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450 lines
14 KiB
HTML
450 lines
14 KiB
HTML
<!DOCTYPE html>
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<html lang="en"><!-- #BeginTemplate "/Templates/Main.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/activity/drawing-squares.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:58:08 GMT -->
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<head>
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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: Drawing Squares</title>
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<meta name="description" content="Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.">
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<div id="stt"></div>
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<div id="adTop"></div>
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<header>
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<div id="hdr"></div>
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<div id="tran"></div>
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<div id="adHide"></div>
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<div id="cookOK"></div>
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</header>
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<div class="mid">
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<nav>
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<div id="menuWide" class="menu"></div>
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<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun"></a></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Activity: Drawing Squares</h1>
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<p>For this activity all you need is a grid of dots, a pencil and your
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brain.</p>
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<h3>Let us discover how many squares you can make on different grids:</h3>
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<p class="center"><img src="images/drawing-squares-grids.svg" alt="grids 1x1, 2x2, 3x3, etc" style="max-width:100%" height="144" width="455"></p>
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<p class="center"><strong>Note:</strong> "1 by 1" means how many <b>sides</b> (not how many dots).</p>
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<h3><br>
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So, let's try drawing in
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some squares and count how many:</h3>
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<h3>1 by 1</h3>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td>Well,
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that's easy, there's just one: </td>
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<td> </td>
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<td><img src="images/squares3.gif" alt="squares3" height="35" width="38"></td>
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</tr>
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</tbody></table>
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<h3>2 by
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2</h3>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td>That seems to be easy too. There are four of them, aren't there?</td>
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<td> </td>
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<td><img src="images/squares4.gif" alt="squares4" height="60" width="62"></td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td style="text-align:right;">But
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wait, that's not the complete answer.<br>
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There's also this bigger
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one:</td>
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<td style="text-align:right;"> </td>
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<td><img src="images/squares5.gif" alt="squares5" height="60" width="65"></td>
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</tr>
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</tbody></table>
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<p>That makes
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five squares altogether - four
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1 by 1 squares and one
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2 by 2 square</p>
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<h2>Your Turn !</h2>
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<h3>3 by 3</h3>
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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>Over to you now.</p>
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<p class="right">Here's the grid:</p></td>
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<td> </td>
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<td><img src="images/squares6.gif" alt="squares6" height="88" width="88"></td>
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</tr>
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</tbody></table> <div class="center80">
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<p><strong>Hint:</strong> For
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the 3 by 3 case, you will expect to get 1 by 1 squares, 2 by 2 squares
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and 3 by 3 squares. How many of each?</p>
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</div><p><br></p>
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<p><input onclick="printImg('images/squares-grid.gif')" value="You could also print this bigger grid" type="button"><br><img src="images/squares-grid.gif" alt="squares grid" height="325" width="326">
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</p>
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<p><br></p>
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<p> </p>
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<p><strong>Now you can start to fill in a table:</strong></p>
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<div style="overflow:auto;">
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<table cellspacing="2" cellpadding="2" border="1">
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<tbody>
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<tr>
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<td style="width:100px;"> </td>
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<td align="center" width="100"> How Many<br>
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1 by 1 squares</td>
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<td align="center" width="100"> How Many<br>
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2 by 2 squares</td>
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<td align="center" width="100"> How Many<br>
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3 by 3 squares</td>
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<td align="center" width="100"> How Many<br>
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4 by 4 squares</td>
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<td align="center" width="100"> How Many<br>
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5 by 5 squares</td>
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<td align="center" width="100"><strong>Total</strong></td>
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</tr>
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<tr>
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<td align="center" height="25">1 by
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1 Grid:</td>
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<td style="text-align:center;">1</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td style="text-align:center;">1</td>
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</tr>
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<tr>
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<td align="center" height="25">2 by
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2 Grid:</td>
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<td style="text-align:center;">4</td>
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<td style="text-align:center;">1</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td style="text-align:center;">5</td>
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</tr>
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<tr>
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<td align="center" height="25">3 by
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3 Grid:</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td align="center" height="25">4 by
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4 Grid:</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td align="center" height="25">5 by
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5 Grid:</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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</tbody>
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</table>
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</div>
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<p> </p>
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<h2>Did you notice anything about the numbers in the table?</h2>
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<p>They are all <a href="../numberpatterns.html">square
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numbers</a>:</p>
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<ul>
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<li>1<sup>2</sup> = <b>1</b>,</li>
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<li>2<sup>2</sup> = <b>4</b>,</li>
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<li>3<sup>2</sup> = <b>9</b>,</li>
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<li>etc ...</li>
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</ul>
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<p>and the totals are found
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by adding together square numbers.</p>
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<h3>Formula to The Rescue ... !</h3>
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<p>There is actually a <a href="../definitions/formula.html">formula</a> for adding the first <strong>n</strong> square numbers:</p>
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<p class="center larger">S<sub>n</sub> = n(n+1)(2n+1) / 6</p>
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<div class="example">
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<h3>Example: The number of squares in the <strong>5 by 5</strong> case</h3>
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<p>Try <a href="../algebra/substitution.html">substituting</a> <strong>n = 5</strong> into the formula:</p>
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<div class="so"> S<sub>n</sub> = n(n+1)(2n+1) / 6 </div>
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<div class="so"> S<sub>5</sub> = 5 × (5+1) × (2×5+1) / 6 </div>
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<div class="so"> S<sub>5</sub> = 5 × 6 × 11 / 6 </div>
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<div class="so"> S<sub>5</sub> = 55 </div>
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</div>
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<h3> </h3>
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<h2>So, we seem to have solved the question. Yipee!</h2>
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<h3>But wait ... there's more!</h3>
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<p>I said you would need to use your brains. Let's go back to the 2 by 2 case:</p>
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<h3>2 by
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2</h3>
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<p>There is another square too, this one:</p>
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<blockquote> <img src="images/squares7.gif" alt="squares9" height="61" width="65"></blockquote>
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<div class="center80">
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<p><strong>Why is it a <a href="../quadrilaterals.html">square</a>? </strong>It has four equal sides and four right angles, so that's a square.</p>
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</div>
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<p>So, that makes
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six squares altogether.</p>
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<p>Four
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1 by 1 squares, one
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2 by 2 square and one
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<b>x by x</b> square.</p>
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<div class="center80">
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<p>What is the value of <b>x</b>? We can use <a href="../pythagoras.html">Pythagoras'
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Theorem</a> to find it:</p>
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<div class="so"> x<sup>2</sup> = 1<sup>2</sup> + 1<sup>2</sup> = 1 + 1 = 2 </div>
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<div class="so"> So x = √2 </div>
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</div>
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<p>So, we have <strong>four
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1 by 1 squares</strong>, <strong>one
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2 by 2 square</strong> and <strong>one √2
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by √2 square</strong>.</p>
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<h2>Your Turn !</h2>
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<h3>3 by 3</h3>
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<ul>
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<li>Are there any more squares?</li>
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</ul>
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<p><strong>YES</strong>! Can you find them?</p>
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<h3>4 by 4 and 5 by 5</h3>
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<p>Also try the 4 by 4 grid, and the 5 by 5 grid</p>
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<p>As you proceed, you will find squares like these:</p>
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<p class="center"><img src="images/squares8.svg" alt="tilted squares" style="max-width:100%" height="274" width="512"></p>
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<p> </p>
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<h2>What are the lengths of the sides of these squares?</h2>
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<p>You can use <a href="../pythagoras.html">Pythagoras'
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Theorem</a> to work that out yourself</p>
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<h3>In each case, how many do you get of each one?</h3>
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<p>Here is a table to help you:</p>
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<div class="simple">
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<table border="1">
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<tbody>
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<tr>
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<td> </td>
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<td style="text-align:center;">How Many<br>
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1 by 1 </td>
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<td style="text-align:center;">How Many<br>
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2 by 2 </td>
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<td style="text-align:center;">How Many<br>
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3 by 3 </td>
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<td style="text-align:center;">How Many<br>
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4 by 4 </td>
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<td style="text-align:center;">How Many<br>
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5 by 5 </td>
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<td style="text-align:center;">How Many<br>
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√2 by<br>
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√2 </td>
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<td style="text-align:center;">How Many<br>
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√5 by √5 </td>
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<td style="text-align:center;">How Many<br>
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√8 by √8 </td>
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<td style="text-align:center;">How Many<br>
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√10 by √10</td>
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<td style="text-align:center;">How Many<br>
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√13 by √13 </td>
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<td style="text-align:center;">How Many<br>
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√17 by √17</td>
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<td style="text-align:center;"><strong>Total</strong></td>
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</tr>
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<tr>
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<td height="30">1 by 1 Grid:</td>
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<td style="text-align:center;">1</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td style="text-align:center;">1</td>
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</tr>
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<tr>
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<td height="30">2 by
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2 Grid:</td>
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<td style="text-align:center;">4</td>
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<td style="text-align:center;">1</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td style="text-align:center;">1</td>
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<td style="text-align:center;"> </td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td style="text-align:center;">6</td>
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</tr>
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<tr>
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<td height="30">3 by
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3 Grid:</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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</tr>
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<tr>
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<td height="30">4 by
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4 Grid:</td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
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<td><br></td>
|
||
<td><br></td>
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||
<td><br></td>
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||
<td><br></td>
|
||
</tr>
|
||
<tr>
|
||
<td height="30">5 by
|
||
5 Grid:</td>
|
||
<td><br></td>
|
||
<td><br></td>
|
||
<td><br></td>
|
||
<td><br></td>
|
||
<td><br></td>
|
||
<td><br></td>
|
||
<td><br></td>
|
||
<td><br></td>
|
||
<td><br></td>
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||
<td><br></td>
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||
<td><br></td>
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||
<td><br></td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
</div>
|
||
<p> </p>
|
||
|
||
|
||
<h2>Advanced</h2>
|
||
|
||
<p>Can you find a formula to calculate the number of squares that have
|
||
lengths that are square roots?</p>
|
||
<p>Can you then find a formula for the total number of squares in
|
||
each case?</p>
|
||
<p>Are there any more squares lurking in there that we've missed?</p>
|
||
|
||
|
||
<h2>Conclusion</h2>
|
||
|
||
<p>What started off as seeming to be a simple exercise turned out to
|
||
be quite complex. You really <b>do</b> need to use your brains to think this
|
||
one through, but it's a challenging and rewarding exercise.</p>
|
||
<p> </p>
|
||
|
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