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THE GEOMETER'S SKETCHPAD The Geometer's Sketchpad Version 5.06. The Geometer’s Sketchpad ® is the world’s leading software for teaching mathematics. Sketchpad® gives students at all levels—from third grade through college—a tangible, visual way to learn mathematics that increases STANDARD ACTIVITY TITLE 346 Appendix Exploring Geometry with The Geometer’s Sketchpad © 2012 Key Curriculum Press Standard Activity Title Understand congruence in terms of rigid motions CHAPTER 1: LINES AND ANGLES Polygon Angle Measure Sums 190 Constructing Regular Polygons 193 Constructing Templates for the Platonic Solids 196 Chapter 6: Circles Introducing Circles 201 INSTALLATION GUIDE AND LICENSE AGREEMENT The Geometer’s Sketchpad, Version 5 Installation Guide and License Agreement 3 Key Curriculum Press Software License Agreement DEFINITIONS: • Key means Key Curriculum Press, Inc., 1150 65th Street, Emeryville, CA 94608, USA. • Software means The Geometer’s Sketchpad Version 5 computer program that you received COMPARISON OF SKETCHPAD AND GEOGEBRA Comparison of Sketchpad and GeoGebra page 5 Sketchpad: Every object, even the reflected image point I', can be dragged. GeoGebra: Segments d, f, g, and h, midpoints G and H, the polygon, and reflected image I'

cannot be dragged.

ENDLESS POSSIBILITIES WITH THE GEOMETER’S SKETCHPAD In a third-grade classroom in Philadelphia, students construct fractions along a number line. Down the hall, eighth graders build a visual proof of the Pythagorean Theorem.

Contents Exploring Number and Operations in Grades 3–5 with The Geometer’s Sketchpad Contents iii © 2012 Key Curriculum Press Downloading Sketchpad Documents v EXPLORCONICS FM 1ST12 Exploring Conic Sections with The Geometer’s Sketchpad Contents • iii © 2012 Key Curriculum Press Contents Downloading Sketchpad Documents .. iv Introduction THE RESEARCH BASE FOR THE GEOMETER S SKETCHPAD —EXECUTIVE Copyright © 2009 Key Curriculum Press. All rights reserved. 2 Executive Summary The Geometer’s Sketchpad® is a proven Dynamic Geometry® tool for elementary THE IMPACT OF USING GEOMETERS’ SKETCHPAD ON MALAYSIAN 96 The Impact of Using Geometers’ Sketchpad visualization abilities, and ineffective instruction (Cangelosi, 1996; Noraini, 2006). Poor reasoning skills are also another area of concern among THE GEOMETER'S SKETCHPAD The Geometer's Sketchpad Version 5.06. The Geometer’s Sketchpad ® is the world’s leading software for teaching mathematics. Sketchpad® gives students at all levels—from third grade through college—a tangible, visual way to learn mathematics that increases STANDARD ACTIVITY TITLE 346 Appendix Exploring Geometry with The Geometer’s Sketchpad © 2012 Key Curriculum Press Standard Activity Title Understand congruence in terms of rigid motions CHAPTER 1: LINES AND ANGLES Polygon Angle Measure Sums 190 Constructing Regular Polygons 193 Constructing Templates for the Platonic Solids 196 Chapter 6: Circles Introducing Circles 201 INSTALLATION GUIDE AND LICENSE AGREEMENT The Geometer’s Sketchpad, Version 5 Installation Guide and License Agreement 3 Key Curriculum Press Software License Agreement DEFINITIONS: • Key means Key Curriculum Press, Inc., 1150 65th Street, Emeryville, CA 94608, USA. • Software means The Geometer’s Sketchpad Version 5 computer program that you received COMPARISON OF SKETCHPAD AND GEOGEBRA Comparison of Sketchpad and GeoGebra page 5 Sketchpad: Every object, even the reflected image point I', can be dragged. GeoGebra: Segments d, f, g, and h, midpoints G and H, the polygon, and reflected image I'

cannot be dragged.

Contents Exploring Geometry and Measurement in Grades 3–5 with The Geometer’s Sketchpad Contents iii © 2012 Key Curriculum Press Downloading Sketchpad Documents v PHILOSOPHY AND INSTRUCTIONAL OVERVIEW ConTEnTs Shape Makers l vii QuADRILATERAL EXPLoRATIon 6 Relationships Between Shape Makers 80 Sessions 1 and 2 81 Activity: Relating Shape Makers 81 Sessions 3 and 4

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Contents Exploring Algebra 1 with The Geometer’s Sketchpad Contents iii © 2012 Key Curriculum Press Downloading Sketchpad Documents v Sketchpad Resources vi VERSION 5.04 RELEASE NOTES Version 5.04 Release Notes New Functionality: Sketchpad supports SMART Board color choices for ink by means of pens or buttons. The Windows version also supports multi-touch on newer SMART Boards that incorporate the required hardware. VERSION 4.05 RELEASE NOTES Version 4.05 Release Notes Sketchpad Release 4.05 contains no new functionality, but addresses compatibility issues and bugs with the software and JavaSketchpad. Changes: VERSION 4.06 RELEASE NOTES Version 4.06 Release Notes New Functionality: (Windows and Mac OS X) The new anti-aliased graphics introduced in GSP 4.04 are now enabled, rather than disabled, by default. SKETCHPAD LICENSE AGREEMENT

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THE GEOMETER'S SKETCHPAD VERSION 5.06 _The Geometer’s Sketchpad_® is the world’s leading software for teaching mathematics. Sketchpad® gives students at all levels—from third grade through college—a tangible, visual way to learn mathematics that increases their engagement, understanding, and achievement. Make math more meaningful and memorable using Sketchpad. Review License to Download Review License to Download

* Windows Installer

* Mac Installer

* Combined Mac and Windows Installer * Mac PowerPC (Version 5.04) * Network Install Instructions

* Windows Installer

* Mac Installer

Construct isosceles and equilateral triangles, gain familiarity with _Sketchpad's_ tools, and learn how to change the size and color of

objects.

Go to Constructing Triangles Construct right triangles and parallelograms, create a multi-page sketch, use the drag test, and analyze constructions in a sketch. Go to Properties of Shapes Measure lengths and angles and use the measurements in calculations, create angle marks, and communicate results using Hot Text® Go to Angles in a Triangle Collect measurement data in a table and plot them in a graph, adjust the scale of axes, measure slope, and plot a function to model data. Go to Perimeter and Area Plot a functional relationship and dynamically vary the input value, trace a point, create and animate a parameter, and graph a family of

functions.

Go to Dynamic Algebra Create a moving kaleidoscope by constructing concentric circles, attaching a picture and rotating it, and building an animation button. Go to Rotations and Symmetry Translate a picture by a marked vector, reflect a picture over a line, and then create and apply a custom transformation. Go to Translations and Reflections Construct a square using several different methods, create a Hide/Show button, and create and save a custom tool. Go to Constructing Squares Work with custom tools, construct an illustration of the Pythagorean theorem and then iterate the construction to create a fractal. Go to Pythagorean Theorem Rotate a picture by a marked angle, dilate the picture by a marked scale factor, and then create and apply a custom transformation. Go to Twist and Shrink Define a coordinate system based on a unit circle, use a trigonometric axis, and construct an animation that traces out a sine wave. Go to Sine Wave Tracer Construct a secant line, trace the path of a moving point, create a movement button, and compute and plot the derivative of a function. Go to Tracing the Slope Function Grades 3-5 Grades 6-8 Algebra Geometry Precalculus Calculus JUMP ALONG: FACTOR FAMILIES ON THE NUMBER LINE Students investigate factors from a visual perspective as they find all of the ways a rabbit can take equal-sized jumps to reach a target number. Students write multiplication sentences to represent the rabbit's jumps, and uncover the commutative property of

multiplication.

Download Activity Files (.zip) BUGS IN GROUPS: DIVIDING INTO GROUPS OF EQUAL SIZE Students explore and connect part-whole relationships in multiplication and division as they order a collection of bugs into groups of equal size, and consider the number of groups and the number of bugs left over. Concepts of factors, remainders, and the commutative property of multiplication all arise as students work with

the model.

Download Activity Files (.zip) DIVIDING AND SUBDIVIDING: FRACTIONS ON THE NUMBER LINE Students locate fractions that are less than 1 along a number line using virtual tools and a strategy of dividing and subdividing. The challenge is to find more than one way to locate a given fraction. Download Activity Files (.zip) PLACE-VALUE COUNTER: GET TO THE TARGET Students use their knowledge of place value and their intuitive notions of rounding to operate an interactive numerical counter that works like an odometer. Using buttons that increase or decrease the value of the counter by one, ten, one hundred, one thousand, or more, students work to reach given target values in as few button presses as

possible.

Download Activity Files (.zip) ZOOMING DECIMALS: PRECISION AND PLACE VALUE Students zoom in on a number line to reveal scales calibrated to tenths, hundredths, thousandths, and ten-thousandths. They reason about decimals and place value as they name with increasing precision the location of a point on the number line. Download Activity Files (.zip) RIGHT OR LEFT: ADDING AND SUBTRACTING INTEGERS Students add and subtract integers using animations on a number line and understand that subtracting an integer is the same as adding its

opposite.

Download Activity Files (.zip) CROP AND REFLECT: MIRROR SYMMETRY Students create mirror symmetry lines for a variety of pictures, including the Eiffel Tower, a butterfly, letters of the alphabet, and a human face. Optionally, they can import a picture of their own face into _Sketchpad_ and explore its mirror symmetry. Download Activity Files (.zip) MONDRIAN IN MOTION: PARALLEL AND PERPENDICULAR LINES Students construct parallel and perpendicular lines to create virtual artwork in the style of the painter Piet Mondrian. They also explore the properties of two parallelograms, the square and rectangle. As students manipulate their constructions, they change the orientation of the lines and note that parallel or perpendicular lines need not be oriented vertically and horizontally. Download Activity Files (.zip) MAKING A KALEIDOSCOPE: EXPLORING ROTATIONS Students create virtual kaleidoscopes by rotating quadrilaterals and then animating them. Students learn that rotated figures keep their size and shape; only their orientation changes. Students make a variety of kaleidoscopes, each with a different number of quadrilaterals and amount of rotation. Download Activity Files (.zip) CIRCLES AND SQUARES: TWO UNKNOWNS Students use algebraic thinking as they work with addition statements in which the addends are represented by two symbols. By analyzing the information from two or more statements, students deduce the numerical values of the symbols. Download Activity Files (.zip) RIGHT OR LEFT: ADDING AND SUBTRACTING INTEGERS Students add and subtract integers using animations on a number line, and understand that subtracting an integer is the same as adding its

opposite.

Download Activity Files (.zip) MAKING MEANS: DATA DISTRIBUTION AND AVERAGES Students drag data points on a number line and observe how the mean of their values changes. Students can add, remove, and drag the data points to different locations to create examples of data sets with a

given mean.

Download Activity Files (.zip) MEAN MEETS THE MEDIAN: MEASURES OF CENTRAL TENDENCY Students drag a fixed number of data points to different locations on a number line and observe the effects on the median and the mean. Students can explore the difference in the median that occurs between having an even and an odd number of data points. The emphasis of the activity is on understanding how the mean and median behave depending both on how the data is distributed and on the number of data points. Download Activity Files (.zip) MAKING A KALEIDOSCOPE: EXPLORING ROTATIONS Students create virtual kaleidoscopes by rotating quadrilaterals and then animating them. Students learn that rotated figures keep their size and shape; only their orientation changes. Students make a variety of kaleidoscopes, each with a different number of quadrilaterals and amount of rotation. Download Activity Files (.zip) MELLOW YELLOW: INTERPRETING GRAPHS Students interpret linear piecewise time-distance graphs that represent different stories about a character, Mellow Yellow. They decide whether a given graph corresponds to the motions (walking fast, walking slow, stopping, or going backward) described in the story. Students then create stories based on given graphs and create graphs based on given stories. Download Activity Files (.zip) HIKERS: SOLVING THROUGH MULTIPLE REPRESENTATIONS Students use tables, graphs, and equations to represent and solve a real-world problem about two hikers walking at different speeds in opposite directions along the same trail. Download Activity Files (.zip) QUADRILATERAL PRETENDERS: CLASSIFYING QUADRILATERALS Students drag the edges and vertices of various _Sketchpad_ quadrilaterals to discover which are constructed to have specific characteristics. As they make distinctions on the basis of these characteristics, they deepen their understanding of the definitions of various quadrilaterals, their properties, and the relationships among

them.

balance.

Download Activity Files (.zip) TILING IN A FRAME: MULTIPLYING POLYNOMIALS Students use _Sketchpad_ algebra tiles to multiply polynomials. Using the polynomial factors as dimensions, they build rectangles out of tiles. The area of the completed rectangle represents the product. Download Activity Files (.zip) POINTS LINING UP IN THE PLANE Students are informally and experientially introduced to the relationship between descriptions of coordinate patterns and graphs in the Cartesian plane. Too often students don't really understand the connection between an equation and its graph. This activity fosters the understanding that graphs depict the set of points whose coordinates satisfy an equation. Download Activity Files (.zip)

THE SLOPE GAME

Students acquire an intuitive feel for slope as they construct and play a game in which one player rearranges lines on the screen and the other player tries to match each line with its slope measurement. The game can also be modified to be played by just one person. Download Activity Files (.zip) MELLOW YELLOW: INTERPRETING GRAPHS Students interpret linear piecewise time-distance graphs that represent different stories about a character, Mellow Yellow. They decide whether a given graph corresponds to the motions (walking fast, walking slow, stopping, or going backward) described in the story. Students then create stories based on given graphs and create graphs based on given stories. Download Activity Files (.zip) HIKERS: SOLVING THROUGH MULTIPLE REPRESENTATIONS Students use tables, graphs, and equations to represent and solve a real-world problem about two hikers walking at different speeds in opposite directions along the same trail. Download Activity Files (.zip) BALANCING WITH BALLOONS: SOLVING EQUATIONS WITH NEGATIVES Students use a _Sketchpad_ pan balance model to solve a sequence of equations with positive numbers and variables (represented by weights), and negative numbers and variables (represented by balloons). As the problems increase in difficulty, students go from manipulating the pan balance to solving equations independent of the

balance.

Download Activity Files (.zip) INTRODUCING DYNAGRAPHS Students explore dynagraphs, an alternative to Cartesian graphs, to develop a feel for various types of functional relationships. Download Activity Files (.zip) PARABOLAS IN FACTORED FORM Students plot a quadratic function in factored form, investigate the relationship between the equation and its graph, and use their observations to create functions from various descriptions of their

graphs.

Download Activity Files (.zip) QUADRATIC QUANDARY: FIND THE EQUATION OF A PARABOLA Students convert quadratic functions between standard, vertex, and factored forms. They check their results by changing parameters and comparing the graphs. Download Activity Files (.zip) A RECTANGLE WITH MAXIMUM AREA Students explore areas of rectangles whose perimeter is fixed. They make a conjecture about what type of rectangle has the most area for a given perimeter and check their conjecture by plotting the side length and area of the rectangle on the coordinate plane. Download Activity Files (.zip) TESSELLATIONS THAT USE ROTATIONS Students construct an irregularly shaped tile based on an equilateral triangle, and then use rotation to tessellate the plane with it. Download Activity Files (.zip) MEDIANS IN A TRIANGLE Students construct a triangle and its medians. They observe the concurrence of the medians, measure distances to observe how the centroid divides each median, and make a custom tool for constructing the centroid of a given triangle. Download Activity Files (.zip) QUADRILATERAL PRETENDERS: CLASSIFYING QUADRILATERALS Students drag the edges and vertices of various _Sketchpad_ quadrilaterals to discover which are constructed to have specific characteristics. As they make distinctions on the basis of these characteristics, they deepen their understanding of the definitions of various quadrilaterals, their properties, and the relationships among

them.

Download Activity Files (.zip) MEET THE PARALLELOGRAM: PROPERTIES OF PARALLELOGRAMS Students construct a parallelogram, measure the side lengths and angles, and observe that opposite sides are congruent, opposite angles are congruent, and consecutive angles are supplementary. Then they construct the diagonals, measure the distances from the vertices to the point of intersection, and discover that the diagonals bisect each

other.

Download Activity Files (.zip) MIDPOINT QUADRILATERALS Students connect the midpoints of a quadrilateral to construct a midpoint quadrilateral. They discover that the midpoint quadrilateral is a parallelogram and prove this conjecture. Download Activity Files (.zip) EXTERIOR ANGLES IN A POLYGON Students construct a convex polygon and make a conjecture about the sum of the measures of its exterior angles. They dilate the polygon to approximately a single point to create a visual proof by dilation that the sum of the measures of the exterior angles of a convex polygon is what they conjectured. Download Activity Files (.zip)

CHORDS IN A CIRCLE

Students explore the properties of chords in a circle. They construct a chord and its perpendicular bisector and discover a relationship between the chord's length and its distance to the center of the circle. Then they investigate and write a conjecture about congruent

chords in a circle.

Download Activity Files (.zip) PARALLEL PAIRS: PARALLELOGRAM AND TRIANGLE AREA Students explore the relationship between the areas of parallelograms and triangles using a process called shearing. Students discover that shearing does not affect the area, but changing the lengths of the height and base does. Based on their observations, students write formulas for the area of a parallelogram and the area of a triangle. Download Activity Files (.zip) PYRAMID DISSECTION: SURFACE AREA Students find the surface area of a regular pyramid (a pyramid with a regular polygon base) using a net that appears along a three-dimensional view of the pyramid. They ensure the generality of their results by changing the dimensions and the number of faces of the base. By increasing the number of faces, students extend their results to the surface area of a cone, giving them an informal opportunity to think about limits. Download Activity Files (.zip) VISUAL DEMONSTRATION OF THE PYTHAGOREAN THEOREM Students investigate a visual demonstration of the Pythagorean theorem based on Euclid's proof. They use shearing to modify the squares on the sides of a right triangle to create congruent shapes without changing the areas of the original squares, and then explain why these shapes demonstrate the Pythagorean theorem. Download Activity Files (.zip)

A SINE WAVE TRACER

Students use points on perpendiculars, parallels, and a circle to construct an animation that traces out a sine wave. They explore the sine wave by dragging various points in their construction. Finally, students explore the relationship between a unit circle, its circumference, and the trace of a sine wave. Download Activity Files (.zip) UNIT CIRCLE AND RIGHT TRIANGLE FUNCTIONS Students explore the relationships between the unit circle definitions of trigonometric functions and the right triangle definitions. They then combine the two models and examine the similarities and differences that emerge. Download Activity Files (.zip) PATTY PAPER PARABOLAS Students fold a piece of paper so that the creases outline the shape of a parabola. They compare their parabolas with those of other students to see how the shape depends on the positions of the focus and directrix. Students then do the same construction with _Sketchpad_ and explore how moving the focus or directrix changes the shape of the folds. Finally students are challenged to construct the parabola itself (not just the folds) and to prove that their construction matches the geometric locus definition of a parabola. Download Activity Files (.zip)

ANALYTIC CONICS

Students explore conic sections analytically. They change the parameters in the equation of a conic in standard position and observe changes in the graph, and then explore the general equation of a conic

section.

Download Activity Files (.zip) INTRODUCTION TO VECTORS: WALKING REX Students learn two ways to describe vectors, how to convert between the two descriptions, and how to move vectors around the plane to explore how vectors are independent of specific positions. Download Activity Files (.zip) CARTESIAN GRAPHS AND POLAR GRAPHS Students compare rectangular graphs and polar graphs for functions in the form _y_ = _a_ sin(_bx_) and _r_ = _a_ sin(_bθ_). They find connections between the two types of graphing when they analyze how the period and amplitude of a Cartesian graph correlate with features of the corresponding polar graph. They make predictions as to how changing _a_ and _b_ will affect the polar graph and then check their

predictions.

Download Activity Files (.zip)

INSTANTANEOUS RATE

Students learn about instantaneous rates and derivatives by investigating the rate of change of a door’s angle as it closes. They look at the graph of the angle as a function of time, calculate the average rate of change between two points on the graph, make the time difference between the points smaller and smaller, and discover that the average rate (the slope of the secant) approaches a limiting value: the derivative. Download Activity Files (.zip) PLOTTING THE DERIVATIVE Students approximate the derivative by constructing a secant line between two points on the graph of a function and graphing the value of the secant's slope. They improve the approximation by moving the points that determine the secant closer together. They edit the original function and practice predicting the shape of the derivative graph from the shape of the function graph. Download Activity Files (.zip) ONE TYPE OF INTEGRAL Students explore the concept of definite integral by using a velocity graph to estimate the distance a car travels. They count grid squares beneath the graph, and then make the squares smaller to improve the accuracy of their approximation. They use the same method to approximate definite integrals for several different functions. Download Activity Files (.zip)

TAYLOR SERIES

Students approximate the sine function using a Taylor series. They construct one point corresponding to the first partial sum and iterate to find points corresponding to the subsequent terms. Their final result is a graph drawn to any desired depth. Download Activity Files (.zip) A GEOMETRIC APPROACH TO _EIΠ_ Students explore _e_iπ__. They first explore the limit of (1+_x_/_n_)_n_ for large values of _n_ to find that it approaches _e_x__. They then replace _x_ with _iπ_ and use a geometric approach to multiplication on the complex plane to find the limit and evaluate

_e_iπ__.

Download Activity Files (.zip) BARNSLEY'S FRACTAL FERN Students create several functions that transform a point and then iterate the transformation, choosing randomly among the functions, to create a fractal fern and other fractal shapes. Download Activity Files (.zip) The Geometer's Sketchpad

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