Chapter 16 - Extra Credit Projects

Students always seem to love "extra credit". It gives them an opportunity to raise their grade, to do something creative, and to experience other facets of mathematics besides the information presented in textbooks. Hopefully, these projects can be educational, interesting, and even change their attitudes toward mathematics. Even those students who already like math may find new points of view and new applications in creative projects.

The creative teacher can design educational "extra credit" projects that allow the students to practice their own creativity while learning mathematical concepts. In this chapter, teachers will find ideas for a variety of "hands-on" projects in various areas of mathematics. Many of these projects involve geometric constructions, which can be done using a compass and straightedge, or computer software such as The Geometer's Sketchpad. In either case, these projects provide interesting ways for students to apply constructions and practice their skills.

Getting Blood From a Stone

This was a very popular project with the students. It appealed to them for many reasons: some liked it because it was "different" (not the usual math problem), others liked it because it seemed like a game or puzzle. I liked it because it provided an opportunity for them to truly grasp the nature of a mathematical proof. The rules of writing a mathematical proof are very similar to the rules of "getting blood from a stone". The first thing I needed to do in this project was to explain the old saying "You can't get blood from a stone", which means that nobody can give you anything that they, themselves, do not have.

They were intrigued right from the start, first because of this interesting phrase, and second because they could not imagine what this had to do with mathematics. This, of course, gave me the opportunity to explain the connection between the word game that we would "play" and the concept of proof, which is so important in mathematics.

After explaining what the phrase meant, I proceeded to tell them the "rules of the game", and how these rules relate to mathematical proof:

1) You must start with the word STONE and end up with the word BLOOD by a series of logical steps (much like the steps in a mathematical proof).

2) You must change one letter in each step. Each step must follow directly from the previous step. (In a proof, you must address one concept at a time. Each step in a proof must follow directly from the step before it - including from the 'Given', and lead to the next.)

3) The word in each step must be a real word, and in the dictionary. (In a proof, each step must be true: a definition, postulate, or theorem). I tell the students that if they use an unusual word, they must write the definition.

4) In this game, some people can do this in 10 steps; others may take 15, and both can be correct. There is more than one route, and no one way is necessarily better than another. One may be longer, but the best one is the one is the one that the student "sees" when trying to do the problem! (This is very true of proof; there are dozens of proofs of the Pythagorean Theorem, for example, and even in classroom assignments there is often more than one method.)

5) In this game, some people like to work backwards, or even from both ends toward the middle. (This works very well with proofs, too, and I highly recommend it!)

I usually offer this "extra credit" after we have done an introduction to proofs and I have written a few simple proofs on the board. I start by showing them an example, such as the one below:

Can you get a DOG from a CAT? (Can you go from the word CAT to the word DOG?)

Here are some solutions to this "warmup" problem:

Solution 1: CAT - BAT - BAG - BOG - DOG

Solution 2: CAT - COT - DOT - DOG

I have found this little exercise to be an excellent (and amusing) way to introduce the process of writing proofs - a process that is very different from any mathematics that the students have yet encountered in their educational career. The students enjoy the "game" and from this point on it provides an excellent reminder of the importance of a logical and sequential method for writing proofs.

Here are some examples of student work on this project, including some of their reflections on the experience:

"This was by far the hardest project for me. It took me a very long time to figure out the answer. I decided to work backward, and this helped. I also used the dictionary, and learned a few new words, too! So, here is my solution to getting Blood from a Stone:

Stone

Store

Snore

Snort

Snoot ("a grimace or expression of contempt")

Snoop ("pry into")

Sloop ("a type of boat")

Bloop ("an unpleasing sound")

Blood

I did it!"

And here is a second solution, including the definitions for all of the words

Stone

Shone (Past tense of shine)

Shine (To send out light)

Thine (Possessive case of "thou")

Trine (Threefold, triple)

Brine (Very salty water)

Brane (Algae, seaweed)

Brand (Logo, name)

Bland (Gentle, soothing)

Blond (Light in color)

Blood

"So there is my solution - and I learned some new words too! This was a cool project."

In their reflections about this project, students had many interesting comments to make. Overall, they really enjoyed the project and found it very helpful. As we continued with proofs throughout the school year, students would frequently refer to this project as one that was helpful and interesting.

With regard to this project, Amanda wrote: "The stone to blood project was one of my favorite projects this semester. I spent a lot of time on it, and thought it was very addicting! I just couldn't be satisfied until I figured it out. In the end, I finally found the answer, and I was very proud of myself. This activity helped clarify the process of proofs, to me. When you begin a proof, at first it might seem complicated and maybe impossible to do. The important thing is to try many different possibilities to reach the solution instead of giving up right away. You should always try ways that you think it won't work too, because it might really be the answer. This also showed me that every step in a proof should directly connect with the step before it and the step after it. In proofs you can't skip any steps, like going from STONE to STORK!

Andrew had this to say: "At first I didn't think this problem had anything to do with math. But as I started trying to figure out a way to get from the work stone to the word blood, I realized that the process is very similar to writing a geometry proof! In this project you start with the word stone and then have to change just one letter to get a new word, and then the next. And I suddenly realized that we do the same thing when we try to prove something in geometry. We start with one piece of information and it leads logically to the next, and then that leads to yet another. And each step is just one piece of information, and they are all logically connected. You use what you discover to make the next step. I liked this project. Even though it was difficult, and sometimes I wanted to give up, I just kept going and finally finished. I am so proud of myself because I met the challenge and overcame it - hooray!"

And Laurie wrote: "I think this project laid the foundation for my thinking in this course. By being forced to think of steps, one by one, I learned to think of what I was given and what the result had to be. Then I could think of the steps that needed to be reached before I could accomplish my goal. By adding a few connecting steps, I found I had written my first proof even without knowing it! It took me a long, long time to do, but in the end I was not only proud of myself for finishing, but I had also discovered the key to proofs that has helped me to understand all that we've done so far this year."

Remi wrote a very interesting reflection on her feelings about this project: "This seemed to be the perfect metaphor for geometry - the problem could be solved in a variety of ways, and you can prove it the long way or the short way and both are correct. Whichever one is the 'best' is the one that makes sense to you and is the way that you see it. This was one of the first assignments in the quarter, and it really made a deep impression on me because it gave me an idea of what geometry would be like. It was challenging and took a lot of time, but I feel that I learned various strategies that I have used just about every day in my proofs. Some of these are: working forwards and backwards, trying several starting points (even the middle!) and looking at it from a different angles."

I would like to include, in summary, one last student comment: "Even though this project was all about words and not numbers, it had a lot to do with math. First of all, it clearly demonstrates that there is more than one way to solve a problem. Not only did this problem have many different possible solutions, but it also had different possible approaches to solving. I learned a lot about proofs from it; I really understood the concept of moving from step to step in a logical path. This is absolutely what you have to do in a proof!" Peter

The Nine Dots Problem

This project really brought out the creativity in my students! They came up with some very imaginative solutions, and presented their solutions in interesting and clever ways.

The problem statement is simply this: Draw 3 black dots about an inch apart on a piece of unlined paper. Then draw another 3 more dots, directly below and parallel to the first row of dots, and then a third row. This should look like the drawing below:

Then the instructions to the students said: "Without lifting your pencil from the paper, draw 4 (or less) lines that pass through all nine dots. (You may not trace back over any line.)"

The students found this problem fascinating, and came up with some very clever and unusual solutions. I was amazed with their creativity and their enthusiasm! You can see some of their solutions below, including their written explanations.

"At first, I couldn't figure out how to solve this problem. I knew I could connect all the outside dots in four lines without lifting my pen, but I didn't know how to include the middle dot. Then it suddenly hit me: I had been thinking "inside the box". To solve the problem, you have to "think outside the box", literally! This means you have to think creatively, and look beyond traditional methods - expand your thoughts. You have to let your mind break free."

Adam had exactly the right idea, and so did a number of other students. Their creativity and cleverness amazed me, and it was exciting to see their enthusiasm and their solutions. I have included some of them below, including the students' explanations.

In my Geometry class, we were studying Euclidean Geometry, but I had told the students that there were other geometries. We had taken brief "excursions" into "non-Euclidian" geometries. If you are interested in finding more about these geometries, you can find some very good information at the following website: http://en.wikipedia.org/wiki/Non-Euclidean_geometry, but be sure to come back! Jason drew the following solutions, and wrote a very interesting essay about this problem, below his diagram:

"I had seen this nine dots problem before, and had seen the popular solutions to it. But this time we were encouraged to not only think even further 'outside the box' but also to explain our solutions in words. My own solution involved an alternate form of geometry, called Riemannian, which takes place on the surface of a sphere. I realized that I could use this solution when we discussed map projections in my Social Studies class. I was considering the distortions created by different projections when I recalled the alternative geometries we had discussed in class. Three lines of longitude on a globe could easily be drawn to connect all nine dots without lifting one's hand from the paper (the sphere) as they all intersect at the poles. I learned even more about this concept when we discussed all the different solutions in class. A vast array of answers are possible, more than I had ever imagined, having limited myself to established geometric forms. The one that dealt with perspective has some interesting applications in art, for example. The nine dots problem showed me just how creative one can be when it comes to Geometry!"

And creative, they were! Here is a solution involving some clever folding, by Alison:

She wrote: "I have always liked origami paper-folding, so I thought I would see if I could do something like that with the nine dots problem. I tried to make it as creative as possible, and I really like my solution and I hope you do too. This was such a fun project, and I hope we get a chance to do more projects like this, that let us challenge our minds and have fun, both at the same time!"

This project really did bring out the creativity and humor in my students. The one below was labeled "Fat Dots"!

Jenna drew the following illustration, which she called "One Fat Line":

Probably the most ingenious solution is one that no one picture can truly explain, but I will try. You will have to use your imagination as you read the steps that Ashley wrote to explain her creation. "Take a piece of paper, and draw three short vertical segments. Then draw dot three dots on each segment, one at the top, one in the middle, and one at the bottom as shown in the diagram on the left below. then on the back of the piece of paper, draw two half-circles, one at the top right and the other at the bottom left."

When you are finished, you should have a piece of paper that looks on one side of the paper like the diagram on the left below, and on the other side of the same paper the diagram on the right below. The two diagrams need to line up perfectly for this to work. You can print both and then glue them together "back to back)" or create your own drawing, copying the diagrams back to back on the same piece of paper.

When put together and folded on the dotted lines, your project should look like this:

So, you can see that you can draw one line through all nine dots!

In their reflections on this project, many of the students wrote that they thought it was the most interesting one of all. Some of their solutions involved "thinking outside the box", and thinking three-dimensionally instead of "staying on the paper". We had some very interesting discussions about this, including a digression into other types of geometry than Euclidian geometry. They were fascinated to find out about the geometry of Lobachevsky and of Riemann. I brought a can of "Pringles" (curved potato chips) to class, and passed them out to the students so they could imagine and experience (touch and taste!) geometry on a curved surface. I asked them to draw a "straight" line on the curved surface (which brought much laughter - none of the students were willing to "ruin" their tasty potato chip!).

Dacia wrote in her reflection on this problem "This nine dots extra credit was one of the most interesting discoveries that I was able to do in geometry so far. I was flabbergasted to find so much math in a potato chip; I told my parents about it and they thought it was hilarious and wonderful. I chose this for my portfolio because it covered a wide range of geometric history. This project, while interesting because of the many solutions, was even more interesting because of the history involved - the history of Euclid and of opposing theories. I never knew that people had different interpretations about math! It was fun both in finding solutions to the problem and in learning about the history of math."

Jason had this to say about the project: "My solution involved an alternate form of geometry, know as Riemannian Geometry, after a famous mathematician named Reimann. Georg Friedrich Bernhard Riemann was a German mathematician who made important contributions to analysis and differential geometry, some of them paving the way for the later development of general relativity."

He went on to say "Riemannian Geometry takes place on a sphere, so it is also called "Spherical Geometry". I realized that I could use this solution when we discussed maps in Social Studies. I was considering the distortions created by different projections when I recalled the alternate geometries we had discussed in math class. Three lines of longitude on a globe could easily be drawn to connect all nine dots without lifting one's hand from the paper, as all the lines intersect at the pole. This made me realize that although Euclidian Geometry is most common and is applicable in most cases, there are other forms that are just as useful. The problem of connecting nine dots this way, on the surface of a sphere, showed me just how creative one could be when it comes to geometry!"

You can see Jason's drawing below. He has drawn two possibilities, and says: "On a sphere, the dots could be connected in 2 ways. One way, the 3 lines could be drawn each start and ending at a pole (North Pole and South Pole) as shown below on the left, and the second way to draw it would only use one line, which wraps around the sphere without ever going through a pole as shown below on the right below."

3D Drawing

I feel that it is very important for students to have experience drawing geometric figures. Geometry is a very visible type of mathematics, and the more practice that students have in drawing 2 and 3-dimensional shapes, the better they will understand the mathematics of these figures. In this project, students were given grid paper that would assist them in drawing. There are three types of drawing that can be used in drawing geometric figures: oblique, isometric, and perspective. Teachers and students can find more information about 3D drawing at the following website:

http://mathforum.org/workshops/sum98/participants/sanders/

In this assignment, the students were asked to draw three-dimensional geometric figures such as a cube, a "box " that was not a cube, a cylinder, a pyramid, and a cone. Some drew the figures in oblique, some in isometric, and a few drew then in perspective. The diagram on the left below is an isometric grid, and the diagram on the right is a cube drawn in isometric, to serve as examples.

In this project, the students were asked to create a 3-dimensional, geometric drawing of their own design. They found this project very interesting, and drew some wonderful geometric drawings. Some were quite simple, like the stairs below:

. . . some were more complex:

. . . others were very creative and complex!

...and some were truly amazing (this student was definitely using some sophisticated 3D software):

http://mathforum.org/workshops/sum98/participants/sanders/

Whether or not they were skilled at drawing, the students enjoyed this project, and it helped them to understand geometric shapes. In their reflections on this experience, students made these comments:

"I am proud of this project because it is especially neat and shows a lot of time put into doing it. I think my picture is very creative, and I really liked what I came up with, even though I'm not too much of an artist. Doing this assignment helped me understand the 3D geometry figures better, and I'm sure that will help me in doing volume and surface area math problems. It is so much easier to figure out a volume calculation if you can see it." Jennie

"This assignment was difficult for me at first because I never knew how to draw three dimensional objects, but I always thought it would be cool if I could. The "grid paper" made it so much easier! Now I feel like I can draw anything. Well, maybe with some more practice. When I began to draw the shapes it helped me with the math problems because I had a drawing to go by when figuring out volume etc. I always had trouble with that before but now I'm getting better at it." Michael

"I had to try many times in order to get the 3D thing exactly right. In the end, I really liked what I came up with, mostly since I am not too much of an artist. By doing this assignment, I was able to get practice in drawing planes and shapes. It makes it so much easier to do the math!" I always had trouble drawing before this project." Ben

Merry Christmas!

Website of the Week Reflections

There are so many interesting and educational web sites on the internet. I decided to take advantage of this in my math classes, and offered an "extra credit" opportunity to visit selected web sites, particularly those on the Math Forum. If your students have not been to the Math Forum, then they are really missing something! I offered my students an opportunity to visit the "website of the week", and then write a paragraph or two telling me what they had learned. They enthusiastically complied, and liked this idea so much that they would remind me every week to give them a new one! Here is the web address of the Math Forum, and below that are some of my students comments about what they found there: http://mathforum.org/

"I visited a website called "Ask Dr. Math", and then was connected to various other sites having to do with pi. Although we use pi in math, I never really knew what it stood for. Pi is a fascinating number that has interested mathematicians for centuries. It is a transcendental number and has been know for hundreds of years. Even the Egyptians, Mesopotamians, and others knew of Pi, but some thought that Pi = 3. The reason this number is so phenomenal is because it goes on infinitely and never repeats or stops!" Lily.

"This writing is about the Math Forum website. Wow! I never thought there was so much information about math in this world! I tried to outline the main topics but there were just too many. I think that you could go to the Math Forum every day and learn math and do this for a year and there would still be tons of interesting stuff that you hadn't even gotten to yet! I wish I could go there on my computer during my math test. " Brandon

There are so many pages on the Math Forum website that you could go there once a week for an entire school year and still have hundreds more to go! I am proud to be a contributor to this wonderful website, and would like to offer the web addresses of my pages for teachers and students who are interested in mathematics, particularly geometry:

The first website on this list is http://mathforum.org/~sanders/ which includes five separate sets of web pages: "Creative Geometry", "The Geometry Pages", "Journey to the Center of a Triangle", "Connecting Geometry" and "MathArt Connections"

The other websites that I have on the Math Forum are listed below:

http://mathforum.org/workshops/sum98/participants/sanders/ "3-D Drawing and Geometry"

http://mathforum.org/workshops/sum2000/cathi.html ("Math Art Connections")

http://mathforum.org/sum95/suzanne/hawaii.html ("Tessellations From Hawaii")

Geometry and Art

My geometry students created some beautiful mathematical posters, including the steps of construction for each geometric design. This was a project that the students particularly enjoyed, and they created some very beautiful geometric designs.

The National Council of Teachers of Mathematics published six of these posters, in a beautiful poster book. You can see these posters at the following web page: http://mathforum.org/~sanders/geometry/NCTMposters.html

Here is an example of one of the graphics from that website:

Writing was an important part of each project. The students wrote clear and complete step-by-step instructions for constructing their graphic, as shown below:

Fractal

Step 1: Construct an equilateral triangle.

Step 2: Construct the midpoints of the three sides of the triangle, and then connect these midpoints with segments to form four triangles

Step 3: Construct the midpoints of the three sides of each of the new triangles, and then connnect these midpoints with segments to form four triangles. Notice that in the diagram below, the center large triangle does not contain smaller triangles. This creates a more interesting graphic design, as you saw in the completed fractal graphic above, as does the adding of color to the design.

And, as always in this writing-intensive math class, they wrote reflections on their experiences:

"I had never thought of incorporating mathematics principles with art. This construction required us to understand basic principles of math constructions to create a very cool object called a fractal. I would really like to thank you, Mrs. Sanders, for giving me the opportunity to experiment and create, and for getting me hooked on fractals!" Adam

The graphic above is called a "Sierpinski Gasket" or "Sierpinski Triangle". If you are interested in learning more about this fractal, you will find some very interesting information at the web page below:

http://en.wikipedia.org/wiki/Sierpinski_triangle

You can also see an example of another geometric design below, created by my son, when he was in my Geometry class. My students created a number of beautiful geometric designs, each with detailed written instructions. This mathematical shape is called a Cardiod. You can find out more information about the Cardioid at the following web page: http://mathworld.wolfram.com/Cardioid.html

You can find step-by-step instructions for creating this graphic at the following web address: http://mathforum.org/~sanders/connectinggeometry/Cardioid.html

Betsy Ross and the Six Pointed Star

In this project, just in time for Independence Day during summer school, I offered an extra credit project to design and construct a 6-pointed star. I also asked them to do a bit of research into the origin of this holiday, and of our country's flag. Again, the students rose to the challenge. Some commented that they really thought it was interesting that their teacher "...could find a way to relate everything with math, even a national holiday!" Allison wrote an essay on the history of the flag, constructed a 5-pointed star with compass and straightedge, and found the measures of all the angles: "Betsy Ross is the woman who is famous for making the first U.S. flag. She lived from 1752 to 1836. I have drawn the star below, and labeled it with measures of all the angles:"

Allison also wrote some mathematical information about the star: "When you draw a 5-pointed star, you discover that it is really a pentagon and 5 triangles. The sum of the intererior angles of a pentagon is 540 degrees. Each interior angle of the pentagon is 108 degrees. Each exterior angle is then 72 degrees. So you see that Betsy Ross was quite a mathematician!"

Geometric Constructions

The National Council of Teachers of Mathematics have published many articles on the value of constructions in geometry, whether they are done with compass and straightedge, or using geometry software such as The Geometer's Sketchpad. There are many reasons why constructions should be a part of every geometry class:

Geometric constructions can enrich students' visualization and comprehension of geometry, lay a foundation for analysis and deductive proof, provide opportunities for teachers to address multiple intelligences, and allow students to apply their creativity to mathematics. Writing the construction procedure is an excellent way for students to learn mathematics. When they are asked to explain the steps of a construction, they must put visual and tactile procedures into verbal or written words. The process of creating a clear and complete explanation of the procedure requires the student to clearly understand not only the steps of a construction, but the importance and sequential nature of each step. Writing these steps internalizes them, and the student arrives at "ownership" of the information.

The teacher can help students to understand how to write this type of information by offering some examples. For instance, in explaining how to construct an angle congruent to a given angle, consider the following dialogue between teacher and students:

Teacher: "What is it that makes an angle the "size" that it is? Is it the length of the sides? Are these angles congruent?"

Students: "Yes, the angles are congruent. . . . But one has sides longer than the other! Even so, the angles themselves would be the same number of degrees, even though their sides are not the same length."

Teacher: "What about these angles? Are they congruent?"

Students: "Well, their sides seem to be the same length . . . But one is wider than the other! . . . The angles are not congruent. Even though their sides are the same length. One angle is bigger than the other."

The students may struggle a bit with the terminology, but after some discussion will be able to explain that what determines the "size" of an angle is "how far apart the sides are". Clarifying this, the measure of an angle can be defined as "an amount of rotation". Therefore when we want to "copy" an angle, we must duplicate the amount of rotation, and the lengths of the sides do not matter.

What's wrong here? The students will be quick to point out that though we may have set our compass to the same radius, and measured the same "amount of separation" between the sides of the angle, we are not measuring the new angle in the same "place" as the given angle. Some discussion should follow as to what we mean by "in the same place" and "separation" between the sides. The teacher can elicit from the students the concept that we need to do the operation on each angle (the given and the copy) in exactly the same way, i.e.. in the same "place".

According to the NCTM Standards, "All students need extensive experience reading about, writing about, speaking about, reflecting on, and demonstrating mathematical ideas. It is equally important that students be able to describe how they reached an answer or the difficulties they encountered while trying to solve a problem." Bridging the gap between the vernacular and the mathematical terminology is a rich mathematical experience; writing the instructions is an excellent exercise in communicating mathematics for the students.

What results is a clear understanding of why we need to swing the first arc (from the vertex) and construct an angle in the following way. Rather than arcs, full circles have been drawn in the diagram below as part of a proof of the construction. Writing the proof completes the learning experience: discovering what does and does not work by trial, error and analysis, writing an explanation of the error and communicating in correct mathematical terminology, constructing the angle correctly, and confirming the construction by proof.

Linda, one of my geometry students, wrote this "paragraph proof":

"Radii BE = B'E' and BD = B'D' because circle B is congruent to circle B' as constructed. Radii DE = D'E' because circle E is congruent to circle E' as constructed. Therefore triangle BDE is congruent to triangle B'D'E' by SSS. This makes angle DBE congruent to angle D'B'E' by the definition of congruent triangles."

This connection between construction and proof/verification/logic/meaning of construction is a very valuable part of a geometry course. In writing the proof herself, Linda practiced not only her math skills, but honed her verbal and writing abilities as well.

A continuing theme in the NCTM's Evaluation Standards is the need for multiple sources of information, and the necessity for addressing different learning styles among students. "Students differ in their perceptions and thinking styles. An assessment method that stresses only one kind of task or mode of response does not give an accurate indication of performance, nor does it allow students to show their individual capabilities."

Sketching and constructing geometric figures provides opportunities to relate to different learning styles and to hands-on learners, kinesthetic and visual learners. Discussing and writing the steps of a construction, particularly the more complex ones, provides an opportunity for students to communicate in precise language, and write mathematics. Assessing these activities provides the teacher with more complete insight into students' abilities, on many different levels.

There are additional reasons to emphasize constructions in teaching geometry. Many students enjoy constructions, and find satisfaction in displaying their abilities. They know when they have done the work correctly, and have a tangible, demonstrable result. In my writing-intensive geometry course, I ask the students to write reflections, which are included in a portfolio of their work. This is what they have said:

"Constructing is the most gratifying thing to do in geometry. When it is done right, the lines match up, angles are properly bisected, and everything is perfect in the universe!"

"I chose this construction for my portfolio because it is my favorite part of geometry. I like doing constructions a lot. Doing them is like solving a puzzle. They test your problem solving skills, and I just think of them as fun."

Constructions can be very successfully integrated throughout the course, rather than in a chapter at the end. Constructions can reinforce proof, and provide visual clarity to many geometric relationships. When teaching arithmetic mean and geometric mean, for example, constructions provide a very effective way to visualize the relationship between the arithmetic mean and the geometric mean:

A student constructed this on the Geometer's Sketchpad. Using Sketchpad, this construction can be used as a very effective demonstration: as you drag point D, the right triangle ADC, which is inscribed in the semicircle, changes. AB and BC change also. BD remains the geometric mean, of course, but approaches the arithmetic mean FE. The students clearly see that the only time the geometric mean is equal to the arithmetic mean is when the two given segments are congruent (the trivial case) and the geometric mean never exceeds the arithmetic mean.

My students have done some very interesting projects using these constructions. They used a compass and straightedge (and in later years, Sketchpad) to construct regular polygons, right triangle spirals, geometric curves and fractals, and then they color their geometric designs with colored pencils or pens. An essential part of the project was to write the steps in the construction. This was a very "rich" assignment; involving visualizing, constructing, and writing. Kevin wrote an excellent set of illustrated steps for the conctruction of the fractal below:

When we began to use the Geometer's Sketchpad software, the students were thrilled to find how beautifully they could construct their own designs. They created some very beautiful designs, some of which were published by the National Council of Teachers of Mathematics; you can see them, including the final step of the project above, at the following web page:

http://mathforum.org/~sanders/geometry/NCTMposters.html

"I liked this project because it allowed me to be creative and put to use everything we have learned in the computer lab. Who would have ever thought there was so much art in math! This assignment was the first time I ever felt like you could be really creative in a math course."

The NCTM Standards recommend that "students use computer software based on this dynamic view of transformations to explore properties of translations, line reflections, rotations and dilations, as well as composites of these transformations". These graphic design projects not only help students develop an understanding of the effects of various transformations but also contribute to the development of their skills in visualizing congruent and also similar figures.

The students found this project very rewarding. In their reflections, they discuss their feelings about this work:

"I liked this project because it allowed me to be creative and put to use everything we have learned in the computer lab." Liz

"Who would have ever thought there was so much art in math! This assignment was the first time I ever felt like you could be really creative in a math course." Mark

Some of the students became interested in writing scripts in Sketchpad, and created some graphics using this method. The following script can be used to create equilateral triangles and their midpoints, in producing a fractal:

This "fractal script" produces a fractal such as the one shown below:

In their reflections about their experiences with these projects, my students wrote many interesting and enthusiastic comments. I have included a few here: "wonderful variety", "makes it a more interesting class", "they make math more enjoyable", "provides ways to explore further", "worthwhile and fun", and "I like the way they take what we are learning in math and apply it to real life". Other comments included: "it motivates me to explore math further", and "projects may help by making by making math less left-brained and more applications".

"Mathematics is an independent world created out of pure intelligence."

Wordsworth, William (1770 - 1850)