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.
|How Many Degrees in a Quadrilateral?
(Grades 4 - )
|Lines, Rays, and Segments
|Sum of the Interior Angles of a Polygon
(Grades 4 - )
|Acute, Obtuse, and Straight Angles
(Grades 2 - )
|Guess the Angle
(Grades 2 - )
|Similar Triangles I
|Area and Perimeter
(Grades 3 - )
|Similar Triangles II
|How Many Degrees in a Triangle?
(Grades 4 - )
(Grades 4 - )
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|
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!
Activity: How many degrees in a quadrilateral? (Grades 4- )
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:
|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|
Can you figure out a rule to predict the number of interior degrees in any polygon? What is it?
Activity: Student-made Quiz (grade 7)
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.
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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