Tiling a Bulletin Board with Squares
2021-11-25 03:02
This applet demonstrates tiling a rectangular board with identical squares. It explores the relationship between the rectangle's dimensions and the square's side length, illustrating the Greatest Common Divisor (GCD) in tiling problems. Using polygons, vectors, and sequences, users can dynamically adjust the rectangle's size to observe tiling patterns, connecting number theory with geometry.
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.695.0" app="classic" platform="w" id="fc8ea49e-b857-4ff3-ace5-d912b08f9361" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1000" height="800" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0" orientation="1" />
</panes>
<views>
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="1,1" size="200" window="0,0,400,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1000" window="0,0,400,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="299" tab="ALGEBRA" window="1364,27,374,927" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 | 1011 || 1008 1009 || 6" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="16" visible="false" inframe="false" stylebar="false" location="0" size="227" window="915,562,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="1172,356,965,1038" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1" size="150" window="606,66,1215,941" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 | 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" size="300" window="50,50,500,500" />
</views>
<toolbar show="false" 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="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1000" height="800"/>
<coordSystem xZero="114.3780406377052" yZero="170.18328509729395" scale="17.96422917900963" yscale="17.964229179009596"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="2" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<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" tickDistance="1" tickExpression="1"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1" tickExpression="1"/>
<grid distX="0.2" distY="0.2" distTheta="0.5235987755982988"/>
</euclidianView>
<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" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="point" label="A_0">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="40" y="-32" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="DynamicCoordinates">
<input a0="A_0" a1="If[x(A_0) < 4, 4, round(x(A_0))]" a2="If[y(A_0) > -4, -4, round(y(A_0))]"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<ggbscript onUpdate="SetValue(m, 1)
SetValue(n', floor(g/a))"/>
<coords x="40" y="-32" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="(0, 0)" a1="(x(A), 0)" a2="A" a3="(0, y(A))"/>
<output a0="q1" a1="f" a2="g" a3="h" a4="i"/>
</command>
<element type="polygon" label="q1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<element type="segment" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="0" y="40" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
</element>
<element type="segment" label="g">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="31" y="-2"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="32" y="0" z="-1280"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
<caption val="%v cm"/>
</element>
<element type="segment" label="h">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="-32" y="48"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="0" y="-40" z="-1280"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
<caption val="%v cm"/>
</element>
<element type="segment" label="i">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="-32" y="0" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
</element>
<element type="numeric" label="a">
<value val="1"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="1" max="25" absoluteScreenLocation="true" width="200" x="387" y="26" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Polygon">
<input a0="(0, 0)" a1="(a, 0)" a2="(a, (-a))" a3="(0, (-a))"/>
<output a0="q2" a1="j" a2="k" a3="l" a4="m_{1}"/>
</command>
<element type="polygon" label="q2">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<element type="segment" label="j">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-30" y="-24"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="0" y="1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
<caption val="%v cm"/>
</element>
<element type="segment" label="k">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="1" y="0" z="-1"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
</element>
<element type="segment" label="l">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="0" y="-1" z="-1"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
</element>
<element type="segment" label="m_{1}">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-74" y="7"/>
<labelMode val="3"/>
<selectionAllowed val="false"/>
<auxiliary val="false"/>
<coords x="-1" y="0" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<startStyle val="default"/>
<endStyle val="default"/>
<caption val="%v cm"/>
</element>
<element type="button" label="button1">
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="1"/>
<layer val="0"/>
<labelOffset x="140" y="106"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetValue(a, a+1)
SetValue(m, 1)
SetValue(n', floor(g/a))"/>
<font serif="false" sizeM="1.4" size="10" style="0"/>
<dimensions width="33" height="30" />
<caption val="+"/>
</element>
<element type="button" label="button2">
<show object="true" label="true"/>
<objColor r="255" g="0" b="0" alpha="1"/>
<layer val="0"/>
<labelOffset x="96" y="106"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<ggbscript val="SetValue(a, a-1)
SetValue(m, 1)
SetValue(n', floor(g/a))"/>
<font serif="false" sizeM="1.4" size="10" style="0"/>
<dimensions width="33" height="30" />
<caption val="−"/>
</element>
<element type="numeric" label="m">
<value val="1"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="80" y="8"/>
<labelMode val="1"/>
<slider min="1" max="floor(f / a)" absoluteScreenLocation="true" width="250" x="114" y="88" fixed="true" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="numeric" label="n'">
<value val="32"/>
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-2" y="80"/>
<labelMode val="1"/>
<slider min="1" max="floor(g / a)" absoluteScreenLocation="true" width="250" x="28" y="425" fixed="true" horizontal="false" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<command name="Vector">
<input a0="(a, 0)"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1" y="0" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
</element>
<command name="Vector">
<input a0="(0, (-a))"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="false" label="false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="-1" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
</element>
<expression label="n" exp="floor(g / a) - n' + 1" />
<element type="numeric" label="n">
<value val="1"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Polygon">
<input a0="{(0, 0), ((a * m), 0), ((a * m), ((-a) * n)), (0, ((-a) * n))}"/>
<output a0="q3"/>
</command>
<element type="polygon" label="q3">
<lineStyle thickness="0" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.15000000596046448"/>
<layer val="1"/>
<labelMode val="0"/>
</element>
<command name="Sequence">
<input a0="Segment[((i * a), 0), ((i * a), ((-a) * n))]" a1="i" a2="0" a3="m"/>
<output a0="l1" randomResult="{1, 1}"/>
</command>
<element type="list" label="l1">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<command name="Sequence">
<input a0="Segment[(0, ((-a) * i)), ((a * m), ((-a) * i))]" a1="i" a2="0" a3="n"/>
<output a0="l2" randomResult="{1, 1}"/>
</command>
<element type="list" label="l2">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
<angleStyle val="0"/>
</element>
<expression label="text1" exp=""壁報板的邊長分別為 " + f + " cm 和 " + g + " cm，要在壁報板上鋪滿大小相等的正方形，
每個正方形的邊長最長是多少厘米？"" />
<element type="text" label="text1">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<absoluteScreenLocation x="17" y="29"/>
</element>
<element type="image" label="qsip">
<file name="cf29d7dc047bd16cde4cc1ad9af60587/QSIP (smaller).png"/>
<inBackground val="true"/>
<absoluteScreenLocation x="3" y="781"/> <show object="true" label="false"/>
<objColor r="0" g="0" b="0" alpha="1"/>
<layer val="1"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
</construction>
</geogebra>