Construction of Pascal's Triangle using Vectors and Sequences
2020-03-25 02:17
This example demonstrates how to dynamically construct Pascal's Triangle using GeoGebra commands such as Vector, Sequence, and Flatten. By defining base points and vectors, and applying sequence commands to generate the geometric arrangement of binomial coefficients, it visually illustrates the recursive relationships and symmetry of combinations, making it ideal for algebra and discrete mathemati
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.574.0" app="classic" platform="d" id="4fe4f39c-009f-4477-a628-1595f262e70a" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="2160" height="1380" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.15046296296296297" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="false" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="496" window="100,100,700,550" />
<view id="1" visible="true" inframe="false" stylebar="false" location="1" size="1797" window="100,100,600,400" />
<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="325" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
</views>
<toolbar show="true" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1797" height="1164"/>
<coordSystem xZero="668.0000000000001" yZero="597.0" scale="50.00000000000002" yscale="50.0"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="3"/>
</algebraView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2.0" max="2.0" step="1.0"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="A">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="3.9" y="9.18" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="B">
<show object="true" label="true"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-10.42" y="5.5" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="C">
<show object="true" label="true"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-6.84" y="5.5" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="B" a1="C"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="3.58" y="0.0" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="B"/>
</element>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-8.8" y="7.84" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="D" a1="B"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.6199999999999992" y="-2.34" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="D"/>
</element>
<element type="numeric" label="a">
<value val="8.0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="0" max="20" absoluteScreenLocation="true" width="200.0" x="30.0" y="90.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Sequence">
<input a0="Sequence[A + (i * u) + (j * v), i, 0, j]" a1="j" a2="0" a3="a - 1"/>
<output a0="l1"/>
</command>
<element type="list" label="l1">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Flatten">
<input a0="l1"/>
<output a0="l2"/>
</command>
<element type="list" label="l2">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<element type="numeric" label="n">
<value val="36.0"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="Length[l2]" absoluteScreenLocation="true" width="200.0" x="30.0" y="138.0" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Sequence">
<input a0="Element[l2, i]" a1="i" a2="1" a3="n"/>
<output a0="l3"/>
</command>
<element type="list" label="l3">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Sequence[nCr[j, i], i, 0, j]" a1="j" a2="0" a3="a"/>
<output a0="l4"/>
</command>
<element type="list" label="l4">
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Flatten">
<input a0="l4"/>
<output a0="l5"/>
</command>
<element type="list" label="l5">
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Text[("\large" * l5(i)), l3(i), true, true]" a1="i" a2="1" a3="n"/>
<output a0="l6"/>
</command>
<element type="list" label="l6">
<show object="true" label="true"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<angleStyle val="0"/>
</element>
</construction>
</geogebra>