### by Geri Anderson-Nielsen

Geri is a technology/math consultant. This booklet was first written as a handout for presentations made at the 1998 NCSM/NCTM Annual Meetings.

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.

Our first activity is always free exploration. After a brief description of the software, I say, "Take the next five minutes and discover as many DIFFERENT things this program can do as you can." After five to ten minutes, I bring the class back together and have each student or pair of students show one aspect of the program on the big monitor.

### Activity: Lines, Rays and Segments (Grades 3-5)

We first name the tools: select, point, circle, line segment and have the students try each one:

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.

### Activity: Acute, Obtuse, Straight Angles (Grades 2 - )

We ask the students to make a construction with two rays beginning at the same point. Using the text/hand tool, we ask them to label the control point on each ray and the beginning point for the rays (the vertex of the angle). We show them how to select the three points, the vertex being the second, and to use the "Measure" menu to find the number of degrees in their angle. We then ask them to pull on one of the control points to make the angle bigger and smaller and to notice the angle measure. (They all love to make right angles and straight angles!). We then have a conversation about naming the angle as acute, right, obtuse and straight. (Notice that Sketchpad never measures an angle greater than 180 degrees.)

### Activity: Guess the Angle. (Grades 2 - )

In this activity, the students can take the angle made above, hide (using the "Action Button Hide/Show") the measurement and then ask a classmate whether the angle is acute, obtuse, right or straight or estimate the number of degrees in the angle. After the classmate has made a guess, the student can reveal the exact measure.

### Activity: Area and Perimeter (Grade 3 - )

Although this is an activity commonly done with manipulatives, our third grade teachers wanted to do it with Sketchpad as well to provide the students with both an opportunity to use GSP and at the same time the reinforcement of rectangle area and perimeter measurements.

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.

### Activity: How many degrees in a triangle? (Grades 4- )

This is a usual Sketchpad activity and the exact instructions can be found in the Key Curriculum Press materials. We begin this activity in grade four just after the students have been introduced to the notion of angle measure and using a protractor to determine the measure of a particular angle.

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.

### Activity: Circles I (Grade 4 - )

When Nate was in second grade his teachers asked me to help out with some more math/technology extensions for him. So one afternoon he came up to the lab to meet me and as a "getting to know each other" activity, I brought up the Sketchpad. He started trying out the tools and menus with some input from me. Following is his first exploration - one that I have borrowed from him to use as an introduction to circles.

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!

This can be a self-contained activity or one that leads to "Investigating the sum of the interior angles of a polygon." We ask the students to construct a convex (and here you will need to have a conversation about convex and concave - an interesting discussion to which to return with older students) quadrilateral, pull a vertex to be sure that the construction is secure, and label the vertices. We then ask them to guess how many degrees are in that polygon and give a reason for the guess.

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.

### Activity: What is the sum of the interior angles of a polygon? (Grades 4 - )

This is an activity that we have begun in grade 4 by asking students to extend the quadrilateral angle measure to a pentagon, then hexagon and then see if they can figure out without performing the construction, how many degrees are in the interior of a 21-sided figure. In the pentagon, after measuring and summing the angles, we ask the students to construct all the diagonal possible from one vertex and observe how many triangles there are. This activity can obviously be done easily with paper and pencil by drawing the figures and non-overlapping diagonals. However the dynamic aspect of the Sketchpad appears to add to their involvement, as well as make the activity more accessible to students without the motor skills to make the pencil/paper constructions. Below is a copy of the worksheet for this activity developed by my colleague Judith Knight for her 7th grade Math Analysis class. Investigating the sum of the interior angles of a polygon In order to do this exercise you will need to know:
1. how to construct a polygon using the segment tool.
2. how to label the vertices of the polygon using the text tool.
3. how to calculate the measures of angles using the calculator tool
4. how to write your observations using the text tool.
Triangle: This is review.
1. Construct a triangle.
2. Label the vertices.
3. Measure the angles.
4. Using the calculator, sum the angles.
5. What is the sum? With the select tool (arrow), pull a vertex to change the shape of the triangle. What is the sum now?
6. 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.)
8. 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.
1. 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.
2. Follow Triangle steps 2-5 above.
3. Now construct one diagonal.
4. 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?
6. Print from Print Preview menu.
Pentagon:
1. Construct a pentagon. Follow the instructions for steps 2-5 of Triangle.
2. Now construct two diagonals, each beginning at the same vertex. What added information do you have? Record your results and print.
Now begin to fill out the following table. Construct a hexagon if you need to in order to fill out the information. Do you see a pattern? Suppose you constructed a regular polygon of the types listed above. What would be the measure of each of the angles of the figure? Fill in the last column of the chart with this information.

 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?

Following is a sketch and a quiz constructed by one of our seventh grade students. His class had been studying parallel lines, transversals, and triangle area among other topics. Grade 7 Quiz: Measure all of the angles that are not already measured and then answer the questions.

1. The 20 degree angle is about 11.1% of what angle measurement?
2. 46 degrees is what percent of the sum of the angles of a triangle?
3. What percent of all the angles you measured are acute angles?
4. What percent of all the angles you measured are obtuse angles?
5. What percent of all the angles you measured are right angles?
6. Have you accounted for all of the angles? Prove it numerically.
7. Using 3 letters, name a pair of corresponding angles.
8. Using 3 letters name a pair of supplementary angles.
9. Using 3 letters name a pair of alternate interior angles.
10. If 30% of your day is spent sleeping, what size angle would represent that amount of time on a circle graph?
11. 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?
12. The area of the triangle is given and so is the base measurement. Find the height. Show your work.

### Activity: Similar triangles I (Grade 7)

In this activity the students were asked to measure each of the angles and each of the sides of the triangles. They were then asked to look for a relationship between the two triangles. After they wrote their hypothesis, they were asked to pull one or the other of the triangles and see if their hypothesis held. We then talked about the definition of similarity, one that they had not yet encountered formally.

### Activity: Similar Triangles II (Grade 7)

This is another activity developed by Judith Knight. Instructions for constructing a line parallel to the base of a triangle.
1. Construct a triangle.
2. Construct a point on one side of the triangle.
3. Select the point and a side of the triangle.
4. From the construct menu, choose parallel line.
5. Construct the point at the intersection of the parallel line and the side it intersects.
6. Label the vertices and the points where the parallel line intersects the triangle.
7. Measure the lengths of the sides of the triangle.
8. 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.)
9. Now demonstrate that the large triangle is similar to the smaller triangle - in my sketch that is triangle ABC is similar to triangle BEF.

### Activity: Transformations I (Grade 7)

Again with thanks to our colleague Judith Knight. Our seventh grade program, which strives for interdisciplinary projects, includes the study of Oriental History. This fall Judy discovered the featured description of the symmetry of Oriental carpets on the Math Forum Home Page. She brought her class to the lab where we introduced them to the Transformation tools in the Sketchpad and then had them "turn to" the Math Forum home page where they continued their lesson with the activities presented there. This is just a beginning. Here are some other features of GSP.
1. Transformations (slides, flips, and turns; shrinking and enlarging).
2. Scripts to record work for analysis and assessment of student work.
3. Coordinate axes.
4. 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!
5. Sketches and scripts that come with every GSP supplement.

## References

Albrecht, Marsha et al. Discovering Geometry with the Geometer's Sketchpad. Key Curriculum Press. Berkeley, California, 1997.

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.

## Web Sites

Key Curriculum Press
http://www.keypress.com

The Math Forum
http://mathforum.org