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Source: http://doubledivision.org

  1. <html>
  2. <head>
  3. <title>Teaching Long Division and Double Division</title>
  4. <meta name="description" content="Teaching Double Division can help in teaching long division by reinforcing the principles of division and giving students success with a less frustrating alternative.">
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  18. </style>
  19. <script src="DoubledivisionWholeNumbers.js"></script>
  20. <script>
  21. function StartCalculator(){
  22. document.ddcalculator.divisor.value='37';
  23. document.ddcalculator.dividend.value='85434';
  24. ddStart(0);
  25. ddFinish(0);
  26. document.all.ddExplanation.innerHTML = '<center>&nbsp;<br><span style="background-color: Yellow;">&nbsp;Click <b>Step 1</b> now,</span><br>then click Next Step,<br>Next Step, Next Step,,<br>&nbsp;<br><a href="images/Double-Division-Example-With-Remainder.html" target="example" onclick="newwindow=window.open(\'images/Double-Division-Example-With-Remainder.jpg\',\'popup\',\'height=440,width=475\'); if (window.focus) newwindow.focus(); return false;">(hand written example)</a></center>';
  27. }
  28. </script>
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  31. </head>
  32.  
  33. <body topmargin=20 onLoad="window.name='main'; StartCalculator();"><div align="center">
  34.  
  35.  
  36. <table width="770" border="0" cellspacing="0" cellpadding="0" bordercolor="blue">
  37. <tr>
  38. <td width="360" align="left" valign="top"><img src="images/cleardot.gif" width="360" height="1" alt="" border="0"><br>
  40. <h1>Long Division Teaching Aid, "Double Division"</h1>
  41. Alternative name: "<b>1-2-4-8 Division</b>"<br>
  42. <!-- Alternative name: "<b>Double Double Double Division</b>"<br> -->
  43. Alternative name: "<b>Decimal Double Division</b>"<br>
  44. <script>document.write('<a href="mail'+'to:wilson'+'@'+'silcom.com">');</script>Jeff Wilson<script>document.write('</a>');</script>  
  45. &nbsp;&nbsp;9/29/2005
  46. &nbsp;&nbsp;(<script src="countsessions.asp"></script> site visits)
  48. <p></p>
  49. <p><strong>Teaching Double Division can help in teaching long division by reinforcing the principles of division and giving students success with a less frustrating alternative.</strong>
  50. <p><strong>Double Division does not depend on memorizing the multiplication facts or estimating how many times one number goes into another. It may take 50% longer, but it is far less frustrating and probably easier to understand than Long Division.</strong>
  51. <a href="images/DDivisionDecimals-2.gif" target="example" onclick="newwindow=window.open('images/Double-Division-Example-With-Remainder.jpg','popup','height=440,width=475'); if (window.focus) newwindow.focus(); return false;">example</a>
  53. <!-- <p><strong>Trial and error guessing is eliminated and you can calculate your answer to a remainder, or two any number of decimal places.</strong> -->
  55. <p><table width="100%" cellspacing="0" cellpadding="0"><tr>
  56. <td nowrap>
  57. It's Easy!<br>
  58. <img src="images/cleardot.gif" width="1" height="3" alt="" border="0"><br>
  59. Step 1 - Double, double, double.<br>
  60. Step 2 - Subtract off multiples.<br>
  61. Step 3 - Add up your answer.
  62. </td>
  63. <td><img src="images/cleardot.gif" width="10" height="1" alt="" border="0"></td>
  64. <td align="center">
  65. <table border="0" cellspacing="0" cellpadding="5" bgcolor="FFFFC0"><tr><td align="center" style="FONT-SIZE: 12px;">
  66. <b>From the class room:</b><br>
  67. <img src="images/cleardot.gif" width="1" height="5" alt="" border="0"><br>
  68. <b>(1)</b>&nbsp;&nbsp;<a href="comment-2.html" target="popup" onclick="newwindow=window.open('comment-2.html','popup','width=500,height=440,resizable,scrollbars'); if (window.focus) newwindow.focus(); return false;">"This is actually fun!"</a><br>
  69. <img src="images/cleardot.gif" width="1" height="5" alt="" border="0"><br>
  70. <b>(2)</b>&nbsp;&nbsp;<a href="comment-1.html" target="popup" onclick="newwindow=window.open('comment-1.html','popup','width=500,height=440,resizable,scrollbars'); if (window.focus) newwindow.focus(); return false;">"More right answers than wrong!"</a><br>
  71. <img src="images/cleardot.gif" width="1" height="5" alt="" border="0"><br>
  72. <b>(3)</b>&nbsp;&nbsp;<a href="comment-3.html" target="popup" onclick="newwindow=window.open('comment-3.html','popup','width=500,height=440,resizable,scrollbars'); if (window.focus) newwindow.focus(); return false;">What about 8.268 / 3.5?</a><br>
  73. </td></tr></table>
  74. </td>
  75. </tr></table>
  77. <p><strong>Click "<b>Step 1</b>" on the calculator (above right) for step-by-step instructions.</strong>
  78.  
  79.  
  80.  
  81.  
  82. <p>Double Division may be <b>easier to understand</b> than Long Division because it deals with whole numbers rather than individual digits.
  83. You can see multiples of the divisor being subtracted from the dividend.  
  84.  
  85.  
  86.  
  87. In the example to the right, you see that 37 x 2,000 = 74,000 is being subtracted from the dividend of 85,434.
  88. <a href="images/Double-Division-Example-With-Remainder.html" target="example" onclick="newwindow=window.open('images/Double-Division-Example-With-Remainder.jpg','popup','height=440,width=475'); if (window.focus) newwindow.focus(); return false;">(better notation?)</a>.
  89. With Long Division you'd be subtracting 74 from 85; neither of which are real numbers in the problem.
  90.  
  91. <!-- <p>Long Division multiplication and estimation are good practice, but when combined with the already dificult procedure of long division, it's no wonder why many students have trouble.    
  92. -->
  95. <!-- <p>Double Division may also be easier to remember ten years out of school than Long Division.
  96. <p>And finally, it may be easier for people ten years out of school to remember this method as opposed to long division. -->
  98. <p><strong><b>Extension to decimals</b> (April 2008):</strong><br>
  99. To extend the answer by one decimal place, start again with the remainder times ten.  Repeat this as many times as you'd like, or until a remainder repeats itself.
  100. <a href="images/remainder-to-decimal.html" target="example">Decimal Examples</a><br>
  101. <!-- <b>Streamlined notation:</b> <a href="images/double-division-4-decimal-places.pdf" target="example">(4 places)</a> <a href="images/double-division-16-decimal-places.pdf" target="example">(16 plc rpt)</a> <a href="comment-3.html" target="popup" onclick="newwindow=window.open('comment-3.html','popup','width=500,height=440,resizable,scrollbars'); if (window.focus) newwindow.focus(); return false;">(8.268/3.5)</a> -->
  103.  
  104. <p class=head>DOUBLE DIVISION DISCUSSIONS: <script>document.write('<a href="mail'+'to:wilson'+'@'+'silcom.com">');</script>(others?)<script>document.write('</a>');</script>
  105. <p style="margin-top:5px">
  106. <a href="http://www.teach-nology.com/forum/showthread.php?t=1163&page=2#20" target="forum">Teach-nology.com</a>
  107. &nbsp;&nbsp;<a href="http://www.eslcafe.com/forums/teacher/viewtopic.php?p=20715#20715" target="forum">ESLcafe.com</a>
  108. &nbsp;&nbsp;<a href="http://www.proteacher.net/discussions/showthread.php?t=141923#3" target="forum">ProTeacher.net</a> <a href="http://www.proteacher.org/a/131040_Forgiving+Division.html" target="forum">(2)</a>
  109. <br><a href="http://mathforum.org/t2t/discuss/message.taco?thread=4130&n=8" target="forum">MathForum.org</a> <a href="http://mathforum.org/t2t/discuss/message.taco?thread=2797&n=5" target="forum">(2)</a>
  110. &nbsp;&nbsp;<a href="http://www.ldonline.org/xarbb/topic/13319" target="forum">LDonline.org</a>
  112. <p class=head>LINKS AND EXAMPLES:
  113. <p style="margin-top:5px">Video on Double Division at <a href="#below">LearnZillion.com</a>.
  114. <p style="margin-top:5px">Writen procedure <a href="#below">below</a>, &nbsp;Bigger calculator <a href="DoubleDivisionCalculator.html">here</a>.
  115. <p style="margin-top:5px">Long Division and Double Division <a href="side-by-side.html" target="example">side-by-side</a>.
  116. <p style="margin-top:5px">Decimal examples: <a href="images/remainder-to-decimal.html" target="example">remainder to decimal</a>&nbsp;&nbsp;<a href="images/DDivisionDecimals-1.gif" target="example">neg powers of 10</a><!-- &nbsp;<a href="images/DDivisionDecimals-2.gif" target="example">&nbsp;</a>-->
  117. <p style="margin-top:5px">Webpage on <a href="http://www.mathsisfun.com/long_division2.html" target="example">the long division procedure</a>.
  118. <p style="margin-top:5px">Paper on <a href="http://www.math.technion.ac.il/~ra/englongdivision.doc" target="example">teaching long division</a>.
  119. <!--<p style="margin-top:5px">Similiar to <a href="http://www.math.nyu.edu/~braams/links/em-arith.html" target="example">Chicago Math - Partial Quotients Method</a>.&nbsp;&nbsp;<a href="http://mb.msdpt.k12.in.us/Math/PartialQuotients.wmv" target="example">(movie)</a>-->
  120. <p style="margin-top:5px">This method is a <a href="http://www.teach-nology.com/forum/showthread.php?t=1163&page=2#20" target="forum">God send</a>. This method <a href="http://www.ldonline.org/xarbb/topic/13319" target="forum">sucks</a>!
  121. <p style="margin-top:5px">Funny video on <a href="http://www.youtube.com/watch?v=UIKGV2cTgqA" target="forum">new math</a>.
  122.                    <p style="margin-top:5px">Practice more for free at <a href="http://www.highschooldiplomaonline.org">High School Diploma Online</a>.
  124. <p class=head>Links To DoubleDivision.org:
  125. <p style="margin-top:5px">
  126.  
  127.                <a target="forum" href="http://themathworksheetsite.com/subscr/double_division.html">themathworksheetsite</a>
  128. &nbsp;&nbsp;    <a target="forum" href="http://kahri.blogspot.com/2006/11/new-way-to-teach-long-division.html">Lynn family blog</a>
  129. &nbsp;&nbsp;    <a target="forum" href="http://globofthoughts.blogspot.com/2005/10/long-division-ugh.html">glob of thoughts</a>
  130. &nbsp;&nbsp;    <a target="forum" href="http://littletonpublicschools.net/forStaff/TechnologyResources/HelpfulMathLinks/Grade5MathLinks/tabid/539/Default.aspx">Littleton Public Schools</a>
  131.                        &nbsp;&nbsp;    <a target="forum" href="https://wikis.engrade.com/msbuckleysclass/websites">Ms. Buckley's Class</a>
  132.                        &nbsp;&nbsp;    <a target="forum" href="http://www.refseek.com/directory/mathematics.html">refseek</a>
  133.                        &nbsp;&nbsp;    <a target="forum" href="http://www.homeschoolmath.net/online/division.php">Homeschoolmath.net</a>
  134.                        &nbsp;&nbsp;    <a target="forum" href="http://www.d123.org/olhms/mdoran/Long-Division.cfm">Games and Extra Practice</a>
  135.                        <br>            <a target="forum" href="http://www.mathmammoth.com/preview/Division_2_Contents.pdf">Math Mammoth</a>
  136.                        &nbsp;&nbsp;    <a target="forum" href="http://www.hillside.naugatuck.k12.ct.us/?PageName=TeacherPage&Page=10&StaffID=130014">Hillside Intermediate</a>
  137.                        &nbsp;&nbsp;    <a target="forum" href="http://www.coolinfolinks.com/mathematics.htm">coolinfolinks.com</a>
  138.                        &nbsp;&nbsp;    <a target="forum" href="http://www.ck12.org/user:c2ZveDJAb3N3ZWdvLm9yZw../book/Oswego-City-School-District---Grade-7-Common-Core/section/2.0/">modeling division</a>
  139.                        &nbsp;&nbsp;    <a target="forum" href="http://wcms.white.k12.ga.us/?PageName=TeacherPage&Page=16&StaffID=174001">White County Middle</a>
  140.                        <br>            <a target="forum" href="http://www.livebinders.com/play/play?id=466937">Other ways to do division</a>
  141.                        &nbsp;&nbsp;    <a target="forum" href="http://www.teacherspayteachers.com/Product/Long-Division-Packet-98246">Step by step guide</a>
  143. </td>
  144. <td width="20"><img src="images/cleardot.gif" width="20" height="1" alt="" border="0"></td>
  145. <td width="390" align="right" valign="top"><img src="images/cleardot.gif" width="390" height="1" alt="" border="0"><br>
  147. <table border="0" cellspacing="0" cellpadding="0" class="ddCalculator">
  148. <tr><td>
  149. <table border="0" cellspacing="0" cellpadding="5" bgcolor="#88C981">
  150. <form action="" style="margin: 0; padding: 0;" name="ddcalculator" id="ddcalculator" onSubmit="return false;">
  151. <tr><td align="right" style="font-size: 10px; padding: 1px,3px,0,0;"><script>document.write('Works on IE and FireFox, but&nbsp;<br>may not work on Apple Safari.');</script></td></tr>
  152. <tr><td align="center" style="font-size: 20px;">Double Division Help Calculator</td></tr>
  153. <tr><td align="center">
  155. <table width="100%" border="0" cellspacing="0" cellpadding="0">
  156. <tr>
  157. <td align="center">
  158. <table border="0" cellspacing="0" cellpadding="0" width="1"><tr><td>
  159. Fill in known the values and<br>click Step 1, Next, Next,,,
  160. <table border="1" cellspacing="0" cellpadding="10" bgcolor="White" height="100"><tr><td>
  161. <table border="0" cellspacing="0" cellpadding="0">
  162. <tr>
  163. <td></td>
  164. <td></td>
  165. <td nowrap><input type="text" name="quotient" style="width: 50px; height: 18px; font-size: 12px; margin: 0; padding: 1px;" onKeyUp="OnlyNumbers(this);"> R
  166. <input type="text" name="remainder" style="width: 30px; height: 18px; font-size: 12px; margin: 0; padding: 1px;" onKeyUp="OnlyNumbers(this);"></td>
  167. </tr>
  168. <tr>
  169. <td></td>
  170. <td><img src="ddCalcImages/corner.gif"></td>
  171. <td><img src="ddCalcImages/hline.gif"><img src="ddCalcImages/hline.gif"><img src="ddCalcImages/hline.gif"><img src="ddCalcImages/hline.gif"><img src="ddCalcImages/hline.gif"></td>
  172. </tr>
  173. <tr>
  174. <td><input type="text" name="divisor" style="width: 30px; height: 18px; font-size: 12px; margin: 0; padding: 2px; text-align: right;" onKeyUp="OnlyNumbers(this);"></td>
  175. <td><img src="ddCalcImages/vline.gif"></td>
  176. <td><input type="text" name="dividend" style="width: 50px; height: 18px; font-size: 12px; margin: 0; padding: 1px;" onKeyUp="OnlyNumbers(this);"></td>
  177. </tr>
  178. </table>
  179. </td></tr></table>
  180. </td></tr></table>
  181. </td>
  182. <td align="center">
  183. <table border="0" cellspacing="0" cellpadding="0" width="1"><tr><td>
  184. Each step is explained here, and<br>the step is highlighted below.
  185. <table width="200" height="100" border="1" cellspacing="0" bgcolor="White">
  186. <tr>
  187. <td valign="top">
  188. <span id="ddExplanation">Double Division Help Calculator to help show steps of double division.  (Long Division Help Calculator showing steps of long division - COMING SOON.)</span>
  189. </td>
  190. </tr>
  191. </table>
  192. </td></tr></table>
  193. </td>
  194. </tr>
  195. </table>
  197. </td></tr>
  198. <tr><td align="center" style="padding:0 7px;">
  200. <table width="100%" border="1" cellspacing="0" cellpadding="1"><tr><td style="font-size:10px">
  201. <b>WHAT THE BUTTONS DO:</b><br>
  202. - <b>Step 1</b> => Enter numbers and click this to start the problem.<br>
  203. - <b>Next Step</b> => Does next step of the problem with explanation above.<br>
  204. - <b>Back</b> => Clicking this backs up one step.<br>
  205. - <b>Finish</b> => Click finish and the problem will be completed to the end.<br></td></tr></table>
  207. </td></tr>
  208. <tr><td align="center">
  209. <table width="100%" border="0" cellspacing="0" cellpadding="0">
  210. <tr>
  211. <td style="padding:5px,0,0,5px;">
  212. <input type="submit" value="Step 1" onClick="ddStart();return false;">
  213. <input type="submit" value="Next Step" onClick="ddNext();return false;">
  214. <input type="submit" value="Finish" onClick="ddFinish();return false;">
  215. </td>
  216. <td align="right" style="padding:5px,5px,0,0;">
  217. <input type="submit" value="Back" onClick="ddBack();return false;">
  218. </td>
  219. </tr>
  220. </table>
  221. </td></tr>
  222. <tr><td align="center">
  223. <script>ddDrawPaper(28,25);</script>
  224. </td></tr>
  225. </form>
  226. </table>
  227. </td></tr>
  228. <tr><td align="center" style="padding-top:2px; color:gray;">
  229. <table width="100%" cellspacing="0" cellpadding="0">
  230. <tr>
  231. <td align="center"><script>document.write('<a href="mail'+'to:wilson'+'@'+'silcom.com?subject=double division" style="text-decoration:none;">');</script>feedback<script>document.write('</a>');</script></td>
  232. <td align="center" style="color:gray;">&copy; Copyright 2006<img src="images/cleardot.gif" width="1" height="1" alt="" border="0" name="buttonclick"></td>
  233. <td align="center"><a href="DoubleDivisionCalculator.html" style="text-decoration:none;">bigger one</a></td>
  234. </tr>
  235. </table>
  236. </td></tr>
  237. </table>
  238. <script>
  239. function OnlyNumbers(obj){var nums = /^[0-9]*$/; if(!nums.test(obj.value)) obj.value=''; }
  240. </script>
  241.  
  242. <div align="center" style="margin:20px 0px 0px 0px; padding:5px; background-color: FFFFC0;">
  243. View the log file for this website to see from where people are coming from, how long they stay,
  244. what buttons they press, and what links they click on.
  245. <script>document.write('<'+'a'+' hr'+'ef="log-sorted.txt" target="log">(click here)</a>');</script>
  246. </div>
  248. </td>
  249. </tr>
  250. <tr>
  251. <td colspan="3">
  254. <p class=head>INTEREST GROWING:
  255. <p><img src="images/doubledivision-traffic.gif"><br>
  256.  
  258. <p class=head>MOTIVATION:
  259. <p>The value of teaching manual division is to give people a method they can actually use if they have to, and to teach about mathematics.
  260. <p>It could be argued that if students seldom use long division after leaving school then it might be better for them to have a simpler and more intuitive method at their disposal - one that is easier to remember and understand.  Would someone ten years out of school have an easier time doing long division or Double Division?  -  I'm not sure, but Double Division seems simpler to me.
  261. <p>About teaching math, I'm not sure that long division teaches math very well.  It helps teach people how to follow a long procedure and it gives some practice multiplying and subtracting.  I'm not sure how many people understand what is happening when you bring down the next digit or understand why you have to add a zero to the answer when the "number after subtracting" is less than the divisor.
  262. <p>One problem I see with Double Division is that you have to do more subtraction.  I personally find subtraction harder than multiplication.  The other problem is that (not counting the trial and error) Double Division will usually have more steps.
  264. <p class=head>WHAT'S THE DIFFERENCE?
  265. <p>1) In long division you guess which multiple to subtract, where as in Double Division you pick from four options.  (Any multiple you pick works.  You don't always have to pick the largest possible one; sometimes I pick the easiest one to subtract.)
  266. <p>2) In Double Division you write out the zeros so it's clear how big the numbers are.
  267. <p>3) In Double Division you write your answer on the right side where you have room to accumulate many parts of the answer.  In long division you write the answer on the top and have to get each digit of the answer exactly right - because there is only room there to write one number.
  269. <p></p>
  270. <p>Here is how I see it so far:
  272. <p class=head>REASONS TO TEACH DIVISION:
  273. <p>- to teach mathematics
  274. <p>- a method to actually use in rare instances
  275. <p>- to prepare students for higher math
  277. <p class=head>Double DIVISION ADVANTAGES:
  278. <p>- teaches how division works
  279. <p>- no trial and error: I don't know how many times 372 go into 2711.
  280. <p>- easier to do:  Double, double, double, subtract the best ones, add up the answer.
  281. <p>- gives practice doubling numbers
  283. <p class=head>Double DIVISION DISADVANTAGE:
  284. <p>- more subtraction
  285. <p>- more steps (see note below)
  286. <p>- requires more space on the paper
  287. <p>- incremental difficulty in going on to decimals ?may? be more
  288. <p>- may not lead as directly to polynomial division - arguable
  290. <p class=head>IS Double DIVISION LONGER?
  291. <p>If we assume there is an equal chance of all ten digits being in the answer then on average there will be 1.5X as many "multiple and subtract" steps.  For example a "7" in the answer requires 3 steps, and a zero in the answer requires no steps.
  292. <p>Also remember that the multiply part of the "multiply and subtract" steps is already done for you.  So this part will be faster.  Of course you have to pre-multiple the divisor three times in the beginning.
  293. <p>In the end I think it is longer, but not as much as you might think initially.
  295. <p class=head>FEEDBACK SO FAR - 10/14/2005
  296. <p>The response has been mixed.  A few people have thought doubling the divisor three times was an efficient way to get trial multiples, and liked the idea of not having to guess.  Several people have said this method is no easier than long division.  I think the challenge is comparing the "easiness" of one thing that you know to another thing you are just learning.  But on the other hand, this <a href="http://www.mathsisfun.com/long_division2.html" target="example">explanation of long division</a> is pretty simple.
  297. <p style="background-color: Yellow;">On October 11, 2005 a sixth grade teacher posted
  298. <a href="comment-1.html" target="popup" onclick="window.open('comment-1.html','popup','width=425,height=400,resizable');return false;">this comment</a>
  299. after teaching Double Division to his class.  
  301. <p class=head>STEP-BY-STEP<a name="below">&nbsp;</a>
  302. <p><img src="images/image002.gif" width="723" height="70" alt="" border="0">
  303. <p><img src="images/image004.gif" width="658" height="135" alt="" border="0">
  304. <p><img src="images/image006.gif" width="763" height="138" alt="" border="0">
  305. <p><img src="images/image008.gif" width="730" height="137" alt="" border="0">
  306. <p><img src="images/image010.gif" width="751" height="135" alt="" border="0">
  307. <p><img src="images/image012.gif" width="748" height="137" alt="" border="0">
  308. <p><img src="images/image014.gif" width="761" height="137" alt="" border="0">
  309. <p><img src="images/image016.gif" width="650" height="157" alt="" border="0">
  310. <p><img src="images/image018.gif" width="753" height="287" alt="" border="0">
  311. <p><img src="images/image020.gif" width="749" height="310" alt="" border="0">
  312. <p><img src="images/image022.gif" width="673" height="332" alt="" border="0">
  315. </td>
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