3.05:_Division-_Dots_and_Boxes

<span id="Quotative_Model_of_Division"></span><h2 class="lt-math-9839">Quotative Model of Division</h2>

<p class="lt-math-9839">Suppose you are asked to compute 3906 : 3. One way to interpret this question (there are others) is:</p>

<p class="mt-align-center lt-math-9839">&ldquo;How many groups of 3 fit into 3906?&rdquo;</p>

<section class="box-definition"><span id="Definition"></span><h5 class="box-legend lt-math-9839"><span class="lt-icon-default">Definition</span></h5>

<p class="lt-math-9839">In the <strong>quotative model of division</strong>, you are given a <strong>dividend</strong> (here it is 3906), and you are asked to split it into equal-sized groups, where the size of the group is given by the <strong>divisor</strong> (here it is 3).</p>
</section>

<p class="lt-math-9839"> In our dots and boxes model, the dividend 3906 looks like this:</p>

<p class="lt-math-9839">and three dots looks like this: <img alt="divide2.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10337/divide2.png?revision=1" /></p>

<p class="lt-math-9839">So we are really asking:</p>

<p class="mt-align-center lt-math-9839">&ldquo;How many groups of <img alt="divide2.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10337/divide2.png?revision=1" /> fit into the picture of 3906?&rdquo;</p>

<section class="box-example"><span id=":_3906_.C3.B7_3"></span><h5 class="box-legend lt-math-9839"><span class="lt-icon-default">: 3906 &divide; 3</span></h5>

<p class="lt-math-9839">There is one group of 3 at the thousands level, and three at the hundreds level, none at the tens level, and two at the ones level.</p>

<p class="lt-math-9839">Notice what we have in the picture:</p>

<ul>
    <li class="lt-math-9839">One group of 3 in the thousands box.</li>
    <li class="lt-math-9839">Three groups of 3 in the hundreds box.</li>
    <li class="lt-math-9839">Zero groups of 3 in the tens box.</li>
    <li class="lt-math-9839">Two groups of 3 in the ones box.</li>
</ul>

<p class="lt-math-9839">This shows that 3 goes into 3906 one thousand, three hundreds and two ones times. That is,</p>

<p class="lt-math-9839">\[3906 \div 3 = 1302 \ldotp \nonumber \]</p>
</section>

<p class="lt-math-9839">Let&rsquo;s try a harder one! Consider 402 : 3. Here&rsquo;s the picture:</p>

<p class="lt-math-9839">We are still looking for groups of three dots: <img alt="divide2.png" class="internal" loading="lazy" src="https://math.libretexts.org/@api/deki/files/10337/divide2.png?revision=1" /></p>

<p class="lt-math-9839">There is certainly one group at the 100&rsquo;s level.</p>

<p class="lt-math-9839">and now it seems we are stuck  there are no more groups of three!</p>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share"></span><h5 class="box-legend lt-math-9839"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<p class="lt-math-9839">What can we do now? Are we really stuck? Can you finish the division problem?</p>
</section>

<section class="box-example"><span id=":_402_.C3.B7_3"></span><h5 class="box-legend lt-math-9839"><span class="lt-icon-default">: 402 &divide; 3</span></h5>

<p class="lt-math-9839">Here are the details worked out for 402 : 3. But don&rsquo;t read this until you&rsquo;ve thought about it yourself!</p>

<p class="lt-math-9839">Since each dot is worth ten dots in the box to the right we can write:</p>

<p class="lt-math-9839">Now we can find more groups of three:</p>

<p class="lt-math-9839">There is still a troublesome extra dot. Let&rsquo;s unexplode it too</p>

<p class="lt-math-9839">This gives us more groups of three:</p>

<p class="lt-math-9839">In the picture we have:</p>

<ul>
    <li class="lt-math-9839">One group of 3 in the hundreds box.</li>
    <li class="lt-math-9839">Three groups of 3 in the tens box.</li>
    <li class="lt-math-9839">Four groups of 3 in the ones box.</li>
</ul>

<p class="lt-math-9839">Finally we have the answer!</p>

<p class="lt-math-9839">\[402 \div 3 = 134 \ldotp \nonumber \]</p>
</section>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_2"></span><h5 class="box-legend lt-math-9839"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<p class="lt-math-9839">Solve each of these exercises using the dots and boxes method:</p>

<p class="lt-math-9839">\[62124 : 3 \qquad \qquad 61230 : 5 \nonumber \]</p>
</section>

<section class="box-example"><span id=":_156_.C3.B7_12"></span><h5 class="box-legend lt-math-9839"><span class="lt-icon-default">: 156 &divide; 12</span></h5>

<p class="lt-math-9839">Let&rsquo;s turn up the difficulty a notch. Consider 156 : 12. Here we are looking for groups of 12 in this picture:</p>

<p class="lt-math-9839">What does 12 look like? It can be twelve dots in a single box:</p>

<p class="lt-math-9839">But most often we would write 12 this way, as a ten and 2 ones:</p>

<p class="lt-math-9839">We certainly see some of these in the picture.  There is certainly one at the tens level:</p>

<p class="lt-math-9839"><b>Note:</b> With an unexplosion this would be twelve dots in the tens box, so we mark one group of 12 above the tens box.</p>

<p class="lt-math-9839">We also see three groups of twelve ones:</p>

<p class="lt-math-9839">So in the picture we have:</p>

<ul>
    <li class="lt-math-9839">One group of 12 dots in the tens box.</li>
    <li class="lt-math-9839">Three groups of 12 dots in the ones box.</li>
</ul>

<p class="lt-math-9839">That means</p>

<p class="lt-math-9839">\[156 : 12 = 13 \ldotp \nonumber \]</p>
</section>

<section class="box-example"><span id="Problem_6"></span><h5 class="box-legend lt-math-9839"><span class="lt-icon-default">Problem 6</span></h5>

<p class="lt-math-9839">Use the dots and boxes model to compute each of the following:</p>

<p class="lt-math-9839">\[\begin{split} 13453 &amp; : 11 \\ 4853 &amp; : 23 \\ 214506 &amp; : 102 \end{split} \nonumber \]</p>
</section>

<section class="box-example"><span id="Problem_7"></span><h5 class="box-legend lt-math-9839"><span class="lt-icon-default">Problem 7</span></h5>

<p class="lt-math-9839">Remember that base five numbers are in a 1 &larr; 5 dots-and-boxes system. What are the place values in the 1 &larr; 5 system? Fill in the blanks:</p>

<ol>
    <li class="lt-math-9839">Draw a dots-and-boxes picture of the number $424_{five}$.</li>
    <li class="lt-math-9839">Draw a dots-and-boxes picture of the number $11_{five}$.</li>
    <li class="lt-math-9839">Use the dots and boxes method to find $424_{five} \div 11_{five}$.</li>
    <li class="lt-math-9839">Rewrite the division sentence $424_{five} \div 11_{five} = 34_{five}$ in base ten, and check that it&rsquo;s correct.</li>
    <li class="lt-math-9839">Use dots-and-boxes to find $2021_{five} \div 12_{five}$. <i>Don&rsquo;t convert to base 10!</i></li>
</ol>
</section>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_3"></span><h5 class="box-legend lt-math-9839"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<ul>
    <li class="lt-math-9839">Use dots and boxes to compute these. $$\begin{split} 2130 &amp; :10 \\ 41300 &amp; : 100 \end{split}$$</li>
    <li class="lt-math-9839">What pictures did you use for 10 and for 100?  Can you describe in words what happens when dividing by 10 and by 100 and why?</li>
</ul>
</section>

<span id="The_Standard_Algorithm_for_Division"></span><h2 class="lt-math-9839">The Standard Algorithm for Division</h2>

<p class="lt-math-9839">We used dots and boxes to show that 402 : 3 = 134.</p>

<p class="lt-math-9839">In elementary school, you might have learned to solve this division problem by using a diagram like the following:</p>

<p class="lt-math-9839">At first glance this seems very mysterious, but it is really no different from the dots and boxes method. Here is what the table means.</p>

<p class="lt-math-9839">To compute 402 : 3, we first make a big estimation as to how many groups of 3 there are in 402. Let&rsquo;s guess that there are 100 groups of three.</p>

<p class="lt-math-9839">How much is left over after taking away 100 groups of 3?  We subtract to find that there is 102 left.</p>

<p class="lt-math-9839">How many groups of 3 are in 102? Let&rsquo;s try 30:</p>

<p class="lt-math-9839">How many are left? There are 12 left and there are four groups of 3 in 12.</p>

<p class="lt-math-9839">That accounts for entire number 402. And where do we find the final answer? Just add the total count of groups of three that we tallied:</p>

<p class="lt-math-9839">\[402 : 3 = 100 + 30 + 4 = 134 \ldotp \nonumber \]</p>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_4"></span><h5 class="box-legend lt-math-9839"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<ul>
    <li class="lt-math-9839">Compare the two division diagrams below. In what way are they the same? In what way are they different?</li>
    <li class="lt-math-9839">Also look at the dots and boxes method. In what way is it the same or different from the two diagrams?</li>
</ul>
</section>

<ul>
    <li class="lt-math-9839"><b>Why do we like the standard algorithm? </b>Because it is quick, not too much to write down, and it works every time.</li>
    <li class="lt-math-9839"><b>Why  do  we  like  the  dots  and  boxes  method? </b>  Because it easy to understand. (And drawing dots and boxes is kind of fun!)</li>
</ul>

<span id="Division_with_Remainders"></span><h2 class="lt-math-9839">Division with Remainders</h2>

<p class="lt-math-9839">We saw that 402 is evenly divisible by 3:  402 : 3 = 134.  This means that 403, one more, shouldn&rsquo;t be divisible by three. It should be one dot too big.</p>

<section class="box-example"><span id=":_403_.C3.B7_3"></span><h5 class="box-legend lt-math-9839"><span class="lt-icon-default">: 403 &divide; 3</span></h5>

<p class="lt-math-9839">Do we see the extra dot if we compute 402 : 3 with dots and boxes?</p>

<p class="lt-math-9839">Yes we do! We have one dot left at the end that can&rsquo;t be divided.  This is how it looks in the standard algorithm.</p>

<p class="lt-math-9839">In school, we say that we have a <i>remainder</i> of one and sometimes write:</p>

<p class="lt-math-9839">\[403 \div 3 = 134\; \text{R} 1 \ldotp \nonumber \]</p>

<p class="lt-math-9839">But what does that really mean?  It means that we have 134 groups of three with one dot left over. So</p>

<p class="lt-math-9839">\[402 = 134 \cdot 3 + 1 \ldotp \nonumber \]</p>
</section>

<section class="box-example"><span id=":_263_.C3.B7_12"></span><h5 class="box-legend lt-math-9839"><span class="lt-icon-default">: 263 &divide; 12</span></h5>

<p class="lt-math-9839">Let&rsquo;s try another one: 263 &divide; 12. Here&rsquo;s what we have:</p>

<p class="lt-math-9839">And we are looking for groups like this:</p>

<p class="lt-math-9839">Unexploding won&rsquo;t help any further and we are indeed left with one remaining dot in the tens position and a dot in the ones position. We have 21 groups of twelve, and a remainder of eleven.</p>

<p class="lt-math-9839">\[263 = 21 \cdot 12 + 11 \ldotp \nonumber \]</p>
</section>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_5"></span><h5 class="box-legend lt-math-9839"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<ul>
    <li class="lt-math-9839">Use the dots and boxes method to compute each quotient and remainder: $$\begin{split} 5210 &amp; : 4 \\ 4857 &amp; : 23 \\ 31533 &amp; : 101 \end{split}$$</li>
    <li class="lt-math-9839">Now use the standard algorithm (an example is shown below) to compute each of the quotients and remainders above.</li>
</ul>

<p class="lt-math-9839">\[402 = 134 \cdot 3 + 1 \ldotp \nonumber \]</p>

<ul>
    <li class="lt-math-9839">Which method do you like better: dots and boxes or the standard algorithm method? Or does it depend on the problem you are doing?</li>
</ul>
</section>

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