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288 lines
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Commutative, Associative and Distributive Laws</h1>
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<p class="center"><i>Wow! What a mouthful of words! But the ideas are simple.</i></p>
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<div class="video">H1zsWdHC_V8</div>
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<h2 id="titlecomm">Commutative Laws</h2>
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<p>The "Commutative Laws" say we can <b>swap numbers</b> over and still get the same answer ...</p>
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<p>... when we <b>add</b>:</p>
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<div class="center80">
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<p class="center large">a + b<b> = </b> b + a</p>
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</div>
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<h3 align="center">Example:</h3>
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<p class="center"><img src="numbers/images/commutative-add.svg" alt="Commutative Law Addition" height="80" width="365"></p>
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<p> </p>
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<p>... or when we <b>multiply</b>:</p>
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<div class="center80">
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<p class="center large">a × b<b> = </b> b × a</p>
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</div>
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<h3 align="center">Example:</h3>
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<p class="center"><img src="numbers/images/commutative-multiply.svg" alt="Commutative Law multiplication" height="101" width="188"></p>
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<p> </p>
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<a id="qc"></a>
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<h3>Percentages too!</h3>
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<p>Because <span class="large">a × b<b> = </b> b × a</span> it is also true that:</p>
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<p class="larger center">a% of b<b> = </b> b% of a</p>
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<div class="example">
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<h3>Example: what is 8% of 50 ?</h3>
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<p class="center">8% of 50 = 50% of 8<br>
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= 4</p>
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</div>
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<p> </p>
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<div class="words">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="numbers/images/commute.jpg" alt="commute" height="127" width="300"></p>
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<p>Why <b>"commutative</b>" ... ?</p>
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<p>Because the numbers can travel back and forth like a <b>commuter</b>.</p>
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<div style="clear:both"></div>
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</div>
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<div class="questions">4591, 4599, 4615, 4639, 4647, 4592, 4600, 4616</div>
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<p> </p>
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<div class="video">KBfnkUGeMvI</div>
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<h2 id="titleassoc">Associative Laws</h2>
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<p>The "Associative Laws" say that it doesn't matter how we group the numbers (i.e. which we calculate first) ...</p>
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<p>... when we <b>add</b>:</p>
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<div class="center80">
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<p class="center large">(a + b) + c<b> = </b> a + (b + c)</p>
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</div>
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<p class="center"><img src="numbers/images/associative-add.svg" alt="Associative Law addition" height="68" width="405"></p>
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<p>... or when we <b>multiply</b>:</p>
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<div class="center80">
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<p class="center large">(a × b) × c<b> = </b> a × (b × c)</p>
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</div>
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<h3 align="center"><img src="numbers/images/associative-multiply.svg" alt="Associative Law multiplication" height="107" width="265"></h3>
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<h3>Examples:</h3>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td>This:</td>
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<td nowrap="nowrap"><b>(2 + 4)</b> + 5<b> = 6</b> + 5<b> = 11</b></td>
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</tr>
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<tr>
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<td>Has the same answer as this:</td>
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<td nowrap="nowrap">2 +<b> (4 + 5)</b> = 2 +<b> 9 = 11</b></td>
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</tr>
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</tbody></table>
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</div><br>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td>This:</td>
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<td nowrap="nowrap"><b>(3 × 4)</b> × 5 = <b>12</b> × 5 = <b>60</b></td>
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</tr>
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<tr>
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<td>Has the same answer as this:</td>
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<td nowrap="nowrap">3 ×<b> (4 × 5)</b> = 3 × <b>20</b> = <b>60</b></td>
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</tr>
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</tbody></table>
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</div>
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<h3>Uses:</h3>
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<p>Sometimes it is easier to add or multiply in a different order:</p>
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<div class="simple">
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<div class="example">
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<h3>What is 19 + 36 + 4?</h3>
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<p class="center">19 + 36 + 4 = 19 + <b>(36 + 4)</b> <br>
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= 19 + <b>40</b> = 59</p>
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</div>
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<a id="qa"></a>
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<p>Or to rearrange a little:</p>
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<div class="simple">
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<div class="example">
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<h3>What is 2 × 16 × 5?</h3>
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<p class="center">2 × 16 × 5<b> = (2 × 5)</b> × 16<b> <br>
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= 10</b> × 16 = 160</p>
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</div>
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</div>
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</div>
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<p> </p>
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<div class="questions">4603, 4610, 4627, 4631, 4643, 4654, 4606, 4612</div>
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<p> </p>
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<div class="video">0v-G6OwcKmU</div>
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<h2 id="titledist">Distributive Law</h2>
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<p>The "Distributive Law" is the BEST one of all, but needs careful attention.</p>
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<p>This is what it lets us do:</p>
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<p class="center"><img src="numbers/images/distributive-law.svg" alt="Distributive Law" height="99" width="330"></p>
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<p class="center">3 lots of <b>(2+4)</b> is the same as <b>3 lots of 2</b> plus <b>3 lots of 4</b></p>
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<p>So, the <b>3×</b> can be "distributed" across the <b>2+4</b>, into <b>3×2</b> and <b>3×4</b></p>
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<p>And we write it like this:</p>
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<div class="center80">
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<p class="center larger">a × (b + c) = a × b + a × c</p>
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</div>
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<p>Try the calculations yourself:</p>
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<ul>
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<li>3 × (<b>2 + 4</b>) = 3 × <b>6</b> = <span class="hide">18</span></li>
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<li>3×2 + 3×4 = 6 + 12 = <span class="hide">18</span></li>
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</ul>
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<p>Either way gets the same answer.</p>
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<p>In English we can say:</p>
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<div class="center80">
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<p>We get the same answer when we:</p>
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<ul>
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<li>multiply a number by a <b>group of numbers added together</b>, or</li>
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<li>do each <b>multiply</b> separately then <b>add</b> them</li>
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</ul>
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</div>
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<p> </p>
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<h3>Uses:</h3>
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<p>Sometimes it is easier to break up a difficult multiplication:</p>
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<div class="example">
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<h3>Example: What is 6 × 204 ?</h3>
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<p class="center">6 × 204 = 6×200 + 6×4 <br>
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= 1,200 + 24 <br>
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= 1,224</p>
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</div>
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<p>Or to combine:</p>
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<div class="example">
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<h3>Example: What is 16 × 6 + 16 × 4?</h3>
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<p class="center">16 × 6 + 16 × 4 = 16 × <b>(6+4)</b> <br>
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= 16 × <b>10</b> <br>
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= 160</p>
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</div>
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<p>We can use it in subtraction too:</p>
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<div class="example">
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<h3>Example: 26×3 - 24×3</h3>
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<div class="center">26×3 - 24×3 = <b>(26 - 24)</b> × 3 <br>
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= 2 × 3 <br>
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= 6 </div>
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</div>
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<a id="qd"></a>
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<p>We could use it for a long list of additions, too:</p>
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<div class="example">
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<h3>Example: 6×7 + 2×7 + 3×7 + 5×7 + 4×7</h3>
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<p class="center"><b>6</b>×7 + <b>2</b>×7 + <b>3</b>×7 + <b>5</b>×7 + <b>4</b>×7<br>
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= <b>(6+2+3+5+4)</b> × 7<br>
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= <b>20</b> × 7<br>
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= <b>140</b></p>
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</div>
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<p> </p>
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<div class="questions">5656, 5657, 5658, 5659, 5660, 5661, 3172</div>
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<h2>And those are the Laws . . .</h2>
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<h2> . . . but don't go too far!</h2>
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<p>The Commutative Law does <b>not</b> work for subtraction or division:</p>
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<div class="example">
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<h3>Example:</h3>
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<ul>
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<li>12 / 3 = <b>4</b>, but</li>
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<li>3 / 12 = <b>¼</b></li>
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</ul>
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</div>
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<p> The Associative Law does <b>not</b> work for subtraction or division:</p>
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<div class="example">
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<h3>Example:</h3>
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<ul>
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<li>(9 – 4) – 3 = 5 – 3 = <b>2</b>, but</li>
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<li>9 – (4 – 3) = 9 – 1 = <b>8</b></li>
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</ul>
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</div>
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<p> The Distributive Law does <b>not</b> work for division:</p>
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<div class="example">
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<h3>Example:</h3>
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<ul>
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<li>24 / (4 + 8) = 24 / 12 = <b>2</b>, but</li>
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<li>24 / 4 + 24 / 8 = 6 + 3 = <b>9</b></li>
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</ul>
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</div>
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<h2>Summary</h2>
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<table width="100%" cellpadding="6" border="0">
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<tbody>
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<tr>
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<td class="larger" width="33%" valign="top">Commutative Laws:</td>
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<td width="67%"><span class="large">a + b<b> = </b> b + a<br>
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a × b<b> = </b> b × a</span></td>
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</tr>
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<tr>
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<td class="larger" valign="top">Associative Laws:</td>
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<td><span class="large">(a + b) + c<b> = </b> a + (b + c)<br>
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(a × b) × c<b> = </b> a × (b × c)</span></td>
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</tr>
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<tr>
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<td class="larger" valign="top">Distributive Law:</td>
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<td class="large">a × (b + c) = a × b + a × c</td>
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</tr>
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</tbody></table>
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<p> </p>
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<div class="activity"> <a href="activity/associative-commutative-distributive.html">Activity: Commutative, Associative and Distributive</a></div>
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<div class="related">
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<a href="numbers/index.html">Numbers Index</a>
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<a href="algebra/index.html">Algebra Index</a>
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