If you have any comments, suggestions, and/or additions, please send them via email to Geri.

*The Geometer's Sketchpad* (GSP) begins with a blank screen and a tool bar much like other open-ended computer software such as MS Word, KidPix, HyperStudio, etc. In this case the toolbar obeys the rules of Euclidean geometry. GSP is a powerful tool for discovering geometric properties. The program also contains a ruler, a protractor, a calculator and a simple word processor in addition to graphing possibilities on a Cartesian plane. (All of the illustrations in this document were by copying a sketch from *The Geometer's Sketchpad*.)
The following activities are samples of those used in our Lower and Middle School mathematics classes. In many, but not all, of these classes a technology/math specialist co-teaches with the classroom teacher.

Exploration (Grades All) |
How Many Degrees in a Quadrilateral? (Grades 4 - ) |

Lines, Rays, and Segments (Grades 3-5) |
Sum of the Interior Angles of a Polygon (Grades 4 - ) |

Acute, Obtuse, and Straight Angles (Grades 2 - ) |
Student-Made Quiz (Grade 7) |

Guess the Angle (Grades 2 - ) |
Similar Triangles I (Grade 7) |

Area and Perimeter (Grades 3 - ) |
Similar Triangles II (Grade 7) |

How Many Degrees in a Triangle? (Grades 4 - ) |
Transformations I (Grade 7) |

Circles (Grades 4 - ) |
References |

Using a new screen we ask the students to draw a segment 2 inches long. We then demonstrate how to select the segment and choose "length" from the "Measure" menu. The defaults on our lab computers are not the same. Thus some segments will be measured in centimeters and others in inches. Some in whole numbers, some in tenths and some in hundredths. Great discussion items. We then ask them to pull their segment until it is as close to two inches as they can make it.

Next we show them the ray or line tool and have them draw one, then the other. We then ask the students to think about the difference between the three drawings that they have made. (Sometimes the students have already been introduced to these concepts, sometimes not.) As a class we talk about their observations. Usually someone wants to measure the length of the ray or line and another great discussion follows. We now ask the students to go to the File menu and choose "Print Preview." Here they will see the segment, ray, and line displayed with the usual convention of arrows for infinite continuation. (We always ask the students to use "Print Preview" before printing to assure that their sketch will be on one page.)

At this point we introduce the text tool and ask each student to write about the figures they have drawn. Sometimes we ask them to print out their sketches. Other times just to save them on their diskettes that we can then open and write comments back to them.

The students were presented with these five rectangles on the screen. They were asked to complete the following table and then figure out two more rectangles that have the same area. We provided each pair of students with centimeter cubes to help in their work. We also have a similar set of sketches/worksheets for perimeter.

side a | side b | area | perimeter | |

Rectangle 1 | ||||

Rectangle 2 | ||||

Rectangle 3 | ||||

Rectangle 4 | ||||

Rectangle 5 |

Fill in this chart. Use the centimeter blocks if you want to. What do you notice? Now add two more rectangles that share the same pattern.

We begin by asking the students to construct a triangle. Since they have already explored the tools and menus, they know where to find the "point" and "segment" tools on the tool bar and the "Segment" command from the "Construct" menu. We then ask how they chose to construct this triangle since there are several different ways to be successful. The next step is very important since little hands are at work; we ask the students to grab onto a vertex (new vocabulary word in many instances) of their triangle and pull it to be sure their construction is a triangle!

We show them how to label the vertices using the "hand/text" tool and have a discussion/review of how to name an angle. (This gets tested in the next exercise where they measure the angles of a quadrilateral.) The next step then is to measure the angles using the "Measure" menu. When "angle" doesn't appear as an option, it is an opportunity to review how to select more than one object at a time. We ask the students to estimate mentally the sum of the angles they have measured. (If you want only whole number measurements, be sure to change the GSP Preferences before you begin.) In cases where they have measured one angle twice, the total will often be way off and another opportunity to review naming angles.

At this point most students will call out 180 degrees; however, some will find 179 degrees. And here is the opportunity to talk about rounding, errors introduced by rounding, calculator rounding, etc. We then show them the "Calculate" function in the "Measure" menu and ask them to calculate the sum of the angles of their triangle. Then we ask them to grab once again a vertex, pull it, and notice what happens to their measurements and their sum.

The last step is to record their observations.

Nate constructed a circle and a radius. He then measured the radius. Using the GSP calculator, he determined the circumference of the circle. Then he took that answer, divided it by the radius, and then divided it by 2. "Ah ha," he said. "Pi is 3.14!"

When we introduce this activity in grade 4, the students have already been introduced to p and participated in activities in their math classes. In the lab, we hope soon in the math classroom, we have the students explore using the circle tool and measure menus. They soon discover that GSP will perform all the calculations for area and perimeter by a mere selection of the circle itself. I then introduce Nate's exploration so that they gain more experience with the GSP calculator and it is also a chance to introduce parenthesis notation for grouping to avoid having to make two separate divisions. For a final step before they begin writing, I have the students change the calculation preference setting to thousandths because for many of them "3.14" is it!

They are next asked to measure and sum the angles, then compare the sum to
their guess. In the cases where the student's sum is not 360 degrees, there
is an opportunity to discuss the naming of angles once again. Then they are
asked to record their observations. Finally they are asked to construct one
diagonal and think about what they now observe and record that.

- how to construct a polygon using the segment tool.
- how to label the vertices of the polygon using the text tool.
- how to calculate the measures of angles using the calculator tool
- how to write your observations using the text tool.

- Construct a triangle.
- Label the vertices.
- Measure the angles.
- Using the calculator, sum the angles.
- What is the sum? With the select tool (arrow), pull a vertex to change the shape of the triangle. What is the sum now?
- Using the text tool, make a box on your sketch. Type in your name and findings (i.e. what did you observe about the sum of the degrees in a triangle.)
- Save your work.
- From the File menu, choose "Print Preview." Make sure your work is on one page. Click on scale to fit page if needed. Now print.

- Construct a quadrilateral. Using the text tool, make a box and record how many degrees you believe are in the quadrilateral and why you made your guess.
- Follow Triangle steps 2-5 above.
- Now construct one diagonal.
- Using the text tool, make a box on your sheet. Type in your name and findings (i.e., what did you observe about the sum of degrees in a quadrilateral. What added information did the diagonal give you? How many triangles resulted? Do you think there's a relationship between the number of degrees in the triangles and the number of degrees in the quadrilateral?
- Save your work.
- Print from Print Preview menu.

- Construct a pentagon. Follow the instructions for steps 2-5 of Triangle.
- Now construct two diagonals, each beginning at the same vertex. What added information do you have? Record your results and print.

Name of polygon | Number of sides | Number of triangles made by diagonal(s) | Total Number of interior degrees | Number of degrees for each angle of a regular polygon |

triangle | 3 | 1 | 180 degrees | |

quadrilateral | 4 | |||

pentagon | 5 | |||

hexagon | 6 | |||

7 | ||||

8 | ||||

9 | ||||

10 | ||||

20 | ||||

n |

Can you figure out a rule to predict the number of interior degrees in any polygon? What is it?

- The 20 degree angle is about 11.1% of what angle measurement?
- 46 degrees is what percent of the sum of the angles of a triangle?
- What percent of all the angles you measured are acute angles?
- What percent of all the angles you measured are obtuse angles?
- What percent of all the angles you measured are right angles?
- Have you accounted for all of the angles? Prove it numerically.
- Using 3 letters, name a pair of corresponding angles.
- Using 3 letters name a pair of supplementary angles.
- Using 3 letters name a pair of alternate interior angles.
- If 30% of your day is spent sleeping, what size angle would represent that amount of time on a circle graph?
- If you measure a 100.8 degrees angle on your circle graph, what percent of the graph is used? How many hours in a day is that?
- The area of the triangle is given and so is the base measurement. Find the height. Show your work.

- Construct a triangle.
- Construct a point on one side of the triangle.
- Select the point and a side of the triangle.
- From the construct menu, choose parallel line.
- Construct the point at the intersection of the parallel line and the side it intersects.
- Label the vertices and the points where the parallel line intersects the triangle.
- Measure the lengths of the sides of the triangle.
- Measure the lengths of the sides of the small triangle you just created - in my sketch it is triangle BEF. (You will need to measure distance between two points rather than the length of segments.)
- Now demonstrate that the large triangle is similar to the smaller triangle - in my sketch that is triangle ABC is similar to triangle BEF.

- Transformations (slides, flips, and turns; shrinking and enlarging).
- Scripts to record work for analysis and assessment of student work.
- Coordinate axes.
- Animation. (See Geometry Activities for Middle School Students by Wyatt et al for the best introduction that we've found. We Xerox "Animation Tours" for the students (and teachers) and leave them on their own--a good activity for following written instructions!
- Sketches and scripts that come with every GSP supplement.

Battista, T. *Shape Makers: Developing Geometric Reasoning with The Geometer's Sketchpad*. Key Curriculum Press. Berkeley, California, 1998.

Bennett, Dan. *Exploring Geometry with the Geometer's Sketchpad*. Key Curriculum Press. Berkeley, California, 1993.

Finzer, William F and Dan S. Bennett. "From Drawing to Construction with The Geometer's Sketchpad." *The Mathematics Teacher*. NCTM, May 1995.

Manouchehri, Azita et al. "Exploring Geometry with Technology." *Mathematics Teaching in the Middle School*. NCTM, March-April, 1998.

Serra, Michael. *Discovering Geometry: An Inductive Approach*. Key Curriculum Press. Berkeley, California, 1997.

Usiskin, Zalman. "The Implications of 'Geometry for All'", *NCSM Journal of Mathematics Education Leadership*. October 1997.

Wyatt, Karen Windham et al. *Geometry Activities for Middle School Students with The Geometer's Sketchpad*. Key Curriculum Press. Berkeley, California. 1998.

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