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537 lines
20 KiB
HTML
537 lines
20 KiB
HTML
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<!-- #BeginEditable "Body" -->
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<div class="video">GCfiIBPG7Aw</div>
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<h1 id="title" align="center">Multiplying Negatives</h1>
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<h2>When We Multiply:</h2>
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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> </td>
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<th align="center"> </th>
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<td> </td>
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<td class="larger" align="center" nowrap="nowrap">Example</td>
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</tr>
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<tr>
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<td><img src="images/style/plus.svg" alt="plus" style="vertical-align:middle;" height="44" width="44"> <span class="large"><b>×</b></span> <img src="images/style/plus.svg" alt="plus" style="vertical-align:middle;" height="44" width="44"></td>
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<td style="text-align:center;"> two positives make a positive:</td>
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<td><img src="images/style/plus.svg" alt="plus" height="44" width="44"></td>
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<td class="larger" align="right" nowrap="nowrap">3 × 2 = 6</td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;"> </td>
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<td> </td>
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<td class="larger" align="right" nowrap="nowrap"> </td>
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</tr>
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<tr>
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<td><img src="images/style/minus.svg" alt="minus" style="vertical-align:middle;" height="44" width="44"> <span class="large"><b>×</b></span> <img src="images/style/minus.svg" alt="minus" style="vertical-align:middle;" height="44" width="44"></td>
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<td style="text-align:center;"> two negatives make a positive: </td>
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<td><img src="images/style/plus.svg" alt="plus" height="44" width="44"></td>
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<td class="larger" align="right" nowrap="nowrap"> (−3) × (−2) = 6</td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;"> </td>
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<td> </td>
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<td class="larger" align="right" nowrap="nowrap"> </td>
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</tr>
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<tr>
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<td><img src="images/style/minus.svg" alt="minus" style="vertical-align:middle;" height="44" width="44"> <span class="large"><b>×</b></span> <img src="images/style/plus.svg" alt="plus" style="vertical-align:middle;" height="44" width="44"></td>
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<td style="text-align:center;">a negative and a positive<br>
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make a negative:</td>
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<td><img src="images/style/minus.svg" alt="minus" height="44" width="44"></td>
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<td class="larger" align="right" nowrap="nowrap">(−3) × 2 = −6</td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;"> </td>
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<td> </td>
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<td class="larger" align="right" nowrap="nowrap"> </td>
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</tr>
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<tr>
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<td><img src="images/style/plus.svg" alt="plus" style="vertical-align:middle;" height="44" width="44"> <span class="large"><b>×</b></span> <img src="images/style/minus.svg" alt="minus" style="vertical-align:middle;" height="44" width="44"></td>
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<td style="text-align:center;">a positive and a negative<br>
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make a negative:</td>
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<td><img src="images/style/minus.svg" alt="minus" height="44" width="44"></td>
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<td class="larger" align="right" nowrap="nowrap">3 × (−2) = −6</td>
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</tr>
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</tbody></table>
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<p>Yes indeed, two negatives make a positive, and we will explain <b>why</b>, with examples!</p>
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<h2>Signs</h2>
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<p>Let's talk about <b>signs</b>.</p>
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<p>"+" is the positive sign, "−" is the negative sign.</p>
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<p>When a number has <b>no sign</b> it usually means that it is <b>positive</b>.</p>
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<div class="example">
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<p>Example: <b>5</b> is really <b>+5</b></p>
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</div>
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<p>And we can put () around the numbers to avoid confusion.</p>
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<div class="example">
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<p>Example: <b>3 × −2</b> can be written as <b>(+3) × (−2)</b></p>
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</div>
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<h2>Two Signs: The Rules</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="numbers/images/multiply-like-signs.svg" alt="multiply like signs" height="97" width="328"></p>
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<p><span class="larger">"Two like signs make a positive sign,<br>
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two unlike signs make a negative sign"</span></p>
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<div style="clear:both"></div>
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<div class="example">
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<h3>Example: (−2) × (+5)</h3>
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<p>The signs are <span class="hilite">−</span> and <span class="hilite">+</span> (a negative sign and a positive sign), so they are <b>unlike signs</b> (they are different to each other)</p>
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<p>So the result must be <b>negative</b>:</p>
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<p class="center larger">(−2) × (+5) = −10</p>
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</div>
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<div class="example">
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<h3>Example: (−4) × (−3)</h3>
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<p>The signs are <span class="hilite">−</span> and <span class="hilite">−</span> (they are both negative signs), so they are <b>like signs</b> (like each other)</p>
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<p>So the result must be <b>positive</b>:</p>
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<p class="center larger">(−4) × (−3) = +12</p>
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</div>
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<h2>Why does multiplying two negative numbers make a positive?</h2>
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<p>Well, first there is the "common sense" explanation:</p>
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<div class="center80"><img src="images/art/sage-brocolli.gif" alt="dog does not want brocolli" style="float:left; margin: 15px 15px 0 15px;" height="127" width="250">
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<p>When I say "Eat!" I am encouraging you to eat (positive)</p>
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<p>But when I say "Do not eat!" I am saying the opposite (negative).</p>
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<p>Now if I say "Do <b>NOT</b> not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).</p>
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</div>
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<p>So, two negatives make a positive, and if that satisfies you, then you don't need to read any more.</p>
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<h2>Direction</h2>
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<p>It is all about direction. Remember the <a href="number-line.html">Number Line</a>?</p>
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<p class="center"><img src="numbers/images/number-line.svg" alt="number line" height="47" width="595"></p>
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<p>Well here we have Baby Steven taking his first steps. He takes 2 paces at a time, and does this three times, so he moves 2 steps x 3 = 6 steps forward:</p>
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<p class="center"><img src="images/multiply-p2p3.gif" alt="multiply 2 x 3" height="150" width="588"></p>
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<p>Now, Baby Steven can also step backwards (he is a clever little guy). His Dad puts him back at the start and then Steven steps backwards 2 steps, and does this three times:</p>
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<p class="center"><img src="images/multiply-n2p3.gif" alt="multiply -2 x 3" height="152" width="589"></p>
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<p>Once again Steven's Dad puts him back at the start, but facing the other way. Steven takes 2 steps forward (for him!) but he is heading in the negative direction. He does this 3 times:</p>
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<p class="center"><img src="images/multiply-p2n3.gif" alt="multiply 2 x -3" height="151" width="589"></p>
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<p>Back at the start again (thanks Dad!), still facing in the negative direction, he tries his backwards walking, once again taking two steps at a time, and he does this three times:</p>
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<p class="center"><img src="images/multiply-n2n3.gif" alt="multiply -2 x -3" height="153" width="588"></p>
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<p>So, by walking backwards, while facing in the negative direction, he moves in the positive direction.</p>
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<p class="large">Try it yourself! Try walking forwards and backwards, then again but facing the other direction.</p>
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<h2>Play With It</h2>
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<p>But maybe you would like to see it in action? Use the sliders below:</p>
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<div class="script" style="height: 260px;">
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numbers/images/number-line-mult.js
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</div>
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<h2>More Examples</h2>
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<div class="example">
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<h3>Example: Money</h3>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td style="text-align:right;">Sam gives you three $10 notes:</td>
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<td> </td>
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<td>+3 × +10 = you gain $30 </td>
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</tr>
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<tr>
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<td style="text-align:right;">Sam gives you three $10 debts:</td>
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<td> </td>
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<td>+3 × −10 = you lose $30</td>
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</tr>
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<tr>
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<td style="text-align:right;">Sam takes three $10 notes from you:</td>
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<td> </td>
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<td>−3 × +10 = you lose $30 </td>
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</tr>
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<tr>
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<td style="text-align:right;">Sam takes three $10 debts from you:</td>
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<td> </td>
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<td>−3 × −10 = you gain $30</td>
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</tr>
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</tbody></table>
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</div>
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<div class="example">
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<h3>Example: A Video of People Running</h3>
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<p>People Run Forward, Video Normal:</p>
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<p class="so">All normal, people running forwards: +1 × +1 = +1</p>
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<p><br>
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People Run Forward, but Video in <b>Reverse</b>:</p>
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<p><span class="so">Looks like people running backwards: +1 × −1 = −1</span></p>
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<p><br>
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People Run <b>Backward</b>, Video Normal:</p>
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<p class="so">You see people running backwards: −1 × +1 = −1</p>
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<p><br>
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People Run <b>Backward</b>, but Video in <b>Reverse</b>:</p>
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<p class="so">Looks like people running forwards: −1 × −1 = +1</p>
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</div>
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<div class="example">
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<h3>Example: Tank Levels Rising/Falling</h3>
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<p>The tank has 30,000 liters, and 1,000 liters are taken out every day. What was the amount of water in the tank <b>3 days ago</b>?</p>
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<p>We know the amount of water in the tank changes by −1,000 every day, and we need to subtract that 3 times (to go <b>back 3 days</b>), so the change is:</p>
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<p class="center">−3 × −1,000 = +3,000</p>
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<p>The full calculation is:</p>
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<p class="center">30,000 + (−3 × −1,000) = 30,000 + 3,000 = 33,000</p>
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<p>So 3 days ago there were 33,000 liters of water in the tank.</p>
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</div>
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<h2>Multiplication Table</h2>
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<p>Here is <b>another way</b> of looking at it.</p>
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<p>First have a play with this (explanations below):</p>
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<div class="script" style="height: 460px;">
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numbers/images/mult-grid.js?min=-5&max=5
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</div>
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<p>Start with the <a href="tables.html">multiplication table</a> (just up to 4×4 will do):</p>
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<style class="text/css">
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table.thin {
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border: 1px solid #aaaaff;
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border-spacing: 0px;
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}
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table.thin th {
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border: 1px solid #aaaaff
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}
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table.thin td {
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border: 1px solid #dddddd;
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text-align: center;
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}
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</style>
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<table class="thin" align="center">
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<tbody>
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<tr style="text-align:right;">
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<th width="30">×</th>
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<th width="30">1</th>
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<th width="30">2</th>
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<th width="30">3</th>
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<th width="30">4</th>
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</tr>
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<tr style="text-align:right;">
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<th>1</th>
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<td class="bgb1">1</td>
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<td class="bgb1">2</td>
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<td class="bgb1">3</td>
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<td class="bgb1">4</td>
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</tr>
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<tr style="text-align:right;">
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<th>2</th>
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<td class="bgb1">2</td>
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<td class="bgb1">4</td>
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<td class="bgb1">6</td>
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<td class="bgb1">8</td>
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</tr>
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<tr style="text-align:right;">
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<th>3</th>
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<td class="bgb1">3</td>
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<td class="bgb1">6</td>
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<td class="bgb1">9</td>
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<td class="bgb1">12</td>
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</tr>
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<tr style="text-align:right;">
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<th>4</th>
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<td class="bgb1">4</td>
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<td class="bgb1">8</td>
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<td class="bgb1">12</td>
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<td class="bgb1">16</td>
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</tr>
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</tbody></table>
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<p class="center larger">Now see what happens when we head into <b>negatives</b>!</p>
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<p>Let's go <b>backwards</b> through zero:</p>
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<table class="thin" align="center">
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<tbody>
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<tr style="text-align:right;">
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<th width="30">×</th>
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<th width="30">1</th>
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<th width="30">2</th>
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<th width="30">3</th>
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<th width="30">4</th>
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</tr>
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<tr style="text-align:right;">
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<th>-4</th>
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<td class="bga1">-4</td>
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<td class="bga1">-8</td>
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<td class="bga1">-12</td>
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<td class="bga1">-16</td>
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</tr>
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<tr style="text-align:right;">
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<th>-3</th>
|
||
<td class="bga1">-3</td>
|
||
<td class="bga1">-6</td>
|
||
<td class="bga1">-9</td>
|
||
<td class="bga1">-12</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>-2</th>
|
||
<td class="bga1">-2</td>
|
||
<td class="bga1">-4</td>
|
||
<td class="bga1">-6</td>
|
||
<td class="bga1">-8</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>-1</th>
|
||
<td class="bga1">-1</td>
|
||
<td class="bga1">-2</td>
|
||
<td class="bga1">-3</td>
|
||
<td class="bga1">-4</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>0</th>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>1</th>
|
||
<td class="bgb1">1</td>
|
||
<td class="bgb1">2</td>
|
||
<td class="bgb1">3</td>
|
||
<td class="bgb1">4</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>2</th>
|
||
<td class="bgb1">2</td>
|
||
<td class="bgb1">4</td>
|
||
<td class="bgb1">6</td>
|
||
<td class="bgb1">8</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>3</th>
|
||
<td class="bgb1">3</td>
|
||
<td class="bgb1">6</td>
|
||
<td class="bgb1">9</td>
|
||
<td class="bgb1">12</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>4</th>
|
||
<td class="bgb1">4</td>
|
||
<td class="bgb1">8</td>
|
||
<td class="bgb1">12</td>
|
||
<td class="bgb1">16</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>Look at the "4" column: it goes <b>-16, -12, -8, -4, 0, 4, 8, 12, 16</b>. Getting 4 larger each time.</p>
|
||
<p>Look over that table again, make sure you are comfortable with how it works, because ...</p>
|
||
<p>... now we go <b>further to the left</b>, through zero:</p>
|
||
<table class="thin" align="center">
|
||
<tbody>
|
||
<tr style="text-align:right;">
|
||
<th width="30">×</th>
|
||
<th width="30">-4</th>
|
||
<th width="30">-3</th>
|
||
<th width="30">-2</th>
|
||
<th width="30">-1</th>
|
||
<th width="30">0</th>
|
||
<th width="30">1</th>
|
||
<th width="30">2</th>
|
||
<th width="30">3</th>
|
||
<th width="30">4</th>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>-4</th>
|
||
<td class="bgb1">16</td>
|
||
<td class="bgb1">12</td>
|
||
<td class="bgb1">8</td>
|
||
<td class="bgb1">4</td>
|
||
<td>0</td>
|
||
<td class="bga1">-4</td>
|
||
<td class="bga1">-8</td>
|
||
<td class="bga1">-12</td>
|
||
<td class="bga1">-16</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>-3</th>
|
||
<td class="bgb1">12</td>
|
||
<td class="bgb1">9</td>
|
||
<td class="bgb1">6</td>
|
||
<td class="bgb1">3</td>
|
||
<td>0</td>
|
||
<td class="bga1">-3</td>
|
||
<td class="bga1">-6</td>
|
||
<td class="bga1">-9</td>
|
||
<td class="bga1">-12</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>-2</th>
|
||
<td class="bgb1">8</td>
|
||
<td class="bgb1">6</td>
|
||
<td class="bgb1">4</td>
|
||
<td class="bgb1">2</td>
|
||
<td>0</td>
|
||
<td class="bga1">-2</td>
|
||
<td class="bga1">-4</td>
|
||
<td class="bga1">-6</td>
|
||
<td class="bga1">-8</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>-1</th>
|
||
<td class="bgb1">4</td>
|
||
<td class="bgb1">3</td>
|
||
<td class="bgb1">2</td>
|
||
<td class="bgb1">1</td>
|
||
<td>0</td>
|
||
<td class="bga1">-1</td>
|
||
<td class="bga1">-2</td>
|
||
<td class="bga1">-3</td>
|
||
<td class="bga1">-4</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>0</th>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
<td>0</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>1</th>
|
||
<td class="bga1">-4</td>
|
||
<td class="bga1">-3</td>
|
||
<td class="bga1">-2</td>
|
||
<td class="bga1">-1</td>
|
||
<td>0</td>
|
||
<td class="bgb1">1</td>
|
||
<td class="bgb1">2</td>
|
||
<td class="bgb1">3</td>
|
||
<td class="bgb1">4</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>2</th>
|
||
<td class="bga1">-8</td>
|
||
<td class="bga1">-6</td>
|
||
<td class="bga1">-4</td>
|
||
<td class="bga1">-2</td>
|
||
<td>0</td>
|
||
<td class="bgb1">2</td>
|
||
<td class="bgb1">4</td>
|
||
<td class="bgb1">6</td>
|
||
<td class="bgb1">8</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>3</th>
|
||
<td class="bga1">-12</td>
|
||
<td class="bga1">-9</td>
|
||
<td class="bga1">-6</td>
|
||
<td class="bga1">-3</td>
|
||
<td>0</td>
|
||
<td class="bgb1">3</td>
|
||
<td class="bgb1">6</td>
|
||
<td class="bgb1">9</td>
|
||
<td class="bgb1">12</td>
|
||
</tr>
|
||
<tr style="text-align:right;">
|
||
<th>4</th>
|
||
<td class="bga1">-16</td>
|
||
<td class="bga1">-12</td>
|
||
<td class="bga1">-8</td>
|
||
<td class="bga1">-4</td>
|
||
<td>0</td>
|
||
<td class="bgb1">4</td>
|
||
<td class="bgb1">8</td>
|
||
<td class="bgb1">12</td>
|
||
<td class="bgb1">16</td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>We can follow along a row (or column) and the values change consistently:</p>
|
||
<ul>
|
||
<li>Follow the "4" row: it goes <b>-16, -12, -8, -4, 0, 4, 8, 12, 16</b>. Getting 4 larger each time.</li>
|
||
<li>Follow the "-4" row: it goes <b>16, 12, 8, 4, 0, -4, -8, -12, -16</b>. Getting 4 smaller each time.</li>
|
||
<li>etc...</li>
|
||
</ul>
|
||
<p>So it all follows a consistent pattern! </p>
|
||
<p>Also see this at <a href="numbers/multiplication-table-real.html">Real Multiplication Table</a>.</p>
|
||
|
||
<h2>What About Multiplying 3 or More Numbers Together?</h2>
|
||
<p>Multiply two at a time and follow the rules.</p>
|
||
<div class="example">
|
||
<h3>Example: What is (−2) × (−3) × (−4) ?</h3>
|
||
<p>First multiply (−2) × (−3). Two like signs make a positive sign, so:</p>
|
||
<p class="center">(−2) × (−3) = +6</p>
|
||
<p>Next multiply +6 × (−4). Two unlike signs make a negative sign, so:</p>
|
||
<p class="center">+6 × (−4) = −24</p>
|
||
<p><b>Result: (−2) × (−3) × (−4) = −24</b></p>
|
||
</div>
|
||
<a id="q"></a>
|
||
<p> </p>
|
||
<div class="questions">330, 1615, 1616, 1617, 3447, 3448,331, 1618, 3170, 3171</div>
|
||
|
||
<div class="related">
|
||
<a href="timestable.html">Multiplication Tables Test</a>
|
||
<a href="worksheets/multiplication.html">Multiplication Worksheets</a>
|
||
<a href="positive-negative-integers.html">Adding and Subtracting Positive and Negative Numbers</a>
|
||
<a href="number-line.html">Number Line</a>
|
||
</div>
|
||
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