# Begin with GeoGebra 1

Software and s/w Development

Dec 14, 2013 (7 years and 7 months ago)

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1

Begin with GeoG
ebra 1

Core Elements

Getting Started

Installing/enabling Geogebra

Installation WITH Internet access

Installation WITHOUT Internet access

GeoGebra version 4

GeoGebra version 5

Learning Phases

Introduction

of GeoGebra

User
Interface/Main Window of GeoGebra

B
asic

Use of GeoGebra Toolbar

Drawing without Mathematics

Construction Protocol

CheckBox to Show/Hide Objects

Numeric Foundations

Creating dynamic worksheets, mathlets

Using GeoGebra Animation

Geometry buttons/tools:
characteristics and concepts

Basic

geometric construct
ion
s, connection between geometry and algebra

Linear functions, polynomials

of 1
st

degree

of 2
nd

degree

-

statistics

Using a powerful markup language

Fa
mous
patterns and
problems:
Sierpinski triangle
,
Fibonacci se
r
i
es
, normal distribution

and others.

2

Getting started

Crea
te a new folder called GeoGebra_Intro

(or similar) on your desktop or as a folder in your filestructure. It is a
good
stra
tegy to save all files in a separate

folder so they are easy to find later on.

Installation/enabling GeoGebra

Different ways ar
e available to install and start
GeoGebra. Go to the page

to find those ways.

The current version (October 2011) of GeoGebra is GeoGebra 4
. The latest release of GeoGebra 4 is still a Be
ta
version. The release notes are found on the link
http://www.geogebra.org/en/wiki/index.php/Release_Notes_GeoGebra_4.0

Installation WITH Internet access

There are a c
ouple of webbased versions avail
able
:
WebStart
,
AppleStart

and
GeoGebraPrim
.

GeoGebra
WebStart

Open the scrollist in the upper right corner of the download webpage and select the prefe
rred tool/installation
language.

Click on the button called
WebStart
.

In this case the
Java Network Launching Protocol

(
JNLP
)

or
Java Web
Start

functionality is used.
Java Web Start provides a platform
-
independent, secure, and robust deployment
technology. It enables developers to deploy full
-
featured applications to end
-
use
rs by making the applications
available on a standard Web server.

The software is automatically installed on your computer. You only need to confirm all messages that might appear
with
OK

or
YES
.

Using GeoGebra WebStart has several advantages for you
provided that you have an Internet connection available
for the initial installation:

You don’t have to deal with different files because GeoGebra is installed automatically on your computer.

You don’t need to have special user permissions in order to use
GeoGebra WebStart, which is especially useful for
computer labs and laptop computers in schools.

Once GeoGebra WebStart was installed you can use the software off
-
line as well.

Provided you have Internet connection after the initial installation, GeoGebra
WebStart frequently checks for
available updates and installs them automatically. Thus, you are always working with the newest version of
GeoGebra.

When the installation is ready you have got the following short
-
cut icon on the de
sktop.

GeoGebra
AppletStart

Open the scrollist in the upper right corner of the download webpage and select the preferred tool
/installation
language.

Click on the button called
Applet
Start
.

GeoGebra is opened and run as an ordinarie Java Applet.

3

GeoGebra GeoGebraPrim

Open the scrollist in the upper right corner of the download webpage and select the preferred tool/installation
language.

Click on the button called

GeoGebraPrim
.

A corresponding
.jnlp
-
file is then available.

This is a “stripped” version
of GeoGebra and the restrictions can be found on the link
http://www.geogebra.org/en/wiki/index.php/Release_Notes_GeoGebra_4.0

Installation
WITHOUT Internet access
: Offline Installers

You need to have the installation media or installer file

GeoGebra.exe

(
Windows platform), “One click installers” or
“GeoGebra.z
ip

(Linux platform).

offline installer

http://www.geogebra.org/cms/en/installers

will open
.

Copy the installer fil
e for your preferred platform from the storage device into a created

folder

(with a suitable
name)

Current

version for Windows platform:
G
eoGebra
-
Windows
-
Installer
-
4
-
X
-
X
-
X
.exe
.

Double
-
click the GeoGebra installer file and follow the instructions of the installer wizard.

When the installation is ready you have got the following short
-
cut icon on the desktop.

GeoGebra version 5

Find the latest notes about the currently developed GeoGebra 5.0 beta version ( June 2012) in the document

http://wiki.geogebra.org/en/Release_Notes_GeoGebra_5.0

You can run the GeoGebra 5.0 beta version directly here:

http://www.geogebra.org/webstart/5.0/ge ... jogl1.jnlp

If you have trouble with that, try this one which uses JOGL2

http://www.geogebra.org/webstart/5.0/ge ... jogl2.jnlp

Java OpenGL

(
JOGL
) is a wrapper library
that allows

OpenGL

to be used in the

Java programming language.

Learning P
hases

An
important idea in the material “Begin with GeoGebra” is built on three learning phases:

-

collaboration

phase with construction protocol and jointly adapted worksheets and work
-
outs with step
-
by
-
step guidance in discus
sion with teacher/instructor

-

elaboration

phase with discovery/self
-
study/self
-
reviewed worksheets and work
-
outs, typically performed
as investigation of additional concepts, parallel concepts or attack concept/problem from another angle

-

exploration phase/”
e
-
learning
” supported by interactively
modifyable worksheets/work
-
outs to foster
experimental as well as discovery learning to strengthen and confirm the understanding and use of
concepts, patterns and models.

Every GeoGebra construction can be exported as a Web Page (html),
known as a
Dynamic Worksheet. Computer on local base or
interact with it!

Those three phases will be practiced through the “Begin with GeoGebra” material.

Introduction

of GeoGebra

GeoGebra is
a
user
-
friendly and inter
active
software

for mathematics learning

that dynamically combines

geometry, algebra, and calculus and
also
CAS

(
Computer Algebra System)
in
GeoGebra 4 and 5.

On the one hand, GeoGebra is an interactive geometry system
, the geometry
view
. You can do constructions with
points, vectors, segments, lines, and conic sections as well as functions while changing them dynamically
afterwards.

On the other hand,
commands,
equations and coordinates can be entered directly
, the algebra view
. Thu
s,
GeoGebra has the ability to deal with variables for numbers, vectors, and points. It finds derivatives and integrals of
functions and offers commands like Root or Vertex.

The algebra view is connected to an Input fie
ld to make the
direct textual input.

These two views are characteristic of GeoGebra: an expression in the
algebra

view

corresponds to an object in the
geometry

view
and vice versa

and the views are toggled in real time.

The third view is the
calculus

onality. This wi
ll be handled

later on.

4

And there is even a fourth view:
CAS

Computer Algebra System for symbol handling introduced in GeoGebra 4
and 5. This will also
be handled in a separate chapter later on
.

User Interface
/Main Window

of GeoGebra

Start GeoGebra with

a d
ouble
-
click on the GeoGebra WebStart

icon, GeoGebra Installer

icon
, link to Applet Start or
. The GeoGebra tool opens the following standard
/main window
with a common type
of layout for user interface and main page window.

Ge
oGebra’s

user interface/standard main page
consists of a
graph
ics wi
ndow and an algebra view opened for usage. The calculus view and the CAS view are hidden when the
GeoGebra interface/main page is opened. Those view
s

are open from the toolbox, examined l
ater on.

The User Interface/Main Window for GeoGebra 5.0

This interface/main windows
is started with a Perspective Menu Window from which you can easily switch between
different views, without selecting each individually.

You can choose between 5 different standard perspectives:

Algebra & Graphics
:

The

Algebra View

and the

Graphics View

with axes are shown.

Basic Geometry
:

Only the

Graphics View

without axes or grid is displayed.

Geometry
:

Only the
Graphics View

with grid is shown.

:

The

and the
Graphics View

are displayed.

CAS & Graphics
:

The

CAS View

and the

Graphics View

are displayed.

Basic Use of
GeoGebra

Toolbar

Activate a tool by clicking on the button showing the corresponding icon.

Open a toolbox by clicking on the lower part of a button and select another tool from this toolbox.

You don’t have to open the toolbox every time you want to select a tool. If the icon of the desired tool is already
shown on the button it can b
e activated directly.

Toolboxes contain similar tools or tools that generate the same type of new object.

=
j敮u=t
qoolbar=

=

=
䅬来Ara=噩敷
=
=
fnput=ci敬d
=
=
fnput=lptions

5

Check the toolbar help in order to find out which tool is currently activated and how to operate it.

Drawing without Mathematics

Doubl
e
-
click
on any of the GeoGebra WebStart

icon,

GeoGebra Installer

i

The
GeoGebra tool opens the following standard window.

If you don’t have Swedish as the tool language
in the GeoGebra window

(the right picuture above)
, click on the
toolbar tab
Option

and activate
Languge
-
> R
-
Z
-
> Swedish

in the opened scrollist
. Then the tool language i
s
changed to Swedish as in the left picture above.

Open the
View
tab and uncheck

the Axis,
check the Grid alternatives
. Close the Algebra View
.

Then you get a
Geomety View with a Drawing Pad

Select

the
Geometry
tool

New Point

to create a Point
A

6

Select

the

Geometry

tool “Line through two points” and click the red triangle in the low right corner of the tool
icon.
Select the “Segment betwe
en two points”.

Use the mouse cursor (cross mark) to draw a “chair”.

Activate the points of
the chair with a right
-
click and select “Show Label”. The chair is labeled like this

Open the Algebra V
iew. The coordinates for the five points on the

chair

are presented as Free Objects
.

Activate the seat on the chair with the mouse cursor. This is name
d as the segment a or Segment [A,B]. In the
same way the ba
ck, left and right leg are presented
as Segment

name and value
as Dependent Objects in the
Algebra View
.

7

GeoGebra distinguishes between free and dependant objects. While free objects can be
directly modified either
using the mouse or the keyboard, dependant objects adapt to changes of their parent objects. Thereby, it is
irrelevant in which way (mouse or keyboard) an object was initially created!

Construction Protocol

Select

Construction Pro
tocol under the
View

tool
.

In the construction protocol you can see in what order the objects have been constructed
.
At the bottom of the
construction protocol table there are a set of navigation button that can be used to display the construction
sequence of the objects.

The buttons are easily recognized from a common recorder.

CheckBox to Show/Hide Objects

A common use of the CheckBox tool in GeoGebra is to allow objects to hidden or revealed.

We connect a checkbox
“Show chair” to the chair.

Select the tool “CheckBox to Show/Hide Object”. Click on the Grahics View on a optional position.

A checkbox dialog
window is opened.

Write “Show chair” in the Caption textfield and select all objects for the chair in the “Select objects in construction

or choose from list” scrollist. Press “Apply” button.

8

The checkbox “Show chair” is checked. When you uncheck the checkbox with the mouse, the chair is hidden!

In the algebra view there is a Boolean variable created with a
variable
name in alphabetic o
rder
and a value
true
.
When
the checkbox “Show chair” is unchecked, the Boolean variable gets the value
false
.

Exercise

Construct a stick man
.

Exercise

Make a pentagram.

Start with the tool Regular Polygon (Pentagon). Connect all the corners on the pentagon.

Hide the pentagon object.

Congruent constructions

T
wo sets of points

are called
congruent

if, and only if, one can be t
ransformed into the other by an isometry,
i.e., a
combination of translations, rotations and reflections
.

An isometry of the plane

is a linear transformation which
preserves length.

The Euclidean geometry
(Euclidean geometry, see chapter

Geometry buttons/tools: characteristics and concepts

later on)
include five types of isometrics:
translation, rotation, reflection, glide reflection, identity. Reflection or
mirror isometrics can be combined to produce any isometrics.

Mirroring in a line

A point and its mirror point have the same perpendi
cular distance to the line.

Open the View tab and uncheck the tool Axes or
right
-
click anywhere in the drawing pad and uncheck
Axes
.

Enter a line between the points A and B.

Enter a free point C.

Use the tool

Reflect Object in Lin”
. C
lick on the point C
and then on the line. The mirror point C' is created.

In order to distinguish between the free point C (the point you can drag) and the dependent point C', you can
change the look of the points. Right
-
click on C and choose
Object Properties
. Change colour
under the tab Colour.
Change the size and the appearance under the tab Style.

Put a trace on both points by right
-
clicking on them and checking
Trace On
. Draw a picture by dragging the point C.

You can erase the picture drawn by zooming in or out, use the
mouse wheel or the tools in the tool bar.

An image has a
rotational symmetry

if you can rotate the image around some point and get the same image. An
image has a
reflection symmetry

if you can reflect the image in some line and get the same image.

Transla
tion

T
he red arrow is called a vector. A vector has a direction and a length. If you check the check box you can see that
all the vertices of the polygon are translated along the same vector. The gray arrows are all parallel.

You make a vector in GeoGebra
by using the tool Vector between Two Points
.

You make a translation by using the tool Translate Object by Vector
. Click on the object you want to translate
and then on the vector. The object itself is not translated but a translated copy of the object is created.

9

Rotation

In order to rotate an object you need an angle.

Make two segments with one common endpoint A.

Use the tool Segment between Two points
.

Use the tool Angle
. Click on one of the segments, then on the other segment. An angle called
α

appears (
α

is
the first letter in the Greek alphabet
).

Create a geometrical object, a circle or a polygon.

Use the
tool Rotate Object around Point by Angle
. Click on the geometrical object; then on the point A; then on
the angle
α
. You can click either in the drawing pad or in the algebra view.

Parallel and perpendicular

In the Euclidean geometrical

theory

(Euclidean geometry, see chapter “
Geometry buttons/tools: characteristics
and concepts

later on),
there is
only a small collection of self
-
evidently true
axioms

and derive, in a logically
sound manner, the consequences of these, known as
theorems
.

In Geo
Gebra we have a collection of buttons/tools
which correspond to theorems among those “Parallel Line” and “Perpendicular Line”.

Create three optional points with the tool “Points”.

Choose the tool

Move

. Move the three points!

Select the tool

Parallel
Line

. Click on the point C and then on the blue line; a black line appears. Select the

Move

tool again and move the three points. Describe in detail how the two lines are related.

Click on

Reset Construction

in the upper right corner.

Select the too
l

Perpendicular Line

. Click on the point C and then on the blue line; a black line appears. Select the

Move

tool again and move the three points. Describe in detail how the two lines are related.

Click on

Reset Construction

in the upper right corner.

Select the tool

Perpendicular Line

. Click on the point B and then on the blue line; a black line appears. Move the
points!

Using the Input Field

GeoGebra offers algebraic input and commands in addition to the geometry

too
ls. Every tool has a matching
command and therefore, could be applied wit
hout even using the mouse.

GeoGebra offers more commands than geometry tools. Therefore, not every command has a cor
responding
geometry tool!

Check out the list of commands next to t
he input field

(in the lower right corner of the main window)

and look for
commands whose corresponding tools

Use
the Input Field

and construct the chair again with commands.

Input: A = (0,

0). The point A is created.

Input: (0,

2). The point B is created. If a specific

name is not given the objects are not named in alphabetical
order.

Input: C = (2,

2)

Input: D = (0,
-
2)

Input: E = (2,
-
2)

Input: Segment[A,B]

Input: Segment[B,C]

Input: Segment[A,D]

Input:
Segment[B,E]

The same “chair” as before is constructed. The commands are instantly visualized in the geometry view.

Properties of objects

Change properties of objects in order to improve the construction’s appearance (e.g. colors, line thickness, auxiliary
objects dashed,…).

Right click on the chair objects (points, segments) and select the Objects Properties alternative. A
properties
windo
w is opened.

10

Draw text

Tool Slider
-
> ABC

Insert Text

Enter the desired

text into the appearing window.

Numeric Foundation
s

and numbers

and their connections, pattern
s

and change
s
.
this
main track
, the numbers.

with the Number Line

Double
-
click on any of the GeoGebra WebStart icon, GeoGebra Installer icon or links to GeGebra4/GeoGebra5. The
GeoGebra tool opens the standard window.

Prepare a horizontal
number line.

Right click on the Graphics View/Drawing Pad and select the Graphics View properties. Hide the yAxis (uncheck
the
“Show
yAxis

checkbox) and give the xAxis the range from min =
-
10 to max = 10.

Use sliders to
show and
modify
a

variable

and

a variable

value
.

Activate/click on the
Slider

tool in the toolbar and locate and click the mouse cursor on the geometry view/drawing

Right click on the slider icon

and open the Object Properties. Set the slider range with the
Min and M
ax values,
e.g.
the default values
-
5 and 5

and Increment 1 (to get integer values)
.

As all objects the slider variable name is given
the small letter
a

in alphabetic order. Activate/click the
Move

tool in the toolbar

and then use the mouse
to change the value for

the slider/variable
with the knob/pin on the slider. Click on the
slider with the mouse cursor and drag the slider to an optional location on the drawing pad.

Create another slider/variable
b

with the same value range as the slider/variable
a
.

The algebra view is open so the
sliders/variables a and b with the current values are presented as Free Objects.

Create the point
Origo

for the 0 on the number line

as a reference for the start of the number system
. Use the
command

11

Origo = (0, 0)

in t
he Input field
or
activate and drag the tool

to the value 0 on th
e

number line.

Visualize the variable

a

with the startpoint
A

and the endpoint
B

with an arrow (vector)
with the name aVector
along the number line
. Put those objects one
unit above the
numberline, y
-
coordinate = 1
, to get a better
visualibility
.
Write the following commands in the Input field:

A = (0,1)

B = A + (a,
0)

aVector = Vector[A,B]
or use the tool “Vector between two Points”, subtool to “Line between to

Points”

The purpose
with the arrow /
vector is to get a visualization of the signed number
a

so t
he name aVector
can be
hidden. Activate the Move tool and right click on the aVector object and uncheck “Show Label”.

Use the mouse to change the

value for the

slider/variable

a

w
ith the knob/pin on the slider

and observe that the
value is visualized with the arrow/vector length.

Don’t forget to activate the “Move” tool.

Visualize the variable b

with the startpoint
C

and the endpoint
B

with an arrow (vector) with the n
ame b
Vector
along the number line. Put those objects

one

more

unit above, y
-
coordinate = 2, to keep the
visualibility. Write the
following commands in the Input field:

C = B + (
0,1)

D = C + (b
,
0)

bVector = Vector[C,D
]
or use the tool “Vector between two Points
”,
a
subtool to “Line between t
w
o

Points”

The purpose with the arrow /
vector is to get a visualiza
tion of the signed number
b

so t
he name b
Vector can be
hidden. Activate the Mo
ve tool and right click on the b
Vector object and uncheck “Show Label”.

Use the mouse to change the value for the slider/variable
b

with the knob/pin on the slider and observe that the
value is visualized with the arrow/vector length.

12

Visualize the sum a + b.

The slider/variable a and b can now both be changed with the mou
se to visualize the
sum of a + b.
The x co
ordinate for the point D is
the sum of a and b, sum = a + b.

Project the
x coordinate of D,
Sum, and a projection line of D on the number line.
The name Sum must have a capital
letter S because it is the
name of a point
in

GeoGebra.
Use the command
s

Sum = (x(D),0)

Segment[D, Sum]

or use the
tool”
Segment between
two
Points

, a subtool to “Line between two Points”

Uncheck the label for the segment
(default)
name c.

There is n
o need to visualize the name c for the moment.

To get a continuation of the visualized addition, make a segment line also between B anc C

Segment[B, C]

Uncheck the label for the segment (default) name d. There is no need to visualize the name d for the moment.

The get an even more visualization make the segment lines between

B and C and D and Sum. Activate
the “Move”
tool and wright click on the segmen
t objects B and C and D and Sum. Open “Object Properties”, select the “Style”
tab and choose a “dashline style” in the scrollist.

Insert the algebraic expression of the addition

Use the subtool “Insert Text” to the tool “Slider”

Activate the “Inser
t Text” tool and click on the drawing pad. A separate Edit window is opened in which (already
defined) objects can be choosed

with the Object button. Click the Object button and select the variable a

from the
scrollist
. A dynamic textfield for a is
shown in the Edit window and the value is shown in the Preview window
.
Press
OK button. The dynamic value of the variable a is shown on the drawing pad. Activate the “Move” tool and drag the
dynamic
value
of
a

to a suitable location on the drawing pad. C
hange the slider/variable a value to see that the
dynamic text value is following.

13

Now we want the + operator in the a + b expression. This is a static text and this is made with a quote expression
in GeoGebra (like a static S
t
ring in Java).

Use the til
l
tool
“Insert Text” again.

Write

the string

“ + “ in the Edit Window. Press OK. Activate the “Move” tool
and drag the static “value”, operator
+
,

to a suitable location on the drawing pad.

Use the tool “Insert Text” again for the dynamic value of the
object/variable b.

Activate the “Move” tool and drag
the dynamic value of
b

to a suitable location on the drawing pad.
Change the slider/variable b

value to see that the
dynamic text value is following.

Use the tool “Insert Text” again for the static text

for the assignement operator
=
.

Now we need the dynamic value for the a + b sum. We need to create a variable for that, e.g.
sum
. The value for
sum is the x coordinate

for D, see above. Be aware, the object
Sum

above
(with a capital letter S)
is a point! Now
we need a variable
sum

(with a small letter s)
. Wr
i
te the following command in the Input field:

sum = x(D)

14

Use the tool “Insert Text” again for the dynami
c value of the object/variable
sum
.

Activate the “Move” tool and drag
the dynamic val
ue of
sum

to a suitable location on the drawing pad. Change the slider/variable
a

and
b

value to see
that the dynamic text value
s are

following.

Decorate the algebraic expression and the geometric visualization.

To even more increase the visualization decorate the
a

slider/variable and attached arrow/vector with
a

blue color,
the
b

slider/variable and attached arrow/vector with a red color and the point
Sum

and the variable
sum

with a green
color.

Activate the too
l Move and wright click on the specific objects. Open the Color tab and select the color from the
color palett.

Also for the dynamic values for
a
,
b

and
sum

open the Text tab for those Obje
ct Properties and select
suitable font (e.g. Very Large, B, …)
.

For the arrow/vec
tor objects a higher Style/Line Thickness

(e.g. 7) can be
chosen.

Visualizing Integer Subtraction with the Number Line

The calculus rules for addition and subtraction are of course built in GeoGebra for the commands including the
oper
ator + and
-
. (Geogebra uses the development software
Java and all its foundations.)

Delete the object D from the Algebra View and write the subtraction command

D = C

(b,0)

(The main change.)

Rewrite the commands

bVector = Vector(C, D)

(Hide the
bVector o
bject label
.
)

Difference

= (x(D), 0)

(
Point Difference
.
)

Segment(D, Difference
)

(Hide the Segment object label.)

difference

= x(D)

(Variable difference
.
)

Change
the text object
+

to the static subtraction operation
-

and update the dynamic text object for the variable
difference
.

Decorate the objects with color and style as the for the visualize addition case.

15

Visualizing

Integer Mul
t
iplication

of Natural Numbers

Double
-
click on any of the GeoGebra WebStart icon,

GeoGebra Installer icon or links to GeGebra4/GeoGebra5. The
GeoGebra tool opens the standard window.

Use sliders to show and modify a variable and a variable value
.

Hide the Axes view.

Activate/click on the
Slider

tool in the toolbar and locate and c
lick the mouse cursor on the geometry view/drawing

Right click on the slider icon

and open the Object Properties. Set the slider range with the Min and Max values, e.g.
the default values 1 and 10 and Increment 1 (to get integer values). Set the slider width = 500. As all objects the
slider variable name is given the small letter
a

in
alphabetic order.

Change the slider variable name to
factor1
.

Click on the slider with the mouse cursor and drag the slider to an optional location at the bottom of the drawing

Create a point
A
.

Put in the left lower corner on the drawing pad.

Create

a horizontal segment from A with the tool “Segment

with a G
iven Length from Point”.
The segment

object

is
named a.
Give the segment length

the slider variable
name
factor1
. The endpoint of the segment is automatically
called
B

in alphabetic order.

With the slider variable factor1 the segment a (segment AB) can now be given different
length values from 1 to 10.

Draw vertical lines through A and B, perpendicul
ar to segment a
.

Use the tool “Perpendicular Line”.

Activate the
segment object a and then t
he point object A and B with the mouse.

The perpendicular line objects are named b
and c.

Create another slider/variable
fac
tor2

with the same value range as the slider/variable
factor1
. Set this slider
orientation to Vertical. Click on the slider with the mouse cursor and drag the slider to an optional location to the
left side of the drawing pad.

Use the tool “Circle with Center and Radius” to connect the line b to the slider variable factor2. Activate the tool
with the mouse
and click the mouse on the point A. The circle object is named d.
Give the circle the slider

variable factor2.

Use the tool “Intersect Two Objects” to create a point C
as the intersection betw
een the circle

object

d and
line
object
b. Activate the tool “Intersect Two Objects” with the mouse and click the mouse on this intersection. The
intersect
ion point will be called C automatically.

b

The picture is cut down here.)

c

The picture is cut down here.)

c

16

Use the tool “Parallel Line” to create a line through C parallel to segment a. Activate the tool “Parallel Line” with the
mouse and click the mouse on the se
g
ment a and then on the point C. This line object is nam
ed e.

Use the tool “Intersect Two Objects” to create a point D as the intersection between the line object e and line object
c. Activate the tool “Intersect Two Objects” with the mouse and click the mouse on this intersection. The
intersection point will b
e called D automatically.

Use the tool “Polygon” to create the polygon/square ABDC.

Right click on the specific object, uncheck “Show Object” in order to h
ide all line objects b, c e, circle object d and
segmen
t

object a.
Uncheck “Show Label” in order to

h
ide labels of the (polygon) segment objects.

Change the sliders to factor1 = 10 and factor2 = 10.

Divide the polygon/square into 10x10 segments/parts using the slider variable factor1 and factor2.

Use
the Sequence command. Open the built
-
in Command
list with the tab to the right of the Input field. Select “All

Make a sequence list of segments between A and C

(vertical segments)
and then bet
ween A and B

(horizontals
segments)
. Write the following commands in the Input field.

17

Sequence[Segment[A+i*(1,0)
, C+i*(1,0)], i, 1, factor1]

Sequence[Segment[A+i*(0,1
)
, B+i*(0,1)], i, 1, factor2]

The following segment grid is created. Change the slide variables factor1 and factor2 to check that the number of
segments in the grid is changed.

Insert the algebraic expression of the addition

Use the subtool “Insert Text” to the tool “Slider”

Activ
ate the “Insert Text” tool and click on the drawing pad. A separate Edit window is opened in which (already
defined) objects can be choosed with the Object button. Click the Object button and select the variable
factor1

from the scrollist.

A dynamic te
xtfield for a is shown in the Edit window and the value is shown in the Preview window. Press OK
button. The dynamic value of the variable a is shown on the drawing pad. Activate the “Move” tool and drag the
dynamic value of
factor1

to a suitable locatio
n on the drawing pad. Change the slider/variable a value to see that
the dynamic text value is following.

Now we want the * operator in the
factor1 * factor2

expression. This is a static text and this is made with a
quote expression in GeoGebra (like a static S
t
ring in Java).

Use the till tool “Insert Text” again. Write the string

“ *

“ in the Edit Window. Press OK. Activate the “Move” tool
and drag the stat
ic “value”, operator
*
,

to a suitable location on the drawing pad.

Use the tool “Insert Text” again for the dynami
c value of the object/variable
factor2
.

Activate the “Move” tool and
drag the dynamic value of
factor2

to a suitable location on the drawing

factor2

value to see that the dynamic text value is following.

Use the tool “Insert Text” again for the static text for the assignement operator
=
.

Now we need the

dynamic value for the
factor1 * factor2

product
. We need to create a variable for that, e.g.
product
.

Write the following command in the Input field:

product = factor1 * factor2

The following picture visualize the multiplication

7 * 8 = 56

18

Decorate the algebraic expression and the geometric visuali
zation.

To even more increase the visualization decorate the
factor1

slider/variable with a blue color, the
factor2

slider/va
riable

wit
h a red color and the variable product

with a green color.

Activate the tool Move and wright click on the specific
objects. Open the Color tab and select the color from the
color palett. Al
so for the dynamic values for
factor1
,
factor2

and
product

open the Text tab for those Obje
ct
Properties and select suitable font (e.g. Very Large, B, …).

The points A, B, C and D

can optional be hidden.

19

Fractions

A fraction is a number that describes part of a whole number. Because f
ractions are numbers just like 7, 2 or 99
,
they
can
live on a number line.

A fraction is made up of a numerator and a denominator:

The numerator tells how many of those parts you have.

The denominator tells how many parts each unit interval has been cut into.

GeoGebra

Create sliders for the nominator
n
and denominator
d
.

Open the tool “Insert Text”
and use the
Latex Formula , see chapter “
Using a powerful marku
p

language

to
visualize the fraction

Visualize the denominator d with a blue arrow (Vector object) and the fraction

with w red arrow

(Vector object).

If the nominator >

denominator we have an improper fraction. This fraction can be changed into a mixed number
with a whole part and a
part
fraction with a nominator < denominator.

A
fraction c
an be "reduced", like 21 and 7

in the picture above and
have at least one common
factor (other than 1).

This GCD, Greatest Common Divisor, can be calculated in GeoGebra with the function
GCD(), g = GCD(n,d)
.

Reduce the nominator

n

and denominator
d

with
n1

= n/g
and
d1 = d/g

In GeoGebra the whole part can be calculated with the built
-
in function
floor()
,
whole = floor(n
1
/d
1
)
,

and the
remaining

nominator
part = n
1

whole * d
1
.

When those expressions are written in the Input field, use the “Keep Input”
, Alt+
Enter to get this
feature
.

The
value of g will then not effect the value of n and calculate a new value for g.

See
the “Geo
Gebra Documen
tation”
:

Enter: evaluates the current row depending on the selected tool in the toolbar: =, numeric,
keep

input
. Ctrl+Enter
switches bet
ween Numeric and Evaluate. Alt + Enter switches between Keep Input and Evaluate.

Use the Latex Formula to
visualize the origin fraction, the reduced fraction and the mixed fraction.

20

ing

fractions

1. Find common
deno
minator for the fractions

2.
Rename one or both fractions with the common denominator

4. Reduce and get mixed form.

Subtract
ing

fractions

To subtract fractions similar steps as for addition fractions are required
:

1. Find common
deno
minator for the fractions

2.
Rename one or both fractions with the common denominator

3. Subtract nominators.

4. Reduce and get mixed form.

Multiply
ing

fractions

It is a three
-
step process to multiply mixed numbers:

1.
Convert mixed numbers into fractions

2.
Multiply across

3.
Simpl
ify: reduce and rename

Dividing

fractions

It is a four
-
step process to divide mixed numbers:

1.
Convert mixed numbers to impropers

2.
Flip second fraction and change divison to multiplication

3.
Multiply across

4.
Simplify: reduce and rename

21

Geometry buttons/tools
: characteristics and concepts

http://en.wikipedia.org/wiki/Euclid

about Euclid of Alexandria, 300 BC,
a
great mathematician
whose life we
know
but
whos
e

work

has give us
a single, logically coherent
framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that
remains the basis of mathematics 23 centuries later.

He construct
ed a

geometrical theory using a minimum of assumptions. That is to say he wanted to assume only a
small collection of self
-
evidently true
axioms

and derive, in a logically sound manner, the consequences of these,
known as
theorems
. His assumptions were tha
t it is possible to

1.
draw a straight line from any point to any point,

2.
extend a finite straight line indefinitely in a straight line,

3.
draw a circle with any center and any diameter.

In Geogebra t
he first of these
assumption
is implemented by the button
, “Segment between two points”,
and
the first two combine to provide the button
, “Line through two points”
. We shall mostly consider indefinite
straight lines. The third is the button
, “Circle with center through point”
. W
hen Euclid talks of "any diameter"
he means
any previously

constructed

length. That is, he may use two existing points to open his compasses
against. He does not mean any diameter we can imagine or p
erhaps define algebraically.

Furthermore, he assumed that

4.
all right angles are equal to one another, and

5.
that if a straight line falling on two straight lines make the interior angles on the same side less than two right
angles, the two straight lines, if extended indefinitely, meet on that side on which t
he angles are less than the two
right angles.

This last assumption is sometimes known as the
parallel postulate
.

We note in passing that one strength of GeoGebra is the conjunction of algebraic and geometric views of
mathematics. For example, one can type
in the equation of a parabola as
y

=
x
2
, then place a point on the curve
and drag it around. The tangent line can
be illustrated using the button “Tangents”
, and the equation of this line
recovered from the alge
bra window.

GeoGebra provides many
buttons/tools

which have nothing to do with geo
measurement

(“Distance”, “Angle”, “Polygon (area of)
,

“Relation between two objects”. Those

concepts are
not considered within Euclidean Constructions.

Then we try to identif
y a set of
axiomatic buttons/tools

from which all the others can be constructed. If the
functionality of another button
/tool

can be constructed by axioms or previously constructed button
s then it can be
called
a
theorem button
.

Consider the following collection of buttons/tools: “Line through two points”, “Circle
with center and through point”, “Conic through five points”. These three provide one way to create all three of the
classic identifiable objects in plane geometry, the
line, circle and a conic section. Notice that the first two are special
cases of the last.

In addition we may place a "New point"
either unconstrained in space or to be constrained on one of these
objects. We may also find the intersection of two objects
and this button places new point(s) there. This latter
operation applies to any two of line, circle and conic and it returns between zero and two new points, which are
automatically assigned names. The intersection button is needed so that points can be c
reated which are related to
two existing objects
. Without this, or something similar, there is no way to establish relationships between
objects. Notice that implicit in identifying a point of intersection is an assumption of continuity.

The
n

we have a col
lection
of buttons/tools which correspond to theorems. Those are: “Perpendicular
B
isector”,

Perpendicular L
ine”, “Parallel

L
ine”, “
Midpoint or Centre
”, “
Angular B
isector”, “
Circle with Centre and R

Semicircle through Two P
oints”, “
Circle through T
hre
e P
oints”, “
Mirror Object in P
oint”, “
Mirror Object at L
ine”,
“Tangents”, “
Polar or Diameter L
ine”.

In the case of the Circle with center and radius,
, GeoGebra expects the
user to type in an algebraic distance. Hence the status of this button as a pur
ely geometric theorem is questionable.
In order to include this button we would need to include the "Distance tool", or use GeoGebra's "Segment between
two points" which returns the length of the segment.

22

sic arithmetic system by identifying the length of a line segment
with a number, starting with an arbitrary agreed unit. Then, in a systematic way, to construct the geometric
counterparts of the arithmetic operations such as addition, multiplication, and s
o on. It is not at all clear which
operations and hence numbers are
constructible

in this way. This combination of algebra and geometry is a classical
topic and the basic geometric constructions for additio
n, multiplication.

In GeoGebra a facility for user
-
defined “buttons/tools" is provided. This allows a construction to be encapsulated as
a new button, providing the opportunity for the above, or other, constructions to be implemented. Furthermore, an
interface is provided in which existing buttons can be
"switched off" or the order rearranged. This allows

an
application in GeoGebra to be configured
with a web page containing only a small number of buttons, and from
thes

to demonstrate a particular construction. For example, given only
,
and
,
show how we can make the button equivalent to
. While this removes the complaint "we already have a
button for this" it also
removes the freedom

to make choices of their own about the any mutual dependencies.

Basic

geometric constructs, connection betw
een geometry and algebra

Bisect an Angle

This is a most common geometric construct
ion
. Use GeoGebra to do it.

Double
-
click on any of the GeoGebra WebStart icon, GeoGebra Installer icon or links to GeGebra4/GeoGebra5. The
GeoGebra tool opens the standar
d window.

Hide the Axes view.

Use the tool “New Point” and create a point A

Use the tool “Segment between Two Points” and create
segment
s

to a point B

and a point C.
In order to h
ide
the
segments name (a and b),
uncheck “Show Labels”. Those segments
are the angles legs.

Use the tool “Angle” and draw the angle BAC by clicking on the points B, A and C. Right click on the angle object,
open “Objects Properties”, “Basic” and select “Show Label”: Name

Use the circle tool “Compas”. Activate the tool, click

on point A and point B and click on point A to fix the the center
of the circle to point A.
The circle with the radius = length of
segment AB (optional made shorter than
segment AC)
is intersecting the segment AC.

Use the tool “Intersect Two Object” to c
reate/fix the point D.

Create a
segment
object between C and D and
decorate the segment as a dashed line.

Create a point
E
on the
segment CD with an optional location but nearer C than D.

Use the circle tool “Compass”.

Activate the tool and click with the mouse on point D, point E and the again on point
D to fix the center of the circle to point D.

Use the circle tool “Compass” again. Activate the tool and click with the mouse on point D, point E and the again
on point

C to fix the center of the circle to point C.

Use the tool
“Intersect Two Object
” to c
reate/fix the intersection point F between the circles.

23

Use the tool “Ray through Two Points” and create a line between A and F.

Hide all the circles objects, segment

CD and point E with a right click with the mouse on the objects and uncheck
“Show Object”.

Use the tool “Angle” to create the angels DAF and FAC. Activate the “Angle” tool and click with the mouse on point
D, A and F. Activate the “Angle” tool and click
wi
th the mouse on point F, A and C
.

Right click on the name (α and β) and value for the angels, open “Object Properties” and make some decoration for
the angels.

The values of the angles α and β are the same.

Use the tool “Move” and move the point B
and C. As an example you can get a right angel divided into two angles
with the value 45º.

Eqvilateral Triangles

Another classic geometry construction is eqvilateral triangels.

Here is an abstract from
http://en.wikipedia.org/wiki/Euclid%27s_Elements

that gives an overview of
Euclid's
Elements

a collection
of 13 books written by

the

Greek mathematician
300 BC.

24

A proof from Euclid's
Elements

that, given a line segment, an equilateral triangle exists that includes the segment
as one of its sides. The proof is by construction: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε
centered on the points Α and Β, and taking one intersect
ion of the circles as the third vertex of the triangle.

Again GeoGebra is perfect for th
i
s construction.

Double
-
click on any of the GeoGebra WebStart icon, GeoGebra Installer icon or links to GeGebra4/GeoGebra5. The
GeoGebra tool opens the standard win
dow.

Hide the Axes view.

Use the “New Point” tool to create a point A and point B.

Use the Circle tool “Compass” to draw a circle with the radius = distance between A and B. Activate the tool
“Compass” and click with the mounse on point A and B and the
n on A again to make A as the circle center.

Repeat the same scenario for a similar circle but center on the point B.

Create/fix the upper intersection point between the circles with the tool “Intersect Two Objects”.

Create segment AB, AC and BC wit
h the tool “Segment between Two Points”.

Hide the Circle Object.

Use the tool “Angle” to control the eqvilateral triangle angels. Activate the toll “Angle” and click on the point B, A
and C to get the angle BAC.

25

Theorem of Pythagoras

This is a classic
connection between a geometry construction and algebraic expression.

Make a right
-
angled triangle using these tools:

“Line through Two Points”,

“Perpendicular Line”,

“New Point” and

“Polygon”
.

Use the tool

Angle

to show the right angle. Move the
points! The triangle should remain right
-
angled. The points
of the polygon should be placed in a counterclockwise order.

In order to demonstrate Pythagoras' theorem we must show the square of the hypotenuse and the sum of the
squares of the shorter sid
es. We hence introduce two variables to store these values.

There is a standard way of writing subscripts and superscripts in GeoGebra; this way of writing is used in a number
of mathematics programs.

Superscripts are written using ^

x^2 is shown like
this
x
2

or use the symbol table opened by the α icon in the
right end of the Input field

Subscripts are written using _

c_1 is shown like this
c
1

You write a ^ by pressing
Shift
Alt
^
, the character ^ may not show up until you press the next character or

spac
ebar.

Use the I
nput
field

at the bottom of the window to s
tore the square of the hypotenuse in the variable
2

and the sum of the squares of the shorter sides in
2
+c
2
.

O
bserve the values of

and
I the algebra view
as you move the points of the triangle.

26

Ar
ea

of a triangle

Another fundamental connection between geometry and algebra is calculation of the area of a triangle.

Open GeoGebra in standard view with algebra view
, input field and coordinate axes (
View

Create 2 horizontal lines: Write e.g. y=

-
1
(line name a)
and y = 4
(line name b)
in the Input field.

Create two points (tool “New Point”) on line y =
-
1
(
point

name

A and B)
and one point on the line y = 4

(
point
name
C)
. Connect the point
s

with
the tool “Segment between Two Points 2”

(sepment name c, d and e)
.

Dra
w

a perpendicular line between th
e point on line y = 4 (
point C) and the line y =
-
1.

Use the tool “Intersect Two Objects” for the intersection point between the perpendicular
line
and line y =
-
1 (
point
D).

Use the tool “Segment between Two Points 2” between point C and D.

Open the “Objects Properties” for the
segment CD and make the style as “dashed line”.

Hide the lines y =
-
1

(line a),
y = 4

(line b) and the perpendicular line. R
ight click the objects and
uncheck the
“Show Object”.

Rename the segment between A and B to
the
name
b

or
base

and the segment between C and D to
the name
h

or
height
.
Check the “Object Properties”/“Show label”/Name & Value” for the segment b and h.

Wri
te the triangle area formula in the Input field: Area = base*height/2.

Use the “Insert Text” tool to create a text “Area =” and the Area objects value as a Latex Formula.

Move the point
C along the (now hidden) line b (y = 4) and the points A and B along t
he (now hidden) line a (y =
-
1). Observ how
the “height” is following and can be both “inner” and “outer” of the triangle.

Area of a parallelogram

Area of rectangular prisma

Area of a cylinder

Area of a cone

Area of a pyramide

Area of a sphere

27

Making a demonstration

28

Li
near functions, polynomial

of 1
st

degree

T
he term
linear function

is someti
mes used to mean a first
-
degree polynomial function of one variable
. These
functions are
known as "linear" because they ar
e precisely the functions whose graph in the Cartesian coordinate
plae

is a straight line.

Such a function can be written as

y

=
k
x

+
l

(
y

y
1
)

=
k
(
x

x
1
)

a
x

+
b
y

+
c

= 0

The form y = kx + l is called
slope
-
intercept form
, where
k and l

are real constants

and
x

i
s a real variable. The
constant k is often called the slope or gradient, while l is the y
-
interept
, which gives the point of intersection
between the graph of the function and the
y
-
axis. Changing k

makes the line

steep
er or shallower, while changing l

moves the line up or down.

The form y

y
1

= k(x

x
1
)
is called
point
-
slope

form

where k is the is the slope and (x
1
,y
1
) is a given point on
the line.

The form ax + by + c

= 0
is called the
standard form

where a, b and c

are real values, coefficients.

Slope
-
intercept form

Ope
n GeoGebra in standard view with algebra view
, input field and coordinate axes (
View

Create sliders for k and l.

Write y = k*x + l in the input field.

Exercise the line with
different values for k and l on the slider
s
.

Point
-
slope form

Open GeoGebra in standard view with algebra view
, input field and coordinate axes (
View

Create a point
A = (x1, y1)

Write
x1 = x(A) in Input field

Write
y1 = y(A) in Input field

Create a slider for k.

Write y

y1
= k*(x

x1)
in the I
nput field.

29

Standard form

Open GeoGebra in standard view with algebra view
, input field and coordinate axes (
View

Create 3 sliders a, b and c.

Write a*x + b*y + c = 0 in the Input field.

Make a checkbox for the line ax + by + c = 0

Write the “slope
-
intercept form” translation y =
-

x
-

of the standard form
ax + by + c = 0 in the Input field.
Make a checkbox for the line y =
-

x
-

.

Change the values of a, b and
c and toggle between the two checkboxes.

Check the algebraic expression for y =
-

x
-

for the value b = 0!

rat
ic funct
ions, polynomial

of 2
nd

degree

A quadratic equation is an equation of a polynomial of degree two. When graphed, a quadrati
c equation makes a
parabola
with a vertical

symmetric axis

or
“mirror line”
.

A quadratic function can be expressed in three formats:

f(x) = ax
2

+ bx + c

is called the
general form
,

f(x) = a(x

x
1
)(x

x
2
)

is called the
factored form
, where
x
1

and
x
2

are the roots of the quadratic equation,

f(x) = a(x

x
0
)
2

+ y
0

is called the
vertex form

(or
standard form
), where
x
0

and
y
0

are the x and y
coordinates of the vertex, respectively.

30

To convert the
general form

to
factored form
, one needs only the quadratic formula to determine the two roots
x
1

and
x
2
.

To convert the
general form

to
standard form
, one needs a process called completing the square.

To convert the
factored form (or standard form)

to
general form
, one needs to
multiply, expand and/or
distribute the factors.

General

form

The general

form of a quadratic equation is
f(x) = ax
2

+ bx + c

or

y = ax
2

+

bx + c

where

a
,
b

and
c

are
constant coefficients and
a≠0
.

Open GeoGebra in standard view with algebra view
,
input field and coordinate axes (
View

Create 3 sliders a, b and c.

Write
f(x) = a*x^2 + b*x + c

or

y

= a*x^2 + b*x + c

in the I
nput field.

Exercise the line
with different values for a, b and c
l on the slider
s
.

In this example a = 1, b =
-
2 and

c =
-
3
.

Roots of

the equation
ax
2

+ bx + c = 0

Graphic solution

Use the tool “Intersect Two Objects” to find the intersection A and B between the parabola y

= a*x^2 + b*x + c

for the current values a = 1, b =
-
2 and c =
-
3: y = x
2

2x

-
3 and the x
-
axi
s: y = 0. The x
-
coordinates for those
intersection
s

A and B are the roots to the equation: x
2

2x

3 = 0.
x =
-
1 and x = 3

31

The discriminant of a quadratic equation is used to determine if a quadratic equation has real or complex roots. The
expression for the discriminant is
b
2

4ac

If the discriminant is positive, the quadratic equation has two real roots. If the discriminant

equation has one real root. If the discriminant is negative, the quadratic equation has two complex roots.

In “
Begin
with GeoGebra 3

, chapter “
GeoGebra CAS solving equations

there is a complete description of the quadratic
formula

and the use of GeoGebra to solve equations.

Exercise

Give the sliders
new values for a, b and c
, giving the discriminant value zero (one real “double” root) and negative
value (no real roots = no intersection between the parabola and the x
-
axis)

Intercept form

The intercept form or factored

form of a parabolic equation is
y = a(x
-
x
1
)(x
-
x
2
)

where
x
1

is one x
-
intercept of the
x
2

is the other x
-
intercept, and
a

indicates how steep the sides of the quadratic equation are. If
x
1

= x
2
, the quadratic equation intercepts the x
-
axis only once. Not all quadratic equations can be described using
the x
-
intercept form.

Open GeoGebra in standard view with algebra view
, input field and coordinate axes (
View

Create 3 sliders a, x
1

and x
2
.

In the Imput field, write the expression y = a*(x

x_1)*(x

x_2)

Vertex form

The
vertex form

of a parabolic equation is
y
-
y
0

= a(x
-
x
0
)
2
. The vertex of the quadratic equation is at the point
(x
0
,y
0
)
.
a

shows how steep the sides of the quadratic equation are. Click on the points on the sliders in manipulative
4 and drag them to change the figure.

Open GeoGebra in standard view with algebra view
, input field and coordinate axes (
View

Create 3 slid
ers a, x
0

and y
0
.

32

Using a powerful marku
p

language

LaTeX

is a document markup language

and document preparation system for the
TeX

typesetting

program. The
term LaTeX refers only to the language in which documents are written, not to the editor used to write those
documents. In order to create a document in LaTeX,

a .tex file must be created using some form of
text editor
.
While most text editors can be used to create a LaTeX document, a number of editors have been created specifically
for working with LaTeX.

In Geogebra
there is a
LaTeX

Formula editing scrollist
built in in the tool “Insert Text”. If the
LaTeX

Formula is
checked the edit window can be used as a
LaTeX

enabled editor.

This textelement in the preview window can then be configured with tool “Insert Text” Object Properties in
GeoGebra.

The
\

(back
-
slash

character) is heavily used in the LaTeX script language.

Some important LaTeX commands are explained in following table. Please have a look at any LaTeX documentation
for further information.

LaTeX input

Result

a
\
cdot b

a

b

\
frac{a}{b}

ab

\
sqrt{x}

x

\
sqrt[n]{x}

x

n

\
vec{v}

v

\
overline{AB}

AB
−−−

x^{2}

x
2

a_{1}

a
1

\
sin
\
alpha +
\
cos
\
beta

sin
α
+cos
β

\
int_{a}^{b} x dx

baxdx

\
sum_{i=1}^{n} i^2

ni
=1
i
2

http://en.wikipedia.org/wiki/Help:Displaying_a_formula#Basics

LaTeX
FormulaText[Object]

A special feature in LaTeX is
the command
FormulaText[
Object
]
. This command gives a LaTeX formula for the
Object.

In GeoGebra the Object can be an expression invoked in the Input field. e.g. a quadratic function. In this example
y = ax
2

+ bx + c.

This object will
get a
n

ordinary GeoGebra name in alphabetic
order, e.g.
e
.

The name can b
e renamed
as all GeoGebra objects, e.g. to the name
polynomial
.

The three coefficients a, b and c are implemented as sliders.

In the Edit window in the “Insert Text”

tool, check LaTe
X Formula. Between the start
-
end
\$ \$

symbol
for the LaTeX
expression, open the Objects scrolllist at the bottom of the Edit window and select the object
e
.
The text object e
is then translated to LaTeX and presented in the Graphics view

where the “Insert T
ool” has been positioned.

The text object for
y = ax
2

+ bx + c

with the current values for a, b and c is presented in LaTeX format in
Preview window.

33

The text object e:

y = 2x
2

x

1 can be configured with the properties
Text

and
Colo
r under “Object Properties”.

The coefficients a, b and c can be given new values by the sliders and the current text object for the polynomial is
presented.

In the Edit window in the tool “Insert text” you can also create a LaTeX static text object writt
en
between “ “ characters, e.e “Polynomial 2
nd

degree:”.

Another way to create the LaTeX formula for the polynomial expression e, is to write the command
“” +
Formula[e]

or

“” + FormulaText[polynomial]

in the tool “Insert text” Edit window.