an introduction to geogebra.pdf

8/9/2019 An Introduction To GeoGebra.pdf

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An Introduction To GeoGebra

Table of Contents

1. How to Get Started with GeoGebra ......................................................................................................................3

2. GeoGebra 4.0 (August 2011)................................................................................................................................. 4

3. ntrodu!tion to the GeoGebra4.0 nterfa!e..........................................................................................................."

4. Getting to #now The Gra$hi!s %iew.....................................................................................................................&

'. Changing How Things oo* GeoGebra+s ,b-e!t ro$erties ..........................................................................11

". Algebra 1 and Algebra 2 Things..........................................................................................................................14

/. Geoetr Things..................................................................................................................................................2/

. $loring Conditional Hide and Show................................................................................................................3'

&. Creating Custo Tools........................................................................................................................................3

10. re5Cal!ulus and Cal!ulus Things.................................................................................................................. 40

11. Creating 6nai! 7orsheets..........................................................................................................................4/

12. Custoi8ing the Toolbar................................................................................................................................... 4&

13. 9seful ins and nforation* :our GeoGebra to5do ist.............................................................................. '1

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1. How to Get Started with GeoGebra

If you have not downloaded GeoGebra, open your browser and go to the GeoGebra Websitehttp://wwwgeogebraorg

I rarely have any proble!s using the W"#$TA%T on the networ&ed co!puters at !y school, norin individual wor&shops with participants 'owever, if you are having trouble, let !e &now, and Iwill supply on off(line installer for you to use

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ou want to clic& on the Thread *in& shown below

4lic& on the webstart lin& shown below This will install GeoGebra- #eta

The only thing you are as&ed to do is to report any bugs on the GeoGebra- foru!

%e!e!ber5 After August 21, you will follow the procedure in $ection 1to get GeoGebra-started by going to the GeoGebra Website http://wwwgeogebraorg, clic&ing on the 6ownloadbutton, then clic&ing on the Webstart button on the ne7t page

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3. ntrodu!tion to the GeoGebra4.0 nterfa!e

The dyna!ic !athe!aticssoftware GeoGebraprovides si7 differentviews of !athe!aticalob9ects as shown in thefigure at right Three ofthese views theGraphics ;iew 2, The4o!puter Algebra ;iew,and the 4onstruction


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4. Getting to #now The Gra$hi!s %iew

The purpose of this first activity is to practice using the Tool #ar and to get co!fortable wor&ing

with ob9ects in the Graphics ;iew

1 To begin with, hide theAlgebra View There are three waysyou can do this:

ou can go to the ;iew ?enu @ Algebra,

or 4trl $hift A

or

ou can clic& on the s!all icon in the upper rightcorner of the Algebra ;iew Window

2 Be7t, show the Coordinate Grid . There are two ways youcan do this:

ou can go to the ;iew ?enu @ Grid,or

Bow that the Algebra ;iew is closed, you can clic& onthe s!all grid icon in the upper left corner

A:> 6raw a $i!ture with GeoGebra

Cse the !ouse and various tools fro! the Tool#ar to draw figures on the drawing pad .egs=uare, rectangle, house, tree,D To accessthe tools, you !ust clic& on the s!all triangle toopen the tools, then select the tool you want to

use

on!t forget to read the toolbarhelp if you don!t know how touse a tool.

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"hings to try...

Try changing the colors and siFes of different ob9ects4lic& on an ob9ect, then e7plore these !ini(tools


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'. Changing How Things oo* GeoGebra+s ,b-e!t ro$erties

"very ob9ect that you create in GeoGebra has its own PROPERTIES or attributes .color,thic&ness, labels, algebraic representation that can be changed in the PROPERTIES DIALOGwindow

How to a!!ess the ,b-e!t ro$erties

ou have already seen one way tochange the ob9ect properties4lic&ing on the s!all button in theupper right corner of the Graphics;iew will display an b9ect


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'owever there are four other wasthat youcan access a !ore detailed b9ect


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Cnder the $tyle Tab, notice how the 3illing isset to 28JK ou should e7plore the *ine $tylepull(down !enu and the 3illing ?enu, as wellas the Inverse 3illing chec& bo7 It+s o&youwill not hurt anything5

Cnder the Algebra Tab, the e=uation for! can

be selected

These are the tabs that you will use !ostoften We will e7plore the Advanced Tab, ti!eper!itting Anyone who would li&e to e7plorethe $cripting Tab, I can help you individually

"hings to try...

$elect different ob9ects fro! the list on the left hand side and e7plore the availableproperties tabs for different types of ob9ects

$elect several ob9ects in order to change a certain property for all of the! at the sa!eti!e 'old the Ctrl(&ey .?ac$: Ctrl(clic& pressed and select all desired ob9ects

$elect all ob9ects of one type by clic&ing on the corresponding heading $how the value of different ob9ects and try out different label styles 4hange the properties of certain ob9ects .eg color, style,D

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". Soe 9seful Short!uts

There are a handful of things you will do over and over in GeoGebra Theses things are soco!!on, that there are &eyboard shortcuts to !a&e doing this things easier 'ere is a short listof useful shortcuts

To ?ove the Graphics ;iew, hold down the shift &ey, then clic& and hold and drag your!ouse

To Loo! in or out, you can scroll the scroll wheel on your !ouse This is ctrl / on alaptop ou can also right(clic&, hold, and drag a selection bo7 in the graphics view ouwill Foo! in onto this bo7

To Cndo so!ething you have done, you can press the &ey co!bination 4trl(F ften ti!es, you will need to enter


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/. Algebra 1 and Algebra 2 Things

ie a Gra$hing Cal!ulator

GeoGebra can be used as a graphingcalculator 'owever, better than a graphingcalculator, GeoGebra can graph i!plicitfunctions .such as x = 3 and ine=ualities.such as x + y < 3

pen a new window showing the Algebra ;iewand the Graphics ;iew Graph the followinge=uations and ine=ualities, then graph anything

of your choosing:

x 2 (x 3)2+ y2< 4 y x2 4 y > (x 1)(x)(x + 2) x2+ x y + y2= 1 g(x) = (x 1)(x + 2)(x + 3)

1)


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Tra!ing to a S$readsheet

pen a new window, showing the Graphics ;iew and the $preadsheet ;iew

1 Graph the line y = 2x 12 Cse the


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Creating A =ando ine

1 In the Input *ine, type in the following, one at a ti!e, pressing "nter after each

m = Ra!"m#et$ee%&22

= Ra!"m#et$ee%&**

y = m x + .note space between ! and 7 ou can also use the U todenote !ultiplication

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nhan!ing the =ando ine

We are going to ta&e this rando! line and create a 6yna!ic Wor&sheet that students could useto practice finding the e=uation of a line #egin by hiding the Algebra ;iew .;iew ?enu @

Algebra and showing the Grid .;iew ?enu @ Grid

1 Go to the b9ect


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> In the 4hec& #o7 window that appears, giveyour 4hec& #o7 a na!e .4aption, and selectthe ob9ects you want to be controlled by the4hec& #o7 by either clic&ing on the ob9ects inthe Graphics ;iew, or by using the


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Creating a 6nai! 7orsheet

A 6yna!ic Wor&sheet is a webpage with aGeoGebra Applet that students interact with

ou need to do nothing special 4reating thiswebpage is as easy as saving your GeoGebrafile

1 To create a 6yna!ic Wor&sheet, go tothe 3ile ?enu @ "7port @ 6yna!icWor&sheet as Webpage A window li&ethe one below will appear

2 After you have filled in the Title,

Author .C, and any te7t orinstructions you would li&e toappear on the webpage, clic& onthe Advanced Tab

) $ave the file so!ewhere whereyou will find it easily5

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The finished product Cpload this file to your teacher webpage, wi&i, or blog

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inear rograing

*inear progra!!ing is a staple of Algebra 2 curricula GeoGebra can be used to e7plore thesolutions of these proble!s with its dyna!ic capabilities

4onsider the proble! fro! http://wwwpurple!athco!/!odules/linprog)ht!

A calc#lator co%pan& prod#ce a cienti'ic calc#lator and a grap(ing calc#lator.Long)ter% projection indicate an e*pected de%and o' at leat +,, cienti'icand -, grap(ing calc#lator eac( da&. eca#e o' li%itation on prod#ctioncapacit&/ no %ore t(an 0,, cienti'ic and +1, grap(ing calc#lator can be %adedail&. To ati'& a (ipping contract/ a total o' at leat 0,, calc#lator %#c( be(ipped eac( da&.

I' eac( cienti'ic calc#lator old re#lt in a 20 lo/ b#t eac( grap(ing

calc#lator prod#ce a 23 pro'it/ (ow %an& o' eac( t&pe (o#ld be %ade dail& to%a*i%i4e net pro'it5

*et*be the nu!ber if scientific calculators produced, and &be the nu!ber of graphingcalculators produced

The constraints are as follows:

*V 1--&V H-

* 6 2--

& 1E-* $ & V 2--

I want to !a7i!iFe (2* 8&sub9ect to the given constraints

*et+s solve this using GeoGebra

1 #egin by showing the Algebra ;iew, the Graphics ;iew, and the $preadsheet ;iew $howthe 4oordinate A7es

2 We need to change the scales on the a7es and label the

a7es To do this, we need to access the b9ect


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) %egardless of the !ethod youuse to access the Graphicssettings, clic& on the Graphics tabat the top of the window ?a&esure you clic& on the #asic tabunderneath that tab Cse thesewindow settings as a place tostart We can custo!iFe ourwindow settings later

4lic& on the 7A7is tab, and !a&ethe changes shown in the figure

8 4lic& on the yA7is tab and !a&ethe sa!e types of changes there

Cnder both of these tabs, you !ay wantto consider chec&ing the bo7 4lic& 4*$" when finished!a&ing desired changes

E In the Input *ine, type in the following constraints:

1 < x < 2

, < y < 1-

x + y > 2

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H All of your ine=ualities are shaded the sa!e color To changehow the different regions are shaded, we need to access theb9ect


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inear rograing (An Alternati@e Solution)

$o!e fol&s will not care for graphing the ine=ualities, and will rather the students construct thefeasible region 'ere is another way to solve this proble!

1 Cse the sa!e settings as before Graph each of the following e=uations one at a ti!e

x = 1 y= , x = 2 y= 1- x + y = 2

2 Cse the Intersection


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Changing araeters 9sing Sliders

Creating Sliders

ne of the beautiful things about GeoGebra is the ease withwhich you can create a slider Another beautiful thing aboutGeoGebra is that any co!!and that ta&es a nu!ber as aninput .for e7a!ple: Circle%9"it ra!i:sor0"lyg"%9"it 9"it :mer "; si!escan becontrolled by a slider

There are two ways to create a slider:

1 #egin by activating the $liderTool To use the tool, clic&

anywhere is an open space in thegraphics view

In the window that appears, giveyour slider a na!e .no spaces Ifyou are using the slider to controla rotations, select the Anglebutton 6eter!ine the interval foryour slider Cnder the $*I6"%tab, you can control the siFe ofthe slider

2 Another way to create a slideris by entering the na!e of theslider in the Input *ine andpressing "nter The slider will appear in the Algebra ;iew4lic& on the s!all e!pty circle to un(hide the slider nceyou have created the slider, e7plore its ob9ect properties

Things to Tr...

4reate a slider then enter each of these in the inputline: Circle%(12) 0"lyg"%() (1&1) ayl"r0"ly"mial% si(x) 1 ( 2)

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araeters of a inear uation

ou have already created a file that produces a line with a rando! slope and a rando! y(intercept, enhanced the file by adding color, a te7t bo7, and a chec& bo7, then created a6yna!ic Wor&sheet for your students to practice We will now create a file that can be used for

the IB$T%C4TIB of writing e=uations of lines#egin by opening a new GeoGebra file $howi the Algebra ;iew and the Graphics ;iew $howthe Grid and the A7es in the Graphics ;iew

1 4reate sliders mand 2 In the Input *ine, enter lie y = m x +

6rag the sliders and observe the changes in the graph of the line

nhan!ing the Constru!tion

) 4reate the intersection point of the line and the y(a7is byactiviating the intersection point tool, then clic&ing on the lineand the y(a7is ou can use the co!!and?tersect%lie y.xis #y default, this will be na!ed

A

4onstruct a point at the origin #y default, this will bena!ed #

8 4onstruct a seg!ent fro! A to # Cse the seg!ent+s

b9ect


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$loring araeters of a Buadrati!olnoial 9sing Sliders

*et+s e7pand on the linear and construct

;(x) = a x@2 b 7 c by using sliders to !odifythe para!eter values

#egin by opening a new GeoGebra window, showingthe Algebra ;iew and Graphics ;iew, and showingthe A7es and Grid

1 4reate three sliders, a, , and c2 In the Input *ine, enter ;(x) = a x@2 + x + c.re!e!ber to leave spaces or use

the U sy!bol for !ultiplication) 4reate a Te7t #o7 that shows the e=uation of the

parabola

6rag the sliders and observe what happens "nhanceyour construction as you see fit

Things to tr...

4onstruct the verte7 of the polyno!ial so it!oves with the parabola

"nter 7xtrem:m% ; in the input line Whatdoes it doK

"nter R""t% ; in the input line What does itdoK

"nter aget% ; ( 1 ;(1) ) in theinput line What does it doK

"nter 6erivative% ; in the input line, and drag the sliders "nter ?tegral% ; in the input line, and

drag the sliders 4hange the setting on your sliders as shown in

the picture "nter8act"r%;

in the Input *ine

and drag the sliders What do you thin&K "7port this as a 6yna!ic Wor&sheet

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Gra$hing a olnoial 9sing =oots

It !ay be useful to graph a polyno!ial using it roots pen a new window $how the Algebra;iew and the Graphics ;iew $how the coordinate a7es

1 Cse the point tool to place three points on the 7(a7is #ydefault, these points will be labeledA, , and C

2 In the input line, enter ;(x) = ( x x(.) ) ( x x(#) ) ( x x(C) )

) ou !ay need to change the scale of the y(a7is byclic&ing, holding, and dragging the y(a7is until you cansee the entire graph

Things to Tr...

"nhance your construction by adding color, linethic&ness, and a te7t bo7 that displays the functione=uation

Try the co!!and 7x9a!% ;(x) and display theresult in a te7t bo7 in the Graphics ;iew

4reate a =uartic with four roots

4reate a =uadratic using roots, and construct the line of sy!!etry

Try so!e of these co!!ands: 7xtrem:m%;(x)and ?;lecti"0"it%;(x)

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Creating a a!toring ra!ti!e A$$let

"very student in Algebra 1 will need to practice factoring =uadratics at so!e ti!e or anotherThe following steps will wal& you through how to create your own 03actoring


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H 4reate another Te7t #o7, this ti!e, selecting onlyob9ect f

We will need to insert a 4hec& #o7 to show and hideour answer Activate the 4hec& #o7 Tool

1-Give your chec& bo7 a caption.so!ething li&e 04hec& Answerand clic& on the ob9ects in theGraphics ;iew that you want tobe controlled by the 4hec& #o7

11'ide the Algebra ;iew Cnder the ptions ?enu @3ont $iFe select a font siFe of 2- Arrange the te7tbo7es and the chec& bo7 close to each other in theupper left of the Graphics Window, then re(siFeyour window

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12Cnchec& the 4hec& #o7 soyour answer is hidden"7port this file to a6yna!ic Wor&sheet Typeinstructions in the bo7 thatwill appear above theapplet

1)4lic& on the AdvancedTab The only thing youshould chec& on this pageis the $how Icon to %eset4onstruction

1 $ave the resulting ht!l fileso!ewhere where you canfind it easily This file is

what you would upload toyour teacher webpage orwi&i

Things to Tr...

?odify this so the leadingcoefficient is not 1

?odify this so it is a3actoring a 6ifference ofTwo $=uares practice, or a3actoring a


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Creating a a!toring A$$let* A 6ifferent 7a

1 pen a new GeoGebra window $how the Algebra ;iew and the Graphics ;iew 'ide thecoodinate a7es

2 4reate two rando! integers a and b as before

) 4reate the function ;(x) = (x+a)(x+), and hide the graph as before

4reate the function g(x) = sim9li;y% ;(x) , and hide the graph as before

8 4ontinue as before fro! $tep >

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The Seuen!e Coand

If the Algebra ;iew is not open, open it by ;iew ?enu @ Algebra

1 Type each of these co!!ands one at a ti!e in the Input *ine, pressing "nter after each#" 4A%"3C* T


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ibrar of Algebrai! un!tions

Apart fro! polyno!ials there are different types of functions available inGeoGebra .eg trigono!etric functions, absolute value function, e7ponential

function 3unctions are treated as ob9ects and can be used in co!bination withgeo!etric constructions

To access the list of function,clic& on the list icon in thelower left corner of thewindow

ou can clic& on the individualfunction you wish to use, youyou can 9ust type it in yourself

To close the function window,9ust clic& on the list icon again

!ursion into hsi!s* Su$er$osition of Sine 7a@es

$ound waves can be !athe!atically represented as a co!bination of sine waves "very !usical

tone is co!posed of several sine waves of for! &( t)=asin t+ )

The a!plitude ainfluences the volu!e of the tone while the angular fre=uency [ deter!ines thepitch of the tone The para!eter Q is called phase and indicates if the sound wave is shifted inti!e

If two sine waves interfere, superposition occurs This!eans that the sine waves a!plify or di!inish eachother We can si!ulate this pheno!enon withGeoGebra in order to e7a!ine special cases that alsooccur in nature

The lower case Gree& *etters and other special sy!bolscan be found in the Input *ine when you begin to type

To insert a subscript, use anunderscore

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Gra$hing Trig un!tions

pen a new widow, show the Algebra ;iew and the Graphics ;iew

1 #egin by creating three sliders: a_1 FG1 andHG12 "nter the sine function:g(x)= aG1 si(FG1 x + HG1)) 4reate three !ore sliders:a_1 FG1 andHG1 "nter another sine function:

I(x)= aG2 si(FG2 x + HG2)8 4reate the su! of both functions:

s:m(x) = g(x) + I(x)

Be7t, we want to change the a7es so they wor& better with Trig3unctions %ight(clic& in on open space, and in the !enu thatappears, select Graphics

In the Graphics


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. Geoetr Things

lotting a Gra$h of Two %ariables

*et+s construct an e=uilateral triangle and plot a graph of the area vs the side length

1 #egin by opening a new GeoGebra window $how the Algebra ;iew, the Graphic ;iew,and the G%A


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In the Algebra ;iew, the sides of the e=uilateral triangle asdisplayed as a andc The area of the triangle isdisplayed as 9"ly1

8 4lic& in the Graphics 1 Window *oo& for the


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Changing the Ste$s in a Constru!tion* The Constru!tion roto!ol.

*ets construct a rectangle, and then e7plore how we !ight be able to change the order of thesteps in the construction using so!ething called The 4onstruction


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1-%ight(clic& on one of the colu!n na!es in thetable 4lic& on 4aption and clic& on#rea&point The #rea&points will allow you to0Group steps of a construction to be shownall at one ti!e

11As shown below left, I have selected threebrea&points, therefore, !y construction willonly have three steps $teps 1() will beperfor!ed together, as will steps (H andsteps (11 #efore this happens, I need toright(clic& on one of the colu!n headers, andselect $how nly #rea&points .below right

Things to tr...

Try creating a construction of an e=uilateral triangle and changing the order of the steps

Try creating a construction of a s=uare and changing the order of the steps

Thin& of how you could use this with students

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olgons and i

#egin by turning off the labeling Go to ptions ?enu @ *abeling @ Bo Bew b9ects

1 4reate point .=()and point #=(1)

2 6raw a circle 4ircle Tool with center A and radius point # or byusing the co!!and Circle% . # / ou !ay want toad9ust your a7is scale by Foo!ing in

) Cse the $lider Tool to create a slider nwith incre!ent 1 Activate the tool, clic& in an openspace, then !odify the slider!enu as shown

%otate # about A by angle )>-]/n ou can use the

%otate Tool, or you can type in the co!!andR"tate%# 3JKL . into the Input #ar%e!e!ber Alto

8 Cse the %egular


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Transforation b Eatri!es

If the $preadsheet ;iew is not open, open it by ;iew ?enu @ $preadsheet ;iew

$et the labeling properties to label Bew


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Transforations 9sing ages

nserting an age in GeoGebra

With GeoGebra, users have had the ability to inserti!ages into the Graphics ;iew for !any years Inorder to insert an i!age, be certain you have ani!age saved on your co!puter to use I find studentsli&e to see !y picture distorted5

To insert an i!age, begin by activating the InsertI!age Tool 4lic& in on open space in the Graphics;iew This will set the lower left corner .corner 1 of your i!age

When you insert an i!age, the corners are nu!beredcountercloc&wise beginning with the lower left corner

#y attaching the corners to 3ree


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=esi8ingF =efle!tingF and 6istorting a i!ture

In this activity you will to learn how to resiFe an inserted picture to a certain siFe and how toapply transfor!ations to the picture in GeoGebra

1 #egin by finding a picture online and saving it so!ewherewhere you can easily find it

2 pen a new GeoGebra window and hide the Algebra ;iew) Insert the picture using the Insert


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=otating ages 9sing a Slider

1 #egin by opening a new GeoGebra window, hidingthe Algebra ;iew, setting the *abeling ptions to Bew


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6ilating ages using a Slider

1 #egin by inserting an i!age, and attaching the i!ageto three points

2 4onstruct a slider Ba!e it Scale8act"r

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&. $loring Conditional Hide and Show

This is so!ething that can be anywhere We will wal& through a couple of e7a!ples that willshow you how to !a&e ob9ects appear and disappear based on so!e other ob9ect

1 #egin with a new GeoGebra window $how theAlgebra ;iew and the Graphics ;iew *eave thecoordinate a7es visible

2


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Self5Che!ing =efle!ted oint

These steps will wal& you through the construction of a file that &ids could use to practiceidentifying the reflection of a point in a line

1 #egin by pening a new GeoGebra file 'ide the Algebra ;iew $et point capturing to3i7ed on Grid $et labeling options to Bew


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1) Access the b9ect properties of the Te7t #o7that says Good Wor&5 Go to the AdvancedTab

In the 4onditions to $how b9ect line,type in .'= =# QQ a Typing Y Y is how you test for two things to be e=ualTyping in `` is the logical AB6

#y typing this, we are telling the te7t bo7 to appearonly when point # is dragged to point A+, and whenthe 4hec& #o7 is chec&ed 6 BT 4*$" T'"WIB6W "T5

14lic& on text2on the left side of the b9ect


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Conditional Coloring

ou are going to construct a circle and one point that is not the center or radius point of thecircle 6epending where the point is located with respect to the lines will deter!ine the color of

the point

1 #egin by opening a new window $how the Algebra;iew and the Graphics ;iew 'ide the coordinatea7es $et the *abeling ptions to Bew


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10. Creating Custo Tools

This is probably !ost useful in Geo!etry, especially when you haveto repeat a nu!ber of constructions over and over 4reatingcusto!s tools is easy to do *et+s !a&e a 4usto! Tool that willconstruct the centroid of a triangle .ignoring the fact that there I$ acentroid co!!and already5

1 #egin with a new GeoGebra window 'ide the Algebra ;iewand the coordinate a7es $et the *abeling ptions to Bew


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Bow that our construction is co!plete, we want to !a&e atool that will construct the centroid point 3 for us

> Cnder the Tool ?enu, select 4reate Bew Tool

E In the window that pops up, you can select yourutput b9ects .for us, point 3 fro! the pull

down !enu, or you can clic& on the utputb9ects in the Graphics ;iew

H 4lic& on the Input b9ects Tab The Inputb9ects are selected auto!atically for you

4lic& on the Ba!e ` Icon Tab ou need togive your tool a recogniFable na!e ou shouldalso provide so!e Tool 'elp 'ints that will help

others use the tool ou do not need to fill inthe 4o!!and Ba!e that is done for you

Things to Tr...

4reate a custo! tool for the other threeclassical triangle centers

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11. re5Cal!ulus and Cal!ulus Things

araetri! uations

1 pen a new GeoGebra Window $how the Algebra ;iew, the Graphics ;iew, and thecoordinate a7es

2


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ie!ewise un!tions

1 pen a new GeoGebra Window $how the Algebra ;iew, the Graphics ;iew, and thecoordinate a7es

2


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=estri!ting the 6oain

ou can li!it the graph of a function to a particular do!ain by using the 8:cti"%co!!and

1 #egin by opening a new GeoGebra Window


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7oring 7ith %e!tors

*et+s e7plore so!e of the ways vectors can be constructed and used inGeoGebra

1 #egin by opening a new window $how the Algebra ;iew, theGraphics ;iew, the coordinate a7es, and the coordinate grid $etthe *abeling ptions to A** B"W #^"4T$

2 Cse the


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Eore 7oring 7ith %e!tors

1 pen a new GeoGebra Window, setting up your wor&space li&e before In the input line,type in a = (3&1)and = (34) We are using lower(

case letters for our labels When you do this you are creatingvectors

2 In the Input *ine, enter SeA:ece% a + &* *

) This created a list of vectors displayed in the Algebra ;iew'ide these vectors by clic&ing on the s!all circle

In the Input *ine, enter ie%(3 1) What is happening hereK *oo& at thee=uation of the line 'ow are the para!eters of the line related to vector bK

Things to Tr...

4reate a 6yna!ic Wor&sheet illustrating the parallelogra! !ethod for adding two vectors

4reate a 6yna!ic Wor&sheet illustrating how the diagonals of a parallelogra! can bee7pressed as the su! and the difference of two vectors

Graph the function ;(x) = (x&1)(x&2)(x+1)


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Co$le ubers

1 #egin by opening a new GeoGebra window $how theAlgebra ;iew, the Graphics ;iew, and the coordinate a7es

2 Cse the Loo! In Tool to Foo! in on the origin until the y(a7isdisplays fro! ab out () to )

)


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> In the Input *ine, enter the following one at a ti!e #e certain !ove points around toobserve the drag behavior of the results Botice that your results area na!ed with a X

. + # # C . # # L C C L # . # C

E "nter ..in the input line 6rag pointAto the point .-,1 What does that tell you aboutthe value of i K

H 6elete


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Buadrati! -


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H 4lic& on the %ow A header %ight(clic& on the header,and select b9ect 1


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olar Gra$hing lotting olar oints

There are two parts to polar graphing The first part plotting polar points is easy The secondpart graphing functions r. is 9ust

slightly !ore difficult 'owever, once youset up the polar graphing file, you willhave it forever5

1 *et+s begin by setting up a polarcoordinate grid Access theGraphics ;iew b9ect


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8 It !ay be helpful to !a&e thepositive 7(a7is stand out 4lic& onthe #asic Tab, and select #*6on the *ine $tyle pull(down!enu

> To plot points in polar for!, you plot points as you usually do, but you use a se!i(coloninstead of a co!!a Try plotting the following points by typing the! into Input *ine one ata ti!e %e!e!ber Alto for the degree sy!bol, and Altp for \

( 3 Y 4*K ) ( 4 Y ) ( * Y 1 ) ( * Y 12*K )

What is happeningK

E 4reate a slider Try plotting these points, then dragging the slider

( Y 4*K ) ( 4 Y ) ( c"s() Y ) ( Y )

>)


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olar Gra$hing Gra$hing olar un!tions

As things stand right now, we need to 0tric& GeoGebra into graphing polar functions 'owever,tric&ing GeoGebra into doing this is fun, and will test your understanding of polar functions

$uppose we wanted to graph the polar function r(Z)=1+4c"s(*Z) We can+t .yet 9ust type it inand graph it GeoGebra only recogniFes*and &as variable used for graphing functions

Bow, the tric&

1 #egin by setting up your Graphics ;iew coordinate plane in the sa!e way for plottingpolar points

2 Graph the function r(x)=1+4c"s(*x) 'ide thegraph of the function

) Cse the 4urve co!!and to graphC:rve% r(t) c"s(t) r(t) si(t) t 29i

;iola5 Bowwhy does this wor&K That is for you to thin& about5

Things to Tr...

"nhance your s&etch by adding colors

Add a slider and try graphing

C:rve% r(t) c"s(t) r(t) si(t) t If you 9ust did the one before this, plot the point (r() c"s() r() si())/

6rag the slider

Add a slider and e7plore different functions r(x) = c"s( x)

"7plore different functions r(x)/

>


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ntrodu!ing 6eri@ati@es The Slo$e un!tion

*et+s trace the slope function for a given function as anintroduction to the derivative

1 pen a new GeoGebra file $how the Algebra ;iewand 4oordinate A7es

2 "nter the polyno!ial ;(x) = x@2L2 + 1/

) Assu!ing you rena!ed your slope as %, define a new 8


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$loring olnoials

*et+s e7plore a cubic polyno!ial in a couple of different waysWe will first use the traditional co!!ands, then we will usethe new 3unction Inspector Tool

1 #egin by opening a new GeoGebra docu!ent $howthe Algebra ;iew and coordinate a7es

2 "nter the cubic polyno!ial;(x)=/*x3+2x2+/2x1

) 4reate the roots of polyno!ial ': R = R""t% ; /If there are !ore than one root, GeoGebra will producesubscripts for their na!es if you type inR = (e/g/ R1 R2 R3)/

4reate the !a7i!u! and !ini!u! of polyno!ial ':7 = 7xtrem:m% ;

8 4reate the inflection point of polyno!ial ':? = ?;lecti"0"it% ;

$loring olnoials 9sing the un!tionns$e!tor

1 #egin by opening a new GeoGebra docu!ent $how theAlgebra ;iew and coordinate a7es "nter the sa!epolyno!ial as before

2 Activate the 3unction InspectorTool The 3unction Inspector hasonly been fully activated for abouta !onth I a! curious to hear ifyou li&e it5

>>


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Talor Series

3or a given function '.*, you can use GeoGebra to find a Taylor polyno!ial appro7i!ation of '.*at*9c

1 #egin by opening a newGeoGebra window, showing the

Algebra ;iew and the Graphics;iew 6rag the Algebra ;iew ontop of the Graphics ;iew byclic&ing on the header of the

Algebra ;iew, holding anddragging to the top and center ofthe GeoGebra window This willallow you to see the entire Taylor


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8 Access the Graphics ;iew ou should now have so!ething that loo&s li&ethe figure at right

E In the Input *ine, type in the co!!andayl"r0"ly"mial% ;(x)x(.)

H $lowly dragthe slider

Things to Tr...

"nhance your s&etch by changing colors

pen the $preadsheet ;iew and create a table of values for f.7 and for the Taylor


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ower SusF 9$$er SusF and Tra$e8oidal Sus

ou will now create a dyna!ic wor&sheet that illustrates how lower, upper, and trapeFoidal su!s

can be used to appro7i!ate the area between a function and the *(a7is, which can be used tointroduce the concept of integral to students

1 pen a new GeoGebra file $how the Algebra ;iew,and coordinate a7es $et labelingoptions to Bew "nter:l"$ers:m = "$erS:m%; x(.) x(#) /

E "nter:tra9eX"i!als:m = ra9eX"i!alS:m%; x(.) x(#)

H "nter:itegral = ?tegral%; x(.) x(#)

4reate a 6yna!ic Te7t #o7 for each of thesu!s constructed in steps 8(H

>


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1-4reate a chec&bo7 to hide and show each ofthe four su!s created in steps 8(H as well asthe te7tbo7 that goes along with it ou shouldhave four chec&bo7es when done, eachcontrolling one su! and one te7t bo7

When you create your 4hec& #o7, 9ust clic& onthe ob9ects you want to be controlled by thechec& bo7 I clic&ed on uppersu! in the Algebra;iew, and I clic&ed on the Te7t #o7 in theGraphics ;iew

Things to Tr...

Arrange the ob9ects on your screen so si!ilar ob9ects are close together and enhanceyour construction by adding color to the various ob9ects 4onsider !a&ing everything

associated with one particular su!, say, the Cpper $u!, all one color

"7port this to a 6yna!ic Wor&sheet

E-


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ewtonIs Eethod

This ne7t construction will de!onstrate the power of the $preadsheet If the $preadsheet ;iew isnot open, open it 4lose the Algebra view if it is open 4hange the *abeling options to Bo Bew

b9ects

1 $tart by graphing your favorite cubic function,then ad9ust the a7es .shift(drag on the a7is ifneed be so you can see the entire function

2


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Cobweb Gra$h

This is another e7a!ple of the power of the $preadsheet;iew, in that each individual cell is an Input *ine and a

variable at the sa!e ti!e

1 #egin by opening a new GeoGebra window,showing the Graphics Window and the$preadsheet ;iew $et the *abeling ptions toBew


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12. Creating 6nai! 7orsheets

We have already e7portedour GeoGebra 3iles to6yna!ic Wor&sheets Thissection will show you whatthe things on the

Advanced Tab will do

We will start with the CserInterface, and the 4heco7es available there

9ser nterfa!e

6o you want the ?enubar, or Toolbar, or Toolbar help,or Input #ar to show in your appletK If so, you shouldchec& the appropriate bo7

%C*" 1: $how Toolbar and $how Toolbar 'elp should be chec&ed together%C*" 2: "nable $ave,


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un!tionalit

The first two chec& bo7es are self(e7planatory I rarely

chec& the first one .this will allow the user to changeob9ect properties and the li&e I usually chec& thesecond one .so!eti!es labels get it the way

%C*" ): Whenever you create a %ando! file, you!ust chec& $how Icon To %eset 4onstruction This will allow the user to create a new proble!!ade possible by the %ando!#etween co!!and

%C*" : 4hec&ing 6ouble(clic& pens Application Window allows the user to double clic& on theapplet, have it open in a GeoGebra window, and !a&e the file their own Cseful, but be carefulusing it with students

%C*" 8: I have never used the last bo7 Try it, though ou !ay li&e it

%C*" >: If you have properly resiFed your GeoGebrawindow, you will never need to !ess with thesenu!bers

If you use ?oodle, or if you have a class wi&i or ablog, the 4*I


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13. Custoi8ing the Toolbar

4usto!iFing the Tool #ar is so!ething I found usefulin Geo!etry 3or e7a!ple, I !ay want !y students toconstruct a parallel line through a given point, but withthe


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) I li&e to begin by opening up the ?ove ToolIcon by clic&ing on the s!all ne7t to the?ove Tool Icon, then 9ust re!oving A** thetools in the category I do this for each toolThis way, I can select only those tools I want touse

f course, this is 9ust !e ou !ay discover abetter !ethod that suites you5

These are the tools I have included in !y


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14. 9seful ins and nforation* :our GeoGebra to5do ist

1 %e!e!ber to visit the GeoGebra Website: http://wwwgeogebraorg

2 %e!e!ber to register on the foru!: wwwgeogebraorg/foru!/

) %e!e!ber to visit GeoGebra on 3aceboo&: wwwfaceboo&co!/geogebra

%e!e!ber to wor& through the tutorials for beginning to advanced users:http://!athallageswordpressco!/geogebra/

8 %e!e!ber to watch the video tutorials on GeoGebra+s ouTube 4hannel:http://wwwyoutubeco!/user/GeoGebra4hannel

> %e!e!ber to visit the GeoGebra Wi&i:

http://wwwgeogebraorg/en/wi&i/inde7php/?ain

an introduction to geogebra.pdf

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