If you (or students) are not familiar with Geometer’s Sketchpad, this is the book for you. This book is designed to introduce middle school, high school students, and teachers to the basic tools of Geometer’s Sketchpad. This handbook is a teaching tool. It is intended to supplement, not replace, the school’s geometry curriculum.

During the first or second day of your work sessions, students should carefully read part one, establishing a good study program, and then move on to complete the free play tour. When possible, illustrations are provided to help visualize the various Sketchpad tools and to enhance written instructions. The tours and investigations will encourage you to work together in pairs and small groups to build on previous knowledge and apply knowledge to new information. Students learn well working in groups. When students get stuck on something, they say so, and other students help. This kind of learning is efficient, natural, and well- known.

The Geometer’s Sketchpad, or Sketchpad, is a construction tablet on which one draws models of geometric shapes, transforms them, colors them, measures them, and animates them. The models invite students to explore, represent, solve problems, construct, discuss, investigate, describe, and predict. Implicit to these functions is the ability to build mathematical models of simple and complex ideas. The Sketchpad allows students to engage in “doing mathematics,” which is emphasized in the National Council of Teachers of Mathematics (NCTM) Standards and CORE Mathematics Curriculum.

A transformation is a movement of a geometric figure to a new position. One kind of transformation is a reflection.

Reflections using Geometer’s Sketchpad

A key word in connection with reflection is mirror. In order to reflect a selected object in a line, the line must first be selected and marked using Transform | Mark Mirror – refer to Stickman 1 below.

Draw Mr. Stickman on the left of the segment. Reflect Mr. Stickman in a line. Then reflect that reflection in a second line. Now vary the lines. Save your work – refer to Stickman 2 below.

A reflection is sometimes called a flip. The figure is flipped across or over or in a certain line. Observe that triangle ABC is reflected across the y-axis to make ABC. The corresponding points of the two triangles have the same y-coordinates. Their x-coordinates are opposites – refer to Reflection 1 below.

The figure that results from a reflection is called the image of the first figure under a reflection. RSTU is the image of RSTU under a reflection in the line with equation y = 2. Corresponding points of the two figures are the same distance from the line of reflection, but on opposite sides – refer to Reflection 2 below.

Reflections do not change the shape or size of a figure, so the reflection image is congruent ( is the symbol used for congruency) to the original figure (the pre-image). In the examples above, ABC ABC, and RSTU RSTU.

rx = 0(ABC) = ABC. Reflection over the line x = 0 (the y-axis).

ry = 0(RSTU) = RSTU Reflection over the line y = 0 (the x-axis).

Reflection over an axis changes coordinates of points as follows:

x-axis: ry = 0(x,y) = (x,-y)

y-axis: rx = 0(x,y) = (-x,y)

1) The diagram shows ABC. If ABC is reflected across the x-axis to produce ABC, what will be the coordinates of Point A? First use the appropriate reflection notation formula to solve the problem, then check your computations by using Sketchpad to reflect ABC across the x-axis – refer to reflection 5 below.

2) The diagram below shows parallelogram P and its image under a certain reflection – refer to reflection 6 below.

Which describes the reflection?

A rx = 1(P) = P

C ry = 2(P) = P

B rx = 2(P) = P

D ry = 4(P) = P

3) The diagram below shows DEF - – refer to reflection 7below.

Without using Sketchpad Reflect command, draw D’E’F’, the result of the reflection described by ry = -1(DEF).

4) Look at triangle PQR on the coordinate grid below. Refer to Rotation 8 below.

PQR will be rotated 180 about the origin to make PQR. What will be the coordinates of the image vertices of the triangle? Remember to first use the rotation notation to confirm your answers by using Sketchpad to perform the indicated rotation.

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