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Applied Project: Connect Geometry Formula

         Thitiya Pathakkhinang

     University of Nevada, Las Vegas
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       The Geometers' Sketchpad (GSP) is an award-winning, dynamic construction and

exploration tool that enables students to explore and understand mathematics in ways that are

simply not possible with traditional tools or with other mathematics software programs (The

Sketchpad Story, 2008). The key advantage of the GSP is that students can construct an object

and then explore its mathematical properties by dragging the object with the mouse. All

mathematical relationships are preserved, allowing students to examine an entire set of similar

cases in a matter of seconds, leading them by natural course to generalizations (Bennett, 2006).

Plus, GSP is designed with an intuitive interface for ease of use, so that students will be focused

on             the              activity,               not            the              navigation.
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TOPIC: Review Geometry Formula

LEVEL: Geometry: Grade 8 – 12

CLASS TIME: One period (55 minutes)

TEACHING MATERIALS: Geometer sketchpad Software


       From algebra, students know how to write equations to represent lines and parabolas.

After finish this lesson students will be able to:

       1. Make a connection between distance formula and circle formula when the center of a

           circle is on coordinate (h, k) or an origin.

       2. Make conjectures from prior knowledge that the Pythagorean Theorem can be used to

           develop equations for circles.

       3. Students will be able to restatement of the Pythagorean Theorem in coordinate

           geometry is the distance formula, which tells how to find the distance between two

           points whose coordinates are known.


       Use of The Geometer's Sketchpad as an investigative mathematical tool plays an integral

part in the activities by providing informal, concrete experiences for students as they explore

abstract geometric concepts. The lesson addresses two of the topics which the NCTM Standards

suggest should receive increased attention in geometry classrooms:

       1. Use geometric models to gain insights into, and answer questions in, other areas of

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        2. Computer-based explorations of figures.

        3. Establish the validity of geometric conjectures using deduction, prove theorems, and

            critique arguments made by others


        1. Construct a segment with both of the end points is the origin.

        2. Measure the length of the segment by using its formula then check the answer with

            the program.

    Guide students by asking what will happen if we construct a circle by using one of the end

points of a segment as a center of the circle? How do you see the segment inside the circle?

        3. Have students calculate an equation of the circle by hand and shear with their

            classmate to see if they have the same answer if not find down why? They then check

            the answer with the program.

    Students will then go on to investigate of how the equation of the circle will change if they

move the center to an origin (0, 0).

        4. Have students construct a right triangle inside the circle by using the center of the

            circle as its vertex.

    Ask students that how does an equation of a circle when its center is an origin relate to

Pythagorean Theorem formula.

        Bring students to the attention that since they could conclude that a distance formula;

d² = (x-x1 ) + (y-y1 )², relate to a circle by imagine that the point (x 1 , y1 ) is the center of a circle,

and all the points on the circle are a distance d away from it. If rename the center (h, k) and

rename the distance r, they can see that circle is all the points (x, y) that are this distance away

from (h, k).
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    If the center is at the point (0, 0), the points satisfying the equation are all the distance r from

the origin – that is, they all lie on a circle centered at the origin and having radius r.

          The equation of that circle is x 2 + y2 = r2 , which is the Pythagorean Theorem. Have

students use a compass to draw a circle centered at the origin with a certain radius and then ask

him or her to identify a few points on that circle and substitute them into the equation. At the

end of the lesson challenge students to investigate about how other math formula relate to each



          I circulate during the lesson to answer questions and to make "just the right comments" to

force students to examine their work. i.e. "How does this equation compare with the distance

formula?" or " Why is the general equation of a circle is (x – h) ² + (y – k) ² = r²?" or "What

numbers do h, k, and r represent?" This interaction provides an opportunity to observe the

criteria students use as they investigate.

This time also provides me with an opportunity to observe:

            ability to follow directions

            ability to work in a group setting

            spatial reasoning abilities

            problem-solving and data analysis strategies

          This lesson gives me some insight as to which students have a workable understanding of

the concept of construct vs. a drawing. The difficulty I had five minutes before the lesson was

we could not used a computer lap like I plan to, so I used LCD projector to guide my students

through the lesson and have they investigate by hand. However, students had fun discovering

through the lesson. Next time I have to make sure that the computer lap is really available. As an
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educator I learned how Geometer sketchpad program could help students move seque ntially

through the geometric cognitive levels as defined by the van Hiele paradigm.
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       Larson, R., Boswell, L., & Stiff, L. (2005). McDougal Littell GEOMETRY. Evanston, IL:

McDougal Littell.

       The Sketchpad Story. (2008). Retrieved April 23-26, 2008, from The Geometer's

Sketchpad Resource Center Web site:


       Bennett, D. (2006). Exploring Geometry with the Geometer's Sketchpad (1st ed.).

Emeryville, CA: Key Curriculum Press.

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