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346 lines
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HTML
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<title>Activity: Clocks and Angles</title>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Activity:
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Clocks and Angles</h1>
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<p><i>This
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activity is about <a href="../time-clocks.html">Analog clocks</a> and the angles made by the
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hands of the clock. You can find out more about angles and how they're measured on the
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page <a href="../geometry/degrees.html">Degrees
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(Angles)</a>.</i></p>
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<p> </p>
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<h3> What is the angle between the hands of a clock at 1 o'clock?</h3>
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<p class="center"> <img style="width: 251px; height: 239px;" alt="clock1" src="images/clock1.jpg"></p>
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<p> At 1 o'clock the minute hand
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(red) points to the 12 and the
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hour hand (blue) points to the 1.
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So we need to find the angle between
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the 12
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and the 1.</p>
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<h3> How many of this angle are there in a complete turn?</h3>
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<p class="center"> <img style="width: 277px; height: 275px;" alt="clock2" src="images/clock2.jpg"></p>
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<p> There are 12 of them in a complete turn (360°), so each
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one
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must be
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<strong>360°
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÷ 12
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= 30°</strong>
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</p>
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<p> So the angle between
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the
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hands
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of a clock at 1 o'clock is
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<strong>30°</strong>
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.</p>
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<div class="center80">
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<p>
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<strong>Note:</strong>
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</p>
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<ul>
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<li>It doesn't
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matter whether we are talking about 1 am or 1
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pm, the answer is exactly the same for both.<br>
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<br>
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</li>
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<li>The angle between the hands at
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1 o'clock could also be given as the reflex angle 330°, but we will
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always give the smaller (acute or obtuse) angle.</li>
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</ul>
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</div>
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<h4> <br>
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</h4>
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<h3>What is the angle
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between the hands of a clock at 2:30? </h3>
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<p class="center"> <img style="width: 242px; height: 252px;" alt="clock3" src="images/clock3.jpg"></p>
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<div class="center80">At 2:30 the minute hand
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(red) points to the 6 and the hour hand
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(blue) points halfway between the 2 and the 3.</div>
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<h3> <br>
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So how many lots of 30° do we have this time?</h3>
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<p class="center"> <img style="width: 275px; height: 275px;" alt="clock4" src="images/clock4.jpg"></p>
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<ul>
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<li>The angle between the 5 and the 6 is
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<strong>30°</strong>
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</li>
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<li>The angle between the 4 and the
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5 is
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<strong>30°</strong>
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</li>
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<li>The angle between the 3 and the
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4 is
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<strong>30°</strong>
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</li>
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<li>The remaining angle is
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<strong>½ × 30° = 15°</strong>
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</li>
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</ul>
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<p> So the angle
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between the hands of a clock at
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<strong>2:30 = 30° + 30° + 30° + 15° = 105°</strong>
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</p>
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<p> </p>
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<h2>Your Turn</h2>
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<h3> Complete the following table (give the smaller angle in each case):<br>
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</h3>
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<table cellspacing="2" cellpadding="2" border="1">
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<tbody>
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<tr>
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<td width="70"><strong>Time</strong></td>
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<td style="text-align: center;" width="70">1:00</td>
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<td style="text-align: center;" width="70">2:30</td>
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<td style="text-align: center;" width="70">7:00</td>
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<td style="text-align: center;" width="70">10:30</td>
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<td style="text-align: center;" width="70">11:20</td>
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<td style="text-align: center;" width="70">3:40</td>
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<td style="text-align: center;" width="70">5:15</td>
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<td style="text-align: center;" width="70">8:45</td>
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</tr>
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<tr>
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<td height="40"><strong>Angle</strong></td>
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<td style="text-align: center;">30°</td>
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<td style="text-align: center;">105°</td>
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<td style="text-align: center;"><br></td>
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<td style="text-align: center;"><br></td>
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<td style="text-align: center;"><br></td>
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<td style="text-align: center;"><br></td>
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<td style="text-align: center;"><br></td>
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<td style="text-align: center;"><br></td>
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</tr>
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</tbody>
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</table>
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<p>Check your answers at the bottom of the page.</p>
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<p> </p>
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<h2>More Complicated Times </h2>
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<p>Finding the angle between the hands of a clock is easy as long
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as we don't use complicated times. </p>
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<p>For example finding the angle
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between the hands at 9:37 is much more difficult. You can try
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that one if you wish, but it's probably too hard.</p>
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<h3> <br>
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Example: At
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what times is the angle between the hands of a clock equal to
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30°?</h3>
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<p> Notice that the question asks for 'times'.
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There are many possible
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answers. Some of them are easy to find, others much more difficult.</p>
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<p>Here are two easy answers:</p>
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<table align="center" border="0">
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<tbody>
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<tr>
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<td><img alt="clock5" src="images/clock5a.jpg"></td>
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<td><img alt="clock5" src="images/clock5b.jpg"></td>
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</tr>
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<tr align="center">
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<td class="large">1 o'clock</td>
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<td class="large">11 o'clock</td>
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</tr>
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<tr>
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<td> </td>
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<td> </td>
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</tr>
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<tr>
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<td colspan="2"><strong>But what about
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this one?</strong></td>
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</tr>
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<tr>
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<td colspan="2" align="center"><img alt="clock6" src="images/clock6.jpg"></td>
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</tr>
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<tr>
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<td colspan="2" class="large" align="center">4:15 PM</td>
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</tr>
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</tbody></table>
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<p class="center"> </p>
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<p> At first glance, it looks like this might be a 30° angle also,
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but at 4:15, the hour hand has already moved
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a quarter of the way between the 4 and the 5. </p>
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<p>So the
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angle is
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<strong>30°
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+ ¼ × 30° = 30° + 7½° = 37½°</strong>
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.</p>
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<p> This might be a more accurate answer:</p>
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<p class="center"> <img style="width: 247px; height: 259px;" alt="clock7" src="images/clock7.jpg"></p>
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<p> Can you find more 30° angles like this one?</p>
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<p> </p>
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<h2>At what times of the day do the hands of a clock lie in a straight
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line?</h2>
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<p>
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<strong>In other words the angle between them is 180°?</strong>
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</p>
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<p> One obvious answer is 6 o'clock:</p>
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<p class="center"> <img style="width: 247px; height: 261px;" alt="clock8" src="images/clock8.jpg"></p>
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<h3> </h3>
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<h3>But what other answers could there be?</h3>
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<p> 9:15 is not correct for a similar reason that 4:15 didn't give us exactly a
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30° angle ... the hour hand has moved on beyond the 9.</p>
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<p> This seems to be a very difficult question to answer, but there is an
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easy way.</p>
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<h3> How many times between 6:00 am and 6:00 pm do the hands make a
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straight line?<br>
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</h3>
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<p> There must be at least one point time for each hour:</p>
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<ul>
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<li>one between 7:00 am and 8:00 am, </li>
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<li>one between 8:00 am and 9:00 am, </li>
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<li>one between
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9:00 am and 10:00 am, </li>
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<li>etc, up to ...</li>
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<li>one between 4:00 pm and 5:00 pm</li>
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</ul>
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<p> That makes 11 equal parts, and so:</p>
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<div class="so">12 Hours / 11 </div>
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<div class="so">= 1 + <sup>1</sup>/<sub>11</sub> hours</div>
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<div class="so">= 1 hour + <sup>60</sup>/<sub>11</sub> minutes</div>
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<div class="so">= 1 hour 5 <sup>5</sup>/<sub>11</sub> minutes </div>
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<div class="so">= 1 hour 5 ½ minutes (approximately)</div>
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<p>The next time after 6:00 am that the hands make a straight line is about
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<strong>7:05½ am</strong>
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:</p>
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<p class="center"> <img style="width: 247px; height: 261px;" alt="clock9" src="images/clock9.jpg"></p>
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<h2>Your Turn!</h2>
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<table cellspacing="2" cellpadding="2" border="1">
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<tbody>
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<tr>
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<td width="70"><strong>Hours</strong></td>
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<td style="text-align: center;" width="70">7am - 8am</td>
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<td style="text-align: center;" width="70">8am - 9am</td>
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<td style="text-align: center;" width="70">9am - 10am</td>
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<td style="text-align: center;" width="70">10am - 11am</td>
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<td style="text-align: center;" width="70">11am - 12pm</td>
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<td style="text-align: center;" width="70">12pm - 1pm</td>
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<td style="text-align: center;" width="70">1pm - 2pm</td>
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<td style="text-align: center;" width="70">2pm - 3pm</td>
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<td style="text-align: center;" width="70">3pm - 4pm</td>
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<td style="text-align: center;" width="70">4pm - 5pm</td>
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</tr>
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<tr>
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<td height="40"><strong>Time</strong></td>
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<td style="text-align: center;"><strong>7:05½ am</strong></td>
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<td style="text-align: center;"> </td>
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<td style="text-align: center;"><br></td>
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<td style="text-align: center;"><br></td>
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<td style="text-align: center;"><br></td>
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<td style="text-align: center;"><br></td>
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<td style="text-align: center;"><br></td>
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<td style="text-align: center;"><br></td>
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<td style="text-align: center;"><br></td>
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<td style="text-align: center;"><br></td>
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</tr>
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</tbody>
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</table>
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<p class="center"> You might
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like to check your answers on the page <a href="../time-clocks-analog-digital.html">Analog
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and Digital Clocks Animation</a></p>
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<h2> </h2>
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<h2>Can you work out the times of the day when the hands of a clock make a
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right angle?</h2>
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<p> (Hint: there are 22 of them)</p>
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<p> </p>
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<p> </p>
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<h3> Answer to the earlier exercise:</h3>
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<table cellspacing="2" cellpadding="2" border="1">
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<tbody>
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<tr>
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<td><strong>Time</strong></td>
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<td style="text-align: center;">1:00</td>
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<td style="text-align: center;">2:30</td>
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<td style="text-align: center;">7:00</td>
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<td style="text-align: center;">10:30</td>
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<td style="text-align: center;">11:20</td>
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<td style="text-align: center;">3:40</td>
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<td style="text-align: center;">5:15</td>
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<td style="text-align: center;">8:45</td>
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</tr>
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<tr>
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<td><strong>Angle</strong></td>
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<td style="text-align: center;">30°</td>
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<td style="text-align: center;">105°</td>
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<td style="text-align: center;">150°</td>
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<td style="text-align: center;">135°</td>
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<td style="text-align: center;">140°</td>
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<td style="text-align: center;">130°</td>
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<td style="text-align: center;">67½°</td>
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<td style="text-align: center;">7½°</td>
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</tr>
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</tbody>
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</table>
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<p> </p>
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<div class="related">
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<a href="../time-clocks-analog-digital.html">Analog and Digital Clocks Animation</a>
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<a href="index.html">Activity Index</a>
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