5612
/www/nrich/html/content/id/5612/
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2011-10-03T16:18:36
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<h4><em><em>If you are a teacher, click <a href="http://nrich.maths.org/5612&part=note">here</a> for a version of the problem suitable for classroom use, together with supporting materials. Otherwise, read on...</em></em></h4>
<p> </p>
<p>If you had to work out $23 \times 21$ how would you do it?<br></br>
What if you needed to work out $246 \times 34$?</p>
<p>Here are eight videos showing four different methods for working out the two multiplications. Can you make sense of them?</p>
<p><strong>Grid Multiplication</strong></p>
<p> </p>
<div class="toggle"><br></br>
$23 \times 21$<br></br>
<br></br>
<iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/fFs7PkHSHAI?rel=0" width="420"></iframe><br></br>
<br></br>
$246 \times 34$<br></br>
<br></br>
<iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/SymxnhU-XwM?rel=0" width="420"></iframe></div>
<p> </p>
<p><strong>Column Multiplication</strong></p>
<p> </p>
<div class="toggle"><br></br>
$23 \times 21$<br></br>
<br></br>
<iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/2UsNzORBqLw?rel=0" width="420"></iframe><br></br>
<br></br>
$246 \times 34$<br></br>
<br></br>
<iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/yIsT_xs8Z5U?rel=0" width="420"></iframe></div>
<p> </p>
<p><strong>Multiplying with Lines</strong></p>
<p> </p>
<div class="toggle"><br></br>
$23 \times 21$<br></br>
<br></br>
<iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/YlCm_GxGndQ?rel=0" width="420"></iframe><br></br>
<br></br>
$246 \times 34$<br></br>
<br></br>
<iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/mqt6ccw2UM4?rel=0" width="420"></iframe></div>
<p> </p>
<p><strong>Gelosia Multiplication</strong></p>
<p> </p>
<div class="toggle"><br></br>
$23 \times 21$<br></br>
<br></br>
<iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/bz0HpaZc7Y0?rel=0" width="420"></iframe><br></br>
<br></br>
$246 \times 34$<br></br>
<br></br>
<iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/GI_YlTOjG1M?rel=0" width="420"></iframe></div>
<p> </p>
<p>Once you have watched the videos, make up some multiplication calculations of your own and have a go at answering them using all four methods. Check that you get the same answer each time!</p>
<p> </p>
<p><strong>Here are some questions to consider:</strong></p>
<p>Why does each method work?</p>
<p>What do the methods have in common?</p>
<p>What are the advantages and disadvantages to each method?</p>
<p> </p>
<p><br></br>
<strong>Extension challenge</strong></p>
<p>Here is a video of another multiplication method, one where no writing down is needed along the way. Can you make sense of the video, and explain how this method works?</p>
<p> </p>
<div class="toggle"><iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/6-mpspx6ICg?rel=0" width="420"></iframe>
<p> </p>
</div>
<p> </p>
<p> </p>
<p> </p></mdoxml>
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<mdoxml version="1.0"><p><span class="editorial">It's interesting to see lots of different ways of doing the same calculation - different people seem to prefer different methods, and this was just a handful of potential ways of doing this sum.</span></p>
<p><span class="editorial">Holly, from Hymers College, gave some great explanations of the different methods:</span></p>
<p>The <strong>grid multiplication</strong> method works by splitting up the numbers (e.g. 23 into 20 and 3) to times them together more easily (as it is much easier trying to do 20x20 than 23x21). After you times every digit in the first number by every digit in the second number, you simply add them all up.</p>
<p><strong>Column multiplication</strong> works in a similar way to the grid multiplication method, as the column method splits the numbers up in a similar fashion. </p>
<p>With the sum 23x21: <br></br>
you do 1x3, and put the answer in the units column,<br></br>
then 1x2, and put the answer in the tens column (you are basically doing 1x20 and carrying over from the units to the tens).<br></br>
For the next line, as you are really doing 20x3, you put a 0 in the units column, and multiply 2x3; the 6 in the tens column represents 60.</p>
<p>The <strong>line method</strong> works so that after you have drawn all your lines and added them up, they are split into units, tens, hundreds, and so on. This is done by multiplying the units by units (the only way to get units), the tens by units and units by tens (the only ways to get tens), the units by hundreds and the tens by tens (the only ways to get hundreds), and
so forth for any<br></br>
other columns.</p>
<p>The <strong>Gelosia method</strong> still works with the units x units, tens x units, etc. system explained for the line method of multiplication. The squares are halved diagonally in case you need to 'carry' a digit over to the next column; it will simply be added in with the next column automatically, instead of you having to carry it over yourself.</p>
<p><span class="editorial">Alexander from Wilson's School also gave similar explanations, and added his thoughts:</span></p>
<p>There are a few disadvantages to each of the methods:<br></br>
Firstly, in Grid Multiplication, you have to do many multiplication sums, and then add them all together.<br></br>
In column multiplication, the problem is that if you forget to add the zero in the second row, your product will be incorrect.<br></br>
Line multiplication takes a long time to draw and can get very confusing. <br></br>
Gelosia multiplication's disadvantage is that it takes a long time to draw, and if you've got a big sum, you have to carry a lot of numbers.</p>
<p>However, all of the methods have advantages:<br></br>
Grid Multiplication method is very simple, therefore very difficult to get wrong.<br></br>
Column multiplication is good because it doesn't take up that much space and is very quick.<br></br>
In Multiplying with lines you don't have to multiply, you just count.<br></br>
Gelosia multiplication is good because it carries most of the numbers that have to be carried for you.</p>
<p><span class="editorial">Chloe from Landau Fort Academy performed the same calculations on a different sum to make sure she had understood the methods, and added her thoughts on what she had done:</span></p>
<p>Grid method: easy, but takes ages to draw.</p>
<p>Column method: easy, but might get mixed up.</p>
<p>Multiplying with lines: doesn't take long once drawn, but it's quite hard!</p>
<p>Gelosia method: makes sense, but you might mix it up, and it takes a while to draw.</p>
<p><span class="editorial">Three students from Westfield Middle School sent us their calculations: Ayesha seemed to like the grid method, Mohammed preferred the Gelosia method and Valentins liked the column method most. Interesting!</span></p>
<p><span class="editorial">Rachel from NLCS Jeju spotted the following clever trick to one of the multiplications:</span></p>
<p>$23$ is $(22+1)$, and $21$ is $(22-1)$,<br></br>
so $23\times 21 = (22+1)(22-1)$.</p>
<p>But we know that $(a+b)(a-b) = a^2 - b^2$,<br></br>
so $23\times 21 = 22^2 - 1 = 484 - 1 = 483$.</p></mdoxml>
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<h3>Why do this problem?</h3>
<div><br></br>
Most of us can carry out long multiplications using a standard method. But do we understand what we are doing? Here is a chance to find out...</div>
<br></br>
<h3>Possible approach</h3>
<div> </div>
<div>Rather than showing the videos from the <a href="/5612">problem</a>, you may choose to study the methods and recreate them on the board at the start of the lesson (in silence, as in the videos, so that students are expected to make sense of it without any explanation offered). The latter approach has the advantage of preserving a record of the four methods, if your board is big enough. The
rest of the lesson follows in the same way however you choose to introduce it.</div>
<div> </div>
<div>"Here are two multiplication calculations that I'd like you to do, using whatever method you like:"</div>
<div>$$23 \times 21$$ $$246 \times 34$$</div>
<div> </div>
<div>Give students a short time to carry out the multiplications, perhaps on individual whiteboards.</div>
<div> </div>
<div>"Now I'm going to show you four methods that could be used to work out those multiplications. Some of you used some of these methods, but there might be methods that you haven't seen before. Watch carefully, and see if you can work out what's going on."</div>
<div> </div>
<div>Show the videos, or recreate the calculations from the videos on the board.</div>
<div> </div>
<div>Then hand out <a href="/content/id/5612/Methodsmain.pdf">this worksheet</a> showing each finished method for $246 \times 34$.</div>
<div>"With your partner, try to recreate the methods and make sense of the different steps. Once you think you have made sense of them, make up a few calculations of your own and work them out using all four methods until you feel confident that you can use each method effectively."</div>
<div> </div>
<div>Towards the end of the lesson(s), bring the class together and invite students to explain why each method works, and to discuss the benefits and disadvantages of each method.</div>
<div> </div>
<div>Finally, <a href="http://video.google.com/videoplay?docid=7106559846794044495" target="_blank">this 2 minute video clip from the film <i>Ma and Pa Kettle</i></a> could be shown to provoke discussion about the misconceptions in Ma and Pa's methods.</div>
<div> </div>
<h3>Key questions</h3>
<p>Will each method always work?<br></br>
Where is the $20 \times 20$? Where is the $20 \times 3$? ...<br></br>
</p>
<h3>Possible extension</h3>
<div> </div>
<div>Challenge students to adapt each method for multiplying decimals.</div>
<div> </div>
<div>Show students this <a href="http://youtu.be/6-mpspx6ICg">video clip</a> of another multiplication method that requires no intermediate writing down, and invite them to make sense of it.</div>
<h3> </h3>
<h3>Possible support</h3>
<p>Offer students <a href="/content/id/5612/Methodssupport.pdf">this worksheet</a> with the methods for $23 \times 21$ to make sense of first, as there are no 'carry' digits so it is clearer to see what is going on.<br></br>
</p>
<p> </p>
<p><em>Stu Cork created this <a href="/content/id/5612/multiplication.ggb">GeoGebra file</a> to use with this problem, which he has kindly given us permission to share.</em></p></mdoxml>
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<p>$23 \times 21$ is the same as</p>
<p>$$(20 \times 21) + (3 \times 21)$$</p>
<p>which is the same as</p>
<p>$$((20 \times 20) + (20 \times 1)) + ((3 \times 20) + (3 \times 1))$$</p>
<p>Can you figure out where each of these four products appears in the different methods?</p>
<p>Can you deconstruct $246 \times 34$ in the same way?<br></br>
</p>
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<p class="editorial">We had many correct solutions from students who recognised that multiplying with lines is a graphical way of representing what most of us do when we carry out standard multiplications on paper.<br></br>
Well done Melissa from All Saints Primary School in Chatham, Luca from Devonshire Primary School and Yachna from Bromet Primary School (and m.a.p. from P.P.S.!).</p>
<p>The spaced out lines correspond to the number of units, tens, hundreds, etc.:<br></br>
241 = 2 hundreds, 4 tens and 1 unit, so it is represented by 2, 4 and 1 lines.<br></br>
<br></br>
There are two sets of each type of line, one for each number.<br></br>
Where they intersect shows the result of multiplying them:<br></br>
e.g. 3 x 2 lines will intersect each other 6 times.<br></br>
So the intersection of 3 hundreds and 2 tens, or 2 hundreds and 3 tens gives you 6 thousands.<br></br>
<br></br>
On the far right section you have the result of multiplying <span style="font-weight: bold;">units by units,</span> so the result is the number of <span style="font-weight: bold;">units.</span><br></br>
<br></br>
Moving to the left, in the next section you have the result of multiplying <span style="font-weight: bold;">units by tens</span> plus <span style="font-weight: bold;">tens by units,</span> so the result is the number of <span style="font-weight: bold;">tens.</span><br></br>
<br></br>
Moving to the left again, in the next section you have the result of multiplying <span style="font-weight: bold;">units by hundreds</span> plus <span style="font-weight: bold;">tens by tens</span> plus <span style="font-weight: bold;">hundreds by units,</span> so the result is the number of <span style="font-weight: bold;">hundreds.</span><br></br>
<br></br>
Moving to the left again, in the next section you have the result of multiplying <span style="font-weight: bold;">units by thousands</span> plus <span style="font-weight: bold;">tens by hundreds</span> plus <span style="font-weight: bold;">hundreds by tens</span> plus <span style="font-weight: bold;">thousands by units,</span> so the result is the number of <span style="font-weight: bold;">thousands.</span><br></br>
<br></br>
Note that as the power of 10 of one of the numbers increases the power of 10 of the other number decreases.<br></br>
<br></br>
As we move to the left we move to the next power of 10: from units to tens to hundreds to thousands to ten thousands...</p>
<p class="editorial">Yachna mentioned that at the end you carry digits to the next section if you have reached 10 or more in any section.<br></br>
<br></br>
Ethan from Stratford Landing School pointed out that "we need to be careful when we have a multiplication like 210 x 769 because there will be a need to write zero lines".</p>
<p>To ensure that all the lines keep their correct place you can represent zeros with dotted lines, so the result of crossing any dotted line with another dotted or plain line will be zero.</p>
<p class="editorial">Luca mentioned that "it works a bit like the grid method because if the sum was 23 x 32 it would have 20 x 30, 20 x 2, 3 x 30 and 3 x 2 but it would be set out differently".</p>
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