The Geometer S Sketchpad

Table of Contents

Introduction to Project2

Parallel & Perpendicular Lines Activity3
Direct Variation Activity8

Quadratic Function Activity13

Absolute Value Functions Activity19

Rectangle with Maximum Area Activity25
Equations of Circles Activity30

Vertical Angles Activity34

Triangle Inequality Activity38
Properties of Reflections Activity42
Isosceles Triangle Activity48
Exterior Angles of a Triangle Activity53

Central Angles & Arc Measures Activity58

Diagonals of a Parallelogram Activity62
Diagonals of a Rectangle Activity67
Diagonals of a Rhombus Activity71
Midpoint Quadrilaterals Activity76
Sum of the Interior Angles of a Polygon Activity80
Right Triangle Trigonometric Ratios85

The Geometer’s Sketchpad

Geometry & Algebra Activities

I choose to do my project on creating activities that can be used in my classroom with the Geometer’s Sketchpad. Prior to this year I had used Sketchpad a few times in Geometry class. The students seemed to enjoy those activities and learned a great deal from them so I decided to work this past year on creating more good activities to use in Geometry as well as a few good activities that can be used in teaching Algebra. It was a great learning experience for me as well. I have learned a great deal about the capabilities of Sketchpad that I did not take the time to learn about before.

The Geometer’s Sketchpad was released in 1991 and has a new upgrade being releases this August. Sketchpad was designed so that students could have a quick method of constructing figures and manipulating them to discover properties of those figures. In Sketchpad students can construct objects and explore properties by dragging the vertices of the object with the mouse. Students can make conjectures about the objects before attempting a proof.

I have split my project up into two main parts, Algebra activities and the Geometry activities. The Algebra activities include activities that encourage students to form conjectures about parallel & perpendicular lines, direct variation, quadratic functions, absolute value functions, and equations of circles. The Geometry activities include activities that encourage students to form conjectures about vertical angles, triangle inequality, reflections, isosceles triangles, the measure of an exterior angle of a triangle, central angles and arc measures, parallelogram, rectangle, rhombus, midpoint quadrilateral, sum of the angles in a polygon, and right triangle trigonometric ratios. After each activity is a sample sketch with answers to questions posed to the students in the activity.

Parallel & Perpendicular Lines

In this activity students discover that the slopes of parallel lines are the same and the product of slopes of 2 perpendicular lines is a -1. This lesson usually takes about 40 – 45 minutes especially if the students have never used Sketchpad before.

The students use Sketchpad first to construct parallel lines and calculate the slope of those lines. Then they move the lines by dragging one point and record the data of the slopes to come to a conclusion about the slopes of parallel lines. This meets the NCTM

Algebra Standard (Understanding patterns and Analyzing change) and the NCTM Measurement Standard (Understanding measurable attributes of objects and processes of measurement).

The students then use Sketchpad to construct perpendicular lines and calculate the slope of each. The students record the data they obtain by dragging those lines as well to come to a conclusion about the slopes of perpendicular lines. This meets the NCTM Communication Standard (Organize their mathematical thinking through communication).

The students work together in pairs so that they can discuss their conclusions. This meets the NCTM Communication Standard(Analyze and evaluate the mathematical thinking and strategies of others).

Parallel & Perpendicular Lines

Before you begin your sketch go to the Edit Menu and select Preferences then under the Text tab and make sure For All New Points is checked then click on OK.

Step 1Using the Straightedge Tool (Left click and hold to select the straightedge that is a line), construct a line in the white space.

Step 2Using the Point Tool create a point in the white space that is not on the line you just created.

Step 3Using the Selection Arrow Tool select the point and the line and from the Construct Menu choose Parallel Line.

Step 4From the Graph Menu select Define Coordinate System. (The parallel lines you constructed will now appear on a coordinate plane.) Select both lines and from the Display Menu choose Line Width then select Thick.

Step 5Next, from the Measure Menu choose Slope.

Your sketch should look similar to the one below.

Step 6Record your initial measurements in the table below labeled Parallel Lines.

Step 7Now click on one of the points on your initial line you constructed and drag so that the line is steeper. Record your measurements in the table.

Step 8If your line had a positive slope initially drag the other point on the initial line so that the slope is now negative and record your data.

Step 9Move the other line that was constructed second and record the measurements.

Step 10Print your sketch

Parallel Lines

Using the data you have collected can you make a generalization about parallel lines?

Step 11From the File Menu select New Sketch.

Step 12Using the Straightedge Tool again construct an oblique line in the white space.

Step 13Using the Point Tool construct a point in the white space not on the line you just constructed.

Step 14Using the Selection Arrow Tool select the point and the line and from the Construct Menu select Perpendicular Line.

Step 15Using the Selection Arrow Tool select both lines and from the Measure Menu choose Slope.

Step 16From the Measure Menu select Calculate and when the calculator appears click on the slope of one line then click on the *sign on the calculator then click on the slope of the other line then click OK.

Step 17Record your initial measurements in the table below labeled Perpendicular Lines.

Step 18Again drag point A and record your measurements.

Step 19Drag point B and record your measurements.

Step 20Drag the point on line j and record your measurements.

Perpendicular Lines

Using your data what generalizations can you make about perpendicular lines?

Activity Objectives

In this activity students should discover that the slopes of lines parallel are the same and that the product of the slopes of 2 lines that are perpendicular is a -1.

Students should be familiar with calculating slope and graphing lines.

Activity Length

This lesson may take longer than usual. Students may not be familiar with Sketchpad so you may need to spend as much as 40 – 45 minutes on this activity. I would suggest pairing students in teams of two to work on this activity.

NCTM Standards

Algebra Standard

• Understand patterns, relations, and functions.
• Analyze change in various contexts.

Measurement Standard

• Understand measurable attributes of objects and the units, systems, and processes of measurement.

Communication Standard

• Organize and consolidate their mathematical thinking through communication.
• Analyze and evaluate the mathematical thinking and strategies of others.

Direct Variation

This activity is designed so that students will discover a direct variation function. In this construction the area of the parallelogram is calculated while the height of the parallelogram varies and the base remains constant. This activity came from Exploring Algebra with The Geometer’s Sketchpad written by Steven Chanan, Eric Bergofsky, and Dan Bennett. This activity takes approximately 45 minutes. It may take longer depending on how familiar the students are with analyzing graphs of functions.

Students will graph the height and area on a coordinate plane. Students will use the graph to describe how the area changes as the height changes. This meets the NCTM Algebra Standard (Understands patterns & functions, and Represents mathematical situations using algebraic symbols).

Students have the Geometer’s Sketchpad calculate the measurement required (base, height, and area of the parallelogram). This meets the NCTM Measurement Standard (Understands the processes of measurement, and Uses appropriate tools to find measurements).

I pair students up so that they must collaborate with one another in forming conjectures and answering question about the activity. They must discuss and write about the mathematics they have discovered through guided questions. This meets the NCTM Communication Standard (Communicate their mathematical thinking clearly to peers and others, Analyze and evaluate the mathematical thinking of others, and Use the language of mathematics to express ideas clearly).

Direct Variation

Make a conjecture about what will happen to the area of a parallelogram if you keep the length of the base constant while varying the height?

Before you begin this sketch go to the Edit Menu and select Preferences then under the Text Tab make sure that For All New Points is checked then click on OK.

Step 1Using the Straightedge Tool construct a segment AB then AC as shown below.

Step 2Using the Selection Arrow Tool select C then segment AB and under the Construct Menu select Parallel Line.

Step 3Click in the white space so that nothing is selected then click on B and segment AC and under the Construct Menu select Parallel Line. Click in the white space once more.

Step 4Select the two lines you constructed in steps 2 and 3 and then press Ctrl + I to construct the intersection D. Click in the white space.

Step 5Select the two lines and press Ctrl + H to hide the lines.

Step 6Select C then D and press Ctrl + L to construct a line segment.

Step 7Click in the white space so that nothing is selected. Select D then B and press Ctrl + L to construct a line segment. Click in the white space once more.

Step 8Select A, B, D, then C and then press Ctrl + P to construct the parallelogram’s interior.

Step 9Select the interior of the parallelogram and using the Measure Menu choose Area.

Step 10Click in the white space so that nothing is selected, then click on segment AB and from the Measure Menu choose Length.

Step 11Using the Text Tool double click on the measurement for AB and change the label to b (for base) and click on OK.

Step 12Using the Selection Arrow Tool click in the white space so that nothing is selected, then select C and segment AB and from the Measure Menu choose Distance.

Step 13Using the Text Tool double click on the distance measurement and label it h (for height) and then click on OK.

Step 14Using the Selection Arrow Tool click in the white space so that nothing is selected then select the measurement labeled h and the area (in that order) and from the Graph Menu choose Plot as (x,y).

Step 15You should have point E plotted on the coordinate plane. Select point E and press Ctrl + T to trace point E as you move C.

Step 16Drag point C closer to segment AB and observe the effect on the plotted point.

Question 1

What does the path of the plotted point tell you about how height and area relate

in a parallelogram whose base is held constant?

Question 2

Write the formula for the area of a parallelogram with base b and height h.

Question 3

Write the same formula as a function using f(x) for area, x for height, and b for

base.

Step 17Press Ctrl + G to plot your function from Question 3. Then click in the white space.

Question 4

How does the function plot relate to the path of the plotted point as you vary the

height of the parallelogram?

Question 5

Why does it make sense that the graph passes through the origin?

Question 6

What should the domain of this function be to accurately represent the

situation?

Activity Objective

In this activity from Exploring Algebra with The Geometer’s Sketchpad (p. 47) the

students will discover direct variation functions and how it is represented

symbolically and graphically.

Activity Length

This activity takes approximately 30 – 40 minutes. Students should be familiar

with functions and function notation and determining the domain of a function.

NCTM Standards

Algebra Standard

• Understand patterns, relations, and functions.
• Represent and analyze mathematical situations and structures using algebraic symbols.

Measurement Standard

• Understand measurable attributes of objects and the units, systems, and processes of measurement.
• Apply appropriate techniques, tools, and formulas to determine measurements.

Communication Standard

• Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
• Analyze and evaluate the mathematical thinking and strategies of others.
• Use the language of mathematics to express mathematical ideas precisely.

Quadratic Functions Activity

The objective of this activity is for the students to discover the effects of a,b, and c in the quadratic function f(x) = a(x – b)2 + c. Students will also discover the vertex form of a quadratic equation. This activity takes approximately 45 minutes to complete.

In this activity students will construct 3 sliders, one for a, b, and c and plot the graphs f(x) = x2, g(x) = (x – b)2 + c , and

h(x) = a(x-b)2 + c. This activity take approximately 45 minutes to complete.

The students will move the sliders for a, b , and c, and compare the graph with the original graph of f(x) = x2 to determine the affects of each variable. This meets the NCTM Algebra Standard

( Understand patterns, relations, and functions & Analyze change in various contexts ).

The students develop conjectures about how a, b, and c affect the graph of the quadratic equation. Then students come to a conclusion concerning the vertex form of a quadratic equation. These discoveries that lead to conjectures meet the NCTM Reasoning & Proof Standard (Make and investigate mathematical conjectures).

The students are working in pairs discussing what they find and developing conjectures together. This collaborative effort meets the NCTM Communication Standard (Use the language of mathematics to express mathematical ideas precisely & Communicate clearly the mathematical ideas to peers, teachers, and others).

Quadratic Functions Activity

Before you begin the sketch go to the Edit Menu and select Preferences and under the Text Tab make sure For All New Points ischecked. Then click on OK.

In Steps 1 – 10 we will be constructing 3 sliders that we will use to observe relationships of the variables a, b, and c in quadratic functions.

Step 1Select the Straightedge Tool and hold down until it becomes a line. Then in the white space construct a line. Your line will have two points A and B on it.

Step 2From the Construct Menu choose Point On Line. Point C will be constructed on your line. Drag C so that it is between A & B.

Step 3Select the Selection Arrow Tool and then select points A, B, and C (in that order) and from the Measure Menu choose Ratio.

Step 4Click in the white space so that the ratio is not highlighted, then click on the line and point B. Then press Ctrl + H to hide the line and the selected point.

Step 5Select point A, then point C and press Ctrl + L to construct a segment. Click in the white space so that no items are selected.

Step 6Using the Text Tool, double click the measured ratio and label a.

Step 7Using the Selection Arrow Tool, first click in the white space, then select the segment, endpoints, and ratio and press Ctrl + C to copy. Then click in the white space so that nothing is selected and press Ctrl + V to paste. Drag the copied objects so that they are below the original.

Step 8Using the Text Tool, double click on the copied measured ratio and label b.

Step 9Using the Selection Arrow Tool click in the white space so that nothing is selected, then press Ctrl + V to paste again. Drag the copied objects so that they are below the slider labeled b. Click in the white space.

Step 10Using the Text Tool, double click on the copied measured ratio and label c.

In steps 11 – 18 we will be graphing quadratic functions.

Step 11Press Ctrl + G and the New Function Calculator will appear.

Step 12To graph y = x2 we use the x key then the ^ key then select 2 on the key pad then OK.

Step 13Using the Selection Arrow Tool, click in the white space and then move the equation of the function next to the curve.

Step 14Select the graph and the equation and from the Display Menu choose Line Width then select Thick. Then from the Display Menu choose Color and select the royal blue color. Then click in the white space.

Step 15Press Ctrl + G and select the following keys in the order given.

(, x, -, the ratio in the white space labeled b, ), ^, 2, +, the ratio in the white space labeled c, then OK.