GSP Teaching Math.pdf

Teaching Mathematics with

Dynamic Geometry ® Software for Exploring Mathematics

Teaching Notes Sample Activities Windows ® /Macintosh ®

Key Curriculum Press Key College Publishing

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 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 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 510-595-7000 [email protected] http://www.keypress.com 10 9 8 7 6 5 4 3 2

0 5 0 4 0 3 02

ISBN 1-55953-582-2

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 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 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 510-595-7000 [email protected] http://www.keypress.com 10 9 8 7 6 5 4 3 2

0 5 0 4 0 3 02

ISBN 1-55953-582-2

Contents Teaching Notes ..................... ................................ ..................... ..................... ...................... ...................... ...................... ...................... ...................... ...................... ....................... ................ 1 The Geometer’s Sketchpad and Changes Changes in Mathematics Teaching ....................................... ....................................... 1 Where Sketchpad Came From ......................... ...................................... .......................... .......................... .......................... .......................... ......................... ................ 2 Using Sketchpad in the Classroom Classroom ............................ .......................................... ............................ ............................ ............................ ........................ .............. 3 A Guided Investigation: Napoleon’s Theorem .............................. .............................................. ................................ .............................. .............. 4 An Open-Ended Exploration: Constructing Constructing Rhombuses ................................. ................................................. ......................... ......... 5 A Demonstration: A Visual Demonstration Demonstration of the Pythagorean Pythagorean Theorem ............................ ............................ 6 Using Sketchpad in Different Classroom Settings ................................ ................................................. ................................. ..................... ..... A Classroom Classroom with One Computer ........................... ......................................... ........................... ........................... ........................... ........................... .................. One Computer and a Projection Device ............................. ............................................ ............................. ............................. ............................. ................ A Classroom with a Handful of Computers ............................. ............................................ .............................. ............................... ..................... ..... A Computer Lab ........................ .................................... ........................ ......................... ......................... ........................ ........................ ........................ ..................... ................... ..........

7 7 7 7 8

Using Sketchpad as a Presentation Tool ............................. ........................................... ............................. ............................. ............................. ............... 8 Using Sketchpad as a Productivity Tool ............................. ........................................... ............................. ............................. ............................. ............... 9 The Geometer’s Sketchpad and Your Textbook ................................ ................................................ ................................ ........................ ........ 10

Sample Activities ...................... ................................. ...................... ...................... ...................... ...................... ...................... ...................... ...................... ...................... ................. ...... 11 Introduction Introduction ........................ .................................... ........................ ........................ ........................ ........................ ........................ ........................ ....................... ...................... ............... .... 11 Angles ....................... .................................. ...................... ....................... ....................... ....................... ....................... ....................... ....................... ...................... ...................... ................... ........ 12 Constructing Constructing a Sketchpad Kaleidoscope Kaleidoscope ............................ ........................................... .............................. ............................. .......................... ............ 13 Properties of Reflection .......................... ....................................... .......................... .......................... .......................... .......................... ......................... ....................... ........... 16 Tessellations Using Only Translations ........................... .......................................... ............................. ............................. ............................. ................. ... 18 The Euler Segment ........................ .................................... ......................... ......................... ........................ ......................... ......................... ....................... ...................... ............... 20 Napoleon’s Theorem ........................ .................................... ......................... ......................... ......................... ......................... ........................ ........................ .................... ........ 22 Constructing Constructing Rhombuses ......................... ...................................... ......................... ......................... .......................... .......................... .......................... ...................... ......... 23 Midpoint Quadrilaterals 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 Introduction to Vectors ............................. ............................................ ............................... ............................... ..................... ...... 44 Leonardo da Vinci’s Proof ......................... ...................................... .......................... .......................... .......................... .......................... ........................ ................... ........ 46 The Folded Circle Construction Construction .......................... ........................................ ........................... ........................... ............................ ............................ ................... ..... 49

© 2002 Key Curriculum Curriculum Press

Teaching Mathematics Mathematics with The Geometer’s Geometer’s Sketchpad •

iii

The Expanding Circle Construction ............................................................................................ Distances in an Equilateral Triangle ............................................................................................ Varignon Area ................................................................................................................................. Visualizing Change: Velocity ........................................................................................................ Going Off on a Tangent ................................................................................................................. Accumulating Area .........................................................................................................................

53 56 60 64 68 71

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

iv

• Teaching Mathematics with The Geometer’s Sketchpad

© 2002 Key Curriculum Press

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.

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.

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 a pproach 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 © 2002 Key Curriculum Press

Teaching Mathematics with The Geometer’s Sketchpad • 1

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, 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 V isual 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 Jac kiw 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.

2 • Teaching Mathematics with The Geometer’s Sketchpad


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 i nto 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 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.



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 investigation 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.

Napoleon s Theorem ’

Name(s):

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

One way to construct the center is to construct two medians and their point of intersection.

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. ’

Select the entire figure; then choose Create New Tool

from the Custom Tools menu in the Toolbox (the bottom tool).

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 ∆ABC.

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 ∆ABC ) or is not attached properly, undo and try attaching it again.

B

7. Use the custom tool to construct equilateral triangles on each side of ∆ABC. 8. Drag to make sure each equilateral triangle is stuck to a side.

A

C

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 ∆ABC.

After students have discussed their findings in pairs or Q1 State what you think Napoleon s theorem might be. small groups, it’s important to discuss them as a large group. Ask students to share Explore More any special cases they’ve 1. Construct segments connecting each vertex of your original triangle discovered, and use your with the most remote vertex of the equilateral triangle on the opposite questions to emphasize side. What can you say about these three segments? 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. ’



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 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.

D B

C A

Method 1:

Properties:

Method 2:

Properties:

Method 3:

Properties:

Method 4:

Properties:


Teaching Mathematics with The Geometer ’s Sketchpad • 5

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 investigation. The purpose of this lesson, though, is as a demonstration 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 the Pythagorean Theorem

All sketches referred to in this booklet can be found in Sketchpad | Samples | Teach ing Mathematics (Sketchpad is

Name(s):

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.

the folder that contains the application itself.) Click on a polygon interior to select it. Then, in the Measure menu, choose Area.

C

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.

a

4. Drag point B onto the line. It should overlap point A so that the two parallelograms form a single irregular shape.

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.

c b

A

B

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. C

C C

a

A

a

c b

A B

To confirm that this works for any right triangle, change the shape of the triangle and try the experiment again.

6 • Teaching Mathematics with The Geometer ’s Sketchpad

Q1

c b

A B

Step 3

a

c b

B

Step 4

Step 5

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


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. Y ou 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 fin dings 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 fig ures 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 chalkb oard 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. © 2002 Key Curriculum Press


• That you move among groups posing questions, giving help if needed, and keeping students on task. • That students’ 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’ it’s perhaps best to have students work in pairs. Students learn best when they communicate about ab out what they’ they’re learning, and students working together can better stimulate ideas and lend help to one another. If you do have students working at their own computers, encourage them to talk about what they’ they’re doing and to compare their findings with those of their nearest neighbor—they should peek over each others’ 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’ You’ll find that Sketchpad’ 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 a nnotate 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’ 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” played” with the click of a button. In other words, action buttons turn sketches into presentations.

The Circle Squared D. Bennett 7.6.01 Given a segment AB, I've constructed a square and a circle with equal areas. First I constructed a square with side length AB. Then I calculated the radius of a circle with the same area. Finally, I translated the center of the square by that quantity and constructed the circle.

m AB = 1.347 in. (m AB)2 = 0.760 in. π

Radius Area

CC' = 0.7 60 in. CC' = 1.8 15 in 2

Area ABA'B'= 1.815 in 2

Text and action buttons make possible presentations without presenters: A sufficiently annotated sketch could Press the action buttons to transform speak for itself when opened by the figure. another user at a time when the sketch creator isn’ isn’t around to explain it. A square me! circle me! presentation, in this context, is not A necessarily designed for a group audience looking at an overhead display. The audience for an annotated A Captioned Sketch sketch might be a fellow f ellow 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 i n captions. 8 • Teaching Mathematics Mathematics with The The Geometer ’s Sketchpad

C

C'

B


Sketchpad’ Sketchpad’s web integration facilities allows you to draw d raw 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’ 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’ page’s Dynamic Geometry illustrations whether they have Sketchpad or not! By browsing through the sample documents that come with Sketchpad you can get g et ideas for different ways sketch captions can be used to communicate mathematically.

Using Sketchpad as a Productivity P roductivity Tool Manual describes how to use the Edit menu to cut, copy, and paste Sketchpad The Reference Manual describes 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 a s 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’ you ’re writing a book or article that will be printed professionally, Sketchpad graphics can even be output on a typesetting machine with very hig h quality results. Lines and rays are truncated when pasted into other programs, just as they are when printed in Sketchpad.

Solve for x and and y:

x

y

y

5 in.

8 cm 10 cm

x

x

y

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


Teaching Mathematics Mathematics with The Geometer Geometer ’s Sketchpad • 9

The Geometer’ 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’ 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 Sk etchpad before you require them to do a proof. Working out constructions using S ketchpad and interacting with diagrams dynamically will deepen students’ 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’ Serra’s Discovering Geometry, 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 Geometry or any other book with a similar approach can be done using Sketchpad. The Discovering Geometry student Geometer ’s Sketchpad Projects and Geometry student text includes ten Geometer’ 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 Geometry with The Geometer’s Sketchpad. Sketchpad. These Sketchpad lessons have the same titles and guide students to the same conjectures as the corresponding Discovering Geometry lessons. Geometry lessons. A collection of Sketchpad documents accompany this book on CD-ROM. The Discovering Geometry Teacher’s Resource Book comes comes with demonstration sketches corresponding to Discovering Geometry lessons. Ancillary Sketchpad materials are also al so available for some secondary texts from other publishers, though for a geometry course, none provide as complete a technology package as Key Curriculum Press’ Press ’s Discovering Geometry combined with The Geometer’ Geometer’s Sketchpad. If you’ you’re using a text other than Discovering Geometry, ask Geometry, ask the publisher whether Sketchpad ancillaries are available. Exploring Geometry with The Geometer’s Sketchpad, Sketchpad, available from Key Curriculum Press, contains more than 100 reproducible activities ac tivities that can be used with any an y text. A CD-ROM with activities for Macintosh and Windows computers c omputers accompany the activities. Many other topic-specific volumes of activities are also available avail able from Key Curriculum Press. 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. year’s worth of activities to cover nearly all Exploring Geometry could Geometry could supply a teacher with a year’ the content of a typical high school geometry course using The Geometer’ Geometer’s Sketchpad. And other activity books could occupy a large part of the year in other mathematics courses, too. We don’ don’t, however, advocate that you abandon other teaching methods in favor of using the computer. It’ 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. na ture. 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’ it’s important that students don’ don ’t get the mistaken impression that mathematics exists only in their books and on their computer screens.

10 • Teaching Mathematics Mathematics with The Geometer ’s Sketchpad


Sample Activities Introduction 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. 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

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

A Rectangle with Maximum Area

Distances in an Equilateral Triangle


Varignon Area

The Golden Rectangle A Sine Wave Tracer From Exploring Algebra with The Geometer’s Sketchpad

From Exploring Calculus with The Geometer’s Sketchpad

Visualizing Change: Velocity Going Off on a Tangent Accumulating Area

Adding Integers 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 y our 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.



Angles

Name(s):

1. Open a new sketch. 2. Construct a triangle. 3. Extend one side by constructing a ray using two vertices. C

D B

A

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.

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



Constructing a Sketchpad Kaleidoscope

Name(s):

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 counter-clockwise 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.




Constructing a Sketchpad Kaleidoscope (continued)

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 6 0 and then click Rotate. (Pick a different factor of 360 if you wish.)

selection marquee

after 60˚ rotation

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.

control point

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.

control point

control point




Constructing a Sketchpad Kaleidoscope (continued)

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.

merged points

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.

Animate Points

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.




Properties of Reflection

Name(s):

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.

A

2. Construct point C to the right of the line. Double-click on the line.

3. Mark AB as d a mirror.

C

C'

B

4. Reflect point C to construct point C´. 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 when you wish to erase your traces.

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. Select points C and C ´. In the Display menu, you’ll see Trace Points checked. Choose it to uncheck it.

Sketch and Investigate: Reflecting Geometric Figures

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

D

E

B

9. In the Display menu, choose Erase Traces.

D'

10. Construct jCDE. Select the entire figure; then, in the Transform menu, choose Reflect.

11. Reflect jCDE (sides and vertices) over AB. d

E'

C

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

A C'

From Exploring Geometry with The Geometer’s Sketchpad



Properties of Reflection (continued)

13. Measure the lengths of the sides of triangles CDE and C´DÉ´. 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 Width is in the Display menu.

You may wish to construct points of intersection and measure distances to look for relation ships between the mirror line and the dashed segments.

14. Measure one angle in jCDE and measure the corresponding angle in jC´DÉ´. Q2

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 counter-clockwise direction? In what direction (clockwise or counter-clockwise) are vertices C´, D´, and E´ oriented in the reflected triangle?

15. Construct segments connecting each point and its image: C to C´, D to D´, and E to E´. Make these segments dashed. 16. Drag different parts of the sketch around and observe relationships between the dashed segments and the mirror line. Q5

D

B

E

D'

E' C A

C'

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. From Exploring Geometry with The Geometer’s Sketchpad



Tessellations Using Only Translations

Name(s):

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

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

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.

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

C'

B

A Steps 1–3

3. Construct the remaining sides of your parallelogram. C

C

C'

B

A Step 4

Select the vertices in consecutive order; then, in the Construct menu, choose Polygon Interior.

C

C

C'

B

A

C'

B

A

Step 5

C

C'

B

A

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 color on every other tile to create a pattern.

C

A

C'

B Steps 8–10




Tessellations Using Only Translations (continued)

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.

C

A

C'

B Steps 11 and 12

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.)




The Euler Segment

Name(s):

In this investigation, you’ll look for a relationship among four points of concurrency: the incenter, the circumcenter, the orthocenter, and 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 Triangle Centers.gsp can be found in Sketchpad | Samples | Custom Tools. (Sketchpad is the folder that contains the application itself.)

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

I

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.

I O

Ce

Ci

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.




The Euler Segment (continued)

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.)

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, drag your triangle around and investigate special triangles. Describe any triangles in which some of the nine points coincide.




Napoleon’s Theorem

Name(s):

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

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).

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 j ABC.

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.

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

B

A

C

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 j ABC. 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?




Constructing Rhombuses

Name(s):

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.

D B

C A

Method 1:

Properties:

Method 2:

Properties:

Method 3:

Properties:

Method 4:

Properties:




Midpoint Quadrilaterals

Name(s):

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

ABCD.. 1. Constru Construct ct quadril quadrilate ateral ral ABCD If you select all four sides, you can construct all four midpoints at once.

B F

2. Constru Construct ct the midp midpoin oints ts of the the sides. sides. 3. Connec Connectt the midpoi midpoints nts to const construct ruct another quadrilateral, EFGH .

G

C

A E

4. Drag Drag vertic vertices es of your your orig origina inall quadrilateral and observe the midpoint quadrilateral.

H D

5. Measure Measure the the four four side leng lengths ths o off this 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. Constru Construct ct a diagon diagonal. al.

B

7. Measure Measure the leng length th and and slope slope of the diagonal.

G

F

C

E

8. Drag Drag vertic vertices es of the the origi original nal 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

A H D

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.

From Exploring Exploring Geometry with The Geometer’ Geometer’s Sketchpad



A Rectangle with Maximum Area

Name(s):

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’ 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. Co Cons nstr truc uctt AB. s Select s AB, 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 .

2. Co Cons nstr truc uctt AC AB s on AB. s .

E

3. Cons Constr truct uct line liness perpendicular to AB s through points A points A and and C.

A

D

C

B

CB. 4. Co Cons nstr truct uct circ circle le CB. 5. Cons Constr truct uct poin pointt D where this circle intersects the perpendicular line. D, parallel to AB. AB 6. Constr Construct uct a line line throug through h point point D, parallel s . E, the fourth vertex of rectangle ACDE 7. Co Cons nstr truct uct poin pointt E, the rectangle ACDE..

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 .

ACDE.. 8. Constr Construct uct polyg polygon on inter interior ior ACDE 9. Measure the area area and perimeter perimeter of this this polygon. polygon. 10. 10. Drag Drag poin pointt C back and forth and observe how this affects the area and perimeter of the rectangle. 11. 11. Measu easure re AC AC and and AE AE.. Q1

Without measuring, state how AB how AB is 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?

From Exploring Exploring Geometry with The Geometer’ Geometer’s Sketchpad


Teaching Mathematics Mathematics with The Geometer Geometer ’s Sketchpad • 25

A Rectangle with Maximum Area (continued)

In Steps 12–14, you’ you’ll explore this relationship graphically. Select, in order, m sAC and Area and 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.

12. Plot the measureme measurements nts for for the the length length of of AC and AC ACDE s and the area of ACDE y). as (x, (x, y ). You should get axes and a plotted point H, point H, as as shown below. 13. 13. Drag Drag poin pointt C to see the plotted point move to correspond to different side lengths and areas. H

Area ACDE = 3.74cm 3.74cm2 Perimeter ACDE = 9.41 cm E

D 2

m AC = 3.69 cm m AE = 1.01 cm

You may wish to select point H and and measure its coordinates.

C

-5

-10

Select point H and and point C ; then, in the Construct menu, choose Locus.

A

B

F

G

14. To see a graph of all all possible possible areas for this this rectangle, rectangle, construct construct the the locus of plotted point H point H as as defined by point C. It should now be easy to position point C so that point H point H is 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. 15. Drag Drag poin pointt C so that point H point H moves 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 Investigate area/perim area/perimeter eter relationsh relationships ips in other polygons. polygons. Make 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 Let AC be be x and let AB let AB be (1/2)P (1/2)P, where P stands for perimeter (a constant). Write an equation for area, A area, A,, in terms of x and P. What value for x (in terms of P) gives a maximum value for A for A?? From Exploring Exploring Geometry with The Geometer’ Geometer’s Sketchpad




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.)

Name(s):

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.

Click on an interior to select it. Then, in the Measure menu, choose Area.

C

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.

a A

4. Drag point B onto the line. It should overlap point A so that the two parallelograms form a single irregular shape.

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.

c b

B

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. C

C C

a

A

a

c b

A

B

To confirm that this works for any right triangle, change the shape of the triangle and try the experiment again.

Q1

c b

A B

Step 3

a

c b

B

Step 4

Step 5

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




The Golden Rectangle

Name(s):

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 a somehow pleasing to the eye, perhaps because they approximate the shape of our field of vision. b For this reason, they’re used often in architecture, especially the classical architecture of ancient a + b b Greece. In this activity, you’ll construct a golden = b a 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 You can use the tool 4/Square (By Edge) from the sketch Polygons.gsp that comes with the program.

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 AD. s 4. Construct circle EC.

B

C B

A

A

Steps 1 – 4

Select the objects; then, in the Display menu, choose Hide Objects.

G

D

F

B

C

G

A

D

F

D

E

Hold down the mouse button on the Segment tool to show the Straight Objects palette. Drag right to choose the Ray tool.

C

E

Steps 5 – 8

Steps 9 – 11

5. Extend sides AD and BC with rays, as shown. 6. Construct point F where AD f intersects the circle. 7. Construct a line perpendicular to AD f through point F. 8. Construct point G where this perpendicular intersects BC. f Rectangle AFGB is a golden rectangle. 9. Hide the lines, the rays, the circle, and point E. BC. 10. Hide AD, s DC, and s s From Exploring Geometry with The Geometer’s Sketchpad



The Golden Rectangle (continued)

11. Construct BG, s GF, s and FA. s Select, in order,

sAF and AB s ; 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.

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.

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.

B

C

G

A

D

F

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.

B

Explore More A 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 ø 2 and 1/ ø . How are these numbers related to ø ? Use algebra to demonstrate why these relationships hold.




A Sine Wave Tracer

Name(s):

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. E C

F

D A

B

2. Construct a circle with center A and radius endpoint C. 3. Construct point D on AB. s Select point D and AB s ; then, in the Construct menu, choose Perpendicular Line.

4. Construct a line perpendicular to AB s through point D. 5. Construct point E on the circle. 6. Construct a line parallel to AB s 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.

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

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.




A Sine Wave Tracer (continued) Select points D and E and choose Edit | Action Buttons | Animation. Choose forward in the Direction pop-up menu for point D.

8. Make an action button that animates point D forward on AB s 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 AB s is the value of the sine of ∠EAD. E C

F B

D A

Q5

5

10

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




Adding Integers Definition: Integers are positive and negative whole numbers, including zero. On a number line, tick marks usually represent the integers.

Name(s):

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

All sketches referred to in this booklet can be found in Sketchpad | Samples | Teaching Mathematics (Sketchpad is the folder that contains the application itself.)

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

8 + 5

2. Study the problem that’s modeled: 8 + 5 = 13. Then drag the two “drag” -1 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?

drag

0

drag

1

2

3

4

5

10

-6

drag

+ -3

drag

-5

-4

-3

-2

-1

0

1

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.

From Exploring Algebra with The Geometer’s Sketchpad



Adding Integers (continued) Q4

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

5 + -5

drag

-5

drag

-4

-3

-2

-1

0

1

2

3

4

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?

4

drag

+ -7

drag

-5

-4

-3

-2

-1

0

1

2

3

4

5

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?




Adding Integers (continued) 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 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.

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.




Points “Lining Up” in the Plane

Name(s):

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. Holding down the Shift key keeps all five points selected.

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.

To measure the coordinates of selected points, choose Coordinates from the Measure menu.

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

To hide objects, select them and choose Hide from the Display menu.

A: (3.00, 3.00) B: (-1.00, -1.00)

A

C: (2.00, -2.00)

2

D: (3.00, -1.00)

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

B

5. Choose Snap Points from the Graph menu. From now on, the points will only land on locations with integer coordinates. Q1

E

E: (1.00, 2.00)

D -2

C

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.

The absolute value of a number is its “positive value.” The absolute value of both 5 and –5 is 5.

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.




Points “Lining Up” in the Plane (continued)

a.

-10

b. 6

6

3

3

-5

5

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-3

-6

-6

6

6

3

3

-5

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-6

-6

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10

f. 6

6

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3

-5

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10

-10

-5

-3

-3

-6

-6

g.

-10

5

d.

e.

-10

-5

-3

c.

-10

-10

h. 6

6

3

3

-5

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10

-10

-5

-3

-3

-6

-6




Points “Lining Up” in the Plane (continued) Backward Thinking

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

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.




Parabolas in Vertex Form

Name(s):

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 All sketches referred to in this booklet can be found in Sketchpad | Samples | Teaching 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.

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.

y = 1.4(x – (0.9))2 – 1.6

2

f(x) = a⋅(x-h)2+k

-5

P

a = 1.4

xP = 2.5

h = 0.9 -2 3. Enter a*(x–h)^ 2+ k and k = -1.6 click OK. 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. 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.

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




Parabolas in Vertex Form (continued) 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.




Parabolas in Vertex Form (continued) 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.

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? ′




Reflection in Geometry and Algebra

Name(s):

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.

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.

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).




Reflection in Geometry and Algebra (continued)

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

A'

A 1

-2

2

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.




Reflection in Geometry and Algebra (continued) 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? A special challenge is to make sure your answers to this question and Q6 work regardless of what quadrants the points are in.

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?




Walking Rex: An Introduction to Vectors

Name(s):

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.

All sketches referred to in this booklet can be found in Sketchpad | Samples | Teaching 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!

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




Walking Rex (continued) 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.




Leonardo da Vinci’s Proof

Name(s):

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 In this figure, you don’t have to construct the square on the hypotenuse.

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

B a

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

b a

c

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. The Action Buttons submenu is in the Edit menu.

c

C

A

H

b

Hide reflection

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. B a

c b

C

A

H

Show reflection

From Pythagoras Plugged In



Leonardo da Vinci’s Proof (continued)

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. a

B c

a

b

b

A

C H

C'

A'

B'

Show reflection Hide rotation

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

′

10. Construct the polygon interior of BA´B´ A and of the two triangles adjacent to it. 11. Select A B, B A, and the three polygon interiors and create a x x Hide/Show action button. Name it “Hide c squared.” ′

′

a

B c

a b

c

b A

C H

C'

A'

B'

Show reflection Hide rotation Hide c squared




Leonardo da Vinci’s Proof (continued) 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.




The Folded Circle Construction

Name(s):

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. 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.

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.

B A

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. Mathematicians would describe your set of creases as an envelope of creases.

B A

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.

From Exploring Conic Sections with The Geometer’s Sketchpad



The Folded Circle Construction (continued) 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.

C

B A crease

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. 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.

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. From Exploring Conic Sections with The Geometer’s Sketchpad



The Folded Circle Construction (continued)

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

The Merge and Split commands

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.

appear in the Edit menu.

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. All sketches referred to in this booklet can be found in

Sketchpad | Samples | Teaching 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.

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 Folded Circle Construction (continued) How to Prove It C

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.

I

E B A

H

23. Add segments CB, BE, and AC to the picture. 24. Label the intersection of CB with the crease line as point D. Questions

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.

Q8

Use a triangle congruence theorem to prove that jBED m

Q9

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

Q10

jCED.

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?




The Expanding Circle Construction 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.

Name(s):

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.

G

H

4

C 2

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.

D 6

A -5

E

F -2

5

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. 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.

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.




The Expanding Circle Construction (continued)

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, x 2 = 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 . All sketches referred to in this booklet can be found in


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.

x a

b

x

14. Use Sketchpad’s calculator to compute the geometric mean of lengths a and b. Compare this value to x.




The Expanding Circle Construction (continued) 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. G

H = (x, y)

D 6

4

2

C A = (0, 0)

E -5

F 5

-2

B (0, -3)

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. ′




Distances in an Equilateral Triangle

Name(s) C

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 A B 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 All sketches referred to in this booklet can be found in

Sketchpad | Samples | Teaching Mathematics (Sketchpad is

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.

the folder that contains the application itself.)

From Rethinking

Proof with The Geometer’s Sketchpad



Distances in an Equilateral Triangle (continued) 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?

a h3

h1 P

h2

a

Although further exploration in Sketchpad might succeed in convincing you even more a 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 validat e 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.

From Rethinking


Proof with The Geometer’s Sketchpad Teaching Mathematics with The Geometer’s Sketchpad • 57

Distances in an Equilateral Triangle (continued)

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.

From Rethinking




Distances in an Equilateral Triangle (continued) 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.

From Rethinking



Varignon Area

Name(s)

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

All sketches referred to in this booklet can be found in


F

B

Conjecture

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

C

E G A H D

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?

B

E

Q5

Q6

0%

A

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

F

C

G H D

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. 25%

From Rethinking

50%

75%

100%




Varignon Area (continued) 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 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 EF s s related to AC? s Why? and HG

B F E

C

A

G H

Q8

Q9

Construct diagonal BD. How are EH s and FG s related to BD? s Why?

D

Use Q7 and Q8 to explain why EFGH must be a parallelogram.

From Rethinking



Varignon Area (continued)

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.)

B F

E

C A

G H D

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.

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?

B F'

F

E

C A G'

G

H D

x C and G x C. 4. Construct F ′

Q13

′

How is jEBF related to j F CF?

B

′

F'

F

E

C A

Q14

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

G

H

G'

D

Q15

How is j HDG related to jG CG?

Q16


′

From Rethinking




Varignon Area (continued) Q17

How is j AEH related to jCF G ?

Q18


Q19

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

′

′

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.)

From Rethinking



Visualizing Change: Velocity

Name(s):

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 6 Home Me are heading. Can knowing the velocity of an object tell you anything else? Are there any 4 Me2d relationships or patterns between position and velocity? 2 In this activity you will start to velocity = 1.08 answer these questions by v moving a point, controlling its 5 10 velocity with a slider. Sketch and Investigate All sketches referred to in this booklet can be found in Sketchpad | Samples | Teaching Mathematics (Sketchpad is the folder that contains the application itself.)

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 time Me2d) represents time, and the y-coordinate (labeled position Me2d) 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. What happens to point Me? Explain.

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.

FutureMe

4

The grid in the figure is for clarification. It will not appear in the sketch.

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 constant. The deltaT slider is set at 1 and the velocity From Exploring

2 Me2d

2

1

Calculus with The Geometer’s Sketchpad



Visualizing Change: Velocity (continued)

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?

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

Start Motion

deltaT = 1.0 0 8

Reset

velocity = 2.00 6

4. Press the Reset button to move point Me2d to time = 0.

4

5. Turn on tracing for point Me2d.

Home

Me

FutureMe

position

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

2

Me2d

time 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?

From Exploring


Calculus with The Geometer’s Sketchpad Teaching Mathematics with The Geometer’s Sketchpad • 65

Visualizing Change: Velocity (continued)

To hide a point, select the point and then choose Hide Plotted Point from the Display menu.

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.

deltaT = 0.01 8

velocity = 2.00 6

Home

Me

4

2

Me2d

time

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 deltaT = 0.01 switch from increasing the velocity to 8 decreasing it? velocity = 1.33

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

Q14

6

Home

Me

4 Me2d

2

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

5

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

From Exploring




Visualizing Change: Velocity (continued) 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?

Home

Me

5

Me2d

10

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?

From Exploring



Going Off on a Tangent

Name(s):

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 Q one point approaches the other, average rate P approaches instantaneous rate (provided that y=f(x) 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. 4

2

2

-2

Sketch and Investigate All sketches referred to in this booklet can be found in


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! Be careful here —the grid is not square!

If you’d like, you can animate point P by selecting it, then choosing Animate Point from the Display menu.

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. P

Q4

y=f(x)

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? From Exploring




Going Off on a Tangent (continued) 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. Go to page 2 of the document. Here the function is f (x) = 4 sin(x). If you want to recenter your sketch, select the origin and move it to the desired location.

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

P y=f(x)

5

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

f(x) = 4⋅s in(x)

6. Move point P slowly along the function to the right until you get to about x = 6. As you move the point, watch the tangent line’s slope so 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.

From Exploring



Going Off on a Tangent (continued)

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.

TangentSlope = 1.34277 f(x) = 4⋅s in(x)

5

P y=f(x)

Q12

Do your answers to Q9 –Q10 hold up?

-5

5

Explore More

Each of the following functions has some interesting problems or characteristics. For each one, change the equation for f (x) by double-clicking on the expression for f (x) and entering in the new expression. Then 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 parameter and enter a new value. f1 (x) = x − 2 f 2 (x) = x 2 − 6x + 8 f 3 (x) = x − 1 Q1

Where does the derivative not exist for f 1(x) and why? (What happens to the tangent line at that point?)

Q2

Answer Q1 for f 2 (x) = x 2 − 6 x + 8 .

Q3

Answer Q1 for f 3 (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?

From Exploring




Accumulating Area

Name(s):

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

∫

4

0

f or

1

y=f(x) 0. 5

2

4

-0.5

4

∫ f (x) dx 0

which is much faster to write! In general, the notation

b

∫ a f (x) dx 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 All sketches referred to in this booklet can be found in


1. Open the document Area2.gsp. You have a function f composed of some line segments and a semicircle connected by moveable points. If you need to evaluate the integral

4

∫ f (x) dx, the

D

6

0

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.

4

2

y=f(x)

So on [0, 2] you have 2

∫ f (x) dx = 0.25πr = 0.25π(2) 2

0

2

=π

C

5

and on [2, 4] you have 4

∫ f (x) dx = 0.5(base)(height) = 0.5(2)(6) = 6 2

so

4

∫ f (x) dx = 6 + π 0

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. From Exploring



Accumulating Area (continued)

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.

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.

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

4

∫ − f (x) dx . 2

4. To check your answer, move point start to point B, press the Reset P button, and then press the Calculate Area button. Q3

What do you think will happen to the area measurement if you switch −2

∫ 4

the order of the integral, in other words, what is

f (x) dx ?

5. To check your answer, move point start to x = 4 and point finish as close as you can get to x = –2, then press the Reset P button. 6. Choose Erase Traces from the Display menu and then press 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

The grid is shown here for comparison. It doesn’t appear in the sketch.

6

∫ f (x) dx.

D

4

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

6

∫ f (x) dx. 4

(Hint: You can do this one quickest by thinking.) 7. Make sure point start is at x = 4 and move point finish to x = 6. Press the Reset P button.

You can also erase traces by pressing the Esc key twice.

5

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

−3

∫ −6 f (x) dx using the process in Q5 –Q7 and

E

check your answer using steps 7 and 8. If you fix your starting point with xstart = –6, you can define a new function, A(xP ) =

xP

∫ −

6

f (x) dx , which accumulates the signed area between

f and the x-axis as P moves along the x-axis. Q9

Why is A(−6) = From Exploring

−6

∫ −

6

f (x) dx = 0 ?




Accumulating Area (continued) 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. 9. Move point start to x = –6 exactly. Now move point finish to x = 9. The measurement AreaP is now the function A(xP ) = To turn off tracing, choose Trace Segment from the Display menu. If you can’t see the new point, scroll or enlarge the window until you do. To enlarge the window, press the Show Unit Points button, resize the window, then press the Set Function button.

xP

∫ −

6

f ( x) dx .

10. Select the line segment that joins point P to the curve. Turn off tracing for the segment. Erase all traces. 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 P button and then the Calculate Area button to move point P along the x-axis and create a trace of the area function. Q11

Q12

Q13

Q14

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

D 5

Why doesn’t the trace become positive as soon as point P is to the right of point B? What is the significance (in terms of area) of the trace’s first root to the right of point start? The second root?

y=f(x)

F

B

C

10

A -5

E

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

14. Turn off tracing for point I and erase all traces. Check the status line to see that point P is selected. If not, click on the point again.

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

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.

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.

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.

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

From Exploring



Accumulating Area (continued)

B. Press the Set Function button to 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. Press the Set Function button to 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. 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 xP

∫ x

∫ −

6

f ( x) dx and

5

D

y=f(x)

A

E

f ( x)dx are related.

start

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.

xP

F start

B

C

finish

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 the 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.

From Exploring




Activity Notes for Sample Activities Extensive Activity Notes are included in all Sketchpad curriculum published by Key Curriculum Press. Activity Notes provide suggestions for presenting an activity and other important supporting material, including what prerequisites the activity might have, potential stumbling blocks for students, and answers to the questions posed in the activity. Just as activities differ in form from book to book, so do Activity Notes. Some books have separate Activity Notes for each activity. Others have chapter notes or have a special section for general notes. The following pages contain sample Activity Notes for most of the preceding sample activities. Notes for the activities from Geometry Activities for Middle School Students with The Geometer’s Sketchpad are not included here because their form made their inclusion impractical. Note that pre-made sketches that are required to do activities can be found in the folder Sketchpad | Samples | Teaching Mathematics. (Sketchpad is the folder that contains the application itself.)


Activity Notes—Teaching Geometry with The Geometer’s Sketchpad • 75

Properties of Reflection (page 16) Prerequisites: None Sketchpad Proficiency: Beginner Activity Time: 30–45 minutes (The first part,

Mirror Writing, takes about 5 minutes and can be done independently of the rest of the activity.) Sketch and Investigate Q1

Point C´ traces the mirror image of the student’s name.

Q2

Reflection preserves lengths and angle measures.

Q3

A figure and its reflected image are always congruent.

Q4

The vertices of the triangle CDE go from C to D to E in a counter-clockwise direction. The vertices of the reflected triangle C ´DÉ´ go from C ´ to D´ to E´ in a clockwise direction.

Q5

The mirror line is the perpendicular bisector of any segment connecting a point and its reflected image.

Tessellations Using Only Translations (page 18) Prerequisites: Students should understand the terms parallelogram, tessellation, and translation. Sketchpad Proficiency: Intermediate Activity Time: 25–35 minutes Presenting the Sketch

Since this activity does not ask for a written response, you might want students to print their tessellations or save them electronically. If they print the design, remind them that color will not print (unless you have a color printer). You might have them print the outline of the tessellation and take it home to color it in and decorate with details. If they save the sketch electronically, remind them to use the Text tool to put their names on the sketch and possibly describe their constructions and tile patterns. Explore More

1.

This is a quick way to introduce the animation button, with potentially exciting results. Have students try it even if you have only an extra minute or two.

2.

The process is the same for a regular hexagon, except that there are three pairs of parallel sides instead of two. So students can make three different irregular edges on adjacent sides, which they then translate across the hexagon to the opposite sides. This tiling is more interesting, but also more complex.

Explore More

1.

2.

Here is one way to perform this construction: Construct a line through the given point, perpendicular to the given mirror line. Then construct a circle from the intersection of the line and the perpendicular to the original point. The other intersection of the circle and the perpendicular is the reflected image of the original point. For an extra challenge, try doing this construction using only the Euclidean tools, that is, not using the menu at all. Reflect a point across a line. Connect the point with its image point. Also connect each of these points with a third point on the line.

76 • Activity Notes—Teaching Geometry with The Geometer’s Sketchpad


The Euler Segment (page 20)

Napoleon’s Theorem (page 22)

Prerequisites: Students should know the names of the different triangle centers: the incenter, circumcenter, orthocenter, and centroid. This activity introduces them to the Euler segment.

Prerequisites: Students should know the terms equilateral triangle, midpoint, median, and centroid.

Sketchpad Proficiency: Intermediate/

Advanced. Students will be using custom tools. Activity Time: 30–45 minutes Required Sketch: Triangle Centers.gsp

Sketchpad Proficiency: Advanced. This is not

a very difficult construction, but is rated for the advanced user because it suggests using a custom tool. This investigation is a good way to introduce using custom tools to speed up a construction. Activity Time: 30–45 minutes

Sketch and Investigate Q1

Q2


The orthocenter, the centroid, and the circumcenter are always collinear.

Q1

Here are some of the observations students could make: In an equilateral triangle, all four points are coincident. In an isosceles triangle, all four points are collinear and lie along the median to the vertex angle. In an acute triangle, all four points lie inside the triangle. In an obtuse triangle, the circumcenter and orthocenter lie outside the triangle. In a right triangle, the orthocenter lies on the vertex of the right angle, and the circumcenter lies on the midpoint of the hypotenuse.

Q3

The circumcenter and the orthocenter are the endpoints of the Euler segment. The centroid lies between them.

Q4

The distance from the orthocenter to the centroid is twice the distance from the centroid to the circumcenter.

Segments joining the centroids of equilateral triangles constructed on the sides of any triangle form another equilateral triangle.

Explore More

1.

The segments that each connect a vertex of the original triangle with the most remote vertex of the equilateral triangle on the opposite side are all congruent.

Explore More

1.

Three of the points are the midpoints of the sides of the original triangle. Three other points are the points where the altitudes intersect the opposite sides of the triangle (the feet of the altitudes). The last three points are the midpoints of the segments connecting the orthocenter with each vertex.

2.

In an equilateral triangle, three pairs of points coincide, reducing the nine points to six. In a right triangle, three points coincide at the right angle vertex, and a pair of points coincide at each of two side midpoints, so the nine points are reduced to five. In an isosceles right triangle, the foot of the third altitude coincides with the midpoint of a side, so the five points in a right triangle are reduced to four.



Constructing Rhombuses (page 23) Prerequisites: Students should be familiar with

the definition of a rhombus and also familiar with various properties of a rhombus. Sketchpad Proficiency: Beginner/Inter-

mediate/Advanced. The more advanced the user, the more construction methods that user will discover. Activity Time: 20–45 minutes (depends entirely

on how long you choose to set students loose) Notes

The more time you give students, the more methods they’ll come up with. Have students drag vertices of their figures to make sure their constructions are correct. Rhombuses that fall apart and can turn into other shapes are underconstrained. Constructions that remain rhombuses but that can’t take on all the shapes of a rhombus are overconstrained. Here are various methods for constructing a rhombus. The first is a popular one that is overconstrained. The rest are neither overconstrained nor underconstrained. Method: Construct circles AB and BA. Construct a rhombus connecting the centers and the two points of intersection of the circles. (This is a special rhombus, composed of two equilateral triangles.) Properties: A rhombus has four equal sides.

Method: Construct a circle AB and then segment BC , where point C is on the circle. Reflect point A across BC s . ABAĆ is a rhombus. Properties: A rhombus has a pair of equal consecutive sides and a line of symmetry through their unshared endpoints. Method: Construct a segment AB (to be a diagonal) and its midpoint, C . Construct a perpendicular through point C . Construct point D on the perpendicular. Reflect point D across AB s . ADBD´ is a rhombus. Properties: The diagonals of a rhombus are perpendicular and are axes of reflection symmetry. Method: Construct a segment AB to serve as half a diagonal. Construct a perpendicular through point B. Construct point C on the perpendicular. Rotate points A and C 180° about point B. ACAĆ ´ is a rhombus. Properties: The diagonals of a rhombus are perpendicular and their point of intersection is a center of 180° rotation symmetry. Method: Construct a circle AB and point C on the circle. Bisect angle BAC . Construct circle BA and point D, the intersection of this circle and the bisector. ABDC is a rhombus. Properties: The sides of a rhombus are equal and the diagonals bisect the angles.

Method: Construct circles AB and BA. Construct circle CA, where point C is a point on circle AB. Construct point D at the point of intersection of circles BA and CA. ABDC is a rhombus. Properties: A rhombus has four equal sides. Method: Construct a circle AB and two radii. Construct a parallel to each radius through the endpoint of the other. Properties: A rhombus has equal consecutive sides and parallel opposite sides. Method: Construct a segment AB and its midpoint C . Construct a line through point C , perpendicular to AB s . Construct circle CD, where point D is on the perpendicular line. Construct point E, the other intersection of the circle with the line. ADBE is a rhombus. Properties: The diagonals of a rhombus bisect each other.



Midpoint Quadrilaterals (page 24) Prerequisites: To make conjectures, students should be able to identify parallelograms and other

special quadrilaterals. To explain their conjectures, students need to know that the segment connecting the midpoints of two sides of a triangle (the midsegment) is parallel to the third side and half as long. Sketchpad Proficiency: Beginner Activity Time: 25–40 minutes

A Rectangle with Maximum Area (page 25) Prerequisites: Students should know the terms rectangle, square, area, and perimeter . Sketchpad Proficiency: Advanced Activity Time: 40–50 minutes. If you are short on

time, you can stop after Q2. This leaves out the exploration of the problem using the maxima and minima of a graph. Sketch and Investigate

Sketch and Investigate Q1

The quadrilateral whose sides connect the midpoints of any quadrilateral is a parallelogram. The measurements support this conjecture because they show that the opposite sides of the midpoint quadrilateral are equal in length and that opposite sides have equal slope (and therefore are parallel).

Q2

A diagonal divides the quadrilateral into two triangles. Two sides of the midpoint quadrilateral are midsegments of these triangles. This means they are both parallel to the diagonal and half as long. If one pair of opposite sides of a quadrilateral are both equal in length and parallel, the quadrilateral is a parallelogram. (Students might construct the other diagonal and use a second pair of triangles to show that the other pair of sides of the midpoint quadrilateral are also equal in length and parallel.)

Q1

As students drag point C , they should notice that the area of the rectangle changes but its perimeter remains constant. Because CB s and CD s are radii of the same circle, the sum of two sides of the rectangle, AC + CD, is equal to AB. Thus, AB is half the perimeter of the rectangle. As long as this length is kept constant, the perimeter of the rectangle will be constant.

Q2

A square is the rectangle with the greatest area for a given perimeter.

Q3

The coordinates of the high point of the graph show the side length and area of the maximum-area rectangle. The side length at this point verifies that the rectangle with the maximum area is a square.

Q4

The low points on the graph show where the area of the rectangle is zero. This happens when AC is zero and when AC = AB.

You might want to discuss with students why the locus graph of (side length, area) of a rectangle is a parabola. Explore More


1.

Regular polygons have maximum area for a given perimeter. Polygons with more sides are more efficient. The circle is the closed planar figure that gives maximum area for a given perimeter.

2.

The area of the rectangle can be represented by the equation A = x[(1/2)P – x]. The graph is a parabola with roots 0 and (1/2) P. So the x-value of the maximum point is (1/4)P. Since the side length of the maximum area rectangle is 1/4 the rectangle’s perimeter, the rectangle must be a square.


Visual Demonstration of the Pythagorean Theorem (page 27)

The Golden Rectangle (page 28)

Prerequisites: Students will appreciate this more

with setting up proportions, such as those used to describe relationships in similar polygons. Students will be more impressed with this activity if they’ve already learned something about golden rectangles and their significance. A fun activity is to have students take a poll to choose the most “popular” rectangle in a group of rectangles, one or two of which are golden. Golden rectangles tend to win hands down.

if they already have some experience with the Pythagorean theorem. Sketchpad Proficiency: Beginner Activity Time: 10–15 minutes. This short

demonstration works well even if you have only a single computer with an overhead viewing device. You might coordinate it with any of the other activities in this section. Required Sketch: Shear Pythagoras.gsp Sketch and Investigate Q1

In this sketch, the squares on the sides of a right triangle are sheared, without changing their areas, so that a shape on the legs is congruent to a shape on the hypotenuse. This shows that the sum of the areas of the original squares on the legs of a right triangle is equal to the area of the original square on the hypotenuse, thus demonstrating the Pythagorean theorem.

Prerequisites: Students should have experience

Sketchpad Proficiency: Advanced. Students

should be familiar with making and using custom tools. Activity Time: 40–55 minutes Sketch and Investigate

is approximately 1.618. Students’ answers will vary depending on their Scalars Precision settings in Preferences.

Q1 ø

Q2

Region DFGC is also a golden rectangle because the custom tool for constructing a golden rectangle creates a shape that fits perfectly inside it.

Explore More

1. The proportion in the 1-by- ø rectangle matches the proportion at the top of the first page of the activity, except that a = 1 and b = ø. So we have: ø/1 = (1 + ø)/ ø. Cross-multiplying gives ø2 = 1 + ø, or ø2 – ø – 1 = 0. The two solutions generated by the quadratic formula are 1+ 5 = 1.618... and 2 1− 5 = –0.618... 2 The positive solution gives the golden ratio. 2. ø2 = 2.618..., which equals ø + 1. This is verified in the quadratic equation in the last problem. 1/ ø = 0.618..., which equals ø – 1. To prove this

algebraically, start with the definition ø/1 = (1 + ø)/ ø. So ø = (1/ ø) + 1. Solve for 1/ ø to get 1/ ø = ø – 1.



A Sine Wave Tracer (page 30)

Adding Integers (page 32)

Prerequisites: Students should be familiar with the Cartesian coordinate system. The activity is most

Student Audience: Pre-algebra/Algebra 1

meaningful to students who have begun studying trigonometry.

Prerequisites: None. This will be a review topic

for most Algebra 1 students. Sketchpad Proficiency: Beginner. Students

Sketchpad Proficiency: Intermediate

manipulate a pre -made sketch.

Activity Time: 30–50 minutes

Activity Time: 20–30 minutes


Required Sketch: Add Integers.gsp. Extra pages

Q1

As you drag point D, point F moves horizontally.

Q2

As you drag point E around the circle, point F moves vertically up and down like a sewingmachine needle.

(not referred to in the activity) include a vertical version of the addition model and a version that doesn’t round to integers. General Notes: This activity works well as an

introduction to integer addition for pre -algebra students, a start -of-the-year refresher for Algebra 1 students, or a supplemental activity for any student having difficulty with the topic. The most important thing is for students to actually study, understand, and use the number -line sketch. Even the strongest students make careless mistakes with integers due to relying too much on verbal rules; having an internalized picture can help. Teachers may need to encourage students who already have some experience with integers to approach this activity with a mind open to fresh perspectives.

Q3

Answers will vary. Students might sketch a path somewhat like the curve below.

Q4

The sketch will look something like this. Also, if students leave the animation running, they will probably get a series of curves like this that will start to fill in the area around the curve.

Q5

The unit circle has a circumference of 2π, about 6.28 grid units.


For the trace to repeat itself without tracing a new curve, the length of AB s must be an integer multiple of the circumference of the circle. The circumference of the circle is 2π , or about 6.28 grid units, so the x-coordinate of point B should be about 6.28.

Q2

It’s similar in that if you ignore all the negative signs, the answer is the same. For example, 3 + 9 = 12 and ( –)3 + (–)9 = (–)12. It’s different in that the answer turns out to be negative instead of positive.

Q3

No, not possible. The reason is that you’re starting to the left of the origin (because the first number is negative) and you’re moving further left (because the second number is negative). You can’t do this and end up to the right of the origin.

Q4

0

Q5

Each must be the opposite of the other.

Q6

Q1 –9

Q6 –3 Q7


If the “bigger” number (longest arrow/ greatest absolute value) is positive, the sum will be positive. If the “bigger” number (longest arrow/greatest absolute value) is negative, the sum will be negative.


Q8

Q9

Since the arrows go in opposite directions, in a way you are actually subtracting values. Completely ignoring negative signs for now, the answer is always the bigger addend minus the smaller addend. For example, 7 + ( –5) = 2 and 7 – 5 = 2. Also, –9 + 8 = –1 and 9 – 8 = 1.

Points “Lining Up” in the Plane (page 35)

a. b. c. d.

defined, but it isn’t a major focus of the activity.

positive negative its opposite positive; negative

Note that while these rules are important, memorizing them isn’t necessary if students remember the number line and arrows. Explore More

1.

a.

It makes sense because all that changes is the order in which you move one way or the other. Moving three steps to the right, then five steps to the left is the same as moving five steps to the left, then three steps to the right.

b.

Yes, it does.

Student Audience: Pre-algebra/Algebra 1 Prerequisites: Familiarity with the Cartesian plane. The term absolute value is used and briefly Sketchpad Proficiency: Beginner. Students

construct points and measure their coordinates. Activity Time: 20–30 minutes Required Sketch: Line Up.gsp General Notes: The purpose of this activity is to

give students an informal and experiential introduction to the relationship between descriptions of coordinate patterns and graphs in the Cartesian plane. Too often, students don’t really “get” the connection between an equation and the graph it produces. It’s important for students to understand that graphs represent the set of points whose coordinates satisfy an equation. This activity attempts to foster that understanding. Thus, any sort of class or group discussion that encourages students to ponder this relationship (“Why do the points ‘line up’ in such regular ways?” “If you could plot not just five, but every point that satisfies the description, what would that look like?”) will deepen the experience. Sketch and Investigate Q1

In each case, the answer shown depicts all possible answers with integer coordinates on the grid provided. The question asks for five answers, so any five of the points shown is a correct response. a.

6

3

-10

-5

5

10

5

10

-3

-6

b.

6

3

-10

-5 -3

-6



c.

Q2

6

3

-10

-5

5

a.

The y-coordinate equals the x-coordinate.

b.

The y-coordinate is one less than the x-coordinate.

c.

The y-coordinate is twice the x-coordinate. (Or, the x-coordinate is one-half the y-coordinate.)

10

-3

-6

d.

d. The y-coordinate is two less than twice the x-coordinate.

6

3

-10

-5

5

e.

The y-coordinate is one-third the x-coordinate. (Or, the x-coordinate is three times the y-coordinate.)

f.

The y-coordinate is always –1 (regardless of the value of the x-coordinate).

g.

The y-coordinate is the opposite of the absolute value of the x-coordinate. (An acceptable alternate answer for students not familiar with the term absolute value might be “The y-coordinate is the ‘negative value’ of the x-coordinate, regardless of whether the x-coordinate is positive or negative.”)

10

-3

-6

e.

6

3

-10

-5

5

10

-3

-6

f.

h. The product of the y-coordinate and the x-coordinate is 6.

6

3

-10

-5

Explore More 5

10

1.

-3

6

3

-10

-5

2. 5

10

5

10

-3

-6

h.

6

3

-10

-5 -3

-6


c. y = 2x f. x + y = 5

Q2: a. y = x b. y = x – 1 c. y = 2x d. y = 2x – 2 e. y = (1/3)x or x = 3y f. y = –1 g. y = –|x| h. xy = 6

-6

g.

Q1: a. y = x b. y = x + 1 d. y = 2x + 1 e. y = –x g. y = |x| h. y = x2

Answers will vary. Here’s how to set up the Movement button (more detailed instructions are on page 2 of Line Up.gsp): Plot the eight destination points using the Plot Point command. Select all 16 points in the sketch in the following order: point A, point A’s destination, point B, point B’s destination, point C, point C’s destination, . . . , point H, point H’s destination. Now, choose Movement from the Action Buttons submenu of the Edit menu. Change the speed and label (on the Label panel) if you’d like, and then click OK. Now hide the eight destination points (using the Hide command in the Display menu).


Parabolas in Vertex Form (page 38) Student Audience: Algebra 1/Algebra 2 Prerequisites: Students need to understand

Reflection in Geometry and Algebra (page 41) Student Audience: Algebra 1

the basic idea of a function and the role that the variables x and y play in the equation and graph of a function. Solving simple linear equations for one unknown after substituting given values for other unknowns is also part of this activity.

Prerequisites: Familiarity with the Cartesian

Sketchpad Proficiency: Intermediate/Advanced

Example Sketch: Reflect.gsp. The pages of this

Activity Time: 40–50 minutes without the

Explore More section. Required Sketch: Vertex Form.gsp Sketch and Investigate Q1

Many possible answers. This is a good calculator activity for substituting an x-value into the right side of a function to determine a y-value. Any small discrepancy is due to Sketchpad’s rounding (depending on the Precision settings in Preferences).

Q2

If a is positive, the parabola opens up; if a is negative, the parabola opens down. The larger the absolute value of a, the “narrower” the parabola. The closer a is to zero, the “wider” the parabola.

Q3

The coordinates of the vertex are ( h, k).

Q4

The parabola moves right and left as h changes (right as h gets bigger, left as it gets smaller). The parabola moves up and down as k changes (up as k gets bigger, down as it gets smaller).

Q5

a. b. c. d. e.

y = 5(x – 1)2 – 1 y = 0.5(x + 4)2 – 3 y = –0.5(x – 5)2 + 2 y = 2(x – 2)2 – 2 y = 4(x + 1)2 + 3

It’s very important that students find the equations of these parabolas using paper and pencil calculations and use Sketchpad to check their answers. Slider accuracy may account for small differences. Q6

x = 3; x = h

Q7

(1, 9)

Explore More

1.

Since x = h is the axis of symmetry, (2h – s, t) is the symmetry point.

2.

The axis of symmetry is perpendicular to the x-axis and passes through the vertex ( h, k).

plane. Sketchpad Proficiency: Beginner/Intermediate Class Time: 30–40 minutes

sketch show the activity after steps 6, 10, 13, and 16. We recommend that you not try to shorten the activity by having students start at one of these junctures as the process of doing the actual geometric work is key to understanding the later algebraic process. General Notes: This activity works well as a

brush-up for students having problems with the coordinate plane, as an introduction to using Sketchpad for both geometry and algebra, and as a preparation for function transformation in the subsequent activity. Before starting this activity, it would be good to brainstorm what students already know about reflection. A useful question to discuss— anticipating questions 4 and 6 —is “If you stand 3 ft. from a mirror, how far do you appear to be from your reflected image?” Construction Tips

Step 4: You may choose instead to leave Fade Traces Over Time unchecked so that traces remain on screen to be examined. In this case, students would periodically need to choose Erase Traces from the Display menu to clear traces from their screens. Step 14: The reason for using Coordinate Distance instead of Length or Distance from the Measure menu is that students may have rescaled their axes at some point (so that coordinate units are different than the distance units of centimeters or inches). If students haven’t touched the unit point, any of the three commands would work here too. Sketch and Investigate Q1

Dragging a reflecting point results in a “mirror” pattern with the two traces mirroring each other across the line. Dragging a line point causes the reflected image point to draw a circle around the other line point. (The radius of this circle is the distance between the other line point and the reflecting pre -image.)



Q2

Q3 Q4

The line is the segment’s perpendicular bisector, meaning that the angle they form is 90 ° and the line cuts the segment in half. (–a, b)

Walking Rex: An Introduction to Vectors (page 44)

2a may be considered an acceptable answer; a more technically correct answer is |2a| or 2|a|. The absolute value signs ensure that the answer will be positive even if a is negative. This is desirable because the distance between two things is always considered to be positive (or 0).

be a first introduction to vectors.

Q5

(c, –d)

Q6

2d, |2d|, or 2|d| (See the answer to Q4 above.)

Student Audience: Algebra 1/Algebra 2 Prerequisites: None. This activity is designed to Sketchpad Proficiency: Beginner. Students work

with a pre-made sketch. Class Time: 20–30 minutes. It may be possible to

do this activity and the follow -up activity, Vector Addition and Subtraction, in one class period. We recommend using two periods, though, so that the material has more of a chance to sink in. Required Sketch: Walk the Dog.gsp

Explore More

Walk the Dog

1.

The coordinates switch places. In other words, the image of a point ( a, b) reflected across the line y = x is the point (b, a).

Q1

The magnitude stays the same. (In other words, Rex is always the same distance—the length of his leash—from the tree.)

2.

These three transformations are equivalent. The coordinates of (a, b) after any of the transformations is (–a, –b).

Q2

a. b. c.

(4.33, 2.50) (0, 5) (3.54, 3.54)

Note that the x-coordinate in a. and both coordinates in c. are approximations—the answers students get may be slightly different. (The exact value of the x-coordinate in a. is

2.5 3 and the exact value of either coordinate in c. is 2.5 2 .) Q3

a. b. c. d.

magnitude: 5; direction: 0° magnitude: 5; direction: 53.13° magnitude: 5; direction: 270° magnitude: 5; direction: 233.13°

Note that the direction values in b. and d. are both approximations—the answers students get may be slightly different. (The exact value for b. is arctan(4/3) and for d. it’s that value plus 180°.) Q4

Rex should move to (–5, 0) to get away from the ladybug when she’s at (5, 0), and ( –3, 4) to get away from her when she’s at (3, –4). In general, vectors (a, b) and (–a, –b) face opposite directions.



Q5

The only things that change when vector j is dragged are the locations of its head and tail. The first method for defining vectors uses magnitude and direction —neither of these changes as j is dragged. The second uses the coordinates of the head when the tail is at the origin. Regardless of where the head and tail actually are, the coordinates of the head still would be the same if the tail were at the origin.

Q6

(82, 84). No matter where you’re standing, Rex is 2 units to the right of you and 4 units “north” of you.

Q7

Walk 2: (83, 81). Drag the vector so that “you” are at the origin. Rex will be at (3, 1), meaning that he is always 3 units to the right of you and 1 unit “north” of you. Walk 3: (74, 83). Drag the vector so that “you” are at (8, 6). Rex is at (2, 9), meaning that he is always 6 units to the left of you and 3 units “north” of you.

Q8

Leonardo da Vinci’s Proof (page 46) Construct Difficulty: Moderate. The construction itself isn’t

difficult, but careful selection is required to make the action buttons work properly. 1.

Make sure students don’t construct a square on the hypotenuse.

5.

Hide/Show is a command in the Action Buttons submenu of the Edit menu. The command produces both a hide and a show button simultaneously. The buttons can be relabeled by double-clicking on them with the Text tool.

Investigate

If students play the buttons in the suggested sequence, they’ll see a sequence of figures like those below:

Walk 2: (86, 82). If the leash were twice as long, Rex would be 6 units to the right of you and 2 units “north” of you. Walk 3: (68, 86). If the leash were twice as long, Rex would be 12 units to the left of you and 6 units “north” of you.

Prove Difficulty: Easy

The initial figure shows two identical right triangles with sides a, b, and c, and two squares with side lengths a and b. None of the transformations performed change the area of the figure. In the final figure, we have two triangles identical to the two triangles in the initial figure, and between them we have a square with side length c. Therefore, the sum of the areas of the two squares in the initial figure ( a2 + b2) must equal the area of the square in the final figure (c2).



The Folded Circle Construction (page 49)

Playing Detective

they’ll need to know the distance definition of an ellipse and the SAS triangle congruency theorem.

In step 21, students are asked to construct the point of tangency to their ellipse. The point of tangency lies at the intersection of the crease line and segment AC .

Sketchpad Proficiency: Intermediate.

How to Prove It

Prerequisites: For students to complete the proof,

Students build a perpendicular bisector line and follow the steps of an extended construction.

Q8

Activity Time: 70–80 minutes Required Sketch: Folded Circle.gsp General Notes: Of all the ellipse construction

methods, this is the one most likely to elicit “oohs” and “aahs.” Once students get the hang of folding their circles, they find it amazing to watch their creases gradually come together in the outline of an ellipse. The equivalent Sketchpad model is equally impressive. Constructing a Physical Model

Q9

Q10

Since jBED ≅ jCED, the corresponding sides BE and CE are equal. If point E traces an ellipse, then AE + BE must be constant. Substituting CE for BE gives: AE + BE = AE + CE = AC = the radius of the circle, which is constant

Explore More

Q1

Points A and B are the foci.

Q2

The ellipse would appear “skinnier” and more elongated.

Q3

The ellipse would appear “fatter” and more like a circle.

Constructing a Sketchpad Model

1.

When point B lies outside the circle, the creases outline a hyperbola. It’s tempting to think that when point B lies on the circle, the creases will form a parabola, but, in fact, they don’t!

2.

∠ AEH = ∠ CED, as they are vertical angles. Since jBED ≅ jCED, we have ∠ CED = ∠ BED. Putting these two equalities together gives ∠ AEH = ∠ BED.

In step 11, students must study the geometry of their crease lines. Specifically, given points C and B, how do you use Sketchpad to construct the “crease” formed when C is folded onto B? (The crease is the perpendicular bisector of segment CB.) As preparation for this construction step, you might ask students to take a fresh sheet of notebook paper, mark two random points, fold one onto the other, then unfold the paper. What is the geometric relationship of the crease line to the two points? Q4

The ellipse becomes “skinnier” and more elongated.

Q5

The ellipse becomes “fatter” and looks more like a circle.

Q6

The creases all pass through the circle’s center, point A.

Q7

The creases outline a circle.


Since the crease line is the perpendicular bisector of segment BC , we have DB = DC and ∠EDB = ∠EDC = 90˚. And, of course, ED = ED. Thus, by the SAS triangle congruency theorem, jBED ≅ jCED.

(A full-page advertisement for elliptic pool tables appeared in the July 1, 1964, issue of The New York Times. Actors Paul Newman and Joanne Woodward made an in-store appearance to promote the game. On a related topic, students can search the Internet for information about the sound reflection properties of “whispering galleries.”) 3.

Points A and B are the foci of the ellipse. BC + CA = (r + p) + (R – p) = r + R, which is constant


The Expanding Circle Construction (page 53)

How to Prove It

Prerequisites: For students to complete the proof,

Q5

( AF )2 = ( AB)( AD) or equivalently, x2 = 3y

Q6

Point G is the reflection of point H across the y-axis. Since the y-axis serves as the parabola’s line of symmetry, point G must also sit on the parabola.

Q7

When AB = s, we have x2 = sy.

they’ll need some information about geometric means (provided in the activity) and the algebraic definition of a parabola. If you omit the proof, there are no prerequisites. Sketchpad Proficiency: Intermediate. Students

construct several perpendicular lines and follow the steps of an extended construction.

Q4 AF =

Explore More

1.

Activity Time: 60–70 minutes Required Sketch: Geometric Mean.gsp Constructing a Sketchpad Model Q1

The curve always passes through point A and is symmetric across the y-axis, but it appears to become wider as point B is dragged downward.

Q2

If you had only a compass and straightedge available, you would need to draw a collection of individual circles, each passing through point B with a center somewhere along the positive y-axis. Then, moving from circle to circle, you would construct the necessary lines to locate points G and H (a pair of points for each circle). Using graph paper could help by providing rough guidelines for the perpendiculars.

x; AD = y

Return to your expanding circle sketch and add segments DE and EB to the construction. As you drag point C , notice that jDEB remains a right triangle as it is inscribed in a semicircle.

The Geometric Mean Q3

The proof below follows the setup on the second page of the sketch Geometric Mean.gsp. As all angles inscribed in a semicircle are right angles, ∠DGE = 90°. Since ∠GDF + ∠DGF = 90° = ∠EGF + ∠DGF, we have ∠GDF = ∠EGF. Thus right triangles DFG and GFE are similar. Based on this similarity, we can write the proportion DF GF

=

FG FE

or equivalently,

a x

=

x b

Cross-multiplying gives x2 = ab.



Distances in an Equilateral Triangle (page 56)

Q1

The sum of the distances remains constant.

Q2

Increasing or decreasing the size of the equilateral triangle increases or decreases the sum. However, for a triangle of any given size, moving point P around inside the triangle doesn’t change the sum.

Q3

The sum of the distances is not constant. (However, if we consider distances falling completely outside the triangle as negative, the result still holds—see the answer to Q11.)

Q4

In an equilateral triangle, the sum of the distances from a point inside the triangle to its sides is constant.

Prerequisites: Area formulas for triangles,

elementary algebra (factorization). Required Sketch: Distance.gsp General Notes: This activity is intended as a first

introduction to proof as a means of explanation. The language you use is crucial in this introductory phase of the study of proof. Students find it much more meaningful if instead of saying, as usual, “We cannot be sure that this result is true for all possible variations, and we therefore have to (deductively) prove it to make absolutely sure,” you instead say, “We now know this result to be true from our extensive experimental investigation. Let us now see if we can explain why it is true in terms of other well-known geometric results—in other words, how it is the logical consequence of other results.” Avoid using the word proof initially; use the word explanation instead to emphasize the intended function of the given deductive argument. The word proof, in everyday language, predominantly carries with it the idea of verification or conviction (which students grasp firmly once they’ve explored a result extensively on Sketchpad), and to use it in an introductory context would implicitly convey this meaning, even if the intended meaning was that of explanation. The verification meaning of proof is, of course, important and will be developed in some of the later activities; at that time, it will become appropriate to start using the word proof for the given deductive arguments. Conjecture

A ready-made sketch is provided, since it is timeconsuming for students to first construct an equilateral triangle. In addition, the actual construction of the sketch plays no part in the specific learning objective of making and explaining a conjecture. Students can, however, drag and measure sides of the triangle to check that it is indeed equilateral. Students will tend to think first that the optimum position for point P is at the center of the equilateral triangle, and it therefore comes as quite a surprise when they later find that the sum of the distances is actually independent of the position of point P.


Explaining Q5

The three sides are all equal, but since their lengths may vary, they are indicated by the same variable, a.

Q6

The areas of the triangles are, respectively: 0.5ah1, 0.5ah2, and 0.5ah3.

Q7

Sum 0.5ah1 + 0.5ah2 + 0.5ah3 = 0.5a(h1 + h2 + h3).

Q8

Area of whole triangle = sum of areas of small triangles. Therefore, if we represent the area of the whole triangle by A, it follows that h1 + h2 + h3 = 2 A/a.

Q9

For an equilateral triangle of fixed size, its area A and its side length a are constant. Therefore, the sum of the distances h1 + h2 + h3 is also constant.

Q10

The sum of the distances is equal to the altitude of the original triangle, say H . This can be explained as follows: 0.5aH = 0.5a(h1 + h2 + h3) ⇔ H = h1 + h2 + h3

Q11

The sum of the distances will remain constant only if there is a common factor 0.5a that can be taken out of the three areas; that is, the triangle must be equilateral.

This result is in fact also true if point P is dragged outside the triangle, but an explanation requires the introduction of directed line segments (distances falling completely outside are considered negative). For example, consider the figure where point P lies outside, as indicated. In this case, the sum of the areas of triangles PAB, PBC, and PCA is not the area of j ABC. To again obtain the area of j ABC, we now have to subtract the area of jPAB from the sum of the other two. Therefore, in this case we have H = h1 + h2 – h3.


C

b. The proof is the same as for the equilateral triangle, since all the sides are equal. The result is generalizable to regular (equisided) polygons. In general, we would have for any equisided n-gon An (n > 2) with side length a that h3

A

B

3. a. The sum of the distances from P to the sides of a parallelogram is constant (see below).

h1 h2 P

In order to make the general formula H = h1 + h2 + h3 work, we therefore need to consider distances as negative if they fall completely outside the triangle. However, considering P outside the triangle may complicate things unnecessarily for students at this stage. You can come back to this result at a later stage to deal with this aspect of it if you wish. Further Exploration

1.

To find the minimum sum for an arbitrary triangle, the point P has to be situated at the vertex opposite the longest side (where the altitude is the smallest).

2.

a. The sum of the distances from P to the sides of a rhombus is constant (see below). Distance(P to segment n) = 2.43 cm Distance(P to segment m) = 0.81 cm Distance(P to segment k) = 1.74 cm Distance(P to segment j) = 3.36 cm Distance(P to segment j) + Distance(P to segment k) + . . . + . . . = 8.33 cm m

B

C P

n

k

A

 A 

∑ni = 1 hi = 2 an   .

Distance(P to segment n) = 2.64 cm Distance(P to segment m) = 2.15 cm Distance(P to segment k) = 1.52 cm Distance(P to segment j) = 2.02 cm Distance(P to segment j) + Distance(P to segment k) + . . . + . . . = 8.33 cm m

B

C

n

h4

h2

b. The sum h1 + h2 is constant, since the distance between the two opposite parallel sides is constant. Similarly, h3 + h4 is constant. Therefore, h1 + h2 + h3 + h4 is constant (equal to the sum of the two distances between the pairs of opposite sides). The result is generalizable to any polygon with an even number of sides and opposite sides that are equal and parallel (that is, a parallelo-2n-gon (n > 1)), where the sum of the 2n distances to the sides will be equal to the sum of the n distances between the pairs of opposite sides. Another possibility to consider is the generalization to three dimensions (and more). Since the 3D analog of a triangle is a tetrahedron, students may first want to consider a regular tetrahedron. Instead of working with areas and distances, they will now need to work with volumes and areas. After further reflection, they should realize that the sum of the distances to the four faces of a tetrahedron, with all four faces having the same area a, would also be constant. For example, a point P inside the tetrahedron divides it into four tetrahedra, so that

D

j

h3 h1

(1/3)aH = (1/3)a(h1 + h2 + h3 + h4) ⇔ H = h1 + h2 + h3 + h4 Note that if the area of each face is the same, the height H from each face to the opposite vertex must also be the same, since its volume is constant. But this does not imply that the tetrahedron is necessarily regular.

k P

A

j

D



Varignon Area (page 60)

parallelograms), and angle EFB = angle F ′ FC (directly opposite angles).

Prerequisites: Kite Midpoints activity or

knowledge of the result that the line connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. Properties of parallelograms. Conditions for congruency.

Q15

jHDG is congruent to jG′ CG (SAS).

Q16

Similar to 14.

Q17

′ (SSS). j AEH is congruent to jCF ′G

Required Sketch: Varignon.gsp

Q18

From Q13, we have CF ′ = BE, but BE = AE. Therefore, AE = CF ′. Similarly, from Q17, we have AH = CG′ .

Conjecture Q1

EFGH is a parallelogram. (This is true even for concave and crossed cases.)

Q2

The area of the parallelogram is half that of the original quadrilateral.

Q3

No.

Q4

No.

Q5

The midpoints of the sides of a quadrilateral form a parallelogram.

Q6

Responses will vary.

Q19

jFGC is common to both ABCD and FGG′ F′ . Therefore, the sum of the areas of the triangles is equal to that of FGG′ F , ′ and therefore to that of EFGH .

Present Your Proof

Provides an opportunity for students to synthesize the argument and write it up in a coherent way. Further Exploration

Proving Q7

Also, EH = FG (corresponding sides of translated parallelograms).

B

sEF || AC s || HG, s since E and F are midpoints of sides AB and CB in triangle ABC and H and G are midpoints of sides AD and CD in triangle ADC .

Q8

sEH || BD s || FG s (same reasons).

Q9

sEF || HG s and EH s || FG, s so opposite sides are parallel, and therefore EFGH is a parallelogram. Another way of proving it is to note in Q7 that not only is s EF || HG s , but since both EF and HG are equal to half AC, they are also equal to each other. So one pair of opposite sides are equal and parallel, from which it follows that EFGH is a parallelogram.

F

F’

A

Note: You may also wish to ask your students to prove that the result is also true in the concave and crossed cases. The proofs are similar, except that now one or both diagonals fall outside. Q10

There are four triangles lying outside EFGH, namely AEH, DHG, CGF, and BFE.

Q11

The sum of the areas of these triangles must be equal to the area of EFGH .

Q12

The translated quadrilateral is congruent to EFGH (property of translation), so it is also a parallelogram with area equal to that of EFGH .

Q13

jEBF is congruent to jF ′ CF (SAS).

Q14

FB = FC (F is midpoint of BC ); FE = FF ′ (corresponding sides of translated


E

C H

G

G’ D

This proof is a little tricky. In the concave case, only three triangles, namely AEH, DHG, and BFE, fall within ABCD. The remaining triangle CGF now falls outside ABCD. If we use the notation (XYZ) to represent the area of a polygon XYZ, then ( ABCD) = ( AEH ) + (BFE) + (DHG) + (EFGH ) – (CGF ) = ( AEH ) + (BFE) + (DHG) – (CGF ) + (EFGH ). In other words, we now have to prove that ( AEH ) + (BFE) + (DHG) – (CGF ) = (EFGH ). From the translation, EFGH is still congruent to FGG′ F ′ . As before, triangles EBF and F ′ CF, triangles HDG and G′ CG, and triangles AEH and CF ′G ′ are congruent. But if we subtract the area of triangle CGF from the sum of the areas of triangles G′ CG, CF ′ G′ , and F ′ CF, we obtain the area of parallelogram FGG′ F ′ , which is equal to that of EFGH . Q.E.D.


B

Alternative Proof

F

There are several different ways of proving this result. It might be instructive for your students to work through hints such as those given here.

C E

G D

Hints H

1.

Express the area of EFGH in terms of the area of ABCD and the areas of triangles AEH, CFG, BEF, and DHG. B

F C

E G

A

H

D

2.

Drop a perpendicular from A to BD s and express the area of triangle AEH in terms of the area of triangle ABD.

3.

Similarly, express the areas of triangles CFG, BEF, and DHG, respectively, in terms of the areas of triangles CBD, BAC, and DAC, and substitute in 1.

4.

Simplify the equation in 3 to obtain the desired result.

A

Crossed Quadrilaterals

It is also true for the crossed quadrilateral ABCD that EFGH has half its area, as some of your students may have found on Sketchpad. However, the proof is even more tricky and first requires consideration of what we mean by the area of a crossed quadrilateral. Let us now first carefully try to define a general area formula for convex and concave quadrilaterals. It seems natural to define the area of a convex quadrilateral to be the sum of the areas of the two triangles into which it is decomposed by a diagonal. For example, diagonal AC s decomposes the area as follows (see first figure): ( ABCD) = ( ABC ) + (CDA) A

Proof

1.

Using the notation ( XYZ) for the area of a polygon XYZ, we have (EFGH ) = ( ABCD) – ( AEH ) – (CFG) – (BEF ) – (DHG).

B D

2. If the height of j ABD is h, then ( ABD) = 0.5BD ⋅ h and ( AEH ) = 0.5(0.5BD)(0.5h) = 0.25( ABD), or simply, the base and the height are half those of the large triangle. 3.

(EFGH ) = ( ABCD) – 0.25( ABD) – 0.25(CBD) – 0.25(BAC ) – 0.25(DAC ).

4.

(EFGH ) = ( ABCD) – 0.25( ABCD) – 0.25( ABCD) = 0.5( ABCD).

C

A

B

D

C

Further Discussion

You may also want your students to work through an explanation for the concave case, because it is generically different. For example, unless the notation is carefully reformulated (e.g., see crossed quadrilaterals, below), the equation in 1 does not hold in the concave case, but becomes ( EFGH ) = ( ABCD) – ( AEH ) – (CFG) – (BEF ) + (DHG) (see below). However, substituting into this equation as before, and simplifying, leads to the same conclusion.

In order to make this formula work for the concave case as well (see second figure), we obviously need to define (CDA) = –( ADC ). In other words, we can regard the area of a triangle as being positive or negative depending on whether its vertices are named in counter -clockwise or clockwise order. For example: ( ABC ) = (BCA) = (CAB) = –(CBA) = –(BAC ) = –( ACB)



Visualizing Change: Velocity (page 7)

A

C

Prerequisites: Students should be familiar with

the concept of velocity, as well as how to find the slope and equation of a line on a position vs. time plot.

O

B

Sketchpad Proficiency: Beginner/Intermediate.

D

Applying the above formula and definition of area in a crossed quadrilateral (see figure above), we find that diagonal AC decomposes its area as follows: ( ABCD) = ( ABC ) + (CDA) = ( ABC ) – ( ADC ) In other words, this formula forces us to regard the “area” of a crossed quadrilateral as the difference between the areas of the two small triangles ABO and ODC . [Note that diagonal BD similarly decomposes ( ABCD) into (BCD) + (DAB) = –(DCB) + (DAB)]. An interesting consequence of this is that a crossed quadrilateral will have zero “area” if the areas of triangles ABO and ODC are equal.

B G

E F A

C

Using this valuable notation, the result can now simultaneously be proved for all three cases (convex, concave, and crossed) as follows: (EFGH ) = ( ABCD) – ( AEH ) – (FCG) – (EBF ) – (DHG) = ( ABCD) – 0.25( ABD) – 0.25(CDB) – 0.25(BCA) – 0.25(DAC ) = ( ABCD) – 0.25( ABCD) – 0.25( ABCD) = 0.5( ABCD)

Activity Time: 45–55 minutes Required Sketch: Velocity.gsp General Notes: In this activity, students will be

able to visualize the sentence “A particle moves with velocity…” by controlling the velocity themselves, as you might do with a remote control toy car (or your own automobile, for that matter!). Students will observe both the motion of the particle—a point—on a line, as well as a plot of time and position. In the Extension, you will see how to create such a demonstration yourself. Sketch and Investigate

D

H

Students will need to measure coordinates and move objects in the plane.

In the document Velocity.gsp, a point Me2d has been constructed in the plane. The y-coordinate of this point has been used to create a point, Me, on a line at the top of the sketch. In Q1, the position of this point on the line is determined by the y-coordinate of point Me2d. If point Me2d has a y-coordinate of 3, then point Me will be 3 units from point Home (as determined by the scale of the yaxis). In Q2, if the y-coordinate of point Me2d does not change, neither will the location of point Me. The x-coordinate of point Me2d represents time. When students show the velocity control, a point FutureMe will also be shown. This point shows the time and position of point Me2d if the point has constant velocity, as determined by the slider for velocity. In Q3, with the time control set to 0.5, and the velocity control set to 2, point Me2d will move right 0.5 unit and up 1 unit. In Q4, different time settings will simply move point FutureMe along a line with a slope equal to the velocity setting. In Q5, changing the velocity setting will change the slope of the line (not shown) between point Me2d and point FutureMe. At all times, the slope of this line will equal the velocity setting. In Sketchpad, you can make points “chase” each other, and that is what will happen when you press the Animate Time button. Point Me2d will move toward point FutureMe. In Q6 through Q10 , you will see point Me2d move along a line with a



slope equal to the velocity setting. With the velocity set to 0, the point will create a horizontal line, and point Me will not move at all. Even more interesting is what happens when you change the velocity during the “chase.” Point Me2d will try to follow the point FutureMe at all times, even as point FutureMe moves itself. By changing the velocity value, you change the slope between the two points, and the trace will curve down if you decrease the velocity and curve upward when you increase it. You will see this behavior in Q11 through Q14. Increasing the velocity will create a concave up trace, while decreasing it will create a concave down trace. Switching from increasing to decreasing, but leaving the velocity the same sign, will cause your trace to have a point of inflection. Changing from a positive to a negative velocity will create a maximum in your trace maximum, while changing from a negative to a positive will create a minimum. In a very real sense, what you are doing here is tracing the antiderivative of the velocity function you are creating by your movement of the velocity slider as time passes. Explore More

Students can trace a function plot by adjusting the slope of the trace as they go. Extension

To create a similar demonstration, construct a point in the plane ( Me2d). Create a slider to represent time, and a slider to represent velocity. Add the time value to the x-coordinate of Me2d. Then, multiply the time value by the velocity value, and add this to the y-coordinate of Me2d. Plot these two new measurements as a point in the plane (FutureMe). Create a Move button to have Me2d chase FutureMe, and you’re off and moving! To create the particle itself to show motion along a line, create a line using the Line tool, and construct a point on this line (or use the x-axis and the origin, if you wish to model AP questions of this form). The coordinates of this point will serve as your origin. Add the y-coordinate of Me2d to the x-coordinate of your origin, and then plot this result and the y-coordinate of your origin as a point in the plane. A point will be created on your line whose position on the line is determined by the y-coordinate of Me2d, and will move as point Me2d moves.

Going Off on a Tangent (page 59) Prerequisites: Students should be familiar with

representing average rates as secant slopes, and with the idea of the derivative as the limit of the average rate of change. Sketchpad Proficiency: Beginner/Intermediate.

Students will need to measure coordinates and move objects in the plane. Activity Time: 45–55 minutes Required Sketch: Tangents.gsp General Notes: In this activity, students will

visualize the instantaneous rate of change of a function as a tangent line to the plot of the function and use the slope of the tangent line to speculate about the derivative. Sketch and Investigate

In the document Tangents.gsp, point P is a point on a function plot. A tangent line has been constructed to the plot of f through point P by calculating the derivative of f (using the Derivative command) at the x-value of point P and using the derivative value to plot the equation of the tangent line through point P. These calculations are hidden, as the activity assumes only that students are familiar with the derivative as a limit of average rates of change. In Q1, Q2, and Q3, students can move point P to visualize and estimate the instantaneous rate of change of the function (the derivative) by using the grid to estimate the slope of the tangent line. At x = –1, the slope of the tangent line appears to be 10; at x = 0, 1, and at x = 1, –2. In Q4 and Q5, relationships between the increasing behavior of the function and a positive slope of the tangent line and decreasing behavior of the function and a negative slope of the tangent line are visualized. When the derivative is positive, the function is increasing, and when the derivative is negative, the function is decreasing. Q6 introduces the idea that when a function has a maximum or a minimum the derivative equals 0. (You may wish to edit the function f to illustrate that a function’s derivative can equal 0 without a maximum or minimum occurring.) In Q7, the derivative is greatest at both endpoints of the interval [–1, 3]. Q8 repeats

Q4–Q7 for a sine function, while asking students to use what they have seen to visualize the tangent line at different locations on the function plot. By examining where the function is



increasing, decreasing, or 0, students can estimate the value of the derivative.

Accumulating Area (page 120)

In Q9–Q11, students can explore the connection between the slope of the tangent line and the curvature of the function plot. Whenever the tangent line is “under” the function plot, the slope of the line is increasing, and the function plot is concave up.

graphs of functions and area formulas.

Explore More

Required Sketch: Area2.gsp

The three functions listed have a point or points where the derivative is undefined. For the absolute value function, the tangent line can have only a slope of –1 or 1. Both this function and the next exhibit an abrupt change in slope at particular x-values. With the second and third functions, however, the abrupt change is not of the same nature. For both these functions, the slope of the tangent line increases without bound. This can be seen by zooming in and continuing to observe the slope.

General Notes: In this activity, students will

Extension

You can plot points on the function plot and create buttons that will move point P to specific locations. To do this, use the calculator to calculate a value of 1. (All you need to do in the calculator is type in the number 1.) Then calculate f (1) and plot (1, f (1)) as a point in the plane. Finally, create a button to move point P to the newly plotted point. To examine the local behavior of the function near any particular point, you can use the zoom controls included in this sketch. To make this process easier, plot ( a, b) as a point in the coordinate grid. Create a button to move point P to this point. After pressing the Move button you have created, point P will be at the exact location around which the sketch will zoom. Then use the Zoom tools to examine the behavior of the function near the point. As you zoom in, the function plot should become indistinguishable from the tangent line —except in the case of the functions listed in the Explore More section. Note for advanced Sketchpad users: You can’t edit the values of a and b to equal other measurements in the sketch because much of the sketch is dependent on the values of a and b.


Prerequisites: Students should be familiar with Sketchpad Proficiency: Beginner/Intermediate.

Students will measure coordinates and move objects in the plane. Activity Time: 30–50 minutes

informally extend the concept of area using the notation of integrals with a function consisting of geometric figures. Sketch and Investigate

In Q1, students should find that the value of the integral is close to 6 + π. The difference lies in the exact locations of the points start and finish. You can create Move buttons to move these points to the exact locations. If these points are moved in this fashion, the Calculate Area button will give you a result of 9.14159. In Q2, the value of

4

∫ −2 f (x ) dx

is 6 + 2π. In Q3, answers will vary. Using the Area tools will reveal that the value is the opposite of the previous answer, namely –6 – 2π. One way to motivate this property of integrals is to appeal to the way area was calculated in the previous activity, “The Trapezoid Tool.” In Q4, the formula used to calculate the area introduces a negative sign when the “finish” point comes after the “start” point. In Q5 through Q7, the integral consists of two triangles; they are congruent, so the integral will −3 equal 0. In Q8, the value of f (x ) dx is –7.5. −6

∫

In Q9, the notion of the area function is introduced, and the area from –6 to –6 is 0. In Q10, −3 A(–3), which equals f (x ) dx , is –7.5, as above. −6

∫

In Q11 through Q14 the area trace decreases immediately because the value of the area is negative. An area of –8 will accumulate first, so the area plot will not simply become positive at x = –2, but it will increase because the area between the function plot and the x-axis becomes positive at this point. At the places where the area trace crosses the x-axis, there is an accumulated area of 0—the integral from –6 to that point is 0. At the points where the function f crosses the x-axis, the area trace has a maximum or a minimum; it is at


GSP Teaching Math.pdf

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