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275 lines
11 KiB
HTML
275 lines
11 KiB
HTML
<!doctype html>
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<meta http-equiv="content-type" content="text/html; charset=utf-8">
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<!-- #BeginEditable "doctitle" -->
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<title>Activity: Dropping a Coin onto a Grid</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Activity: Dropping a Coin onto a Grid</h1>
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<div class="center80">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/coin-1750.jpg" alt="coin 1750" height="133" width="240"></p>
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<p>A few hundred years ago people enjoyed betting on coins tossed on to the floor ... would they cross a line or not?</p>
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<p>A man (Georges-Louis Leclerc, the <i>Count of Buffon</i>, see "<a href="buffons-needle.html">Buffon's Needle</a>") started thinking about this and worked out how to calculate the <a href="../data/probability.html">probability</a>.</p>
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<p>Now it is your turn to have a go!</p>
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</div>
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<p> </p>
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<p>You will need:</p>
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<div class="simple">
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<table align="center" border="0">
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<tbody>
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<tr>
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<td align="center"><img src="../money/images/currency/us1.gif" style="vertical-align:middle;" alt="us 1 penny" height="47" width="48"><br>
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<img src="../money/images/currency/eu1.gif" style="vertical-align:middle;" alt="1 euro" height="45" width="44"> <img src="../money/images/currency/in5.gif" style="vertical-align:middle;" alt="india 5 rupee" height="55" width="55"></td>
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<td>
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<p>A <b>small round coin</b>,</p>
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<p>such as a US penny, a 1c Euro or 5 Rupee.</p></td>
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</tr>
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<tr>
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<td align="center"><a href="images/coin-grid.gif"><img src="images/coin-grid.gif" alt="grid" height="100" width="100" border="0"></a><br>
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</td>
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<td>
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<p>A sheet of paper with a grid of 30 mm squares.</p>
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<p><input onclick="printImg('images/coin-grid.gif')" value="Print this one" type="button"></p></td>
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</tr>
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</tbody></table>
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</div>
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<h2>Steps</h2>
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<div class="bigul">
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<ul>
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<li>Measure the diameter of your coin: ____ mm
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<ul>
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<li>a US Penny is 19mm, a 1c Euro is 16.25mm, a Rs 5 is 23mm</li>
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</ul></li>
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<li>Also measure the spacing of your grid (it may not print at exactly 30mm): ____ mm</li>
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<li>Put your sheet of paper on a flat surface such as a table top or the floor.</li>
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<li>From a height of about 5cm, drop the coin onto the paper and record whether it lands:
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<table border="0">
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<tbody>
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<tr>
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<td>
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<p><span class="large">A:</span> Completely inside a square (not touching any grid lines)</p>
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<p><span class="large">B:</span> Crosses one or more lines</p></td>
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<td><img src="images/coin-grid-1.gif" alt="grid: coin A inside, and coin B on" height="103" width="188"></td>
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</tr>
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</tbody></table></li>
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</ul>
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</div>
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<p>The exact height from which you drop the coin is not important, but don't drop it so close to the paper that you are cheating!</p>
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<p>If the coin rolls completely off the paper, then do not count that turn.</p>
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<h2>100 Times</h2>
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<p>Now we will drop the coin 100 times, but first ...</p>
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<div class="center80">
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<p class="center larger">... what percentage do you think will land A, or B?</p>
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</div>
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<p>Make a guess (estimate) before you begin the experiment:</p>
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<table align="center" border="1">
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<tbody>
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<tr>
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<td align="right" height="25">Your Guess for "A" (%):</td>
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<td width="80"> </td>
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</tr>
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<tr>
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<td align="right" height="25">Your Guess for "B" (%):</td>
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<td> </td>
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</tr>
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</tbody></table>
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<p><b>OK let's begin</b>.</p>
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<p>Drop the coin 100 times and record <span class="large">A</span> (does not touch a line) or <span class="large">B</span> (touches a line) using <a href="../data/tally-marks.html">Tally Marks</a>:</p>
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<table width="100%" border="1">
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<tbody>
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<tr>
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<td align="center">Coin lands</td>
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<td width="50%">Tally</td>
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<td align="center">Frequency</td>
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<td align="center">Percentage</td>
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</tr>
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<tr>
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<td align="center">
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<div class="large">A</div></td>
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<td height="70"> </td>
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<td align="center"> </td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="center">
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<div class="large">B</div></td>
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<td height="70"> </td>
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<td align="center"> </td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="center"> </td>
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<td align="right">Totals:</td>
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<td align="center">100</td>
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<td align="center">100% </td>
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</tr>
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</tbody></table>
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<p>Now draw a <a href="../data/bar-graphs.html">Bar Graph</a> to illustrate your results. You can create one at <a href="../data/data-graph.html">Data Graphs (Bar, Line and Pie)</a>.</p>
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<ul>
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<li>Are the bars the same height?</li>
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<li>Did you expect them to be?</li>
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<li>How does the result compare with your guess?</li>
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</ul>
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<h2>We Can Calculate What It Should Be ...</h2>
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<p>Here are some positions for the coin to land so it does <b>not quite touch</b> one of the lines:</p>
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<p class="center"><img src="images/coin-grid-2.gif" alt="coin grid inside different positions" height="101" width="404"></p>
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<div class="center80">
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<p>Place your coin on your grid (like above), and then put a mark on the paper where the center of the coin is (just a rough estimate will do).</p>
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</div>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td><img src="images/coin-grid-3a.gif" alt="coin grid radius" height="133" width="162"></td>
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<td>
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<p>See how the coin's center is one radius <span class="large">r</span> away from a line.</p>
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<p>(Read about a Circle's <a href="../geometry/circle.html">Radius and Diameter</a>.)</p></td>
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</tr>
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</tbody></table>
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<div class="center80">
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<p>Make lots of "center marks" then draw a box connecting them all like below:</p>
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</div>
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<p class="center"><img src="images/coin-grid-3.gif" alt="coin grid 30-d" height="133" width="254"><br>
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<span class="larger">d = Coin's diameter (2 × r)</span></p>
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<p class="center large">When a coin's <b>center</b> is within the yellow box it won't touch any line.</p>
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<p>The yellow box is smaller than the grid by <b>two radiuses</b> (= one diameter) of the coin.</p>
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<p>So what are the areas?</p>
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<ul>
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<li>The area of the grid square is 30 × 30 = 900 mm<sup>2</sup></li>
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<li>The area of the yellow box is (30-d) × (30-d) = (30-d)<sup>2</sup> mm<sup>2</sup></li>
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</ul>
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<p>The above calculation was for a 30 mm grid, but we can use <b>S</b> for grid size:</p>
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<ul>
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<li>The area of the grid square is S × S = S<sup>2</sup> mm<sup>2</sup></li>
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<li>The area of the yellow box is (S-d)<sup>2</sup> mm<sup>2</sup></li>
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</ul>
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<div class="example">
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<h3>Example: A 1c Euro (d=16.25 mm) on a 29mm grid (S=29 mm):</h3>
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<p>Grid Square = 29<sup>2</sup> = 841 mm<sup>2</sup></p>
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<p>Yellow Box = (29-16.25)<sup>2</sup> = 12.75<sup>2</sup> = 162 mm<sup>2</sup> (to the nearest mm<sup>2</sup>)</p>
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<p>So you should expect the coin to land <b>not</b> crossing a line of the grid approximately:</p>
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<p class="center larger">"A" = 162 / 841 = 19.3% of the time</p>
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<p>And "B" = 100% - 19.3% = 80.7%</p>
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</div>
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<p>Now do the calculations for <b>your own</b> grid size and coin size.</p>
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<table align="center" border="1">
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<tbody>
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<tr>
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<td align="right" height="25">Grid Spacing <span class="large">S</span> (mm):</td>
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<td> </td>
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</tr>
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<tr>
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<td align="right" height="25">Diameter of Coin <span class="large">d</span> (mm):</td>
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<td width="80"> </td>
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</tr>
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<tr>
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<td align="right" height="25">Area of Grid Square <span class="large">= S<sup>2</sup></span> (mm<sup>2</sup>):</td>
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<td> </td>
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</tr>
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<tr>
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<td align="right" height="25">Area of Yellow Box <span class="large">= (S-d)<sup>2</sup></span> (mm<sup>2</sup>):</td>
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<td> </td>
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</tr>
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<tr>
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<td align="right" height="25">"A" (%):</td>
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<td> </td>
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</tr>
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<tr>
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<td align="right" height="25">"B" (%):</td>
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<td> </td>
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</tr>
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</tbody></table>
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<div class="center80">
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<p class="center larger">How do these theoretical results compare with your experimental results?</p>
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</div>
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<p>It won't be exact (because it is a random thing) but it may be close.</p>
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<h2>Different Sizes of Coin</h2>
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<p>Try repeating the experiment using a different sized coin.</p>
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<ul>
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<li>First calculate the theoretical value ... how does this affect the values for A and B?</li>
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<li>Then do the experiment to see how close it gets.</li>
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</ul>
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<h2>What You Have Done</h2>
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<p>You have (hopefully) had fun running <b>an experiment</b>.</p>
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<p>You have done some geometry, and had some experience calculating areas and probabilities.</p>
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<p>And you have seen the relationship between theory and reality.</p>
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<p> </p>
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<div class="related">
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<a href="buffons-needle.html">Activity: Buffon's Needle</a>
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<a href="index.html">Activity Index</a>
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Copyright © 2020 MathsIsFun.com
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