Gsp Assignment 3

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1.0 INTRODUCTION

1.1 THE GEOMETERS SKETCHPAD

The Geometer's Sketchpad is a popular commercial interactive geometry

software program for exploring Euclidean geometry, algebra, calculus, and other areas

ofmathematics. It was created by Nicholas Jackiw.

Geometer's Sketchpad is used in many secondary mathematics classrooms throughout

the United States and Canada. NCTM (National Council of Teachers of Mathematics) had

identified one of its six principles as a technology principle, stating that "Technology is

essential in teaching and learning mathematics; it influences the mathematics that is taught

and enhances students' learning." Geometer's Sketchpad is one of these examples. The

program comes with program files to help deepen students' understanding of such concepts as

slope, geometric transformations, and arithmetic on integers.

Geometer's Sketchpad includes the traditional Euclidean tools of classical Geometric

constructions; that is, if a figure (such as the pentadecagon) can be constructed with compass

and straight-edge, it can also be constructed using this program. However, the program also

allows users to employ transformations to "cheat," creating figures impossible to construct

under the traditional compass-and-straight-edge rules (such as the regular nonagon). Objects

can also be animated. The program also allows the determination of the midpoint and mid

segments of objects.

Geometer's Sketchpad also allows to measure lengths ofsegments, measures

ofangles, area, perimeter, etc. Some of the tools one can use include; construct function,

which allows the user to create objects in relation to selected objects. The transform function

allows the user to create points in relation to objects, which include distance, angle, ratio, and

others. With these tools, one can create numerous different objects, measure them, andpotentially figure out hard-to-solve math problems.

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GSP Tools and their uses: There is a toolbar on the left side of the GSP window.

The tools names from top to bottom are:

Selection Toolallows selection of items on screen and is the default tool.

Point Toolplaces points in the sketch at the place defined by the mouse position.

Circle Tooldraws circles in the sketch.

Straightedge Toolallows the user to draw lines, segments, rays, etc.

Text Toolallows the user to make labels and add text to the screen.

Custom Toolallows the user to create customized tools.

Time to Start: Experiment with the tools by drawing some points, lines, rays, segments in the

sketch. To change the segment tool into a line tool you must place the cursor the tool and

right click on the mouse, holding the button on the mouse down you drag the cursor to the

right and there will be segments, lines and rays. Once you have the cursor on your choice let

go of the mouse button.

Selection or Highlighting: Click the Selection tool and then click on one of the points you

created on the screen. Notice that the point is now colored pink. This signifies that the object

is selected. If you click in the white space on the screen, notice the point is no longer pink.

Experiment with the selection tool to answer the following questions.

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Questions

a) How to select one or more objects?

_________________________________________________________________________

b) How to deselect objects?

_________________________________________________________________________

c) How do you know when an object is selected?

_________________________________________________________________________

1.2 PYTHAGORAS THEOREM

The Pythagorean Theorem was one of the earliest theorems known to ancient

civilizations. This famous theorem is named for the Greek mathematician and philosopher,

Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a

Greek seaport in Southern Italy. He is credited with many contributions to mathematics

although some of them may have actually been the work of his students.

Pythagoras' theorem states that in any right angled triangle, the square of thehypotenuse is equal to the sum of the squares of the other two sides.

2.0 LEARNING OBJECTIVES AND LEARNING OUTCOMES :

2.1 LEARNING OBJECTIVES:

Students will be taught to understand the relationship between the sides of a right-angled

triangle.

2.2 LEARNING OUTCOMES :

Students will be able to:

i) Identify the hypotenuse of right-angled triangles.ii) Determine the relationships between the lengths of the sides of a right-angled triangle.

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3.0 QUESTIONS

1. How to determine a right-angled triangle?2. How to determine the hypotenuse in a right-angled triangle?3. What are the relationships of lengths of the sides of a right-angled triangle?

4.0 STEPS

4.1 Constructing a right-angled triangle

i) Open Sketchpad, or click on the Sketchpad window to make it active. Choose File | NewSketch to open a blank sketch.

ii) Choose the Staightedge Tool. Construct a segment using either one of two methods:

- Press and hold, drag and release.- Click, move and click again

iii) Use Text tool to label the endpoints of the segmentA andB.iv) Choose the SelectionArrow tool. Click in empty space to deselect all objects.v) Select the segment and one endpoint and choose Construct | Perpendicular Line.

If thePerpendicular Line command is grayed out and therefore not available for use,

make sure you have both a point and the segment selected.

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vi) Drag the endpoints of the segment. Notice that the new line always remains perpendicularto the segment.

vii) Choose the Point tool to build point C.viii) Construct a point C anywhere on the perpendicular line. Make sure the line is highlighted

when you click.

ix) Choose the Arrow tool and drag the new point. It should always stay on the line. If youcan drag it off the line, choose Edit | Undo and try again.

x) Choose the Straightedge tool. Construct two segments to connect the new point to eachendpoint of the original segment. Label the new point with point C.

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xi) Choose the Selection Arrow tool. Select the perpendicular line and choose Display |Hide Perpendicular Line.

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xii) Connect the vertices with segments.

4.2 Constructing squares

i) Open a new sketch and construct a segment. The endpoints of the segment are the firsttwo vertices of your square.

ii) Label the endpoints of the segmentA andB.

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iii) Select pointA andAB. Choose Construct | Perpendicular Line. You should end up witha line through pointA perpendicular toAB.

Now youll construct two lines perpendicular to AB, one passing through point A and one

through point B.

iv) Construct a line through pointB that is perpendicular toAB.

v) Apply the drag testdrag each endpoint and make sure the lines stay perpendicular to thesegment.

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vi) Construct a circle centered at pointA with radius point at pointB.

Your sketch should still contain just two points, A and B.If youve created a third point,

chooseEdit | Undo and try again.

vii) Construct a point at the intersection of the circle and the perpendicular line throughpointA (both the circle and the line will be highlighted). This is the third vertex of your

square. Label it C.

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viii) Select point CandAB. Choose Construct | Parallel Line.

ix) Construct a point at the new intersection formed by two lines. Label this point D. This isthe last vertex of your square.

x) Select the three lines and the circle and then choose Edit | Action Buttons |Hide/Show. A Hide/Show action button is created. Press the button to hide the objects.

Press it again to show the hidden objects.

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xi) Connect the vertices with segments.

Now that you have made a square, you can save your hard work and make more squares

without having to start from scratch every time. SketchpadsCustom tools feature allows

you to teach Sketchpad how to make a square (or any other construction you might

build).

xii) PressHide Path Objects if your circle and lines are visible. Then select only the fourvertices of your square and choose Display | Hide Labels.

xiii) Draw a selection rectangle around the entire square to select it (vertices, sides, andinterior).

xiv) Press and hold the Custom tools icon and choose Create New Tool from the menu thatappears. Name the tool Square and clickOK.

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xv) Press the Custom tools icon to select your new tool. Move the pointer to your sketchwindow and press, drag, and release (or click and click) to make a square. Use your

square tool to make several squares. Try linking squares by building squares on vertices

of other squares.

Sketchpad lists every step of your square construction as easily readable instructions sothat you can remind yourself of how the square was built.

4.3 Use Your Square Tool

Now youll construct squares on each side of your right triangle and investigate their areas.

i) Press and hold the Custom tools icon and choose a Square tool. Click one vertex ofthe right triangle and then another. A square appears.

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ii) If the square appears inside the right triangle, choose Edit | Undo and then click thetwo vertices in the opposite order.

iii) Construct squares on the other two sides of the right triangle. All three squares shouldsit on the outside of the triangle.

iv) Click in empty space to deselect all objects. Select the three square interiors andchoose Measure | Areas.

Now youll find the sum of the areas of the two smaller squares.

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v) Choose Measure | Calculateto open Sketchpads Calculator.

vi) Click one of the two smaller area measurements in the sketch to enter it into theCalculator. Then click the + key, click the other smaller area measurement in the

sketch, and clickOK.

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v) Identify the hypotenuse of right-angled triangles.

vi) Drag the vertices of your right-angled triangle.vii) How does the sum of the areas of the two smaller squares compare to the area of the

larger square?

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b ) Calculate the length of the hypotenuse in the right-angled triangle below.

c) Find the value of x by using the Phythagoras theorem.

d) Find the value of y by using Phythagoras theorem.

7 cm24cm

x

x

13 cm

5 cm

y cm

17 cm

8 cm

2 cm

S

QP

R

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e )

ABCE is a square and CDEF is a rectangle. Find the area of the shaded region.

6.0 CONCLUSION

From the activities and observations that have been made, we can conclude that Pythagoras Theorem

is .

A

D4cm

3cm

E

C

B