The golden ratio appears often in nature: in the proportions of a nautilus shell, for example, and in some proportions in our bodies and faces. A rectangle whose sides have the golden ratio is called a golden rectangle. In a golden rectangle, the ratio of the sum of the sides to the long side is equal to the ratio of the long side to the short side. Golden rectangles are somehow pleasing to the eye, perhaps because they approximate the shape of our field of vision. For this reason, they’re used often in architecture, especially the classical architecture of ancient Greece. In this activity, you’ll construct a golden rectangle and find an approximation to the golden ratio. Then you’ll see how smaller golden rectangles are found within a golden rectangle. Finally, you’ll construct a golden spiral.

Sketch and Investigate

1. Use a custom tool to construct a square ABCD. Then construct the square’s interior.

2. Orient the square so that the control points are on the left side, one above 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 1–4 Steps 5–8 Steps 9–11

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. Rectangle AFGB is a golden rectangle.

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

10. Hide sAD, sDC, and sBC.

Hold the mouse button down on

the Segment tool to show the Straight

Objects palette. Drag

right to choose the Ray tool.

Select the objects; then, in the Display

menu, choose Hide Objects.

a

b

a + bb =

ba

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The Golden Rectangle (continued)

From Exploring Geometry with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 29

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 always golden.

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 a golden spiral.

16. Construct circle CB.

17. Construct an arc on the circle from point B to point D, then hide the circle.

18. Make a custom tool for this construction.

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

Q2 Make a conjecture about region DFGC.

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

Explore More

1. Let the short side of a golden rectangle have length 1 and the long side have length ø. Write a proportion, cross-multiply, and use 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,

sAF and sAB; then, in the

Measure menu, choose Ratio.

Choose Calculate from

the Measure menu to open

the Calculator. Click once on a measurement to

enter it into a calculation.

G

F

CB

A D

Select, in order, the circle and points B

and D. Then choose

Arc On Circle from the

Construct menu. Select the

entire figure; then choose

Create New Tool from the Custom

Tools menu in the Toolbox

(the bottom tool). If your

rectangle goes the wrong way when you use

the custom tool, undo and

try applying it in the opposite

order.

B

A

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From Exploring Geometry with The Geometer’s Sketchpad

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

A Sine Wave Tracer

In this exploration, you’ll construct an animation “engine” that traces out a special curve called a sine wave. Variations of sine curves are the graphs of functions called periodic functions, functions that repeat themselves. The motion of a pendulum and ocean tides are examples of periodic functions.

Sketch and Investigate

1. 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 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 combines these 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 same time as point E is moving around the circle. Sketch the path you imagine below.

Select point D and sAB; then,

in the Construct menu, choose

Perpendicular Line.

Don’t worry, this isn’t a

trick question!

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A Sine Wave Tracer (continued)

From Exploring Geometry with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 31

8. Make an action button that animates point D forward on sAB and point 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 lie on the x-axis. The y-coordinate of point F above sAB is the value of the sine of ∠EAD.

5 10

F

AB

CD

E

Q5 If the circle has a radius of 1 graph unit, what is its circumference in graph units? (Calculate this yourself; don’t use Sketchpad to measure it because Sketchpad will measure in inches or centimeters, not graph units.)

13. Measure the coordinates of point B.

14. Adjust the segment and the circle until you can make the curve trace 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 the circumference of the circle (in graph units)? Explain why you think this is so.

Select points D and E and

choose Edit | Action Buttons

| Animation. Choose forward

in the Direction

pop-up menu for both points.

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From Exploring Algebra with The Geometer’s Sketchpad

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

Adding Integers

They say that a picture is worth a thousand words. In the next two activities, you’ll explore integer addition and subtraction using a visual Sketchpad model. Keeping this model in mind can help you visualize what these operations do and how they work.

Sketch and Investigate

1. Open the sketch Add Integers.gsp from the folder 1_Fundamentals.

2. Study the problem that’s modeled: 8 + 5 = 13. Then drag the two “drag” circles to model other addition problems. Notice how the two upper arrows relate 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 positive numbers? How is it different?

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

Definition: Integers are positive and

negative whole numbers, including zero. On a number line, tick marks

usually represent the integers.

drag

drag

1-1 2 3 4 5 100

+ 5

8

All sketches referred to

in this booklet can be found in

Sketchpad | Samples | Teach-ing

Mathematics (Sketchpad is

the folder that contains the application

itself.)

drag

drag

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

+ -3

-6

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Adding Integers (continued)

From Exploring Algebra with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 33

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 first number 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 one negative number. Have the first number be positive in three problems and negative in three. Also, make sure that some problems have positive answers and others have negative answers. Write your equations below.

Q7 When adding a positive number and a negative number, how can you tell 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

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Adding Integers (continued)

From Exploring Algebra with The Geometer’s Sketchpad

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

Q8 A classmate says, “Adding a positive and a negative number seems more 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 farther from zero.)

Explore More

1. The Commutative Property of Addition says that for any two numbers a 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 that demonstrate this property.

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

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

To commute means to travel back and forth. The Commutative

Property of Addition

basically says that addends can

commute across an addition sign

without affecting the

sum.

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From Exploring Algebra with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 35

Points “Lining Up” in the Plane

If you’ve seen marching bands perform at football games, you’ve probably seen the following: The band members, wandering in seemingly random directions, suddenly spell a word or form a cool picture. Can these patterns be described mathematically? In this activity, you’ll start to answer this question by exploring simple patterns of dots in the x-y plane.

Sketch and Investigate

1. Open a new sketch.

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

3. Measure the coordinates of the five selected points. A coordinate system appears and the coordinates of the five points are displayed.

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

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

Q1 For each problem, drag the five points to different locations that satisfy the given conditions. Then copy your solutions onto the grids on 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 the Shift key

keeps all five points

selected.

To measure the coordinates

of selected points, choose

Coordinates from the

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 from the Display menu.

The absolute value of a

number is its “positive

value.” The absolute

value of both 5 and –5 is 5.

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Points “Lining Up” in the Plane (continued)

From Exploring Algebra with The Geometer’s Sketchpad

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

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)

From Exploring Algebra with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 37

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), their coordinate measurements, and eight action buttons.

Q2 For each letter, press the corresponding button in the sketch. Like the members of a marching band, the points will “wander” until they form 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 an equation.

For example, part b of Q1 (“the y-coordinate is one greater than the x-coordinate”) can be written as y = x + 1. Write an equation for 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 your patterns. Instructions on how to do this are on page 2 of the sketch.

All sketches referred to in

this booklet can be found in

Sketchpad | Samples | Teach-ing Mathematics

(Sketchpad is the folder that

contains the application

itself.)

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From Exploring Algebra with The Geometer’s Sketchpad

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

Parabolas in Vertex Form

Things with bilateral symmetry—such as the human body—have parts on the sides that come in pairs (such as ears and feet) and parts down the middle there’s just one of (such as the nose and bellybutton). Parabolas are the 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 equation form for parabolas that relates to this unique point.

Sketch and Investigate

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 their tips) and watch the equation change accordingly. There’s no graph yet because we wanted you to practice using Sketchpad’s graphing features.

2. Choose Plot New Function from the Graph menu. The New Function dialog box appears. If necessary, move it so that you can see a, h, and k’s measurements.

3. Enter a*(x–h)̂ 2+k and click 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 same as 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 the Graph 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 referred to in

this booklet can be found in

Sketchpad | Samples | Teach-ing Mathematics

(Sketchpad is the folder that

contains the application

itself.)

To enter a, h, and k, click on

their measurements in the sketch. To enter x, click

on the x in the dialog box.

Choose Calculate from

the Measure menu. Click on

the function equation from

step 3. Then click on

xP to enter it. Now type a

close parenthesis—

“)”— and click OK.

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Parabolas in Vertex Form (continued)

From Exploring Algebra with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 39

Exploring Families of Parabolas

By dragging point P, you’re exploring how the variables x and y vary along one particular parabola with particular values for a, h, and k. For the rest of this activity, you’ll change the values of a, h, and k, which will change the parabola itself, allowing you to explore whole families of 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 discuss a’s

sign (whether it’s positive or negative), its magnitude (how big or small 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, focusing on 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 transforms as h changes. How does that compare to the transformation that occurs 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 a point 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)

From Exploring Algebra with The Geometer’s Sketchpad

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

Q5 Write the equation in vertex form y = a(x – h)2 + k for each parabola described. As a check, adjust the sliders so that the parabola is drawn on 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 –3(x – 2)2 – 2; contains the point (0, 6)

e. same shape as the parabola 4(x + 3)2 – 1; vertex at (−1, 3)

Q6 The axis of symmetry is the line over which a parabola can be flipped and still look the same. What is the equation of the axis of symmetry for the parabola y = 2(x – 3)2 + 1? for y = a(x – h)2 + k?

Q7 Just as your right ear has a corresponding ear across your body’s axis of symmetry, all points on a parabola (except the vertex) have corresponding points across its axis of symmetry.

The point (5, 9) is on the parabola y = 2(x – 3)2 + 1. What is the corresponding point across the axis of symmetry?

Explore More

1. Assume that the point (s, t) is on the right half of the parabola y = a(x – h)2 + k. What is the corresponding point across the axis of symmetry? If (s, t) were on the left half of the parabola, what would the answer be?

2. Use the Perpendicular Line command from the Construct menu to construct the axis of symmetry of your parabola. Then use the Reflect command from the Transform menu to reflect point P across the new axis of symmetry. Measure the coordinates of the new point, P′. Are they what you expected?

Note: In this activity, the precision of measurements

has been set to one decimal

place. It’s important to be aware of

this, and to check your answers by

hand, in addition to

adjusting the sliders in the

sketch.

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From Exploring Algebra with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 41

Reflection in Geometry and Algebra

If you’re like most people, you’ve spent at least a little time looking at yourself in the mirror. So you’re already pretty familiar with reflection. In this activity, you’ll add to your knowledge on the subject as you explore reflection from both geometric and algebraic perspectives.

Sketch and Investigate

1. In a new sketch, use the Point tool to draw a point.

2. With the point still selected, choose a color from the Display | Color submenu. Then choose Trace Point from the Display menu. Use the Arrow tool to drag the point around. The “trail” the point leaves is called its trace.

3. If the trace from the previous step fades and disappears, go on to the next step. If the trace remains on the screen, choose Preferences from the Edit menu. On the Color panel, check the Fade Traces Over Time box and click OK.

4. Using the Line tool, draw a line. With the line selected, choose Mark Mirror from the Transform menu. A brief animation indicates that the mirror line has been marked.

5. Using the Arrow tool, select the point. Choose Reflect from the Transform menu. The point’s reflected image appears.

6. Give the new point a different color and turn tracing on for it as well.

7. What will happen when you drag one of the reflecting points? Ponder this a moment. Then drag and see. What do you think will happen when you drag one of the line points? Find the answer to this question too.

Q1 Briefly describe the two types of patterns you observed in step 7 (one when dragging a reflecting point, the other when dragging a line point).

To choose the

Line tool, press and hold

the mouse button

over the current

Straightedge tool,

then drag and release over

the Line tool in the palette that appears.

Starting in this step,

we’ll refer to the two points

defining the line as line

points and the other

two points as reflecting

points.

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Reflection in Geometry and Algebra (continued)

From Exploring Algebra with The Geometer’s Sketchpad

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

8. Select the reflecting points; then choose Trace Points to toggle tracing off.

9. With the two points still selected, choose Segment from the Construct menu. A segment is constructed between the points. Drag the various objects around and observe the relationship between the line and the segment.

Q2 What angle do the line and the segment appear to make with each other? How does the line appear to divide the segment?

From Geometry to Algebra Now that you’ve learned some geometric properties of reflection, it’s time to apply this knowledge to reflection in the x-y plane. You’ll start by exploring reflection across the y-axis.

10. Click in blank space to deselect all objects. Drag one of the line points so it’s near the center of the sketch. With this point selected, choose Define Origin from the Graph menu. A coordinate system appears. The selected point is the origin—(0, 0).

11. Deselect all objects; then select the y-axis and the other line point (the one that didn’t become the origin). Choose Merge Point To Axis from the Edit menu. The point “attaches” itself to the y-axis, which now acts as the mirror line.

2-2

1

2

A' A

12. Select one of the reflecting points and choose Coordinates from the Measure menu. The point’s (x, y) coordinate measurement appears. Drag the point and watch its coordinates change.

13. How do you think the other reflecting point’s coordinates compare? Measure them to find out if you were right.

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Reflection in Geometry and Algebra (continued)

From Exploring Algebra with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 43

Q3 A point with coordinates (a, b) is reflected across the y-axis. What are the coordinates of its reflected image?

14. How does the distance between the two reflecting points relate to their coordinates? Make a prediction. Then select the two points and choose Coordinate Distance from the Measure menu. Were you right?

Q4 A point with coordinates (a, b) is reflected across the y-axis. How far is it from its reflected image?

15. Deselect all objects. Then select the point on the y-axis that was merged in step 11. Choose Split Point From Axis. The point splits from the y-axis.

16. With the point still selected, select the x-axis as well. Then choose Merge Point To Axis from the Edit menu. The x-axis now acts as the mirror line. Drag one of the reflecting points and observe the various measurements.

Q5 A point with coordinates (c, d ) is reflected across the x-axis. What are the coordinates of its reflected image?

Q6 A point with coordinates (c, d ) is reflected across the x-axis. How far is it from its reflected image?

Explore More

1. Plot the line y = x. Split the point from the x-axis and merge it to the new line. What do you notice about the coordinates of the reflecting points?

2. Consider the following transformations (each is separate): a. Reflect a point over the x-axis, then reflect the image over

the y-axis. b. Reflect a point over the y-axis, then reflect the image over

the x-axis. c. Rotate a point by 180° about the origin.

How do these three transformations compare? What would the coordinates of a point (a, b) be after each of these transformations?

A special challenge is to make sure your

answers to this question and Q6 work regardless

of what quadrants the

points are in.

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From Exploring Algebra with The Geometer’s Sketchpad

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

Walking Rex: An Introduction to Vectors

You know, and most everyone over age five knows, that 2 + 2 = 4. No big shock there. But what if you walk 2 miles north, turn around, then walk 2 miles south—how far have you walked? In one sense, you’ve walked 4 miles—that’s certainly what your feet would tell you. But in another sense, you haven’t really gotten anywhere. We could say: 2N + 2S = 0.

Values that have both a magnitude (size) and a direction are called vectors. Vectors are very useful in studying things like the flight of airplanes in wind currents and the push and pull of magnetic forces. In this activity, you’ll explore some of the algebra and geometry behind vectors in the context of a walk with your faithful dog, Rex.

Walk the Dog

1. Open the sketch Walk the Dog.gsp from the folder 5_Transform. Rex’s leash is tied to a tree at the origin of an x-y coordinate system. Rex is pulling the leash tight as he excitedly waits for you to take him on a walk.

Rex’s taut leash is represented by a vector—a segment with an arrowhead. The end with the arrowhead (Rex) is called the head and the other end (the tree) is called the tail. We’ve labeled this particular vector j.

Q1 One way to define vectors is by their magnitude (length) and direction. Which of these two quantities stays the same as you drag point Rex?

Q2 For each description of vector j, find Rex’s coordinates.

a. magnitude = 5; direction = 30º

b. magnitude = 5; direction = 90º

c. magnitude = 5; direction = 225º

All sketches referred to in

this booklet can be found in

Sketchpad | Samples | Teach-ing Mathematics

(Sketchpad is the folder that

contains the application

itself.)

Rex has a head and tail too,

of course, but those have

nothing to do with the vector!

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Walking Rex (continued)

From Exploring Algebra with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 45

Q3 A second way to define a vector is by the coordinates of its head when its tail is at the origin. Use Sketchpad to find the magnitude and direction of the following vectors:

a. vector j = (5, 0) b. vector k = (3, 4) c. vector l = (0, –5) d. vector m = (–3, –4)

Q4 Rex is terrified of ladybugs. Suppose a ladybug is sitting at (5, 0). Where should Rex move to face the opposite direction and be as far from it as possible? What if the ladybug moves to (3, –4)?

Now it’s time to untie the leash from the tree and take Rex for a walk.

2. Go to the second page of Walk the Dog.gsp: Walk 1. Rex is a very determined dog! As you walk him, he pulls the leash taut and always tries to steer you in the same direction (toward an interesting scent perhaps). Rex is still at the head of vector j (where the arrowhead is) and now you’re at the tail.

Q5 Drag vector j around the screen. Explain why, no matter where you drag it, vector j is always the same vector. Use one of the two methods for defining vectors we’ve discussed to support your argument.

Q6 Suppose you stood at the point (80, 80). Where would Rex be standing? Explain how you found your answer. (Don’t scroll or use Sketchpad’s menus—all the information you need is on the screen.)

3. Go to the third and fourth pages of Walk the Dog.gsp: Walk 2 and Walk 3. You’ll see that Rex is heading in different directions on these pages. The information presented on screen is also a little different for each page.

Q7 As in Q6, determine where Rex will be standing when you’re at (80, 80) for Walk 2 and Walk 3. Explain your reasoning in each case.

Q8 Answer Q7 again, this time assuming that you have a leash twice as long and Rex heads in the same directions.

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From Exploring Conic Sections with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 46

Leonardo da Vinci’s Proof

Leonardo da Vinci (1452–1519) was a great Italian painter, engineer, and inventor during the Renaissance. He is most famous, perhaps, for his painting the Mona Lisa. He is also credited with the following proof of the Pythagorean theorem.

Construct

1. Construct a right triangle and squares on the legs.

2. Connect corners of the squares to construct a second right triangle congruent to the original.

3. Construct a segment through the center of this figure, connecting far corners of the squares and passing through C.

4. Construct the midpoint, H, of this segment.

5. This segment divides the figure into mirror image halves. Select all the segments and points on one side of the center line and create a Hide/Show action button. Change its label to read “Hide reflection.”

6. Press the Hide reflection action button. You should now see half the figure.

a

b

c

Show reflection

H

B

C A

7. Mark H as center and rotate the entire figure (not the action buttons) by

180° around H.

8. Select all the objects making up the rotated half of this figure and create a Hide/Show action button. Relabel this button to read “Hide rotation” but don’t hide the rotated half yet.

In this figure, you don’t have

to construct the square on

the hypotenuse. a

b

c

b

a

c

Hide reflection

H

B

C A

The Action Buttons submenu

is in the Edit menu.

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The Folded Circle Construction (continued)

From Exploring Conic Sections with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 47

a

b

c

a

b

Hide rotationShow reflection

C'A'

B'

H

B

C A

9. Construct xA′B and xB′A. Do you see c squared?

10. Construct the polygon interior of BA´B´A and of the two triangles adjacent to it.

11. Select xA′B, xB′A, and the three polygon interiors and create a Hide/Show action button. Name it “Hide c squared.”

a

b

c

a

bc

Hide c squaredHide rotationShow reflection

C'A'

B'

H

B

C A

Investigate From going through this construction, you may have a good idea of how Leonardo’s proof goes. Press all the hide buttons, then play through the buttons in this sequence: “Show reflection,” “Show rotation,” “Hide reflection,” “Show c squared.” You should see the transformation from two right triangles with squares on the legs into two identical right triangles with a square on their hypotenuses. Explain to a classmate or make a presentation to the class to explain Leonardo’s proof of the Pythagorean theorem.

Prove Leonardo’s is another of those elegant proofs where the figure tells pretty much the whole story. Write a paragraph that explains why the two hexagons have equal areas and how these equal hexagons prove the Pythagorean theorem.

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From Exploring Conic Sections with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 48

The Folded Circle Construction

Sometimes a conic section appears in the unlikeliest of places. In this activity, you’ll explore a paper-folding construction in which crease lines interact in a surprising way to form a conic.

Constructing a Physical Model

Preparation: Use a compass to draw a circle with a radius of approximately three inches on a piece of wax paper or patty paper. Cut out the circle with a pair of scissors. (If you don’t have these materials, you can draw the circle in Sketchpad and print it.)

1. Mark point A, the center of your circle.

2. Mark a random point B within the interior of your circle.

3. As shown below right, fold the circle so that a point on its circumference lands directly onto point B. Make a sharp crease to keep a record of this fold. Unfold the circle.

4. Fold the circle along a new crease so that a different point on the circumference lands on point B. Unfold the circle and repeat the process.

5. After you’ve made a dozen or so creases, examine them to see if you spot any emerging patterns.

6. Resume creasing your circle. Gradually, a well-outlined curve will appear. Be patient—it may take a little while.

7. Discuss what you see with your classmates and compare their folded curves to yours. If you’re doing this activity alone, fold a second circle with point B in a different location.

B

A

If you’re working

in a class, have members

place B at different

distances from the center.

If you’re working alone,

do this section twice—

once with B close

to the center, once with B

close to the edge.

B

A

Mathematicians would describe

your set of creases as an

envelope of creases.

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The Folded Circle Construction (continued)

From Exploring Conic Sections with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 49

Questions Q1 The creases on your circle seem to form the outline of an ellipse.

What appear to be its focal points?

Q2 If you were to move point B closer to the edge of the circle and fold another curve, how do you think its shape would compare to the first curve?

Q3 If you were to move point B closer to the center of the circle and fold another curve, how do you think its shape would compare to the first curve?

Constructing a Sketchpad Model Fold and unfold. Fold and unfold. Creasing your circle takes some work. Folding one or two sheets is fun, but what would happen if you wanted to continue testing different locations for point B? You’d need to keep starting with fresh circles, folding new sets of creases. Sketchpad can streamline your work. With just one circle and one set of creases, you can drag point B to new locations and watch the crease lines adjust themselves instantaneously.

8. Open a new sketch and use the Compass tool to draw a large circle with center A. Hide the circle’s radius point.

9. Use the Point tool to draw a point B at a random spot inside the circle.

10. Construct a point C on the circle’s circumference.

11. Construct the “crease” formed when point C is folded onto point B.

12. Drag point C around the circle. If you constructed your crease line correctly, it should adjust to the new locations of point C.

13. Select the crease line and choose Trace Line from the Display menu.

14. Drag point C around the circle to create a collection of crease lines.

15. Drag point B to a different location and then, if necessary, choose Erase Traces from the Display menu.

16. Drag point C around the circle to create another collection of crease lines.

If you don’t want your

traces to fade, be sure the Fade Traces

Over Time box is unchecked on the Color

panel of the

Preferences dialog box.

crease

B

A

C

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The Folded Circle Construction (continued)

From Exploring Conic Sections with The Geometer’s Sketchpad

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

Retracing creases for each location of point B is certainly faster than folding new circles. But we can do better. Ideally, your crease lines should relocate automatically as you drag point B. Sketchpad’s powerful Locus command makes this possible.

17. Turn tracing off for your original crease line by selecting it and once again choosing Trace Line from the Display menu.

18. Now select your crease line and point C. Choose Locus from the Construct menu. An entire set of creases will appear: the locus of crease locations as point C moves along its path. If you drag point B, you’ll see that the crease lines readjust automatically.

19. Save your sketch for possible future use. Give it a descriptive name such as Creased Circle.

Questions

Q4 How does the shape of the curve change as you move point B closer to the edge of the circle?

Q5 How does the shape of the curve change as you move point B closer to the center of the circle?

Q6 Select point B and the circle. Then merge point B onto the circle’s circumference. Describe the crease pattern.

Q7 Select point B and split it from the circle’s circumference. Then merge it with the circle’s center. Describe the crease pattern.

Playing Detective Each crease line on your circle touches the ellipse at exactly one point. Another way of saying this is that each crease is tangent to the ellipse. By engaging in some detective work, you can locate these tangency points and use them to construct just the ellipse without its creases.

20. Open the sketch Folded Circle.gsp. You’ll see a thick crease line and its locus already in place.

21. Drag point C and notice that the crease line remains tangent to the ellipse. The exact point of tangency lies at the intersection of two lines—the crease line and another line not shown here. Construct this line in your sketch as well as the point of tangency, point E.

22. Select point E and point C and choose Locus from the Construct menu. If you’ve identified the tangency point correctly, you should see a curve appear precisely in the white space bordered by the creases.

The Merge and Split commands appear in the

Edit menu.

All sketches referred to in

this booklet can be found in

Sketchpad | Samples | Teach-ing Mathematics

(Sketchpad is the folder that

contains the application

itself.) Select the

locus and make its

width thicker so that it’s

easier to see.

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The Folded Circle Construction (continued)

From Exploring Conic Sections with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 51

How to Prove It The Folded Circle construction seems to generate ellipses. Can you prove that it does? Try developing a proof on your own, or work through the following steps and questions.

The picture at right should resemble your construction. Line HI (the perpendicular bisector of segment CB) represents the crease formed when point C is folded onto point B. Point E sits on the curve itself.

23. Add segments CB, BE, and AC to the picture.

24. Label the intersection of CB with the crease line as point D.

Questions

Q8 Use a triangle congruence theorem to prove that jBED m jCED.

Q9 Segment BE is equal in length to which other segment? Why?

Q10 Use the distance definition of an ellipse and the result from Q9 to prove that point E traces an ellipse.

Explore More

1. When point B lies within its circle, the creases outline an ellipse. What happens when point B lies outside its circle?

2. Use the illustration from your ellipse proof to show that ∠AEH = ∠BED.

Here’s an interesting consequence of this result: Imagine a pool table in the shape of an ellipse with a hole at one of its focal points. If you place a ball on the other focal point and hit it in any direction without spin, the ball will bounce off the side and go straight into the hole. Guaranteed!

3. The sketch Tangent Circles.gsp in the Ellipse folder shows a red circle c3 that’s simultaneously tangent to circles c1 and c2. Press the Animate button and observe the path of point C, the center of circle c3. Can you prove that C traces an ellipse?

EB

A

C

H

I

Remember: An ellipse is the

set of points such

that the sum of the distances

from each point to two fixed points (the

foci) is constant.

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From Exploring Conic Sections with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 52

The Expanding Circle Construction

In this activity, you’ll explore a little-known parabola construction from the tenth century. The method originates from Ibn Sina, a jack-of-all-trades who was a physician, philosopher, mathematician, and astronomer!

Constructing a Sketchpad Model

1. Open a new sketch. Choose Show Grid from the Graph menu. Then choose Hide Grid to remove the grid lines while keeping the x- and y-axes.

2. Label the origin as point A.

3. Choose the Compass tool. Click on the y-axis above the origin (point C) and then below the origin (point B). You’ll create a circle with center at point C passing through point B.

4. Construct point D, the intersection of the circle and the positive y-axis.

5. Construct points E and F, the intersections of the circle and the x-axis.

6. Construct lines through points E and F perpendicular to the x-axis.

7. Construct a line through point D perpendicular to the y-axis.

8. Construct points G and H, the intersections of the three newly created lines.

9. Select points G and H and choose Trace Intersections from the Display menu. Drag point C up and down the y-axis and observe the curve traced by points G and H.

The curve you see is the locus of points G and H as point C travels along the y-axis.

10. Drag point B to a new location, but keep it below the origin. Then, if necessary, choose Erase Traces from the Display menu to erase your previous curve. Trace several new curves, each time changing the location of point B.

For every new location of point B, you need to retrace your curve. Ideally, your parabola should adjust automatically as you drag point B. Sketchpad’s powerful Locus command makes this possible.

In response to those who

advised him to take life easy,

Ibn Sina is said to have replied, “I

prefer a short life with width to a narrow one

with length.” He died

at the age of 58.

5-5

4

2

-2

6G HD

E FA

B

C

If you don’t want your

traces to fade, be sure the Fade Traces

Over Time box is unchecked on the Color

panel of the

Preferences dialog box.

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The Expanding Circle Construction (continued)

From Exploring Conic Sections with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 53

11. Turn tracing off for points G and H by selecting them and once again choosing Trace Intersections from the Display menu.

12. Now select points G and C. Choose Locus from the Construct menu. Do this again for points H and C. You’ll form an entire curve: the locus of points G and H. Drag point B to vary the shape of the curve.

Questions

Q1 As you drag point B, which features of the curve stay the same? Which features change?

Q2 The creator of this technique, Ibn Sina, didn’t, of course, have Sketchpad available to him in the tenth century! How would this construction be different if you used a compass and straightedge instead?

The Geometric Mean It certainly looks like the Expanding Circle method draws parabolas, but to prove why, you’ll need to know a little about geometric means. The geometric mean x of two numbers, a and b, is equal to ab . Equivalently, x2 = ab. Thus the geometric mean of 4 and 9 is

(4)(9) = 6 It’s possible to determine the geometric mean of two numbers geometrically rather than algebraically. Specifically, if two segments have lengths a and b, we can construct—without measuring—a third segment of length ab .

a b

x

x

All sketches referred to in this booklet can be found in Sketchpad | Samples | Teach-ing Mathematics (Sketchpad is the folder that contains the application itself.) 13. Open the sketch Geometric Mean.gsp. You’ll see a circle whose diameter consists of two segments with lengths a and b laid side to side. A chord perpendicular to the diameter is split into equal segments of length x.

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The Expanding Circle Construction (continued)

From Exploring Conic Sections with The Geometer’s Sketchpad

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

14. Use Sketchpad’s calculator to compute the geometric mean of lengths a and b. Compare this value to x. Questions Q3 The second page of Geometric Mean.gsp outlines a proof showing that x is the geometric mean of a and b. Complete the proof. How to Prove It With your knowledge of geometric means, you can now prove that points G and H of the Expanding Circle construction trace a parabola. Since the location of point H changes as the circle grows and shrinks, it’s labeled below as (x, y), using variables as coordinates. To make things more concrete, we’ll assume AB = 3.

5-5

4

2

-2

6G H = (x, y)D

E F

B (0, -3)

A = (0, 0)

C

Questions The questions that follow provide a step-by-step guided proof. You can answer them or first write your own proof without any hints.

Q4 Fill in the lengths of the following segments in terms of x and y:

AF =

AD =

Q5 Use your knowledge of geometric means to write an equation relating the lengths of AB, AF, and AD. Is this the equation of a parabola?

Q6 Give an argument to explain why point G also traces a parabola.

Q7 Rewrite your proof, this time making it more general. Let AB = s.

Explore More

1. Open the sketch Right Angle.gsp. Angle DEB is constructed to be a right angle. Drag point E and observe the trace of point G and its reflection G′. Explain why this sketch is essentially the same as the Expanding Circle construction.

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From Rethinking Proof with The Geometer’s Sketchpad

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

Distances in an Equilateral Triangle

A shipwreck survivor manages to swim to a desert island.

As it happens, the island closely approximates the shape of an equilateral triangle. She soon discovers that the surfing is outstanding on all three of the island’s coasts. She crafts a surfboard from a fallen tree and surfs every day. Where should she build her house so that the sum of the distances from her house to all three beaches is as small as possible? (She visits each beach with equal frequency.) Before you proceed further, locate a point in the triangle at the spot where you think she should build her house.

Conjecture

1. Open the sketch Distance.gsp. Drag point P to experiment with your sketch.

Q1 Press the button to show the distance sum. Drag point P around the interior of the triangle. What do you notice about the sum of the distances?

Q2 Drag a vertex of the triangle to change the triangle’s size. Again, drag point P around the interior of the triangle. What do you notice now?

Q3 What happens if you drag P outside the triangle?

Q4 Organize your observations from Q1–Q3 into a conjecture. Write your conjecture using complete sentences.

C

BA

All sketches referred to in

this booklet can

be found in Sketchpad |

Samples | Teach-ing Mathematics

(Sketchpad is the folder that

contains the application

itself.)

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Distances in an Equilateral Triangle (continued)

From Rethinking Proof with The Geometer’s Sketchpad

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

Explaining You are no doubt convinced that the total sum of the distances from point P to all three sides of a given equilateral triangle is always constant, as long as P is an interior point. But can you explain why this is true?

Although further exploration in Sketchpad might succeed in convincing you even more fully of the truth of your conjecture, it would only confirm the conjecture’s truth without providing an explanation. For example, the observation that the sun rises every morning does not explain why this is true. We have to try to explain it in terms of something else, for example the rotation of the earth around the polar axis.

Recently, a mathematician named Mitchell Feigenbaum made some experimental discoveries in fractal geometry using a computer, just as you have used Sketchpad to discover your conjecture about a point inside an equilateral triangle. Feigenbaum’s discoveries were later explained by Lanford and others. Here’s what another mathematician had to say about all this:

Lanford and other mathematicians were not trying to validate Feigenbaum’s results any more than, say, Newton was trying to validate the discoveries of Kepler on the planetary orbits. In both cases the validity of the results was never in question. What was missing was the explanation. Why were the orbits ellipses? Why did they satisfy these particular relations? . . . there’s a world of difference between validating and explaining. —M. D. Gale (1990), in The Mathematical Intelligencer, 12(1), 4.

Challenge Use another sheet of paper to try to logically explain your conjecture from Q4. After you have thought for a while and made some notes, use the steps and questions that follow to develop an explanation of your conjectures.

a

a

a

h2

h1h3P

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Distances in an Equilateral Triangle (continued)

From Rethinking Proof with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 57

2. Press the button to show the small triangles in your sketch.

Q5 Drag a vertex of the original triangle. Why are the three different sides all labeled a?

Q6 Write an expression for the area of each small triangle using a and the variables h1, h2, and h3.

Q7 Add the three areas and simplify your expression by taking out any common factors.

Q8 How is the sum in Q7 related to the total area of the equilateral triangle? Write an equation to show this relationship using A for the area of the equilateral triangle.

Q9 Use your equation from Q8 to explain why the sum of the distances to all three sides of a given equilateral triangle is always constant.

Q10 Drag P to a vertex point. How is the sum of the distances related to the altitude of the original triangle in this case?

Q11 Explain why your explanation in Q5–Q9 would not work if the triangle were not equilateral.

Present Your Explanation Summarize your explanation of your original conjecture. You can use Q5–Q11 to help you. You might write your explanation as an argument in paragraph form or as a two-column proof. Use the back of this page, another sheet of paper, a Sketchpad sketch, or some other medium.

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Distances in an Equilateral Triangle (continued)

From Rethinking Proof with The Geometer’s Sketchpad

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

Further Exploration

1. Construct any triangle ABC and an arbitrary point P inside it. Where should you locate P to minimize the sum of the distances to all three sides of the triangle?

2. a. Construct any rhombus and an arbitrary point P inside it. Where should you locate P to minimize the sum of the distances to all four sides of the rhombus?

b. Explain your observation in 2a and generalize to polygons with a similar property.

3. a. Construct any parallelogram and an arbitrary point P inside it. Where should you locate P to minimize the sum of the distances to all four sides of the parallelogram?

b. Explain your observation in 3a and generalize to polygons with a similar property.

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From Rethinking Proof with The Geometer’s Sketchpad

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

Varignon Area

In this activity, you will compare the area of a quadrilateral to the area of another quadrilateral constructed inside it.

Conjecture

1. Open the sketch Varignon.gsp and drag vertices to investigate the shapes in this sketch.

Q1 Points E, F, G, and H are midpoints of the sides of quadrilateral ABCD. Describe polygon EFGH.

2. Press the appropriate button to show the areas of the two polygons you described. Drag a vertex and observe the areas.

Q2 Describe how the areas are related. You might want to find their ratio.

Q3 Drag any of the points A, B, C, and D and observe the two area measurements. Does the ratio between them change?

Q4 Drag a vertex of ABCD until it is concave. Does this change the ratio of the areas?

Q5 Write your discoveries so far as one or more conjectures. Use complete sentences.

Q6 You probably can think of times when something that always appeared to be true turned out to be false at times. How certain are you that your conjecture is always true? Record your level of certainty on the number line and explain your choice.

75%25% 50%0% 100% Challenge If you believe your conjecture is always true, provide some examples to support your view and try to convince your partner or members of your group. Even better, support your conjecture with a logical explanation or a

E

F

G

HD

CB

A

All sketches referred to in

this booklet can be found in

Sketchpad | Samples | Teach-ing Mathematics

(Sketchpad is the folder that

contains the application

itself.)

E F

GH

D

C

B

A

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Varignon Area (continued)

From Rethinking Proof with The Geometer’s Sketchpad

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

convincing proof. If you suspect your conjecture is not always true, try to supply counterexamples.

Proving In the picture, you probably observed that quadrilateral EFGH is a parallelogram. You also probably made a conjecture that goes something like this:

The area of the parallelogram formed by connecting the midpoints of the sides of a quadrilateral is half the area of the quadrilateral.

This first conjecture about quadrilateral EFGH matches a theorem of geometry that is sometimes called Varignon’s theorem. Pierre Varignon was a priest and mathematician born in 1654 in Caen, France. He is known for his work with calculus and mechanics, including discoveries that relate fluid flow and water clocks.

The next three steps will help you verify that quadrilateral EFGH is a parallelogram. If you have verified this before, skip to Q10.

Q7 Construct diagonal AC. How are sEF and sHG related to sAC? Why?

Q8 Construct diagonal BD. How are sEH and sFG related to sBD? Why?

Q9 Use Q7 and Q8 to explain why EFGH must be a parallelogram. Work through the steps that follow for one possible explanation as to why parallelogram EFGH has half the area of quadrilateral ABCD. (If you have constructed diagonals in ABCD, it will help to delete or hide them.)

Q10 Assume for now that ABCD is convex. One way to explain why ABCD has twice the area of EFGH is to look at the regions that are inside ABCD but not inside EFGH. Describe these regions.

EF

GH

D

C

B

A

EF

GH

D

C

B

A

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Varignon Area (continued)

From Rethinking Proof with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 61

Q11 According to your conjecture, how should the total area of the regions you described in Q10 compare to the area of EFGH?

3. Press the button to translate the midpoint quadrilateral EFGH along vector EF.

Q12 Drag any point. How does the area of the translated quadrilateral compare to the area of EFGH?

4. Construct xF′C and xG′C.

Q13 How is jEBF related to jF′CF?

Q14 Explain why the relationship you described in Q13 must be true.

Q15 How is jHDG related to jG′CG?

Q16 Explain why the relationship you described in Q15 must be true.

Q17 How is jAEH related to jCF′G′?

Q18 Explain why the relationship you described in Q17 must be true.

Q19 You have one more triangle to account for. Explain how this last triangle fits into your explanation.

G'

F'E F

GHD

C

B

A

G'

F'E F

GHD

C

B

A

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Varignon Area (continued)

From Rethinking Proof with The Geometer’s Sketchpad

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

Present Your Proof Create a summary of your proof from Q10–Q19. Your summary may be in paper form or electronic form, and may include a presentation sketch in Sketchpad. You may want to discuss the summary with your partner or group.

Further Exploration Which part of your proof does not work for concave quadrilaterals? Try to redo the proof so that it explains the concave case as well. (Hint: Drag point C until quadrilateral ABCD is concave.)

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From Exploring Calculus with The Geometer’s Sketchpad

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

Visualizing Change: Velocity

There are many ways to create motion or move an object. You could control where the object is located—its position—by dragging it around, or you could control how fast or slow the object moves—its speed.

Velocity is related to speed but it provides more information. If you know your velocity, you really know two things—how fast you are moving (speed) and the direction you are heading. Can knowing the velocity of an object tell you anything else? Are there any relationships or patterns between position and velocity? In this activity you will start to answer these questions by moving a point, controlling its velocity with a slider.

Sketch and Investigate

1. Open the sketch Velocity.gsp.

You will see a horizontal line and point Me that moves along it. Point Home represents your base point or origin. You will also see the point Me2d. This point represents where you are at any time. The x-coordinate of point Me2d (labeled timeMe2d) represents time, and the y-coordinate (labeled positionMe2d) represents your position or distance from point Me to point Home.

2. Drag point Me2d around the plane, getting used to the way point Me’s position along the line (in other words, distance from point Home) relates to Me2d’s location in the time/position plane (in other words, its coordinates).

Q1 Drag point Me2d horizontally. Explain point Me’s motion.

Q2 You can drag point Me2d any way you’d like, but dragging in certain directions doesn’t make sense given the way time works in our universe. How do you have to drag point Me2d so that it represents a physically possible motion of point Me?

Now we want to bring in velocity and see what effect it has.

3. Press the Show Controls button.

You should see two sliders, one for velocity and one for a time interval. There is also a new point labeled FutureMe. This point is located one time interval away at the position you would reach if your velocity stayed

All sketches referred to in

this booklet can be found in

Sketchpad | Samples | Teach-ing Mathematics

(Sketchpad is the folder that

contains the application

itself.)

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Visualizing Change: Velocity (continued)

From Exploring Calculus with The Geometer’s Sketchpad

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

constant. The deltaT slider is set at 1 and the velocity slider should be set at 2. So point FutureMe should be to the right 1 unit and up 2 units.

Q3 If you change deltaT to 0.5 and keep the velocity the same, what will happen to point FutureMe? Try it and see.

Q4 Move the deltaT slider to various time intervals. Does point FutureMe move in any particular pattern? What happens to point Me or point Me2d when you change just the time interval? Why is that?

Q5 Set deltaT back to 1 and now move the velocity slider to various values. Does point FutureMe move in any particular pattern? What happens to point Me or point Me2d when you change just the velocity? Why is that?

The Start Motion button will start both points moving in relationship to the set velocity and time intervals.

4. Press the Reset button to set time back to 0.

5. Turn on tracing for point Me2d.

6. Set the velocity slider to 2 and the deltaT slider back to 1.

For these first trials, you won’t change the velocity slider once your point is moving. Predict what kind of position trace you’ll get if your velocity (speed and direction) stays the same. Sketch this prediction in the margin.

Q6 Press the Start Motion button and observe point Me’s motion and point Me2d’s corresponding time/position trace. Press the button again to stop the motion. Describe your trace. (Was it what you predicted?)

Q7 Press the Reset button, but do not clear your trace. Instead, change the velocity slider to 0.5 and make point Me2d a different color. Make a prediction, and then press the Start Motion button again. What happened this time? How are your traces different? How are they the same?

Q8 Repeat Q7, but this time set your velocity slider to a negative value. Any idea what will happen? Press the Start Motion button again. What happened this time? How are your traces different? How are they the same?

Q9 What conclusions can you reach about movement and position traces when velocity is constant over a time interval?

Q10 What are the equations for the different traces you see on your screen? What would the equation for the trace be if velocity were set to 0?

Select point Me2d, then

choose Trace Point

from the Display menu. You can also

change the color of your

selected point and trace in

the Color submenu of the Display menu.

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Visualizing Change: Velocity (continued)

From Exploring Calculus with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 65

For the next set of trials, you will change the velocity of point Me while time is changing. The smaller the time interval, the more accurate the trace, so set deltaT as close to 0.1 as possible and hide point FutureMe. You can change the velocity slider to any value you wish, but try each of these suggested experiments as well. For each experiment, draw a little sketch of your trace in the margin. Remember to choose Erase Traces from the Display menu and press the Reset button when you want to start over. It is also a good idea to change the color of point Me2d for each trace.

A. Start with the velocity at a positive value. Increase the velocity, and then decrease the velocity, but keep it positive throughout the experiment.

B. Start with the velocity at a negative value. Increase the velocity, and then decrease the velocity, but keep it negative throughout the experiment. (Remember that –2 to –1 is an increase!)

C. Start with velocity > 2. Decrease the velocity, and then increase it. Again, keep the velocity positive throughout.

D. Start with –1 < velocity < 0. Decrease the velocity, and then increase it, but again, keep the velocity negative throughout.

E. Start with a positive velocity and decrease to a negative value. Then increase the velocity again until you get to 0. Stay at 0 for a while and then increase the velocity again.

Q11 How are the traces in A and B similar? How are they different? What happens to the position trace when you switch from increasing the velocity to decreasing it?

Q12 How are the traces in C and D similar? How are they different? What happens to the position trace when you switch from decreasing the velocity to increasing it?

Q13 How are the traces in A and C similar? How are they different? What about B and D?

Q14 What happened when you changed the velocity from positive to negative? From negative to positive? What happened when you stayed at velocity = 0?

To hide a point, select the point and

then choose Hide Plotted

Point from the Display menu.

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Visualizing Change: Velocity (continued)

From Exploring Calculus with The Geometer’s Sketchpad

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

Q15 For each of the following, describe the position trace that you would get. Then check your answer using the velocity slider.

a. positive and increasing velocity

b. negative and increasing velocity

c. positive and decreasing velocity

d. negative and decreasing velocity

Explore More Go to page 2 of the sketch. Press the Show Path1 button. Using your answers from Q15 for reference, make a trace trying to match the path as closely as you can. During which part of your trace did you have to go the fastest? When did you move the slowest?

Hide Path 1 and press the Show Path2 button. Again, try to match the path as closely as you can.

What is different about Path 2? Which one was easier to trace? Is it possible to trace Path 2’s corners?

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From Exploring Calculus with The Geometer’s Sketchpad

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

Going Off on a Tangent

You can see what the average rate of change between two points on a function looks like—it’s the slope of the secant line between the two points. You have also learned that as one point approaches the other, average rate approaches instantaneous rate (provided that the limit exists). But what does instantaneous rate look like? In this activity you will get more acquainted with the derivative and learn how to see it in the slope of a very special line.

Sketch and Investigate

1. Open the sketch Tangents.gsp.

In this sketch there is a function plotted and a line that intersects the function at a point P. This new line is called the tangent line because it intersects the function only once in the region near point P. Its slope is the instantaneous rate of change—or derivative—at point P:

tangent’s slope = instantaneous rate at P = f ′(xP)

So how do you find this line? Let’s hold off on that for a bit and look at the line’s slope—the derivative—and see how it behaves. Remember, slope is the key!

Q1 Move point P as close as possible to x = –1. Without using the calculator, estimate f ′(–1)—the derivative of f at x = –1. (Hint: What’s the slope of the tangent line at x = –1?)

Q2 Move point P as close as possible to x = 0. Without using the calculator, estimate f ′(0)—the derivative of f at x = 0. (Hint: See the previous hint!)

Q3 Move point P as close as possible to x = 1. Without using the calculator, estimate f ′(1)—the derivative of f at x = 1. (Sorry, no hint this time.)

2. Move point P back to about x = –1. Drag point P slowly along the function f from left to right. Watch the line’s slope carefully so that you can answer some questions. (If you’d like, you can animate point P by selecting point P, then choosing Animate from the Edit menu. )

Q4 For what x-values is the derivative positive? (Hint: When is the slope of the tangent line positive?) What can you say about the curve where the derivative is positive?

All sketches referred to in

this booklet can be found in

Sketchpad | Samples | Teach-ing Mathematics

(Sketchpad is the folder that

contains the application

itself.)

Be careful here—the grid is not square!

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From Exploring Calculus with The Geometer’s Sketchpad

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

Q5 For what x-values is the derivative negative? (Hint: Look at the hint in Q4 and make up your own hint.) What can you say about the curve where the derivative is negative?

Q6 For what x-values is the derivative 0? What can you say about the curve where the derivative is 0?

Q7 For what value or values of x on the interval from –1 to 3 is the slope of the tangent line the steepest (either positive or negative)? How would you translate this question into the language of derivatives?

3. Double click on the equation for f, and change the function to f (x) = 4 sin(x). (If the graph isn’t what you expected, you probably need to change Angle Units on the Units panel of Preferences to radians.)

4. Press the Show Zoom Tools button and use the x-scale slider to change your window to go from –2π to 2π on the x-axis. (You can hide the tools again by pressing the Hide Zoom Tools button.)

5. Move point P so that its x-coordinate is around x = –6.

6. Move point P slowly along the function from left to right until you get to about x = 6. As you move the point, watch the tangent line’s slope so that you can answer the following questions.

Q8 Answer Q4–Q7 for this function. Could you have relied on physical features of the graph to answer these questions quickly?(In other words, could you have answered Q4–Q7 for this function without moving point P?)

There is an interesting relationship between how the slope is increasing or decreasing and whether the tangent line is above or below the curve. Move point P slowly from left to right again on the function, comparing the steepness of the line to its location—above or below the curve.

Q9 When is the slope of the line increasing? Is the tangent line above or below the function when the slope is increasing?

Q10 When is the slope of the line decreasing? Is the tangent line above or below the function when the slope is decreasing?

Q11 Write your conclusion for the relationship between the slope of the tangent line and its location above or below the curve. How would you translate this into a relationship between the derivative and the function’s concavity?

Let’s check whether or not your conclusion is really true. The derivative is the slope of the tangent line, so an easy way to check is to calculate the slope of the line.

If you want to recenter your

sketch, select the origin and move it to the

desired location.

f(x)=4sin(x)

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From Exploring Calculus with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 69

7. Select the tangent line and measure its slope by choosing Slope from the Measure menu. Label it TangentSlope.

8. Move point P slowly along the function again from left to right and watch the values of the measurement TangentSlope.

Q12 Do your answers to Q9–Q10 hold up?

Explore More Each of the following functions has some interesting problems or characteristics. For each one, change the equation for f (x) as you did in step 3 above and answer the questions below. If you need to zoom in at a point, press the Show Zoom Tools button. Remember that (a, b) represents the point you will zoom in on. To change a or b, double-click on the measurement and enter a new value.

1. f1(x) = x − 2

2. f2(x) = x2 − 6x + 8

3. f3(x) = x − 1

Q1 Where does the derivative not exist for f1(x) and why? (What happens to the tangent line at that point?)

Q2 Answer Q1 for f2(x) = x2 − 6x + 8 .

Q3 Answer Q1 for f3(x) = x − 1 .

Q4 How is the function f1(x) = x − 2 different from all the others that you have looked at in this activity, including f 2 and f 3?

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From Exploring Calculus with The Geometer’s Sketchpad

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

Accumulating Area

How would you describe the shaded region shown here? You could say: The shaded region is the area between the x-axis and the curve f (x) on the interval 0 ≤ x ≤ 4. Or, if you didn’t want to use all those words, you could say: The shaded region is

f0

4

∫ or f (x) dx0

4

which is much faster to write!

In general, the notation f (x) dxa

b∫ represents the signed area between the

curve f and the x-axis on the interval a ≤ x ≤ b. This means that the area below the x-axis is counted as negative. This activity will acquaint you with this notation, which is called the integral, and help you translate it into the signed area it represents.

Sketch and Investigate

1. Open the document Area2.gsp. You have a function f composed of some line segments and a semi-circle connected by moveable points.

If you need to evaluate the integral f (x) dx0

4

∫ , the first step is to translate it into the language of areas. This integral stands for the area between f and the x-axis from x = 0 to x = 4, as shown. This area is easy to find—you have a quarter-circle on 0 ≤ x ≤ 2 and a right triangle on 2 ≤ x ≤ 4.

So on [0, 2] you have

f (x) dx0

2

∫ = 0.25πr2 =0.25π(2)2 = π

and on [2, 4] you have

f (x) dx2

4

∫ = 0.5(base)(height) = 0.5(2)(6) = 6

so f (x) dx0

4

∫ = 6 + π

2. To check this with the Area tools, press the Show Area Tools button.

There are three new points on the x-axis—points start, finish, and P. Points P and start should be at the origin. Point P will sweep out the area under the curve from point start to point finish. Point P has not moved yet, so the measurement AreaP is 0.

3. Press the Calculate Area button to calculate the area between f and the x-axis on the interval [start, finish] and to shade in that region.

All sketches referred to in

this booklet can be found in

Sketchpad | Samples | Teach-ing Mathematics

(Sketchpad is the folder that

contains the application

itself.)

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From Exploring Calculus with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 71

Q1 Is the value of the measurement AreaP close to 6 + π? Why isn’t it exactly 6 + π or even 9.142?

Q2 Based on the above reasoning, evaluate f (x) dx−2

4

∫ .

4. Check your answer by moving point start to point B, pressing the Reset button, and then pressing the Calculate Area button.

Q3 What do you think will happen to the area measurement if you switch

the order of the integral, in other words, what is f (x) dx4

−2

∫ ?

5. Choose Erase Traces from the Display menu.

6. Check your answer by moving point start to x = 4 and point finish as close as you can get to x = –2, pressing the Reset button, the Esc key, and then the Calculate Area button.

Q4 What is the area between f and the x-axis from x = 4 to x = –2?

Now, what happens if your function goes below the x-axis? For example,

suppose you want to evaluate f (x) dx4

6

∫ .

Q5 Translate the integral into a statement about areas.

Q6 What familiar geometric objects make up the area you described in Q5?

Q7 Using your familiar objects, evaluate f (x) dx4

6

∫ . (Hint: You can do this one quickest by thinking.)

7. Move point start to x = 4 and point finish to x = 6. Press the Reset button.

8. Choose Erase Traces from the Display menu, and then press the Calculate Area button to check your answer. Does the result agree with your calculation?

Q8 Evaluate f (x) dx−6

−3

∫ using the process in Q5–Q7 and check your answer using steps 5 and 6.

If you fix your starting point with xstart = –6, you can define a new

function, A(xP) = f (x) dx−6

xP

∫ , which accumulates the signed area between f and the x-axis as P moves along the x-axis.

Q9 Why is A(−6) = f (x) dx−6

−6

∫ = 0?

Q10 What is A(–3)?

To get an idea of how this area function behaves as point P moves along the x-axis, you’ll plot the point (xP, A(xP)) and let Sketchpad do the work.

Before you move the point, check the

Status Line to make sure you have selected

the right point. If you

haven’t, click on the point

again.

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From Exploring Calculus with The Geometer’s Sketchpad

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

9. Move point start to x = –6 exactly. Now move point finish to x = 9.

The measurement AreaP is now the function A(xP) = f (x) dx−6

xP

∫ .

10. Select the line segment that joins point P to the curve. Turn off tracing for the segment.

11. Select measurements xP and AreaP in that order and choose Plot as (x, y) from the Graph menu.

12. Give this new point a bright new color from the Color submenu of the Display menu. Turn on tracing for this point and label it point I.

13. Press the Reset button and then the Calculate Area button to move point P along the x-axis and create a trace of the area function.

Q11 Why does the area trace decrease as soon as point P moves away from point start?

Q12 Why doesn’t the trace become positive as soon as point P is to the right of point B?

Q13 What is the significance (in terms of area) of the trace’s first root to the right of point start? The second root?

Q14 What is significant about the original function f ’s roots? Why is this true?

14. Turn off tracing for point I and Erase Traces.

15. Select points I and P and choose Locus from the Construct menu.

The locus you constructed should look like the trace you had above. The advantage of a locus is that if you move anything in your sketch, the locus will update itself, whereas a trace will not.

There are quite a few familiar relationships between the original function f and this new locus—including the ones suggested in Q11–Q14. See if you can find some of them by trying the experiments below.

A. Move point B (which also controls point C) to make the radius of the semicircle larger, then smaller.

B. Move point B back to (–2, 0). Now move point A around in the plane. (Make sure to stay to the left of point B.) Try dragging point A to various places below the x-axis, and then move point A to various places above the x-axis.

C. Move point A back to (–6, –4). Now move point D around in the plane. (Make sure to stay between point C and point E.) Drag point D to various places above the x-axis, and then drag point D to various places below the x-axis.

To turn off tracing, choose

Trace Segment from the

Display menu.

Double-click on the point with the Text tool to label it.

Be sure to keep points A, B, C,

D, E, and F lined up in

that order from left to right.

If point C moves to the

right of point D, the line

segment CD will no longer

exist.

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From Exploring Calculus with The Geometer’s Sketchpad

© 2002 Key Curriculum Press Teaching Mathematics with The Geometer’s Sketchpad • 73

D. Follow step C with points E and F.

Q15 List the various patterns that you found between the two functions or in the area function alone. How many patterns were you able to find? Any conjectures about the relationship between the two functions?

Explore More Will the area function’s shape change if you move point start to a value other than x = –6?

1. Select point start and move it along the x-axis.

Q1 Does the area function’s shape change when your starting point is shifted along the x-axis? If so, how? If not, what changes, and why?

Q2 Write a conjecture in words for how the two area functions

f (x)dx−6

xP

∫ and

f (x)dxxstart

xP

∫ are

related.

2. Make a new shape for your area function by moving one or more points—point A, B, D, E, or F. Then move point start again along the x-axis.

Q3 Does your conjecture from Q2 still hold? Write your conjecture in integral notation.

3. Fix point start at the origin. Move point P to the left of the origin but to the right of point B.

Q4 The following two sentences sound good, but lead to a contradiction. Where is the error?

The semicircle is above the x-axis from the origin to point P, so the area is positive. Point I, which plots the area, is below the x-axis, so the area is negative.

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© 2002 Key Curriculum Press Activity Notes—Teaching Geometry with The Geometer’s Sketchpad • 74

Also Available from Spectrum The activities in this booklet are excerpted from publications of Key Curriculum Press, available in Canada from Spectrum Educational Supplies. Here’s the complete list of Sketchpad activity modules:

Exploring Geometry with The Geometer’s Sketchpad This collection of over 100 activities in an easy-to-use format covers nearly all the concepts studied in high school geometry. The activities include a range of experiences, from guided investigations to open-ended explorations. . Exploring Algebra with The Geometer’s Sketchpad The activities in this collection help students move from the concrete world of pre-algebra to the more abstract world of algebra. By exploring many of the core Algebra I concepts in a dynamic mathematical environment, students gain a deeper understanding of and greater fluency with the material. Exploring Conic Sections with The Geometer’s Sketchpad These investigations help students experience the connection between the geometry and algebra of ellipses, hyperbolas, and parabolas. Students construct physical models before constructing the corresponding Sketchpad model.

Exploring Calculus with The Geometer’s Sketchpad These activities help students visualize and experiment with the fundamental concepts of calculus—from working with limits to sketching a slope field. Activities can be done with a class, in small groups, or individually.. Pythagoras Plugged In: Proofs and Problems for The Geometer’s Sketchpad This book guides students through a variety of proofs and applications of the theorem that has fascinated mathematicians throughout history. Constructing and dynamically manipulating figures provides insights that no static illustration can offer. Geometry for Middle School Students with The Geometer’s Sketchpad This collection of imaginative activities and projects, written by a team of experienced middle school teachers, covers the range of the middle school geometry curriculum. For additional supplemental curriculum modules, visit the Spectrum web site at http://www.spectrumed.com/index.html. For additional activity ideas, visit The Geometer’s Sketchpad Resource Center at http://www.keypress.com/sketchpad/

1150 65th Street Emeryville, California 94608 USA 510-595-7000 http://www.keypress.com/

125 Mary Street

Aurora, Ontario L4G 1G3 1-800-668-0600

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