8351
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<mdoxml version="1.0"><p><em>For this problem, you will either need multilink cubes or these sets of number base sheets to cut out: <a class="pdflink" href="/content/id/7490/Base%20Three.pdf">Base Three</a> <a class="pdflink" href="/content/id/7490/Base%20Four.pdf">Base Four</a> <a class="pdflink" href="/content/id/7490/Base%20Five.pdf">Base Five</a> <a class="pdflink" href="/content/id/7490/Base%20Six.pdf">Base
Six</a>.</em></p>
<p> </p>
<p>In the video below, Charlie and Alison show how you can make rectangles using sets of squares, sticks and units.</p>
<p> </p>
<video class="video" controls="controls" height="315" src="video1.mp4" tabindex="0" type="video/mp4" width="430"></video>
<p><a href="/content/id/8351/7490%20Factorising%20with%20Multilink.mp4">Factorising with Multilink</a></p>
<p><br></br>
Can you make a rectangle to represent $x^2 + 7x + 12$?<br></br>
Can you do it in more than one base?</p>
<p>Watch the video below to see how Charlie and Alison tackled this question:</p>
<p> </p>
<p><a href="/content/id/8351/7490%20Factorising%20with%20Multilink%20Part%202.mp4">Factorising with Multilink Part 2</a></p>
<p><strong>Take 1 square, 12 sticks, and some units in your chosen base, and make some rectangles that will work in all bases.</strong></p>
<p><em>Charlie and Alison made $x^2 + 7x + 12$ into a rectangle with length $x+4$ and width $x+3$.</em></p>
<p>What do you notice about the dimensions of your rectangles?</p>
<p>Imagine you had 1 square, 100 sticks and lots of units. What can you say about the dimensions of the rectangles it is possible to make?</p>
<p> </p>
<p><strong>Now take 1 square, some sticks, and 12 units in your chosen base, and make some rectangles that will work in all bases.</strong></p>
<p>What do you notice about the dimensions of your rectangles?</p>
<p>Imagine you had 1 square, lots of sticks and 100 units. What can you say about the dimensions of the rectangles it is possible to make?</p>
<p> </p>
<p><strong>If you had 1 square, $p$ sticks and $q$ units, what can you say about the dimensions of the rectangles it is possible to make?</strong></p>
<p> </p>
<p><strong>Extension</strong></p>
<p>Think about the rectangles it's possible to make if you use two, three, four... squares, some sticks and some units.</p>
<p> </p>
<p><em>Many thanks to Paul Andrews whose ideas inspired this problem.</em></p></mdoxml>
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<mdoxml version="1.0"><h3>Why do this problem?</h3>
<p>When students first meet factorisation, they often don't make the connection between factorising an algebraic expression and breaking a number into factor pairs. This problem introduces factorisation using a visual representation that allows students to make that connection and discover for themselves the properties necessary for a quadratic expression to factorise.</p>
<h3>Possible approach</h3>
<div> </div>
<div>If multilink is available, students can create "squares", "sticks" and "units". Alternatively, these worksheets have sets of different bases that can be cut out and used: <a class="pdflink" href="/content/id/7490/Base%20Three.pdf">Base Three</a> <a class="pdflink" href="/content/id/7490/Base%20Four.pdf">Base Four</a> <a class="pdflink" href="/content/id/7490/Base%20Five.pdf">Base Five</a>
<a class="pdflink" href="/content/id/7490/Base%20Six.pdf">Base Six</a>.</div>
<div> </div>
<div>Ask the students to organise themselves into groups of three or four, and select which base each student will use. If using multilink, give them time to create some squares, sticks and units in their bases.</div>
<div> </div>
<div>"I'd like you each to take one square, seven sticks and twelve units and put them together to make a rectangle."</div>
<div>"In your groups, compare the rectangles you've made. Have you arranged them in the same way? The challenge is to find an arrangement that works in all bases."</div>
<div><br></br>
<em>Take a look at the second video to see how Alison and Charlie resolve their different rectangles into one that works for both of them.</em></div>
<div> </div>
<div>"Now I'd like you to try to make a rectangle using a square, five sticks and eight units."</div>
<div>Give groups some time to work on this. </div>
<div>Bring the class together for discussion. "Did anyone manage to make a rectangle?"</div>
<div>(It is possible to make rectangles for bases 3, 4 and 8, but it's not possible to make the same arrangement for all bases.)</div>
<div> </div>
<div><strong>"Your challenge is to come up with a way of deciding quickly whether a combination of squares, sticks and units can be made into a rectangle that works in all bases."</strong></div>
<div> </div>
<div>"To help you approach this in a systematic way, <a href="/content/id/7490/Factorising%20with%20multilink.pdf">here</a> are three questions to have a go at."</div>
<div> </div>
<div>Circulate, and listen out for interesting insights that will be worth sharing with the whole class.</div>
<div>Finally, bring the class together and invite groups to share their findings.</div>
<div> </div>
<h3>Key question</h3>
<p>What is the connection between the number of sticks you need, the number of units you need, and the dimensions of the rectangle?</p>
<p> </p>
<h3>Possible extension</h3>
<p>Think about the rectangles it's possible to make if you use two, three, four... squares, some sticks and some units.</p>
<p><a href="/7452">Finding Factors</a> consolidates the ideas met here and has factorisations which require negative terms.</p>
<h3><br></br>
Possible support</h3>
<p>Start students off on families of examples such as:</p>
<p>$x^2 + 3x + 2$<br></br>
$x^2 + 4x + 3$<br></br>
$x^2 + 5x + 4$</p>
<p>...</p>
<h3><br></br>
</h3></mdoxml>
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<mdoxml version="1.0"><p>Start with families of examples such as:</p>
<p>$x^2 + 3x + 2$<br></br>
$x^2 + 4x + 3$<br></br>
$x^2 + 5x + 4$</p>
<p>...</p>
<p><br></br>
What do you notice?</p>
<p>Can you explain why?</p></mdoxml>
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Video test
HTML5 Video test
Algebra
Factorisation (algebraic)