3.06:_Number_Line_Model

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<p class="lt-math-9840">Another way we often think about numbers is as abstract quantities that can be measured: length, area, and volume are all examples.</p>

<p class="lt-math-9840">In a measurement model, you have to pick a <i>basic unit</i>.   The basic unit is a quantity &mdash; length, area, or volume &mdash; that you assign to the number one. You can then assign numbers to other quantities based on how many of your basic unit fit inside.</p>

<p class="lt-math-9840">For now, we&rsquo;ll focus on the quantity length, and we&rsquo;ll work with a number line where the basic unit is already marked off.</p>

<span id="Addition_and_Subtraction_on_the_Number_Line"></span><h2 class="lt-math-9840">Addition and Subtraction on the Number Line</h2>

<p class="lt-math-9840">Imagine a person &mdash; we&rsquo;ll call him Zed &mdash; who can stand on the number line. We&rsquo;ll say that the distance Zed walks when he takes a step is exactly one unit.</p>

<p class="lt-math-9840">When Zed wants to add or subtract with whole numbers on the number line, he always starts at 0 and faces the positive direction (towards 1). Then what he does depends on the calculation.</p>

<p class="lt-math-9840">If Zed wants to <i>add</i> two numbers, he walks forward (to the right of the number line) however many steps are indicated by the first number (the first <i>addend</i>). Then he walks forward (to your right on the number line) the number of steps indicated by the second number (the second <i>addend</i>). Where he lands is the <i>sum</i> of the two numbers.</p>

<p class="lt-math-9840">If Zed wants to add 3 + 4, he starts at 0 and faces towards the positive numbers. He walks forward 3 steps, then he walks forward 4 more steps.</p>

<p class="lt-math-9840">Zed ends at the number 7, so the sum of 3 and 4 is 7. 3 + 4 = 7. (But you knew that of course! The point right now is to make sense of the <i>number line</i> <i>model</i>.)</p>

<p class="lt-math-9840">When Zed wants to <i>subtract</i> two numbers, he he walks forward (to the right on the number line) however many steps are indicated by the first number (the <i>minuend</i>).  Then he walks <i>backwards</i> (to the left on the number line) the number of steps indicated by the second number (the <i>subtrahend</i>).  Where he lands is the <i>difference</i> of the two numbers.</p>

<section class="box-example"><span id=":_11_.E2.80.93_3"></span><h5 class="box-legend lt-math-9840"><span class="lt-icon-default">: 11 &ndash; 3</span></h5>

<p class="lt-math-9840">If Zed wants to subtract 11 &ndash; 3, he starts at 0 and faces the positive numbers (the right side of the number line). He walks forward 11 steps on the number line, then he walks backwards 3 steps.</p>

<p class="lt-math-9840">Zed ends at the number 8, so the difference of 11 and 3 is 8.   11 &ndash; 3 = 8.  (But you knew that!)</p>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share"></span><h5 class="box-legend lt-math-9840"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<ul>
    <li class="lt-math-9840">Work out each of these exercises on a number line.  You can actually pace it out on a life-sized number line or draw a picture: $$4 + 5 \qquad 6 + 9 \qquad 10 - 7 \qquad 8 - 1$$</li>
    <li class="lt-math-9840">Why does it make sense to walk forward for addition and walk backwards for subtraction? In what way is this the same as &ldquo;combining&rdquo; for addition and &ldquo;take away&rdquo; for subtraction&rdquo;?</li>
    <li class="lt-math-9840">What happens if you do these subtraction problems on a number line? Explain your answers. $$6 - 9 \qquad 1 - 7 \qquad 4 - 11 \qquad 0 - 1$$</li>
    <li class="lt-math-9840">Could you do the subtraction problems above with the dots and boxes model?</li>
</ul>
</section>

<span id="Multiplication_and_Division_on_the_Number_Line"></span><h2 class="lt-math-9840">Multiplication and Division on the Number Line</h2>

<p class="lt-math-9840">Since multiplication is really repeated addition, we can adapt our addition model to become a multiplication model as well.  Let&rsquo;s think about 3 &times; 4.  This means to add four to itself three times (that&rsquo;s simply the definition of multiplication!):</p>

<p class="lt-math-9840">\[3 \times 4 = 4 + 4 + 4 \ldotp \nonumber \]</p>

<p class="lt-math-9840">So to multiply on the number line, we do the process for addition several times.</p>

<p class="lt-math-9840">To multiply two numbers, Zed starts at 0 as always, and he faces the positive direction. He walks forward the number of steps given by the second number (the second <i>factor</i>).   He repeats that process the number of times given by the first number (the first <i>factor</i>). Where he lands is the <i>product</i> of the two numbers.</p>

<section class="box-example"><span id=":_3_.C3.97_4"></span><h5 class="box-legend lt-math-9840"><span class="lt-icon-default">: 3 &times; 4</span></h5>

<p class="lt-math-9840">If Zed wants to multiply 3 &times; 4, he can think of it this way:</p>

<p class="lt-math-9840">\[\begin{split} 3&amp; \qquad \qquad \qquad \times \\ \downarrow &amp; \\ \text{how many times}\; &amp; \text{to repeat it} \end{split} \begin{split} 4&amp; \\ \downarrow &amp; \\ \text{how many steps}\; &amp; \text{to take forward} \end{split} \nonumber \]</p>

<p class="lt-math-9840">Zed starts at 0, facing the positive direction. The he repeats this three times: take four steps forward.</p>

<p class="lt-math-9840">He ends at the number 12, so the product of 3 and 4 is 12. That is, 3 &times; 4 = 12.</p>

<p class="lt-math-9840">Remember our quotative model of division: One way to interpret 15 : 5 is:</p>

<blockquote>
<p class="lt-math-9840">How many groups of 5 fit into 15?</p>
</blockquote>

<p class="lt-math-9840">Thinking on the number line, we can ask it this way:</p>

<blockquote>
<p class="lt-math-9840">Zed takes 5 steps at a time.  If Zed lands at the number 15, how many times did he take 5 steps?</p>
</blockquote>

<p class="lt-math-9840">To calculate a division problem on the number line, Zed starts at 0, facing the positive  direction.   He  walks  forward  the  number  of  steps  given  by  the  second number (the <i>divisor</i>).  He repeats that process until he lands at the first number (the <i>dividend</i>). The number of times he repeated the process gives the <i>quotient</i> of the two numbers.</p>

<section class="box-example"><span id=":_15_.C3.B7_5"></span><h5 class="box-legend lt-math-9840"><span class="lt-icon-default">: 15 &divide; 5</span></h5>

<p class="lt-math-9840">If Zed wants to compute 15 : 5, he can think of it this way:</p>

<p class="lt-math-9840">He starts at 0, facing the positive direction.</p>

<ul>
    <li class="lt-math-9840">Zed takes 5 steps forward.  He is now at 5, not 15.  So he needs to repeat the process.</li>
    <li class="lt-math-9840">Zed takes 5 steps forward again.  He is now at 10, not 15.  So he needs to repeat the process.</li>
    <li class="lt-math-9840">Zed takes 5 more steps forward. He is at 15, so he stops.</li>
</ul>

<p class="lt-math-9840">Since he repeated the process three times, we see there are 3 groups of 5 in 15.   So the quotient of 15 and 5 is 3.    That is, 15 : 5 = 3.</p>

<section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share_2"></span><h5 class="box-legend lt-math-9840"><span class="lt-icon-default">Think / Pair / Share</span></h5>

<ul>
    <li class="lt-math-9840">Work out each of these exercises on a number line.  You can actually pace it out on a life-sized number line or draw a picture: $$2 \times 5 \qquad 7 \times 1 \qquad 10 : 2 \qquad 6 : 1$$</li>
    <li class="lt-math-9840">Can you think of a way to interpret these multiplication problems on a number line? Explain your ideas. $$4 \times 0 \qquad 0 \times 5 \qquad 3 \times (-2) \qquad 2 \times (-1)$$</li>
    <li class="lt-math-9840">What happens if you try to solve these division problems on a number line? Can you do it? Explain your ideas. $$0 : 2 \qquad 0 : 10 \qquad 3 : 0 \qquad 5 : 0$$</li>
</ul>
</section>

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