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3.01:_Introduction
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<p class="lt-math-9835">When learning and teaching about arithmetic, it helps to have mental and physical <i>models</i> for what the operations mean. That way, when you are presented with an unfamiliar problem or a question about why something is true, you can often work it out using the model — this might mean drawing pictures, using physical materials (manipulatives), or just thinking about the model to help you reason out the answer.</p> <section class="box-exercise"><span id="Think_.2F_Pair_.2F_Share"></span><h2 class="box-legend lt-math-9835"><span class="lt-icon-default">Think / Pair / Share</span></h2> <p class="lt-math-9835">Write down your mental models for each of the four basic operations. What do they actually <i>mean</i>? How would you explain them to a second grader? What pictures could you draw for each operation? Think about each one separately, as well as how they relate to each other:</p> <ul> <li class="lt-math-9835">addition</li> <li class="lt-math-9835">subtraction</li> <li class="lt-math-9835">multiplication, and</li> <li class="lt-math-9835">division.</li> </ul> <p class="lt-math-9835">After writing down you own ideas, share them with a partner. Do you and your partner have the same models for each of the operations or do you think about them differently?</p> </section> <p class="lt-math-9835">Teachers should have lots of mental models — lots of ways to explain the same concept. In this chapter, we’ll look at some different ways to understand the four basic arithmetic operations. First, let’s define some terms:</p> <section class="box-definition"><span id="Definition"></span><h2 class="box-legend lt-math-9835"><span class="lt-icon-default">Definition</span></h2> <p class="lt-math-9835"><strong>Counting numbers</strong> are literally the numbers we use for counting: 1, 2, 3, 4, 5… These are sometimes called the <i>natural numbers</i> by mathematicians, and they are represented by the symbol <img alt="\mathbb N" height="12" src="http://pressbooks.oer.hawaii.edu/mathforelementaryteachers/wp-content/ql-cache/quicklatex.com-3d313ef7e09081e296c054186eaa3a47_l3.svg" title="Rendered by QuickLaTeX.com" width="13" />.</p> <p class="lt-math-9835"><strong>Whole numbers</strong> are the counting numbers together with zero.</p> <p class="lt-math-9835"><strong>Integers</strong> include the positive and negative whole numbers, and mathematicians represent these with the symbol <img alt="\mathbb Z" height="12" src="http://pressbooks.oer.hawaii.edu/mathforelementaryteachers/wp-content/ql-cache/quicklatex.com-d0cff47bbd7da68fe23acee1af34cf1e_l3.svg" title="Rendered by QuickLaTeX.com" width="12" />. (This comes from German, where the word for “number” is “zählen.”)</p> </section> <p class="lt-math-9835">We already have a natural model for thinking about counting numbers: a number is a quantity of dots. Depending on which number system you use — Roman numerals, base ten, binary, etc. — you might write down the number in different ways. But the quantity of dots is a counting number, however you write it down.</p>
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