I am a math teacher, so it is not a surprise that out of three exemplary pieces of software we were introduced to during TEP203 class I chose Geometer's Sketchpad to review. Before even starting with the software we read a couple of articles, one of which emphasized the importance of constructing sketches vs drawing. The first activity required us to build a triangle and find the sum of its angles. Then we worked on reflections, and finally, constructed certain quadrilaterals that you can see on the Portfolio page.

What is the Geometer's Sketchpad?

This software is made for math teachers and students from 5^th grade to college. It can be used in pre-algebra, algebra, trigonometry, pre-calculus, calculus, and of course, geometry classes. With Geometer's Sketchpad students can build sketches which obey certain mathematical rules. Then, bu dragging elements of the sketch with mouse or animating the sketch, students can observe the changes, and come up with general rules, that can be proven later.

Let's take a look at the Geometer's Sketchpad window on the right. It has menu items on thetop and buttons on the left. The buttons allow user to construct circle, points, segments, name them, and move objects. You can see a triangle made of segments. These are the basic construction elements. In this example I used menu Measure to find lengths of sides of the triangle ABC and measure of angle BAC. Next to the circle you can see the curcumference, which was measured with the same menu. Below you will find more examples, in which other menu tiems were used to construct parallel or perpenducular lines, hide objects, create animation buttons etc.

Before showing additional examples I would like to include my reflection that was written after I worked with the sketchpad. For most of the students in my class it was the first time when they saw this program. For me, however, it was the second experience working with it. When I encountered the program first time in different class, our instructor expected us to know a little about it so we were given minimal instructions on what to do. And, surprise, I was lost! I didn't know the difference between drawing and constructing sketches, so after half an hour of drawing a perfect sketch, I moved one point, and it was ruined. As I mentioned eralier on this page before we even started working with Geometer's Sketchpad, we read a couple fo articles, one of which described the similar situation and explained the importance of constructing sketches, not just drawing them. So take a look at my reflection after working with Geometer's Sketchpad in Mr. Jerry Balzano's class.

Geometer's Sketchpad Activity

Discuss the Sketchpad experience.
This was my second experience in using the Sketchpad, so I wasn't as confused as first time. I already knew I needed to actually construct lines and make sure that they have particular properties. Fortunately, the article we read and then Mr. Balzano showed us how to make a rhombus – one of the most challenging of quadrilaterals – so the rest of the shapes was easy to make. With the second activity too, Mr. Balzano showed us how to make the reflections, so it was easy too.
The only tricky thing was to figure out what needs to be measured in the third part. We had to decide when measuring angles would be enough, and when we have to measure lengths of sides as well.

Reflect on past experience learning Geometry in High School.
Geometry was the most favorite subject of mine in the high school, so I didn't feel that we missed something (like Geometer's Sketchpad) in particular. Then again I went to school in another country and our educational system is different. Here in America it is more like play and games, and kids expect from the teacher that he or she is going to entertain them. In my country school is hard work, and math is a lot of practice. I liked that approach and I absolutely don't feel bad for myself or other kids that we didn't use such things as Geometer's Sketchpad or Mira, or any other manipulative.

What are the pros and cons of using Geometer's Sketchpad in High School?
I am going to use Geometer's Sketchpad next year: I will be teaching geometry, so it is my intension to use it very actively through the year. We have a set of graphing calculators for every student in class, and the calculators now have Geometer's Sketchpad installed, so it makes it convenient to use the program. I am sure students will like it and will feel better about geometry in general since they will get to use technology. For many of them computers or calculators are friends, while math is the enemy. If you have a friend next to you, it is easier to conquer the enemy. Thus in my opinion, students will have positive attitude towards math in general while using Geometer's Sketchpad.
However, as I understood, Geometer's Sketchpad allows students to check different properties and generalize them, but not prove them. So when we will be switching to old pen and paper in order to prove something, I expect the sight of disappointment from the students. I can think of only that as the negative effect of the Geometer's Sketchpad in my school, which gives us the opportunity to use the software. But in other schools it might be difficult to use Geometer's Sketchpad because they don't have computers or graphing calculators. And, even if the school has an opportunity to get computers, in order to use this software, every student should work at their own computer. This means that math class shouls also be a computer lab which makes a problem. So, it seems that Texas Instruments made a very smart move by installing Geometer's Sketchpad on their graphing claculators. They are portable, so no computer lab is required in order for studnets to work with the software.

Questions I have about Geometer's Sketchpad.
I was thinking for such a long time about using GSP in my classroom that probably I already found answers to my questions. It is hard for me to come up with one now. It seems to me that even though students will be explained that they have to construct sketches instead of drawing them, they will at first draw them anyways. So the question that arrives is how to make it natural for users to construct. Maybe by making buttons in addition to a menu?
Last school year I used a math/art project for pre-calculus students. They had to come up with a picture and then use different graphs of functions (lines, parabolas, hyperbolas, ellipses, logs, exponents…) to draw it. I am thinking whether this can be done with Sketchpad since it seems the program allows using coordinate system?
And, I am also curious about questions from the handout: can we do proofs with Geometer's Sketchpad? And do students learn math while doing "fun" activities?

After writing the reflection I had an opportunity to work with Geometer's Sketchpad some more, so I already found answers to some of these questions I had.

Interdisciplinary activities.
I am a believer in math/art projects. As I mentioned earlier math is the enemy for many students, but almost everybody likes art, so it is a good way to disguise math by asking kids do some art project.
The second part of our today activity would be fun for kids, and probably, middle school students can stop there after they saw what happens to a shape after reflection. They will realize it is like when you look at yourself in a mirror – that's how they explained it to me last year (7th grade). But then later on, students can be asked to do some additional measurements to see what qualities remain the same and what changes. That is basically the same thing as textbooks tell students, but they will figure it out themselves. Once again, the only thing that will have to be done "by hand" is to prove what they observed.

Work Samples

Here are some additional examples of what can be done with Geometer's Sketchpad. After our class I already saw that we can construct simple shapes just like the hexagon below. On the picture you can see that it was constructed, but if you move your mouse on the picture it will change to a hexagon inscribed in a circle - all the extra detailes are hidden and the hexagon was moved and rotated, but the properties remained.

I knew that Geometer's Sketchpad allows to graph functions, so I took liberty of figuring out how it is done. You can see that the program is measuring coordinate of point A which lies on a line. On the bottom left corner there is a motion controller: it lets user to see what happens to the line if constant m is changing. Move your mouse to the image and you will see how line changed with different value of m.

During 2005-2006 school year one of the classes I taught was pre-calculus. I was trying to find something different from plain lectures, but it is not as easy as for middle-school classes. After I learned about the following feature, I thought it can be included in such class. On the picture you can see the construction made in order to trace point F. This point is connected with D and E and when they move: D from left to right, E counterclockwise around the circle, the point F traces sinusoid. Move your mouse to the picture to see that. The placement of point B allows point F repeat the same trajectory.

The last snapshot I would like to share demonstrates Leonardo Da Vinci's proof of Pythagorean Theorem. First, the right triangle ABC with squares on the legs was constructed. Then the reflection line and midpoint H were made. You also can see buttons that allow to hide or show reflection and rotation. Move your mouse to the picture to see reflection being hidden.

After the reflection was hidden, the remaining part was rotated 180⁰ around midpoint H. Finally, a couple more lines were made, and the interiors of square and triangles were built. After you roll over the mouse, you can see that the yellow square is the square of the hypotenuse.

If you are interested in learning more about Geometer's Sketchpad, you can visit the Key Curriculum Press web site to read about this software, download free trial versio, buy it, check different activities etc.