tag:blogger.com,1999:blog-70333362207072517142024-03-06T00:06:43.095+08:00Truly Singaporean Singapore MathematicsA blog about Mathematics and Mathematics Education in Singapore, as well as Mathematics Education in general. Written for students, parents, educators and other stakeholders in Singapore, and around the world.
More information <a href="http://genuinesingaporemaths.blogspot.com/2012/01/about-this-blog.html">here</a>. Check out my <a href="http://lefouque.edublogs.org/">Education and Technology blog</a>. Follow me on <a href="https://twitter.com/#!/lefouque">Twitter</a>.lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.comBlogger199125tag:blogger.com,1999:blog-7033336220707251714.post-6979370896884456072018-10-12T02:09:00.002+08:002018-10-13T17:46:03.263+08:00[STEP3_2018_Q2] Sequence of Functions & Mathematical Induction<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDgg8ehY740yZS8uG-7GAJ8vkXxmJOtKGQxhs2rcW2-4QjLyxje8WTThMa9e1TbCXgUXvoSTVvISvWd1dDfTFZr0CafkhZDS_34xXAwOcod8RzVkwVNFr5pxIgbPgkqq5Il3n6CZyU2k23/s1600/STEP3+---+q.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="915" data-original-width="1600" height="366" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDgg8ehY740yZS8uG-7GAJ8vkXxmJOtKGQxhs2rcW2-4QjLyxje8WTThMa9e1TbCXgUXvoSTVvISvWd1dDfTFZr0CafkhZDS_34xXAwOcod8RzVkwVNFr5pxIgbPgkqq5Il3n6CZyU2k23/s640/STEP3+---+q.png" width="640" /></a></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">This is a challenging problem on a sequence of functions, appearing as a
</span><a href="https://www.admissionstesting.org/for-test-takers/step/about-step/" style="font-size: 12pt;">Sixth
Term Examination Papers (STEP)</a><span style="font-size: 12pt;"> question.</span><span style="font-size: 12pt;">
</span><span style="font-size: 12pt;">STEP is the entrance exam for </span><st1:placetype style="font-size: 12pt;" w:st="on">University</st1:placetype><span style="font-size: 12pt;">
of </span><st1:placename style="font-size: 12pt;" w:st="on">Cambridge</st1:placename><span style="font-size: 12pt;"> and the </span><st1:place style="font-size: 12pt;" w:st="on"><st1:placetype w:st="on">University</st1:placetype> of <st1:placename w:st="on">Warwick</st1:placename></st1:place><span style="font-size: 12pt;">
undergraduate mathematics.</span><span style="font-size: 12pt;"> </span><a href="https://www.admissionstesting.org/Images/302045-courses-requiring-step.pdf" style="font-size: 12pt;">Some
colleges and university departments may also require STEP</a><span style="font-size: 12pt;">.</span></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
The student does not need to know that the question involves a Rodrigues
type of formula. However great facility
in symbolic manipulation including algebra and calculus is needed, as this is
what would be expected of students in a rigorous course involving mathematics
or a related discipline.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
The first part is done via Differentiation using the Product Rule and
the Chain Rule.<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAQ8Q4Rf1pJvbFyMid9i8gE0LmHA6D-U_pegML29k_-UwJnx5FcpBFf-zwRHZQTKaWejHRLtTy2kwGtSbzk9-rgUWVuSreI7zX9l8_UBnlyaX8rcT77xmr6b_4T-m3366AnLQncBdG3f_u/s1600/STEP3+---+s1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="313" data-original-width="1600" height="124" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAQ8Q4Rf1pJvbFyMid9i8gE0LmHA6D-U_pegML29k_-UwJnx5FcpBFf-zwRHZQTKaWejHRLtTy2kwGtSbzk9-rgUWVuSreI7zX9l8_UBnlyaX8rcT77xmr6b_4T-m3366AnLQncBdG3f_u/s640/STEP3+---+s1.png" width="640" /></a></div>
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<span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">For the mathematical induction proof in part (ii), the following is a
rather standard way to begin.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">You should
do this even if you do not feel confident about the proof.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">Just write it down, and worry later.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">Say something like “RTP” (required to prove)
or “to be proven” so as not to give the impression that you are making unproven
assertions or making circular arguments, like what modern journalists and
political activists are prone to do.</span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1mld-X1Atc9dRpo08FNFdbl3c90aRBdF1lDZ3d68AC-X8dmPBAF0hDccgmu3kcyjbHvK3510-3G5exAomCPMYDwQ_93Uaat6eDZTXagh_wO8wE1qBnjE6R4SyKj-KTHK6pyyWKsng4eH7/s1600/STEP3+---+s2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="197" data-original-width="1600" height="78" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1mld-X1Atc9dRpo08FNFdbl3c90aRBdF1lDZ3d68AC-X8dmPBAF0hDccgmu3kcyjbHvK3510-3G5exAomCPMYDwQ_93Uaat6eDZTXagh_wO8wE1qBnjE6R4SyKj-KTHK6pyyWKsng4eH7/s640/STEP3+---+s2.png" width="640" /></a></div>
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<br /></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
The starting case is usually easier to handle. Just follow your nose and differentiate using
the Product Rule and the Chain Rule with
<i>n</i> = 1.<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipi6dGIwh-GwLNMDcjjQSO0N88vnbYNyO-z9KSIazKCQYuZ4czAbKjf2yzON7n7aCrF4d_5dnR9ji7EIfgp-Sc6MKNyf9oLCIb1MGqsrDBeA3Icgi09mzPtQEgbJHjxyY2485zE0hnjY_z/s1600/STEP3+---+s3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="241" data-original-width="1600" height="96" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipi6dGIwh-GwLNMDcjjQSO0N88vnbYNyO-z9KSIazKCQYuZ4czAbKjf2yzON7n7aCrF4d_5dnR9ji7EIfgp-Sc6MKNyf9oLCIb1MGqsrDBeA3Icgi09mzPtQEgbJHjxyY2485zE0hnjY_z/s640/STEP3+---+s3.png" width="640" /></a></div>
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<span style="font-size: 12pt;"> </span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">The next part, the induction step, is the
most challenging part.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">The trick is to
be clear about what is required and be observant.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">There are no derivatives in the final
required expression, and yet you should know that the earlier part of the
question serves as a hint that you must use derivatives.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">Using the induction hypothesis [IH], we end
up with an expression that has two derivatives, which is a pain to do by hand.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">So we repeatedly make use of [1] to convert
back to some expression involving the function sequence, but not involving
derivatives.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">After some cancellation and
simplification we finally complete the step.</span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEmvigz1E4IXd6pi2UPAa4FoDFrTr27hyVmbFHWd9p9OlRk_92W45hVxQZ34wZFdqSBx7p5YWAOkDic2gJ1KXnV6vd5fetzSnYFSIelklk_7oCxNquk0LjF69AO8ZEEU-TVpp0IssfLS0z/s1600/STEP3+---+s4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="560" data-original-width="1600" height="224" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEmvigz1E4IXd6pi2UPAa4FoDFrTr27hyVmbFHWd9p9OlRk_92W45hVxQZ34wZFdqSBx7p5YWAOkDic2gJ1KXnV6vd5fetzSnYFSIelklk_7oCxNquk0LjF69AO8ZEEU-TVpp0IssfLS0z/s640/STEP3+---+s4.png" width="640" /></a></div>
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<span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">The following is the standard type of conclusion for mathematical
induction proofs.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">Just remember to write
it in and earn the marks allocated.</span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzHu0CZHwd_HHFGuSAq8_qfTmVHOlRUzGOsUTuaiaeT19cKpcTBT-_GaMbUd6UQXmT9FS-rGbT7kDGiby73PEIOU7sUAw1n1lIclC9pOeeU8KUNqsHJdh6oNrXwh2wCDR79rd2rmJXlSqF/s1600/STEP3+---+s5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="124" data-original-width="1600" height="48" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzHu0CZHwd_HHFGuSAq8_qfTmVHOlRUzGOsUTuaiaeT19cKpcTBT-_GaMbUd6UQXmT9FS-rGbT7kDGiby73PEIOU7sUAw1n1lIclC9pOeeU8KUNqsHJdh6oNrXwh2wCDR79rd2rmJXlSqF/s640/STEP3+---+s5.png" width="640" /></a></div>
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<span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">The last part is again challenging.</span><span style="font-size: 12pt;">
</span><span style="font-size: 12pt;">The key to solving it is to observe that</span><span style="font-size: 12pt;">
</span><span style="font-size: 12pt;">‘</span><i style="font-size: 12pt;">x</i><span style="font-size: 12pt;">’</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">does not appear explicitly in the desired
final expression.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">So we proceed to try
to eliminate the term that contains</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">‘</span><i style="font-size: 12pt;">x</i><span style="font-size: 12pt;">’.</span><span style="font-size: 12pt;">
</span><span style="font-size: 12pt;">Examining the LHS would suggest the types of terms that we need formulas
for, and upon subtraction, will kill off the term that contains</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">‘</span><i style="font-size: 12pt;">x</i><span style="font-size: 12pt;">’.</span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj83F7wWXXGxty9NGNraZQUY8NKuZRBh5cO112gqsdSY_iwv_a65nnL7I0gOrGiwfDtW5hcIk0lwPfzSnKMLwoahdE9dDtJV5CcLDVxSvf8wenVfrEjF9zYff0jI3SClRgkYbJQ1pFm3ija/s1600/STEP3+---+s6.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="434" data-original-width="1600" height="172" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj83F7wWXXGxty9NGNraZQUY8NKuZRBh5cO112gqsdSY_iwv_a65nnL7I0gOrGiwfDtW5hcIk0lwPfzSnKMLwoahdE9dDtJV5CcLDVxSvf8wenVfrEjF9zYff0jI3SClRgkYbJQ1pFm3ija/s640/STEP3+---+s6.png" width="640" /></a></div>
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<br /></div>
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<span style="font-size: 12pt;">
Ta da! Done finally!<o:p></o:p></span></div>
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<br /></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
To recap: the strategies used to
solve this question is <b><i>observation</i></b>, <b><i>anticipation</i></b> (know what
you want at the ‘end of the rainbow’) and <b><i>elimination</i></b> (get rid of the unwanted
term). Needless to say, you would also
need to be thoroughly familar with the standard ‘A’ level Further Maths stuff
involving differentiation using the Product Rule and the Chain Rule, sequences
and mathematical induction.<br /><span style="font-size: 12pt;"> </span></span><br />
<table border="0" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; margin-left: 5.4pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 480; width: 631px;">
<tbody>
<tr style="height: 15.45pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0; mso-yfti-lastrow: yes;">
<td style="height: 15.45pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 437.85pt;" valign="top" width="584"><div class="MsoNormal">
<span style="font-size: 12pt;">You are invited
to join my group </span><a href="https://www.facebook.com/groups/451579655028901/" style="font-size: 12pt;"><span style="color: black;">Effective and Elegant Mathematics</span></a><span style="font-size: 12pt;"> on
Facebook.</span></div>
</td>
</tr>
</tbody></table>
</div>
<div class="MsoNormal">
<br /></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">This article is suitable for<o:p></o:p></span></b></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">GCE
‘A’ Level Further Mathematics<o:p></o:p></span></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">Students<b> </b>doing<b> </b></span><span style="font-size: 8pt;">STEP and/or students applying to study
undergraduate mathematics in <st1:city w:st="on">Cambridge</st1:city> / <st1:city w:st="on">Oxford</st1:city> / <st1:place w:st="on"><st1:city w:st="on">Warwick</st1:city></st1:place><o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses calculus and sequences<o:p></o:p></span></div>
<span style="font-size: 8pt;">* any learner who is interested</span>lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com1tag:blogger.com,1999:blog-7033336220707251714.post-31679672347238886582017-04-26T03:53:00.001+08:002017-04-26T03:59:04.257+08:00[Pri5 20170426FEM] Baking Éclairs and Macaroons<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Problem / Question</span></b><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdmWJBls2a8uVDGx9qQw-EZ56edsIL6on3dJAzmaq98BOIAe-NSqeZqH_d59dTp0DZDvw8Kb950rneSH8Zmtanw6N-jeXYr5ntkc6gq634RR6UKz9BntdWiWaVvtpidioqTJYpk9DfIgou/s1600/eclairs+and+macaroons+---+q.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="140" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdmWJBls2a8uVDGx9qQw-EZ56edsIL6on3dJAzmaq98BOIAe-NSqeZqH_d59dTp0DZDvw8Kb950rneSH8Zmtanw6N-jeXYr5ntkc6gq634RR6UKz9BntdWiWaVvtpidioqTJYpk9DfIgou/s640/eclairs+and+macaroons+---+q.jpg" width="640" /></a></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"> This problem for primary 5 from one of my
acquaintances on Facebook, considered to be of intermediate level difficulty (in
<st1:country-region w:st="on"><st1:place w:st="on">Singapore</st1:place></st1:country-region>). But it looks rather challenging to draw all
those bar diagrams, doesn’t it?<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"> Here is my quickie solution without
explicit algebra and without bar diagrams.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Solution<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"> For convenience, we use 6 circle units for
Eclairs and 6 square units for Macaroons.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">Suppose there
were half as many Eclairs and Macaroons, then there would be 15 more
Eclairs. So 3 square units add 15 can be
changed to 3 circle units.<o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiH8hTjxm-GDwf2ugJn8wuB0fVdsGWb4QMC9URWHr_znNyYxzLGVJZ-lSK_m7D2AExGIUx2lH_gf2dxgb8gdQ2iTYixaGwwM__IvWM4xkLt30jM0NsrMUQGaLTtvihjtnXAXRwSUTNLk6SG/s1600/eclairs+and+macaroons+---+s0.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="568" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiH8hTjxm-GDwf2ugJn8wuB0fVdsGWb4QMC9URWHr_znNyYxzLGVJZ-lSK_m7D2AExGIUx2lH_gf2dxgb8gdQ2iTYixaGwwM__IvWM4xkLt30jM0NsrMUQGaLTtvihjtnXAXRwSUTNLk6SG/s640/eclairs+and+macaroons+---+s0.png" width="640" /></a></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"> Add 15 to the 17 and change 3 square units
to 2 circle units. We deduce that 5
circle units is the same as 85. From
here we can easily figure out the rest.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12pt;">Ans:</span></b><span style="font-size: 12pt;"> 102 éclairs.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Comment<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"> The problem can be solved by bar
diagrams. However, there are many ways
to skin the cat. For more good stuff, please join my Facebook group “<a href="https://www.facebook.com/groups/451579655028901/">Effective and Elegant
Mathematics</a>”.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="color: #3366ff; font-size: 12.0pt;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html">Heuristics
Used</a><o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H02. Use a
diagram / model<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H05. Work
backwards<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H06. Use
before-after concept<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H08. Make suppositions<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H09. Restate
the problem in another way<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H11. Solve part
of the problem<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">Suitable Levels</span></b></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">Primary School / Elementary School
Mathematics<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
</div>
<div class="MsoNormal">
<span style="font-size: 8.0pt;">* any precocious or independent learner who is
interested<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"><br /></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"><br /></span></div>
<br />lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-48717513454289188782017-01-01T21:44:00.001+08:002017-01-01T21:44:22.603+08:00[Enrich20170101SQT] Calculating Square Roots by Hand<div class="MsoNormal">
<b>Introduction</b></div>
<div class="MsoNormal">
Happy New
Year to our readers! I wish this year
will be a fruitful one for everybody.<o:p></o:p></div>
<div class="MsoNormal">
Today, I
will illustrate how to calculate square roots by hand, using 54 756 as
an example. It is similar to long
division, but has some modifications.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>Solution<o:p></o:p></b></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUOjhhE_mGNH56LvLi7xKP8hqcgFRJQsQY9L17MNw6z13Qf_cncor2BfnQqG9t38GtoA3ui7ELKy1PRNU5xhDif2XHqg2WHkUC0iUWEoRhqcW1hZ22lQLngGuFV7qe6N04cgoSlNUW4PM8/s1600/square+roots+00.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="194" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUOjhhE_mGNH56LvLi7xKP8hqcgFRJQsQY9L17MNw6z13Qf_cncor2BfnQqG9t38GtoA3ui7ELKy1PRNU5xhDif2XHqg2WHkUC0iUWEoRhqcW1hZ22lQLngGuFV7qe6N04cgoSlNUW4PM8/s320/square+roots+00.jpg" width="320" /></a></div>
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<br /></div>
<div align="center" class="MsoNormal" style="text-align: center;">
<br /><!--[endif]--></div>
<div class="MsoNormal">
Starting
from the right, pair up the digits.<o:p></o:p></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5pGfseH02sojWT_vsKoGub6igK5RLjTA5__nT3WEOd65iKHZ8Ntf4jkeEJf5K7UyJ_rDtKEfNHfBjRQwnm7pne1QUl4ih4LJWBFNxQKj6GqGe14zXSv8Wp0sLLRR6Qppehpsm-UHK4Qxq/s1600/square+roots+01.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="230" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5pGfseH02sojWT_vsKoGub6igK5RLjTA5__nT3WEOd65iKHZ8Ntf4jkeEJf5K7UyJ_rDtKEfNHfBjRQwnm7pne1QUl4ih4LJWBFNxQKj6GqGe14zXSv8Wp0sLLRR6Qppehpsm-UHK4Qxq/s320/square+roots+01.jpg" width="320" /></a></div>
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<br /></div>
<div align="center" class="MsoNormal" style="text-align: center;">
<br /><!--[endif]--></div>
<div class="MsoNormal">
2×2 =
4 is the nearest perfect square to 5. Subtract
and bring down the next two digits, giving 147.<o:p></o:p></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_IlkJn-O4SP61VbntB6cmuMO9Nk-ShoqYEq5u9_P6iqdJSk0CmwZf1SjRGfFnaqyT4IGZZ19MNgU1FLhNiTm2sYUMLbpsCTDTPTH1B8APiC4AkkU2SRVCnyexRW8uxpt8M5LZc0kcCVdR/s1600/square+roots+02.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="255" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_IlkJn-O4SP61VbntB6cmuMO9Nk-ShoqYEq5u9_P6iqdJSk0CmwZf1SjRGfFnaqyT4IGZZ19MNgU1FLhNiTm2sYUMLbpsCTDTPTH1B8APiC4AkkU2SRVCnyexRW8uxpt8M5LZc0kcCVdR/s320/square+roots+02.jpg" width="320" /></a></div>
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<br /></div>
<div align="center" class="MsoNormal" style="text-align: center;">
<br /><!--[endif]--></div>
<div class="MsoNormal">
Double the digit
2
to get 4. Think: ? × 4?
gives 147 or nearest possible value. We have 3×43 = 129.<o:p></o:p></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsBlZ1spsPCFou4wKgXvI3o2tKxjnTlkR5kgg2-Cf4plJaeZwKg_ykjfzZ8XdMBZTUxoUdmv1Gm_oSIdPsxylWn4Erd0SZ8LlZJb0CZO8EoM4PWLgdu-EGXPiKgeZhhSZkFiL1I6v2HFnA/s1600/square+roots+03.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="317" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsBlZ1spsPCFou4wKgXvI3o2tKxjnTlkR5kgg2-Cf4plJaeZwKg_ykjfzZ8XdMBZTUxoUdmv1Gm_oSIdPsxylWn4Erd0SZ8LlZJb0CZO8EoM4PWLgdu-EGXPiKgeZhhSZkFiL1I6v2HFnA/s320/square+roots+03.jpg" width="320" /></a></div>
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<br /></div>
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<br /><!--[endif]--></div>
<div class="MsoNormal">
Subtracting
and bringing down the next two digits gives
1856. Replicate the digit 4 on
the left and double the digit 3, giving
46.<o:p></o:p></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgh3eXjAogbmvTL8E6C_vt4qXs-CHpeKU2UxqCo1KHTcP1MtrS1dpvMOIyiwq1jY0yQDkT0rKladsEhcjjuMjhkHTU298oDT35YbmdplnvklDBi5vnL4NcN1G6bdi5EF5EuqDeRt4EkeD-M/s1600/square+roots+04.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgh3eXjAogbmvTL8E6C_vt4qXs-CHpeKU2UxqCo1KHTcP1MtrS1dpvMOIyiwq1jY0yQDkT0rKladsEhcjjuMjhkHTU298oDT35YbmdplnvklDBi5vnL4NcN1G6bdi5EF5EuqDeRt4EkeD-M/s320/square+roots+04.jpg" width="266" /></a></div>
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<br /><!--[endif]--></div>
<div class="MsoNormal">
Now think: ? × 46?
gives 1856 or nearest possible value. It turns out that 4 × 464 gives exactly 1856.
We are done! The square root
of 54 756 is
234.<o:p></o:p></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>How does it work?</b></div>
<br />
This
relies on the algebraic identity (10<i>a</i> + <i>b</i>)²
= 100<i>a</i>² + 20<i>ab</i> + <i>b</i>², the right-hand
expression is equal to 100<i>a</i>²
+ (20<i>a</i> + <i>b</i>)<i>b</i>. For example, at stage 4, we have <i>a</i> =
23, <i>b</i>
= 4 and
(20<i>a</i> + <i>b</i>) = 464.<o:p></o:p><br />
<div class="MsoNormal">
Did you
learn something today?<o:p></o:p></div>
<br />
<br />
<br />lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com1tag:blogger.com,1999:blog-7033336220707251714.post-51150261985716120562016-11-06T09:00:00.000+08:002016-11-06T15:08:32.966+08:00[AM_20161105ITFF] False Friends in Integration (Calculus)<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Question</span></b><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4uV0qaggleLwnVd9xU9YKNHHULWDfHnJkWsWaCvy4UMWZtlUsvKxLb4q_TdFHLoXfiwc1wq065moSMPh851hlTvNomaNIC_TOtQ24Z1QBuZl8ejon03OmopC3QtQ0zPOJgDG_1K7T3u-U/s1600/integration+false+friends+-+q.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="164" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4uV0qaggleLwnVd9xU9YKNHHULWDfHnJkWsWaCvy4UMWZtlUsvKxLb4q_TdFHLoXfiwc1wq065moSMPh851hlTvNomaNIC_TOtQ24Z1QBuZl8ejon03OmopC3QtQ0zPOJgDG_1K7T3u-U/s640/integration+false+friends+-+q.png" width="640" /></a></div>
<div align="center" class="MsoNormal" style="tab-stops: 36.0pt right 432.0pt; text-align: center;">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12pt;">Introduction<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
False friends are words in two languages that look/sound alike, but
differ significantly in meaning. Do you
know that there are also false friends in mathematics? Can you distinguish and explain the
difference between the two integrals?<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12pt;">Solution<o:p></o:p></span></b></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi049tkkzb4Sq6bi8QNAxY4HTV4JVrLO5VEz3mdbMOruqlqMsrMyt2yy6uKEcCzcgEeGciPtJ_VGvu3Kzsw8w28SOXZxJyhdKFwxBkYVQGhGE_9KJ8XaqiByKI_YnJuSK-bkQuk-Ar684Qb/s1600/integration+false+friends+-+s.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="70" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi049tkkzb4Sq6bi8QNAxY4HTV4JVrLO5VEz3mdbMOruqlqMsrMyt2yy6uKEcCzcgEeGciPtJ_VGvu3Kzsw8w28SOXZxJyhdKFwxBkYVQGhGE_9KJ8XaqiByKI_YnJuSK-bkQuk-Ar684Qb/s400/integration+false+friends+-+s.png" width="400" /></a></div>
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<br /></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
The integrand on the left has the <u>variable</u> <i>x</i> as the base and the <u>constant</u> e as
the index. So we integrate it using the
Power Law.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
By contrast, for the integrand on the right, the base e is a
<u>constant</u> whereas the index is the <u>variable</u> <i>x</i>. Integrating
e to the power of <i>x</i> is the eeeeeeeeeeeeeeeasiest. You just get back the same thing, plus the
arbitrary constant of course.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12pt;">Remark<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
Many students make the mistake of trying to apply the Power Law for the exponential.
As a learner of mathematics, one needs
to cultivate the <i>habit</i> of being <i>observant</i> and <i>paying attention to detail</i>. This is part of developing one’s <i><a href="http://genuinesingaporemaths.blogspot.sg/2012/01/mathed000001-proposed-new-framework-for.html" target="_blank">identity</a></i>
and <i>character</i> which is important in
life.<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">Suitable Levels<o:p></o:p></span></b></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">GCE ‘O’ Level Additional Mathematics<o:p></o:p></span></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">GCE ‘A’ Levels (revision)<o:p></o:p></span></div>
<div class="MsoHeader">
<span style="font-size: 8pt;">* revision for IB Mathematics HL &
SL (revision)<o:p></o:p></span></div>
<div class="MsoHeader">
<span style="font-size: 8pt;">* Advanced Placement (AP) <st1:place w:st="on"><st1:city w:st="on">Calculus</st1:city> <st1:state w:st="on">AB</st1:state></st1:place>
& BC<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* University / College Calculus<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses that involve integral
calculus<o:p></o:p></span></div>
<br />
<div class="MsoNormal">
<span style="font-size: 8.0pt;">* whoever is interested<o:p></o:p></span></div>
lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-28450198369414047772016-11-03T22:58:00.000+08:002016-11-05T17:51:15.298+08:00[Enrich20161103NRP] The Napkin Ring Problem<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Problem</span></b><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOaEVbNrNRpQxEsF5qJl4Rr8ovv9h7H2uxloIGITCkbfcb5QxgYcd8T9w_oy52MkiudgWuRQ2e-3HUXomOjVA2atnad9F22Qf6TnpNmEaEMd-qmkMpuRPf4AeWx5rQ6qQUMLVTMpRcbAaC/s1600/Napkin+Ring+Problem+1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="356" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOaEVbNrNRpQxEsF5qJl4Rr8ovv9h7H2uxloIGITCkbfcb5QxgYcd8T9w_oy52MkiudgWuRQ2e-3HUXomOjVA2atnad9F22Qf6TnpNmEaEMd-qmkMpuRPf4AeWx5rQ6qQUMLVTMpRcbAaC/s640/Napkin+Ring+Problem+1.jpg" width="640" /></a></div>
<div align="center" class="MsoNormal" style="tab-stops: 36.0pt right 432.0pt; text-align: center;">
<br /></div>
<div class="MsoNormal" style="background: white;">
<span style="color: #1d2129; font-size: 12.0pt;"> Two rings are made by
drilling a cylindrical hole through a small sphere and a hole through the large
sphere, such that the resulting rings have the same height (2<i>h</i>).<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #1d2129; font-size: 12.0pt;">
Which ring has the larger volume of remaining material?</span><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Solution<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
The answer is: both rings have the same volume. How can we know?</span><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
There is a way to show this using <i>integration</i>. But <i>calculus</i>
is not necessary.</span><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCOdd2bAW5XEuuYghHSI1MNP29LgQrZprQ_8ahzUTHgnmn3TkJtIHnXapPeTP4NzgzZUKpYzYOt8h6I9M2huAoAxzA5Fyk3PPm48Lw4luSz2iVZaiOee63zsBk_cAWLWLqkEEOL150nAQw/s1600/Napkin+Ring+Problem+2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCOdd2bAW5XEuuYghHSI1MNP29LgQrZprQ_8ahzUTHgnmn3TkJtIHnXapPeTP4NzgzZUKpYzYOt8h6I9M2huAoAxzA5Fyk3PPm48Lw4luSz2iVZaiOee63zsBk_cAWLWLqkEEOL150nAQw/s640/Napkin+Ring+Problem+2.jpg" width="640" /></a></div>
<div align="center" class="MsoNormal" style="tab-stops: 36.0pt right 432.0pt; text-align: center;">
<br /></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
Let <i>r</i> be the radius of <i>any</i> chosen sphere and let <i>a</i> be the radius of the cylindrical hole. By Pythagoras’ Theorem, <i>h</i>²
= <i>r</i>² – <i>a</i>². Consider a cross-section
of the ring sliced a distance <i>x</i>
from the centre of the sphere, perpendicular to the axis of the
cylindrical hole. The outer radius of
this cross section is the square root of <i>r</i>² –
<i>x</i>². Hence the area of the material in the
cross-section is<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
</span><i><span style="font-family: "symbol"; font-size: 12pt;">p</span></i><span style="font-size: 12pt;"> [(<i>r</i>²
– <i>x</i>²) – <i>a</i>²] = </span><i><span style="font-family: "symbol"; font-size: 12pt;">p</span></i><span style="font-size: 12pt;"> (<i>r</i>²
– <i>a</i>² – <i>x</i>²) = </span><i><span style="font-family: "symbol"; font-size: 12pt;">p</span></i><span style="font-size: 12pt;"> (<i>h</i>²
– <i>x</i>²)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">Note that <i>r</i> does not appear in the formula. That means the cross-section does not depend
on <i>r</i>. <span style="background: yellow; mso-highlight: yellow;">A bigger (or smaller) sphere would have the same cross-sectional area <i>for each</i> distance <i>x</i> away from the centre</span>. By <i><a href="https://en.wikipedia.org/wiki/Cavalieri%27s_principle" target="_blank">Cavalieri'sPrinciple</a></i>, the other sphere will have the same volume!<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
By the way what is this volume? It
is the same as that of a sphere without hole i.e. where <i>a</i> =
0 and
<i>r</i> = <i>h</i>. This works out to be <sup><span style="background: lime; mso-highlight: lime;">4</span></sup><span style="background: lime; mso-highlight: lime;">/<sub>3</sub></span></span><i><span style="background: lime; font-family: "symbol"; font-size: 12pt;"> p</span></i><span style="background: lime; font-size: 12pt;"> <i>h</i>³</span><span style="font-size: 12pt;">, where <i>h</i> is half the height of the ring.</span><span style="color: navy; font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="color: #3366ff; font-size: 12.0pt;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html">Heuristics
Used</a><o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H02. Use a
diagram / model<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H09. Restate
the problem in another way<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H11. Solve part
of the problem<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H13* Use
Equation / write a Mathematical Sentence<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">Suitable Levels<o:p></o:p></span></b></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">GCE
‘A’ Levels H2 Mathematics (Number patterns, with algebra)<o:p></o:p></span></div>
<div class="MsoHeader">
<span style="font-size: 8pt;">* revision for IB Mathematics HL & SL<o:p></o:p></span></div>
<div class="MsoHeader">
<span style="font-size: 8pt;">* Advanced Placement (AP) <st1:place w:st="on"><st1:city w:st="on">Calculus</st1:city>
<st1:state w:st="on">AB</st1:state></st1:place> & BC<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* University / College Calculus<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses that involve volumes and
Pythagoras’ Theorem<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 8.0pt;">* any learner who is interested<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 8.0pt;"><br /></span></div>
<div class="MsoNormal">
<span style="font-size: 8.0pt;"><br /></span></div>
<div class="MsoNormal">
<span style="font-size: 8.0pt;"><br /></span></div>
<div class="MsoNormal">
<span style="font-size: 8.0pt;"><br /></span></div>
<br />
<div class="MsoNormal">
<br /></div>
lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-41116566537536845112016-10-22T00:27:00.000+08:002016-10-22T00:27:26.857+08:00[Pri1_20161021DIV] Labelling as a strategy for Division<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Introduction<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
The <st1:country-region w:st="on"><st1:place w:st="on">Singapore</st1:place></st1:country-region>
mathematics syllabuses are very well designed, especially the primary school
syllabus. Fundamental concepts and
skills are introduced before going on to complex calculations and problem
solving. At primary 1, pupils learn the
idea of multiplication and division of small numbers by grouping (or partitioning). They are not made to recite the times tables
meaninglessly.<o:p></o:p></span></div>
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<span style="font-size: 12pt;">
Division is easy if the number of things in each group is known. You just keep on circling the known number of
objects until everything is circled.
However, if the number of groups is required but the number of things in
each group is not given, and if the objects are not arranged in a convenient
way, the task can be a bit more challenging.
Remember: they have not memorised the multiplication tables yet.<o:p></o:p></span></div>
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<b><span style="font-size: 12.0pt;">Problem / Question</span></b><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY3OJyJPCsSL_Ei3qHeTBp_D7zAaiwuemp1FIbS_XHFl3JXPB8NCVkttdRfVFST2zp6yZCeFz_6ClwiuYJeIp6r1GpdzGVttyw-mxhWHgI59gVj4VFIpiXmId7y772fbLPzxQPD_4loMJI/s1600/dividing+equally.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="386" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgY3OJyJPCsSL_Ei3qHeTBp_D7zAaiwuemp1FIbS_XHFl3JXPB8NCVkttdRfVFST2zp6yZCeFz_6ClwiuYJeIp6r1GpdzGVttyw-mxhWHgI59gVj4VFIpiXmId7y772fbLPzxQPD_4loMJI/s640/dividing+equally.png" width="640" /></a></div>
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<b><span style="font-size: 12.0pt;">Solution (Suggested)<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
One way to solve this problem is to label the fish 1, 2, 3, 1, 2, 3, ...
in a cyclic fashion, assigning fish to each of the three friends one at a time,
thereby ensuring that each person gets the same number of fish. Start with “1” somewhere on the left, “3” on
the right and “2” somewhere in the middle.
Assign the next “1” close to the previous “1”, the next “2” close to the
previous “2” and the next “3” close to the previous “3”. So all the 1s are close together, the 2s are close
together and the 3s are close together.
After all the fish have been labelled, the partitioning (or grouping)
becomes obvious.<o:p></o:p></span></div>
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<b><span style="color: #3366ff; font-size: 12.0pt;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html">Heuristics
Used</a><o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H02. Use a
diagram / model<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H04. Look for
pattern(s)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H09. Restate
the problem in another way<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">Suitable Levels<o:p></o:p></span></b></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">Primary School / Elementary School
Mathematics<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8pt;">* any precocious or independent learner who is
interested</span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8pt;"><br /></span></div>
<span style="font-size: 12pt;"><br /></span>
<span style="font-size: 12pt;"><br /></span>
<span style="font-size: 12pt;"><br /></span>lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-89069501214765328632016-06-08T03:40:00.000+08:002016-06-08T03:40:05.387+08:00The passing of a Great Giant - Jerome Bruner<a href="https://en.wikipedia.org/wiki/Jerome_Bruner" target="_blank">Jerome Bruner</a>, a great psychologist, has died recently. We thank him for his theories that guided the development of Singapore Mathematics education.<br />
<br />
One of his greatest theories that is useful for children's learning is the <b>Concrete-Pictorial-Abstract</b> (Enactive-Iconic-Symbolic) approach. <br />
<br />
It is impossible to use only words to explain simple mathematical concepts to a young child, since those concepts cannot be further explained using words. You will run out of words to explain! What definitely does not work is to start from use just words and they don't "get it", then scold them for being stupid.<br />
<br />
They have to learn buy touching and playing with things, then from looking at pictures and then using abstract words or reasoning.<br />
<div>
<br /></div>
lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-78825218525344499632016-02-27T21:33:00.001+08:002016-02-27T21:33:12.131+08:00[S1_20160227FZCK] Factorisation by Chunking<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Problem / Question</span></b><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiImE-J0UkXt_382QLByXxsw-ERABxSA0x_LzQtLmRorwOl61gSCM69kUm8e0EAVdkIl_p_TtIaiQasY3MQKyL-NywqFWqdTVJgSzNIpZfZp558_ZN9Vf7f7Ha239JmGEBrlay3BFfQuQ74/s1600/BLG+factorisation+-+q.png" imageanchor="1"><img border="0" height="78" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiImE-J0UkXt_382QLByXxsw-ERABxSA0x_LzQtLmRorwOl61gSCM69kUm8e0EAVdkIl_p_TtIaiQasY3MQKyL-NywqFWqdTVJgSzNIpZfZp558_ZN9Vf7f7Ha239JmGEBrlay3BFfQuQ74/s640/BLG+factorisation+-+q.png" width="640" /></a></div>
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<b><span style="font-size: 12.0pt;">Solution<o:p></o:p></span></b></div>
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<span style="font-size: 12pt;"><!--[if gte vml 1]><v:shape id="_x0000_i1026"
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<span style="font-size: 12pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgE_RjXCegiVZSIpM2ke4KPFN2BS9-6CbDNELi31npcGi6K9MJJz2fy9AzTkOG8xIzQYMdmhWgpO81M4P5SVOmMnFz_o8ggkDrmd8LuD-tEE607ni3Mi1hPQcrpZXJEx8T0xCoqlN-nY72a/s1600/BLG+factorisation+-+s.png" imageanchor="1"><img border="0" height="109" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgE_RjXCegiVZSIpM2ke4KPFN2BS9-6CbDNELi31npcGi6K9MJJz2fy9AzTkOG8xIzQYMdmhWgpO81M4P5SVOmMnFz_o8ggkDrmd8LuD-tEE607ni3Mi1hPQcrpZXJEx8T0xCoqlN-nY72a/s320/BLG+factorisation+-+s.png" width="320" /></a></span></div>
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<b><span style="font-size: 12.0pt;">Commentary<o:p></o:p></span></b></div>
<div class="MsoNormal" style="tab-stops: 36.0pt 72.0pt 117.5pt; text-align: justify; text-justify: inter-ideograph;">
<span style="font-size: 12pt;"> Here I illustrate the usefulness of
chunking to <i>factorise</i> (AmE: <i>factor</i>) an algebraic expression. Observe that 3<i>a</i> –
2<i>b</i> is a repeated part of the expression. I call it a “chunk”. To make it clear, I rewrite (3<i>a</i>
– 2<i>b</i>)² as (3<i>a</i> – 2<i>b</i>)(3<i>a</i> – 2<i>b</i>) so that you can see it as two copies of the
same chunk. I highlight in yellow one
copy of (3<i>a</i> – 2<i>b</i>) from each of
<br />
(3<i>a</i> – 2<i>b</i>) (3<i>a</i> – 2<i>b</i>)
and -3(3<i>a</i> – 2<i>b</i>). The remaining stuff are highlighted in blue
and green. Take out the yellow chunk as
common factor by writing it out on the left in the third line, shown in yellow.
You can pull out the common factor by
writing it out to the right if you want, but here I chose to put it on the
left. The result would be equivalent
anyway. Once you have written out the
common factor, you write out the other
stuff (shown highlighted in blue and green) into another other bracket.<o:p></o:p></span></div>
<div class="MsoNormal" style="tab-stops: 36.0pt 72.0pt 117.5pt; text-align: justify; text-justify: inter-ideograph;">
<span style="font-size: 12pt;"> Once you understand how it works, you can
actually do the second line mentally and write down the answer straightaway. Chunking is a very useful technique in
mathematics. Here are some more examples
of the technique of chunking: <a href="http://genuinesingaporemaths.blogspot.sg/2015/05/s220150519xfch-chunking-for.html">(1)</a>,
<a href="http://genuinesingaporemaths.blogspot.sg/2015/10/s220151029qfcs-chunking-and-piggy-back.html">(2)</a>,
<a href="http://genuinesingaporemaths.blogspot.sg/2015/05/s220150501axc-2011-chunking-and.html">(3)</a>.<o:p></o:p></span></div>
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<b><span style="color: #3366ff; font-size: 12.0pt;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html">Heuristics
Used</a><o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H04. Look for
pattern(s)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H10. Simplify
the problem<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H11. Solve part
of the problem<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">Suitable Levels<o:p></o:p></span></b></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">Lower Secondary Mathematics (Sec 1 ~ grade 7)<o:p></o:p></span></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">GCE ‘O’ Level “Elementary” Mathematics<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses that involve algebra and
factorisation (factoring)<o:p></o:p></span></div>
<span style="font-size: 8pt;">* any learner who is interested</span><br />
<span style="font-size: 8pt;"><br /></span>
<span style="font-size: 8pt;"><br /></span>
<span style="font-size: 8pt;"><br /></span>
<span style="font-size: 8pt;"><br /></span>
<span style="font-size: 8pt;"><br /></span>lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-47804822067217136432016-02-23T23:54:00.006+08:002016-02-25T12:29:37.582+08:00[Pri20160223FPDM] Pernicious Portion Problem? Shift Happens! <div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Problem / Question</span></b><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
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<span style="font-size: 12.0pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpu3jw1IIQhy_VrRk9Fvxy5NnkeqhQkZHf6yZjDtAhViCFiBLw6xCROLSmnNwguNiVl54ApnJ_3pZXl2YBNwroNOLECmYJaCV1hRBBmRLeXguJWEsKbUrVt_F3IBSXbXtGEQ-sOF6nQ3XW/s1600/fraction+shift+-+q.jpg" imageanchor="1"><img border="0" height="308" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpu3jw1IIQhy_VrRk9Fvxy5NnkeqhQkZHf6yZjDtAhViCFiBLw6xCROLSmnNwguNiVl54ApnJ_3pZXl2YBNwroNOLECmYJaCV1hRBBmRLeXguJWEsKbUrVt_F3IBSXbXtGEQ-sOF6nQ3XW/s400/fraction+shift+-+q.jpg" width="400" /></a></span></div>
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<b><span style="font-size: 12.0pt;">Strategy<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"> This seems to be a confounding question on decimals. </span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">What shall we do with the triangles?</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">Is there a short cut?</span></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
</span><br />
<div class="MsoNormal">
<span style="font-size: 12pt;">
Yes! What you can do is to <i>imagine</i> putting the two triangles
together to form a rectangle. And then
the solution becomes easy! This is
because the area is unchanged and hence the proportion of the shaded area is
unchanged, is the same as before. We can
make use of fractions and convert it to a decimal.<o:p></o:p></span></div>
<span style="font-size: 12pt;"><br /></span></div>
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<span style="font-size: 12pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbZyQDcc1MMD2KSA0mr28hMT_1X7MbMF4BeRTbLYwgj7J49S-ZOb6bvtnq_WBmJ_f7ahZIACgYGIy42wdm8dSypsvoYeYlfWu7YHv5bRL7zqf3AHGrEJiHVBNAaQRqm5iklWjinD_USw8v/s1600/fraction+shift+-+s2.jpg" imageanchor="1"><img border="0" height="112" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbZyQDcc1MMD2KSA0mr28hMT_1X7MbMF4BeRTbLYwgj7J49S-ZOb6bvtnq_WBmJ_f7ahZIACgYGIy42wdm8dSypsvoYeYlfWu7YHv5bRL7zqf3AHGrEJiHVBNAaQRqm5iklWjinD_USw8v/s320/fraction+shift+-+s2.jpg" width="320" /></a></span></div>
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<b><span style="font-size: 12.0pt;">Solution<o:p></o:p></span></b></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidks_yRpIrHHBKQ0Ji-lsc-zKU_F03iAMaX9VIie0US0qJTQQEJpIO8_KyQaEpBCsqpodhYEipmjSe8YdWgXBbuwBXN5UOBIl14TOKQeQocIcZTAxs0vuB_W7V3dkupCqGv-GPUT5ZT58C/s1600/fraction+shift+-+s3.png" imageanchor="1" style="font-size: 12pt;"><img border="0" height="44" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidks_yRpIrHHBKQ0Ji-lsc-zKU_F03iAMaX9VIie0US0qJTQQEJpIO8_KyQaEpBCsqpodhYEipmjSe8YdWgXBbuwBXN5UOBIl14TOKQeQocIcZTAxs0vuB_W7V3dkupCqGv-GPUT5ZT58C/s640/fraction+shift+-+s3.png" width="640" /></a></div>
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<br /></div>
<div class="MsoNormal">
<b><span style="color: #3366ff; font-size: 12.0pt;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html" target="_blank">Heuristics Used</a><o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H02. Use a
diagram / model<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H04. Look for
pattern(s)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H09. Restate
the problem in another way<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H10. Simplify
the problem<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 10pt;">Suitable Levels<o:p></o:p></span></b></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 10pt;">* </span></b><span style="font-size: 10pt;">Primary School Mathematics<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;">* other syllabuses that involve fractions
and decimals<o:p></o:p></span></div>
<span style="font-size: 10pt;">* any learner who is interested</span><br />
<span style="font-size: 10pt;"><br /></span>
<span style="font-size: 10pt;"><br /></span>lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-66448401822277461702016-02-22T13:49:00.001+08:002016-02-22T14:03:30.796+08:00[Maths Education] Mathematical Journalling<div class="MsoNormal">
<div style="text-align: justify;">
<span style="font-size: 12pt;"> Nowadays, I see some schools / textbooks asking
students to search the internet and to write on a certain problem on their
“mathematics journal”. It's so very
guided. It's so artificial. The questions should come from the learners
themselves, out of their own curiosity.
The learn then seeks to answer their own questions. The journal can serve as to document and
summarise their process of learning.</span></div>
</div>
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<div style="text-align: justify;">
<br /></div>
</div>
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<div style="text-align: justify;">
<span style="font-size: 12pt;"> The best maths journals are
self-initiated. Great mathematician Karl
Friedrich Gauss and renowned scientist Richard Feynman kept math journals on
their own accord, not because some teacher told them to do it. <o:p></o:p></span></div>
</div>
<div class="MsoNormal">
<div style="text-align: justify;">
<br /></div>
</div>
<div class="MsoNormal">
<div style="text-align: justify;">
<span style="font-size: 12pt;"> When I was a student, I borrowed books
from the National Library on things out of the normal curriculum. I kept notes of things I learned. I also did my own investigations. I accidently discovered quadratic equations
when I was in Primary 4. I read
guidebooks, asked my friend's elder brothers and sisters, my Chinese teacher
(!) and other people to find out more. I
did not like factorisation by trial-and-error.
Neither did I like completing the square nor using the quadratic
formula. So I did my own research to
find a sure-fire way to factorise without trial-and-error. I finally managed to find a way, but my
method had an uncanny similarity to the quadratic formula. It was a Pyrrhic victory, but it was
fun! I thoroughly enjoyed it.<o:p></o:p></span></div>
</div>
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<div style="text-align: justify;">
<br /></div>
</div>
<div class="MsoNormal">
<div style="text-align: justify;">
<span style="font-size: 12pt;"> If students need to be told or goaded to
write mathematics journals, then we as educators need to ask ourselves: Why?
What is their conception of mathematics and education? What experiences have they gone through that
lead them to these beliefs?<o:p></o:p></span></div>
</div>
<div class="MsoNormal">
<div style="text-align: justify;">
<br /></div>
</div>
<div style="text-align: justify;">
<span style="font-size: 12pt;"> Some food for thought, eh?</span></div>
<br />
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
<br /></div>
lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com1tag:blogger.com,1999:blog-7033336220707251714.post-21701720259948394732016-02-18T02:33:00.006+08:002016-02-18T14:37:05.795+08:00[P6_20160217RTTU] Books on Bookshelves<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Problem</span></b><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div align="center" class="MsoNormal" style="tab-stops: 36.0pt right 432.0pt; text-align: center;">
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</v:shape><![endif]--><!--[if !vml]--><!--[endif]--><o:p></o:p></span></div>
<div align="center" class="MsoNormal" style="tab-stops: 36.0pt right 432.0pt; text-align: center;">
<span style="font-size: 12.0pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5tiYXEBqBaK-mvAyx57BVP0DLMZ1qa_0Ceo-Yx1Ucm5c_ropOqG4GbHSId4kn4e8TyRHq9qAsI-FpQmqOIjnpFfixRJ2dvopILUiGZW8oKP1bMd9WJWkZlHYXydT-BPlIuG5hSDEwqwuO/s1600/books+on+bookshelf+-+q.jpg" imageanchor="1"><img border="0" height="144" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5tiYXEBqBaK-mvAyx57BVP0DLMZ1qa_0Ceo-Yx1Ucm5c_ropOqG4GbHSId4kn4e8TyRHq9qAsI-FpQmqOIjnpFfixRJ2dvopILUiGZW8oKP1bMd9WJWkZlHYXydT-BPlIuG5hSDEwqwuO/s640/books+on+bookshelf+-+q.jpg" width="640" /></a></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Introduction<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"> Here
we have a numerically challenging problem that involves ratios, and it ultimately
reduces to an algebraic problem with two unknowns. Nevertheless, we are spoilt for choice as
regards to methods of solution:-</span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> (1) Bar Diagram Modelling<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> (2) explicit letter-symbolic Algebra<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> (3) “p” and “u”
(parts and units)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> (4) Distinguished Ratio Units<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> Despite the
fact that Bar Diagram Modelling made “Singapore mathematics” famous, let us
remember that it is only one of the ways of solving problem by diagramming,
which is just one of the eleven Primary School <a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html">heuristics</a>
recommended by the Singapore Ministry of Education.<o:p></o:p></span><br />
<span style="font-size: 12.0pt;"><span style="font-size: 12pt;"> The methods have a lot in common, and they differ mainly in the form of presentation.
However, standard Bar modelling is
impractical under high-stakes high-stress examination conditions for this
problem, not least because one would have to cut the bars into many pieces. One should not cut off one’s feet just so as to
fit the shoes (</span><span lang="ZH-CN" style="font-family: "simsun"; font-size: 12.0pt;">削足适履</span><span style="font-size: 12pt;">), as one Chinese
saying goes. We need to be flexible and open-minded. I present a solution using my
own Distinguished Ratio Units.</span></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Solution<o:p></o:p></span></b></div>
<div class="MsoNormal" style="text-align: center;">
<span style="font-size: 12.0pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg54nH9t1SUGOddOC7H2dE066IvTCqGg1Zs66xHw7l8swsFVGqpyZHO90cKrIIKI2K6lM0csF9TxHWvqsndFf-jh84X_74jnYTfDOVXWn_hUg5jej6DeMWFmlokKettkv8qRmiRwiCHy_K9/s1600/books+on+bookshelf+-+s.jpeg" imageanchor="1"><img border="0" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg54nH9t1SUGOddOC7H2dE066IvTCqGg1Zs66xHw7l8swsFVGqpyZHO90cKrIIKI2K6lM0csF9TxHWvqsndFf-jh84X_74jnYTfDOVXWn_hUg5jej6DeMWFmlokKettkv8qRmiRwiCHy_K9/s400/books+on+bookshelf+-+s.jpeg" width="315" /></a></span></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Ans:</span></b><span style="font-size: 12.0pt;"> 735 books<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Commentary<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> First off, we
need to equalise the numerators of <sup>2</sup>/<sub>5</sub>
and 1<sup>1</sup>/<sub>4</sub> = <sup>5</sup>/<sub>4</sub>
and put them ratio form. This is because the “2” in
the <sup>2</sup>/<sub>5</sub> represents the same quantity as the “5” in
<sup>5</sup>/<sub>4</sub>.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;">We do this adjustment by multiplying the former through
by 5
and the latter through by 2. Thus we deduce that the original number of books in A
and in B are 25 and
8 “heart” units respectively. <br />
Next, we add on the 2
and 3 “triangle” units. By doing a comparison, we can figure out that
1 “triangle” unit must be 45 more
than 17
“heart” units. So 2 “triangle”
units must be equal to 34 “heart” units plus 90. Replacing
the 2
“triangle” units (shown in yellow) with their equivalent, we now know
that 59
“heart” units plus 90 gives 444. This allows us to figure out that 1 “heart”
is actually 6. Thus, we can work out what 1 “triangle”
unit, and then what 5 “triangle” units are
worth.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Final Remarks<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> Due to the
difficulty of the numbers, the solution presented above is about as streamlined
as I can make it to be. <o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> There is
another variation that can be used – equalising the “triangle” units (akin to
the technique of elimination in standard algebra). What we do is we multiply the group with
total 444 by 3 and
to multiply the group with total
489 by 2. This
would give 6 triangle units on each side. Then we can compare the “heart” units and continue
from there. This way of proceeding is
not for those who fear 4-digit numbers.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> If there
are nicer or more elegant ways to tackle this question, I would definitely love
to hear from you.<o:p></o:p></span></div>
<br />
<div class="MsoNormal">
<b><span style="color: #3366ff; font-size: 12.0pt;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html" target="_blank">Heuristics Used</a><o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H01. Act it out<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H04. Look for
pattern(s)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H05. Work
backwards<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H06. Use
before-after concept<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H09. Restate
the problem in another way<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H11. Solve part
of the problem<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 10pt;">Suitable Levels<o:p></o:p></span></b></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 10pt;">* </span></b><span style="font-size: 10pt;">Primary School Mathematics<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;">* other syllabuses that involve whole
numbers and ratios</span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;">* any problem solver who loves a challenge<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;"><br /></span></div>
lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-90141218455311336342016-02-17T02:15:00.000+08:002016-02-25T02:27:42.692+08:00[S1_20160117AFNS] Much Ado About Nothing?<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Problem</span></b><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_RqxhwVef0M7CCm1S7yITzjV-rhONdaNs4ZaUAm807JfYrGF1kqnZ_bw1-US-LPrmUce-aNGQqX-wLKNupIvKWK0qwzYKFkLxL9CZutWaqU4grhJcKRSMjuL0CW21Yh5-Ciju0DSIpQd3/s1600/WEX++Alg+Fractions+-+q.png" imageanchor="1"><img border="0" height="76" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_RqxhwVef0M7CCm1S7yITzjV-rhONdaNs4ZaUAm807JfYrGF1kqnZ_bw1-US-LPrmUce-aNGQqX-wLKNupIvKWK0qwzYKFkLxL9CZutWaqU4grhJcKRSMjuL0CW21Yh5-Ciju0DSIpQd3/s640/WEX++Alg+Fractions+-+q.png" width="640" /></a> </span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Introduction<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
Assuming no typing errors, this is a tricky Secondary 2 question
involving an equation with algebraic fractions.
How to solve it? How to present
the solution?</span><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Solution<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3QnbIdcbKGQoV-VwNJxxeiqEL5GLdxQB3vWx7tuGVGWEhCv0ScByQNEV913zcgvSSSaYlNk59TjKwRF77AO_UkGIt_AKP3hcL64FGfcMGmxpxXlcId2UDc_E3BVQYBrFADmHqoKfFREDg/s1600/WEX++Alg+Fractions+-+s.png" imageanchor="1"><img border="0" height="198" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3QnbIdcbKGQoV-VwNJxxeiqEL5GLdxQB3vWx7tuGVGWEhCv0ScByQNEV913zcgvSSSaYlNk59TjKwRF77AO_UkGIt_AKP3hcL64FGfcMGmxpxXlcId2UDc_E3BVQYBrFADmHqoKfFREDg/s640/WEX++Alg+Fractions+-+s.png" width="640" /></a></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Remarks<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">[1]
</span><span style="font-size: 12.0pt;"> Once the LHS expression has no meaning, it
would not even make sense to continue.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">[2]
</span><span style="font-size: 12.0pt;"> This is a <a href="http://genuinesingaporemaths.blogspot.sg/2016/01/olymlsec20160118pppc-square-proof-by.html" target="_blank">proof by contradiction</a> type of argument.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;">It turns out that not all algebraic equations are <i>soluble</i> (or <i>solvable</i>). </span><span style="font-size: 12pt;">This problem is one case in point. </span><span style="font-size: 12pt;">The
“unknown” </span><i style="font-size: 12pt;">m</i><span style="font-size: 12pt;"> cannot be </span><sup>5</sup><span style="font-size: 12pt;">/</span><sub>2</sub><span style="font-size: 12pt;">
because that would make the expression undefined. But if you substitute any other value, you
always end up with nonsense like “15 = 0”.
So no matter what, there is </span><i style="font-size: 12pt;">no
solution</i><span style="font-size: 12pt;">. In other words, there is
no value of </span><i style="font-size: 12pt;">m</i><span style="font-size: 12pt;"> that you can substitute
into the equation that makes it a true statement.</span></div>
<br />
<div class="MsoNormal">
<b><span style="color: #3366ff; font-size: 12.0pt;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html" target="_blank">Heuristics Used</a><o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H05. Work
backwards<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff; font-size: 12.0pt;">H08. Make
suppositions<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 10pt;">Suitable Levels<o:p></o:p></span></b></div>
<div class="MsoHeader">
<b>* </b>Lower Secondary
Mathematics (Sec 2 ~ grade 8)<o:p></o:p></div>
<div class="MsoHeader">
<b>* </b>GCE ‘O’ Level
“Elementary” Mathematics<o:p></o:p></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;">* other syllabuses that involve algebra<o:p></o:p></span></div>
<span style="font-size: 10pt;">* any learner who is interested in algebra</span><br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10.0pt;"><br /></span></div>
lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-42813933059518755152016-02-08T18:30:00.002+08:002016-02-08T18:30:58.004+08:00Happy Chinese New Year 2016 (#LunarNewYear)<div class="MsoNormal">
<br /></div>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; margin-left: 23.85pt; mso-border-alt: solid windowtext 1.5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 480;">
<tbody>
<tr style="height: 29.25pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0; mso-yfti-lastrow: yes;">
<td style="border: solid windowtext 1.5pt; height: 29.25pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 381.85pt;" valign="top" width="509">
<div align="center" class="MsoNormal" style="mso-layout-grid-align: none; punctuation-wrap: simple; tab-stops: 36.0pt right 432.0pt; text-align: center; text-autospace: none; vertical-align: baseline;">
<span style="background: white; color: #141823; font-size: 64.0pt;">2016<o:p></o:p></span></div>
<div align="center" class="MsoNormal" style="mso-layout-grid-align: none; punctuation-wrap: simple; tab-stops: 36.0pt right 432.0pt; text-align: center; text-autospace: none; vertical-align: baseline;">
<span style="background: white; color: #141823; font-size: 64.0pt;">= 12 × 168</span><span style="font-size: 64.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="mso-layout-grid-align: none; punctuation-wrap: simple; tab-stops: 36.0pt right 432.0pt; text-autospace: none; vertical-align: baseline;">
<br /></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">
Wishing all my readers a Happy Chinese New Year ... or more accurately a
Lunar New Year – many other Asians (e.g. the Japanese and the Koreans) also
celebrate this festival. This year 2016
is mathematically special, because it is the product of 12
and 168. 12 is a
lucky number for Western people (e.g. 12 signs of the zodiac and 12 days of
Christmas) and “8” is especially
auspicious in Chinese because it sounds like</span><span lang="ZH-CN" style="font-family: SimSun; font-size: 12pt;">发</span><span lang="ZH-CN" style="font-size: 12pt;"> </span><span style="font-size: 12pt;">/ </span><span style="font-family: SimSun; font-size: 12.0pt; mso-ascii-font-family: "Times New Roman"; mso-hansi-font-family: "Times New Roman";">發</span><span style="font-size: 12.0pt;"> [<i>fa</i> in
Mandarin,] which means to prosper or to grow. 168 sounds like </span><span lang="ZH-CN" style="font-family: SimSun; font-size: 12pt;">一路发</span><span style="font-size: 12pt;">
[</span><i><span style="font-size: 12.0pt;">yaad
lou faat</span></i><span style="font-size: 12.0pt;"> in Cantonese] which is to
prosper all the way through life.<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
I
think this year will be challenging, because the Fire Monkey could be monkeying
around even more with the economy and world peace. Nevertheless if all human beings can unite
together and learn to think critically, creatively and logically (and <i>mathematics</i> is about all these), the
planet Earth can be a better place. So
best wishes to one and all!<o:p></o:p></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<br /></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<br /></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<br /></div>
lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-92211461083526461072016-02-04T04:44:00.002+08:002016-02-04T15:36:16.405+08:00[EM_20160204PBIE] “Hillarious” Mathematics of Politics?<div class="MsoNormal">
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<b><span style="font-size: 12.0pt;">Problem</span></b><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
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<span style="background: white; color: #141823; font-size: 12.0pt;">Six coins are tossed
to decide a result for either “C” or “S”.
Assuming that the coins are <i>fair</i>,
and that the results of the tosses are <i>independent</i>,
calculate the probability that all the tosses are in favour of “C”.<o:p></o:p></span></div>
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<b><span style="font-size: 12.0pt;">Introduction<o:p></o:p></span></b></div>
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<b><span style="font-size: 12.0pt;">
</span></b><span style="font-size: 12.0pt;">Hot in recent news is the <a href="http://www.theguardian.com/us-news/2016/feb/02/how-hillary-clinton-won-some-iowa-caucuses-with-a-coin-toss" target="_blank">story of the purported six coin tosses that were needed to determine certain county delegates in the race between Hillary Clinton and Bernie Sandersin the state of Iowa</a>. All six
coin tosses were in favour of Hillary Clinton, and the result is so improbable
that some people said it was “hillarious”.<br />
How is the probability
calculated? This is an example of
mathematics in real life events that has the potential to affect the <st1:country-region w:st="on">United States of America</st1:country-region>, and the whole world
(including <st1:country-region w:st="on"><st1:place w:st="on">Singapore</st1:place></st1:country-region>).<o:p></o:p></span></div>
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<b><span style="font-size: 12.0pt;">Solution<o:p></o:p></span></b></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggJwoLT6ZBTtJTYkEhJhOCAwu_tJCkP3WKJLHrXpMhyAb5Wpjkb_R3UxhTMbAutUccF2bQyTLn_tXl8OMaKX_K04nPgB8DyON0WyIDsVeog61Pfyc5PXwsrdbMM0UgyvGQrV72E9dFt2u9/s1600/hilliary.png" imageanchor="1"><img border="0" height="38" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEggJwoLT6ZBTtJTYkEhJhOCAwu_tJCkP3WKJLHrXpMhyAb5Wpjkb_R3UxhTMbAutUccF2bQyTLn_tXl8OMaKX_K04nPgB8DyON0WyIDsVeog61Pfyc5PXwsrdbMM0UgyvGQrV72E9dFt2u9/s640/hilliary.png" width="640" /></a></div>
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<b><span style="font-size: 12.0pt;">Discussion<o:p></o:p></span></b></div>
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<span style="font-size: 12pt;"> In order to make the calculation, we make
two assumptions: <o:p></o:p></span></div>
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<span style="font-size: 12pt;"> (1) that the coins were <i>fair</i>, and <o:p></o:p></span></div>
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<span style="font-size: 12pt;"> (2) that the coin toss results are <i>independent</i>.<br />
So, <span style="background: lime; mso-highlight: lime;">what does it mean that the coins are “fair”?</span> It means that the probability of getting a “heads” is the same as the probability of getting a “tails”,
which means ½ for each.<br />
Coin toss results can be “heads” or “tails”. These are examples of <i>events</i>. An event is
something that can happen or not happen, and we associate a <i>probability</i> with it. The probability is a number that indicates
how likely the event happens. It is between 0
and 1 inclusive.
Zero probability means a practically impossible event. A probability of 1 means
a practically certain event. [The
reason for me using the word “practically” is technical, which I shall not
discuss.] <span style="background: lime; mso-highlight: lime;">If the events do not affect one another (i.e. in our case, the
coin tosses are not affected by the other coin tosses) then the events are said
to be <i>independent</i>.</span> If the events are independent, then we can
simply multiply the individual probabilites together, as above. If the events are dependent, the calculation would
be more complicated.<o:p></o:p></span></div>
<br />
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<span style="font-size: 12pt;"> So are the coin toss results valid? I do not know. All I can say is: <i>improbable</i> does not mean <i>impossible</i>. Mathematics cannot tell whether the above
assumptions (1) and (2) are correct. But
at least I “lay all the cards on the table”, so that astute students of
probability know the <i>basis</i> of these
calculations. It is up to you to decide,
but at least you would have made a mathematically-informed decision. <a href="http://edition.cnn.com/2016/02/02/politics/hillary-clinton-coin-flip-iowa-bernie-sanders/" target="_blank">There could be other twists to the story, which is beyond the scope of this article</a>. This is one of the reasons why you need to
learn <a href="http://genuinesingaporemaths.blogspot.sg/2012/01/mathed000001-proposed-new-framework-for.html" target="_blank">mathematics carefully and think critically, whether or not you would become a mathematician,engineer, teacher or have a mathematics-intensive career</a>.</span><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
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<b><span style="color: #3366ff;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html" target="_blank">Heuristics Used</a><o:p></o:p></span></b></div>
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<span style="color: #3366ff;">H08. Make suppositions<o:p></o:p></span></div>
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<span style="color: #3366ff;">H10. Simplify the problem<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H11. Solve part of the problem<o:p></o:p></span></div>
<br />
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<b><span style="font-size: 8pt;">Suitable Levels<o:p></o:p></span></b></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">GCE ‘O’ Level “Elementary” Mathematics (Number patterns,
with algebra)<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses that involve probability<o:p></o:p></span></div>
<span style="font-size: 8pt;">* anybody in the whole wide world!</span><br />
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lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-21779413703013906092016-02-01T02:30:00.001+08:002016-02-01T04:51:09.032+08:00[OlymLSec20160201PHHE] Pigeonhole Principle and Harry’s emails<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Problem / Question</span></b><span style="font-size: 12.0pt;"><o:p></o:p></span></div>
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<span style="background: white; color: #141823; font-size: 12.0pt;">Handsome Harry has a
secret email account that only four friends know. Today he received 8 emails in that account.
Which of the following is certainly true?<o:p></o:p></span></div>
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<b><span style="background: white; color: #141823; font-size: 12.0pt;">(A)</span></b><span style="background: white; color: #141823; font-size: 12.0pt;"> Harry received two
emails from each friend. <o:p></o:p></span></div>
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<b><span style="background: white; color: #141823; font-size: 12.0pt;">(B)</span></b><span style="background: white; color: #141823; font-size: 12.0pt;"> Harry cannot have
received eight emails from one of his friends.<o:p></o:p></span></div>
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<b><span style="background: white; color: #141823; font-size: 12.0pt;">(C)</span></b><span style="background: white; color: #141823; font-size: 12.0pt;"> Harry received at
least one email from each friend. <o:p></o:p></span></div>
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<b><span style="background: white; color: #141823; font-size: 12.0pt;">(D)</span></b><span style="background: white; color: #141823; font-size: 12.0pt;"> Harry received at
least two emails from one of his friends<o:p></o:p></span></div>
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<b><span style="background: white; color: #141823; font-size: 12.0pt;">(E)</span></b><span style="background: white; color: #141823; font-size: 12.0pt;"> Harry received at
least two emails from 2 different friends.<o:p></o:p></span></div>
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<b><span style="font-size: 12.0pt;">Introduction<o:p></o:p></span></b></div>
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<span style="background: white; color: #141823; font-size: 12.0pt;"> <b> </b>This question is from some Kangaroo Mathematics
Competition, which tests students on logic and not necessarily things from Singapore
Mathematics syllabus. <o:p></o:p></span></div>
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<b><span style="font-size: 12.0pt;">Solution<o:p></o:p></span></b></div>
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<span style="background: white; color: #141823; font-size: 12.0pt;"> <b> (D)</b>
Harry received at least two emails from one of his friends <o:p></o:p></span></div>
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<b><span style="font-size: 12.0pt;">Explanation<o:p></o:p></span></b></div>
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<span style="background: white; color: #141823; font-size: 12.0pt;"> <b> </b>This is an example of the <a href="https://en.wikipedia.org/wiki/Pigeonhole_principle" target="_blank">Pigeonhole Principle</a>. Perhaps the easiest way to understand this is
to <i>imagine</i> an array of pigeonholes
with four columns (one for each of Harry’s friends) and pigeons (representing
individual emails sent from the friends).
In the diagram below, I draw dots instead of pigeons.<o:p></o:p></span></div>
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<span style="background: white; color: #141823; font-size: 12.0pt;"><!--[if gte vml 1]><v:shapetype
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<v:f eqn="if lineDrawn pixelLineWidth 0"/>
<v:f eqn="sum @0 1 0"/>
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<v:f eqn="prod @3 21600 pixelWidth"/>
<v:f eqn="prod @3 21600 pixelHeight"/>
<v:f eqn="sum @0 0 1"/>
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o:title="Pigeonhole"/>
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<span style="background: white; color: #141823; font-size: 12.0pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjodle7fwq5PaDGJkQXrAbVaY0rejg5PDJQCT9Q5C2R1uiv46Unsg-urb2ZBXm-nsjaG79G29q6m0yr7ej_HJHpDb16LSsHQg3Qldy3Seg2BQo65ZFOR5AY9ARa2nbnPcw3Umn0Dl8Zi5dR/s1600/Pigeonhole.png" imageanchor="1"><img border="0" height="108" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjodle7fwq5PaDGJkQXrAbVaY0rejg5PDJQCT9Q5C2R1uiv46Unsg-urb2ZBXm-nsjaG79G29q6m0yr7ej_HJHpDb16LSsHQg3Qldy3Seg2BQo65ZFOR5AY9ARa2nbnPcw3Umn0Dl8Zi5dR/s320/Pigeonhole.png" width="320" /></a></span></div>
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<span style="background: white; color: #141823; font-size: 12.0pt;">As you can see, no
matter how the eight dots / pigeons are placed, at least one of the friends
will have at least two dots. It is <i>not</i> possible for all the friends to have
less than two emails.<o:p></o:p></span></div>
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<br /></div>
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<b><span style="font-size: 12.0pt;">Formal Proof<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="background: white; color: #141823; font-size: 12.0pt;"> We can use a <i><a href="http://genuinesingaporemaths.blogspot.sg/2016/01/olymlsec20160118pppc-square-proof-by.html" target="_blank">proof by contradiction</a></i> argument. </span><span style="background: white; color: red; font-size: 12.0pt;">Suppose it were not true
that Harry received at least two emails from one of his friends</span><span style="background: white; color: #141823; font-size: 12.0pt;">. That would mean each of his 4
friends sent at most one email.
But then the total number of emails would be 4 or
less. This contradicts the given fact
that Harry received 8 emails.
So this state of affairs is not possible. </span><span style="background: white; color: red; font-size: 12.0pt;">Therefore, the <i>opposite</i>
is true. We conclude that Harry received
at least two emails from one of his friends</span><span style="background: white; color: #141823; font-size: 12.0pt;">.<o:p></o:p></span></div>
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<b><span style="font-size: 12.0pt;">Final Remarks<o:p></o:p></span></b></div>
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<span style="background: white; color: #141823; font-size: 12.0pt;"> <b> </b>The <a href="https://en.wikipedia.org/wiki/Pigeonhole_principle" target="_blank">Pigeonhole Principle</a>
is very useful in many situations, including computer science. In general, if you have more objects
(“pigeons”) than there are containers or slots (“pigeonholes”), one of the
containers must have at least two of those objects.<o:p></o:p></span></div>
<br />
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<b><span style="color: #3366ff;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html" target="_blank">Heuristics Used</a><o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H02. Use a diagram / model<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H04. Look for pattern(s)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H05. Work backwards<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H08. Make suppositions<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H09. Restate the problem in
another way<o:p></o:p></span></div>
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<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">Suitable Levels<o:p></o:p></span></b></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">Lower Secondary Mathematics Competition / Olympiad<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses that involve logic,
combinatorics or Pigeonhole Principle</span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* any precocious or independent mathematics
problem solver who is interested<o:p></o:p></span></div>
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lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-62770297834035850822016-01-26T18:27:00.002+08:002016-01-27T02:02:20.439+08:00[S1_20160126FMNT] Largest Common Remainder for Three Divisors<div class="MsoNormal">
<b><span lang="EN-AU" style="font-size: 12.0pt;">Problem / Question</span></b><span lang="EN-AU" style="font-size: 12.0pt;"><o:p></o:p></span></div>
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<span lang="EN-AU" style="color: #003300; font-size: 12.0pt;">Find the greatest 4 digit number that
will divide 63, 45 and 69 so as to leave the same remainder.<o:p></o:p></span></div>
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<b><span lang="EN-AU" style="font-size: 12.0pt;">Solution</span></b><span lang="EN-AU" style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="font-size: 12pt;"> LCM + (smallest
number – 1) = 7245 + 44 = 7289</span><span lang="EN-AU" style="color: red; font-size: 12.0pt;"> </span><span lang="EN-AU" style="color: red; font-family: "wingdings"; font-size: 12.0pt;">J</span><span lang="EN-AU" style="font-size: 12pt;"><o:p></o:p></span></div>
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<b><span lang="EN-AU" style="font-size: 12.0pt;">Remarks</span></b><span lang="EN-AU" style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="font-size: 12pt;"> This question is
from National University of Singapore High School, which caters to students who
are very interested in science and mathematics, and possibly an academic career. Do not be fooled by my short and sweet
solution. The question is actually quite
challenging, and I went by a long way before coming up with this elegant solution. This reminds me of Human Resource managers who think that more
lines of code written by programmers means more work is done. Actually, a lot of hard thinking could be involved
in writing a one-line code that does the same job.<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="font-size: 12pt;"> Research shows that when <i>expert problem solvers</i>
realise that they are stuck, they change tactics. They go back to the <i>drawing board</i>. “Insanity is
doing the same thing over and over again and expecting the different results.”
said Albert Einstein. The five stages of
mathematical problem solving are<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="font-size: 12pt;"> 1. Understanding
the problem<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="font-size: 12pt;"> 2. Planning a
strategy<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="font-size: 12pt;"> 3. Executing the
plan<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="font-size: 12pt;"> 4. Evaluation<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="font-size: 12pt;"> 5. Reflection<o:p></o:p></span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">(which
can even include blogging about it!)</span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="font-size: 12pt;">At the evaluation stage, if one finds that one is not
getting the results, or if the approach is not elegant, one goes back to stage
2 to devise a new strategy. I had tried
using a complicated Chinese Remainder Theorem approach, got the solution after
one page of work, and realised that the problem can be solved very simply.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span lang="EN-AU" style="font-size: 12.0pt;">How does the solution work</span></b><span lang="EN-AU" style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="font-size: 12pt;"> The set of
remainders dividing by </span><span lang="EN-AU" style="color: #003300; font-size: 12.0pt;">63, 45 and 69 </span><span lang="EN-AU" style="font-size: 12pt;"> </span><span lang="EN-AU" style="color: red; font-size: 12.0pt;">repeat themselves in a cycle and
the length of the cycle happens to be the Lowest Common Multiple</span><span lang="EN-AU" style="color: #003300; font-size: 12.0pt;"> (LCM)</span><span lang="EN-AU" style="font-size: 12pt;">. We can see this quite easily: suppose <i>x</i> and <i>y</i>
both give a remainder <i>R</i></span><span lang="EN-AU" style="font-size: 12pt;"> dividing by </span><span lang="EN-AU" style="color: #003300; font-size: 12pt;">63, 45 and 69,
then <i>x</i> – <i>y</i> </span><span lang="EN-AU" style="font-size: 12pt;">gives a remainder of 0 </span><span lang="EN-AU" style="font-size: 12pt;">when divided by </span><span lang="EN-AU" style="color: #003300; font-size: 12pt;">63, 45 and 69,
which means <i>x</i> – <i>y</i> is a common multiple of 63, 45
and 69, of which the lowest is the LCM.</span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="color: #003300; font-size: 12.0pt;"> It is a
routine matter to get the LCM via prime factorisation as follows: 63 = 3<sup>2</sup> × 7, 45 = 3<sup>2</sup> × 5, and 69
= 3 × 23. </span><span lang="EN-AU" style="color: red; font-size: 12.0pt;">We pick the highest power for each occurring
prime and we obtain LCM = 3<sup>2</sup>
× 5 × 7 × 23 = 7245</span><span lang="EN-AU" style="color: #003300; font-size: 12.0pt;">.<o:p></o:p></span></div>
<span lang="EN-AU" style="color: #003300; font-size: 12pt;"> The next thing to note is that remainders
must be less than the divisors. Hence </span><span lang="EN-AU" style="color: red; font-size: 12pt;">the largest <i>common</i>
remainder must be 44, one less than the smallest divisor</span><span lang="EN-AU" style="color: #003300; font-size: 12pt;">. You
cannot have a remainder of 45 when divided by 45,
because you could simply have bumped up the quotient (the result of
division) by 1 and get zero remainder. So
from 7245, 7246, 7247, ... to 7289, you
get common remainders of 0, 1, 2, ...,
44. Once you hit 7290,
the remainder for division by
45 will hit 0
while the remainders for 63 and
69 will be 45 but
this will not be a common remainder. The
next time we get a common remainder will be 2×LCM = 2×7245 = 14490 which is
5 digits long. So within 4
digits, the highest number with a common remainder is 7289,
which gives a common remainder of
44.</span><br />
<br />
<div class="MsoNormal">
<b><span style="color: #3366ff;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html" target="_blank">Heuristics Used</a><o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H04. Look for pattern(s)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H05. Work backwards<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H09. Restate the problem in
another way<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H10. Simplify the problem<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H13* Use Equation / write a
Mathematical Sentence<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">Suitable Levels<o:p></o:p></span></b></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">Lower Secondary Mathematics (Sec 1 ~ grade 7) challenge<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses that involve factors and
multiples or number theory<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* any precocious or independent learner who
loves a challenge<br /><br /><br /><br /><br /><o:p></o:p></span></div>
<span lang="EN-AU" style="color: #003300; font-size: 12pt;">
</span><br />
<span lang="EN-AU" style="color: #003300; font-size: 12pt;"><br /></span>lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-33996139837645651282016-01-25T04:51:00.000+08:002016-01-25T04:51:36.029+08:00[AM_20160125DAHT] Horizontal Tangents via Quadratic Discriminants<div class="MsoNormal">
<b><span lang="EN-AU" style="font-size: 12.0pt;">Problem</span></b><span lang="EN-AU" style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3q9EFm7UvNwq-W1lnoOp01YMfT0VQlNbh4jGaM1Uw6-MvuUT1wCUCMK3hZaciCe1oBTV45PIm2ZhFZ0xIlYmcZI_Hqmqx-hb-5UYk90d8BG40pLd8v7aftVEpeU_rOOuNbz6GPZi6Da4r/s1600/horizontal+tangent+-+q.png" imageanchor="1"><img border="0" height="103" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3q9EFm7UvNwq-W1lnoOp01YMfT0VQlNbh4jGaM1Uw6-MvuUT1wCUCMK3hZaciCe1oBTV45PIm2ZhFZ0xIlYmcZI_Hqmqx-hb-5UYk90d8BG40pLd8v7aftVEpeU_rOOuNbz6GPZi6Da4r/s400/horizontal+tangent+-+q.png" width="400" /></a></div>
<div class="MsoNormal">
<span lang="EN-AU" style="font-size: 12.0pt;"><br /></span></div>
<div class="MsoNormal">
<b><span lang="EN-AU" style="font-size: 12.0pt;">Introduction</span></b></div>
<div class="MsoNormal">
<span lang="EN-AU" style="font-size: 12pt;">
This is a Additional Mathematics textbook problem. This question is of an intermediate level of
difficulty. The general method is by <i>differentiation</i>. The equation of the curve happens to be capable
of being put into a quadratic equation in
<i>x</i>. Hence we can also use the theory of <i>quadratic discriminants</i>. I present both methods of solution.</span><span lang="EN-AU" style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><u><span lang="EN-AU" style="font-size: 12.0pt;">Method 1</span></u></b><b><span lang="EN-AU" style="font-size: 12.0pt;">
(Using differential calculus)</span></b><span lang="EN-AU" style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: center;">
<b><span lang="EN-AU" style="font-size: 12.0pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnqEajwZv3ps4fInPFpoVAADllGsYHSUNNONJlKcIE3XMj-Gj5UiinqnsAmwtgv2hVwoHL0eRkri_IGs7kSO0QlSIHDJOMdtlE05G1XL8WE9hCKCXlZKhTuf3hqhe5AIdQLzHcKfPnzLHT/s1600/horizontal+tangent+-+s1.png" imageanchor="1"><img border="0" height="225" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnqEajwZv3ps4fInPFpoVAADllGsYHSUNNONJlKcIE3XMj-Gj5UiinqnsAmwtgv2hVwoHL0eRkri_IGs7kSO0QlSIHDJOMdtlE05G1XL8WE9hCKCXlZKhTuf3hqhe5AIdQLzHcKfPnzLHT/s400/horizontal+tangent+-+s1.png" width="400" /></a></span></b></div>
<div class="MsoNormal">
<b><span lang="EN-AU" style="font-size: 12.0pt;"><br /></span></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><u><span lang="EN-AU" style="font-size: 12.0pt;">Method 2</span></u></b><b><span lang="EN-AU" style="font-size: 12.0pt;">
(Using quadratic discriminants)</span></b><span lang="EN-AU" style="font-size: 12.0pt;"><o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: center;">
<b><span lang="EN-AU" style="font-size: 12.0pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCrlut6bUxzaVWH1YDxnHel0rl0ePt9aZsMWyM53tdvg9aubRqzcsuiAhkcHedjbu3coerWMTvkniCUWjJokI70j3uOwBYymit0ElTyDxJ50dmIuM7l5rJ9JKM7LrCUYBQ0X7CmThyphenhyphenvngj/s1600/horizontal+tangent+-+s2.png" imageanchor="1"><img border="0" height="230" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCrlut6bUxzaVWH1YDxnHel0rl0ePt9aZsMWyM53tdvg9aubRqzcsuiAhkcHedjbu3coerWMTvkniCUWjJokI70j3uOwBYymit0ElTyDxJ50dmIuM7l5rJ9JKM7LrCUYBQ0X7CmThyphenhyphenvngj/s400/horizontal+tangent+-+s2.png" width="400" /></a></span></b></div>
<div class="MsoNormal">
<br /></div>
<b><span lang="EN-AU" style="color: #3366ff;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html">Heuristics
Used</a></span></b><br />
<div class="MsoNormal">
<span lang="EN-AU" style="color: #3366ff;">H04. Look for
pattern(s)<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="color: #3366ff;">H05. Work backwards<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="color: #3366ff;">H09. Restate the
problem in another way<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="color: #3366ff;">H11. Solve part of
the problem<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="color: #3366ff;">H12* Think of a
related problem<o:p></o:p></span></div>
<div class="MsoNormal">
<span lang="EN-AU" style="color: #3366ff;">H13* Use Equation /
write a Mathematical Sentence<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">Suitable Levels<o:p></o:p></span></b></div>
<div class="MsoNormal">
<b><span lang="EN-AU" style="font-size: 8pt;">* </span></b><span lang="EN-AU" style="font-size: 8pt;">GCE ‘O’ Level Additional
Mathematics<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses that involve differentiation
or quadratic discriminants<o:p></o:p></span></div>
<span style="font-size: 8pt;">* any independent learner who is interested</span><br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;"><br /></span></div>
lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-11426585944898741092016-01-22T13:46:00.001+08:002016-01-24T22:17:56.780+08:00[Maths_Education] The Need to Harness and Transcend Technology<div class="MsoNormal">
<b><span lang="EN-AU">Problem / Question</span></b></div>
<div class="MsoNormal">
<br /></div>
<table border="1" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; border: none; margin-left: 23.85pt; mso-border-alt: solid windowtext 1.5pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 480;">
<tbody>
<tr style="height: 29.25pt; mso-yfti-firstrow: yes; mso-yfti-irow: 0; mso-yfti-lastrow: yes;">
<td style="border: solid windowtext 1.5pt; height: 29.25pt; padding: 0cm 5.4pt 0cm 5.4pt; width: 381.85pt;" valign="top" width="509"><div class="MsoNormal">
<span lang="EN-AU">Help me please! <span style="color: red;">My teacher needs it
tomorrow!</span></span></div>
<div class="MsoNormal">
<span lang="EN-AU">How
many millilitres are there in 3.4 litres?<span style="background: white; color: #141823;"><o:p></o:p></span></span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span lang="EN-AU">Answer<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span lang="EN-AU"> LOL! Well, if she really needs it, <a href="https://www.google.com.sg/search?q=1+litre+millilitres&oq=1+litre+mililitre&aqs=chrome.1.69i57j0l5.6271j0j7&sourceid=chrome&es_sm=122&ie=UTF-8#q=3.4+litre+millilitres" target="_blank">ask her to type the question into Google</a>!</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span lang="EN-AU">Remarks<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span lang="EN-AU"> Actually,
this is not a joke. Many easy questions
in school mathematics are now answerable by Google. In fact, <a href="http://www.wolframalpha.com/input/?i=integrate+e%5E%282x%29*sin%28x%29+dx" target="_blank">Wolfram Alpha</a> is able to
answer mathematically more difficult but routine questions. Wikipedia and YouTube are also useful for learning
mathematics. There are many other good
resources available on the Internet and public libraries. The sad thing is, </span></div>
<table border="0" cellpadding="0" cellspacing="0" class="MsoTableGrid" style="border-collapse: collapse; margin-left: 15.65pt; mso-padding-alt: 0cm 5.4pt 0cm 5.4pt; mso-yfti-tbllook: 480;">
<tbody>
<tr>
<td style="padding: 0cm 5.4pt 0cm 5.4pt; width: 21.25pt;" valign="top" width="28"><div class="MsoNormal">
<span lang="EN-AU">1)</span></div>
</td>
<td style="padding: 0cm 5.4pt 0cm 5.4pt; width: 396.5pt;" valign="top" width="529"><div class="MsoNormal">
<span lang="EN-AU">It seems that our current generation of
kids views homework as a chore to be done for the teacher, not as an
experience to be used for their own learning.
OK, maybe the homework task should have been designed better, to ask
non-Googleable questions, but educators need to be <i>aware</i> of that is happening to our kids.</span></div>
</td>
</tr>
<tr>
<td style="padding: 0cm 5.4pt 0cm 5.4pt; width: 21.25pt;" valign="top" width="28"><div class="MsoNormal">
<span lang="EN-AU">2)</span></div>
</td>
<td style="padding: 0cm 5.4pt 0cm 5.4pt; width: 396.5pt;" valign="top" width="529"><div class="MsoNormal">
<span lang="EN-AU">Our kids do not know how to <i>choose</i> and <i>use</i> the abundantly available technology and resources</span></div>
</td>
</tr>
<tr>
<td style="padding: 0cm 5.4pt 0cm 5.4pt; width: 21.25pt;" valign="top" width="28"><div class="MsoNormal">
<span lang="EN-AU">3)</span></div>
</td>
<td style="padding: 0cm 5.4pt 0cm 5.4pt; width: 396.5pt;" valign="top" width="529"><div class="MsoNormal">
<span lang="EN-AU">Neither are they taught how to do this in
school (do the teachers know it themselves?)</span></div>
</td>
</tr>
<tr>
<td style="padding: 0cm 5.4pt 0cm 5.4pt; width: 21.25pt;" valign="top" width="28"><div class="MsoNormal">
<span lang="EN-AU">4)</span></div>
</td>
<td style="padding: 0cm 5.4pt 0cm 5.4pt; width: 396.5pt;" valign="top" width="529"><div class="MsoNormal">
<span lang="EN-AU">This type of question merely targets the
lower levels of Blooms Taxonomy</span></div>
</td>
</tr>
<tr>
<td style="padding: 0cm 5.4pt 0cm 5.4pt; width: 21.25pt;" valign="top" width="28"><div class="MsoNormal">
<span lang="EN-AU">5)</span></div>
</td>
<td style="padding: 0cm 5.4pt 0cm 5.4pt; width: 396.5pt;" valign="top" width="529"><div class="MsoNormal">
<span lang="EN-AU">Even for these easy questions, children
are <i>unable</i> or <i>unwilling</i> to make the effort to find their own answers, and how are they to
engage in Higher Order Thinking, creative thinking, reasoning etc.? </span></div>
</td>
</tr>
</tbody></table>
<div class="MsoNormal">
<span lang="EN-AU"> As technology evolves and improves and
replaces many jobs, and as our children de-evolve and slacken, when this
generation grows up they will not only face an even more challenging
environment for their careers than today, they may not have the right values,
attitudes and dispositions for living life.<br /><span lang="EN-AU" style="font-size: 12pt;"> The purpose of learning mathematics in
school should be to learn how to think and to serve others. Human students must be educated to use
technology and go beyond technology, to seek answers and to help other people
instead of merely relying from other people for “help”. <a href="http://genuinesingaporemaths.blogspot.sg/2012/01/mathed000001-proposed-new-framework-for.html">This
is one of the reasons why an identity (“learning to <i>be</i> a type of person”) approach to learning mathematics is
important.</a></span></span></div>
<br />lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com1tag:blogger.com,1999:blog-7033336220707251714.post-56140575139617502512016-01-19T21:05:00.003+08:002016-01-24T20:36:34.212+08:00[S1_Expository] Exploring the HCF and LCM with a calculator<div align="center" class="MsoNormal" style="text-align: center;">
<span style="color: #003300; font-size: 8.0pt;"><!--[if gte vml 1]><v:shapetype id="_x0000_t75"
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<div align="center" class="MsoNormal" style="text-align: center;">
<span style="color: #003300; font-size: 8.0pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjBl5EU8xRZiTCV6fiCVh0vsy3cZ0FMZ8sgG_o0-5sKBr2FljQOMVxfNrnn6xsyktjr_we424-jP7eedlF17Jhclh4U6PKVlNZdN_p79wZsZIScLaZYIptBQRnsoIhQRfaf39y4PqW6bHd/s1600/HCF+%2526+LCM+-+q.png" imageanchor="1"><img border="0" height="376" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjBl5EU8xRZiTCV6fiCVh0vsy3cZ0FMZ8sgG_o0-5sKBr2FljQOMVxfNrnn6xsyktjr_we424-jP7eedlF17Jhclh4U6PKVlNZdN_p79wZsZIScLaZYIptBQRnsoIhQRfaf39y4PqW6bHd/s640/HCF+%2526+LCM+-+q.png" width="640" /></a></span></div>
<div class="MsoNormal">
<b><span style="font-size: 12pt;"><br /></span></b></div>
<div class="MsoNormal">
<b><span style="font-size: 12pt;">How does it work?<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">When we reduce a fraction to its lowest
terms, we actually cancel out as many common factors as possible.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">So eventually the numerator and the
denominator of original fraction get cancelled by their </span><i style="font-size: 12pt;">Highest Common Factor</i><span style="font-size: 12pt;"> (HCF), a.k.a. “</span><i style="font-size: 12pt;">Greatest Common Divisor</i><span style="font-size: 12pt;"> (GCD) ” in </span><st1:place style="font-size: 12pt;" w:st="on"><st1:country-region w:st="on">America</st1:country-region></st1:place><span style="font-size: 12pt;">.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">Just as </span><st1:country-region style="font-size: 12pt;" w:st="on">England</st1:country-region><span style="font-size: 12pt;">
and the </span><st1:place style="font-size: 12pt;" w:st="on"><st1:country-region w:st="on">United States</st1:country-region></st1:place><span style="font-size: 12pt;">
are divided by a common language</span><sup>(*)</sup><span style="font-size: 12pt;">, we can divide</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">756</span><span style="font-size: 12pt;">
</span><span style="font-size: 12pt;">by</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">6</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">to get the HCF.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">Why?</span><span style="font-size: 12pt;">
</span><span style="font-size: 12pt;">That is because</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">756</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">was divided by the HCF to get</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">6.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">You
may try a similar trick with the denominators.</span><span style="font-size: 12pt;">
</span><span style="font-size: 12pt;">You get the same conclusion.</span></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"> It is useful to know that the product of <i>two</i> numbers is equal to the product of
their HCF and
LCM. So <span style="background: lime; mso-highlight: lime;">756 × 1386 = HCF × LCM</span>.
Hence we have<o:p></o:p></span></div>
<div class="MsoNormal" style="text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEian_33tfBvdV4uZJDde-sbZz2ZWwOe-Xe1MoRxaHltrYORL7kzk4MYlhh8AuFORd3CWNDWcXHnRhkRdkjpHUDvarezRUnkBf8ls_aWQzCZmMcVSTHkwF-HFXSQuNtA9mCtS11R5TtrQ1ki/s1600/HCF+%2526+LCM+-+R.png" imageanchor="1"><img border="0" height="57" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEian_33tfBvdV4uZJDde-sbZz2ZWwOe-Xe1MoRxaHltrYORL7kzk4MYlhh8AuFORd3CWNDWcXHnRhkRdkjpHUDvarezRUnkBf8ls_aWQzCZmMcVSTHkwF-HFXSQuNtA9mCtS11R5TtrQ1ki/s400/HCF+%2526+LCM+-+R.png" width="400" /></a></div>
<div class="MsoNormal">
<span style="font-size: 12pt;">Observe that the bracketed
number </span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">(</span><sup>756</sup><span style="font-size: 12pt;">/</span><sub>126</sub><span style="font-size: 12pt;">)</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">is equal to</span><span style="font-size: 12pt;">
</span><span style="font-size: 12pt;">6</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">which is the numerator of the
reduced fraction.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">So we do not even need
to make that calculation.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">Just take the
reduced numerator</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">6</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">and multiply that with the original
denominator</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">1386</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">to get the LCM.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">You may try a similar trick with</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">1386</span><span style="font-size: 12pt;">
</span><span style="font-size: 12pt;">and</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">126.</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">You end up multiplying </span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">756</span><span style="font-size: 12pt;">
</span><span style="font-size: 12pt;">by</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">11,</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">which gives the same answer.</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12pt;">Further Exploration<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"> Try the above with different pairs of
numbers. How can you extend this to find
the HCF and LCM of three or more numbers? <o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 10pt;">(*) OK, just kidding. In <st1:country-region w:st="on"><st1:place w:st="on">Singapore</st1:place></st1:country-region>, we sort of follow
British English, but we are flexible. <br />
</span><br />
<b><span style="font-size: 8pt;">Suitable Levels<o:p></o:p></span></b></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">Lower Secondary Mathematics (Sec 1 ~ grade 7)<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses that involve factors,
HCF (GCD) and LCM<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* any interested learner<o:p></o:p></span></div>
<span style="font-size: 12pt;"><br />
<!--[if !supportLineBreakNewLine]--><br />
<!--[endif]--></span>lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-89944025775529433582016-01-18T03:07:00.000+08:002016-01-19T21:59:12.293+08:00[OlymLSec_20160118PPPC] A Square Proof by Contradiction<div class="MsoNormal">
<b>Question</b></div>
<div align="center" class="MsoNormal" style="tab-stops: 36.0pt right 432.0pt; text-align: center;">
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<div align="center" class="MsoNormal" style="tab-stops: 36.0pt right 432.0pt; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh06EfYoCzv8rfji4nuZugexq-OwpFU-RtYyh39dry-5ZRZFMhLq-2lY4sx4HwRDgYBSwubcgzLJoSEE8RburabDrELHbb3iHBoHpT8YlkS_A7euj5ogEMifabN4dBMGqXSsTYuagT2gPSS/s1600/WEX++H08++Make+Suppositions+-+q.png" imageanchor="1"><img border="0" height="112" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh06EfYoCzv8rfji4nuZugexq-OwpFU-RtYyh39dry-5ZRZFMhLq-2lY4sx4HwRDgYBSwubcgzLJoSEE8RburabDrELHbb3iHBoHpT8YlkS_A7euj5ogEMifabN4dBMGqXSsTYuagT2gPSS/s640/WEX++H08++Make+Suppositions+-+q.png" width="640" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>Explanation<o:p></o:p></b></div>
<div class="MsoNormal">
If <i>a</i> + <i>b</i>
= 11, then 2<i>ab</i>
= (<i>a</i> + <i>b</i>)² <span style="color: navy;">–</span> (<i>a</i>² + <i>b</i>²) = 121 <span style="color: navy;">–</span> 100 = 21. But 2<i>ab</i>
is an even number, whereas 21 is
odd. This is a <i>contradiction</i>. So (B) is
impossible. <span style="color: red; font-family: "symbol"; mso-ascii-font-family: "Times New Roman"; mso-char-type: symbol; mso-hansi-font-family: "Times New Roman"; mso-symbol-font-family: Symbol;">©</span></div>
<br />
<div class="MsoNormal">
<b>Remarks<o:p></o:p></b></div>
<div class="MsoNormal">
Short and sweet isn’t it? This uses the square-of-sum identity <span style="background: yellow; mso-highlight: yellow;">(<i>a</i> + <i>b</i>)² </span><span style="background: yellow; color: navy;">=</span><span style="background: yellow;"> <i>a</i>² + 2<i>ab</i> + <i>b</i>²</span>. I used the tactic
of assuming the answer is correct <span style="color: #3366ff;">[H08] </span>and showing that this leads to something nonsensical <span style="color: #3366ff;">[H05]</span>. So the original assumption must be
wrong. This is called “<i>proof by contradiction</i>” or <i><a href="https://en.wikipedia.org/wiki/Reductio_ad_absurdum">reductio ad absurdum</a></i>
(in Latin).<o:p></o:p></div>
<div class="MsoNormal">
By the way, the correct
answer option is (E) from the <a href="https://en.wikipedia.org/wiki/Pythagorean_triple">Pythagorean Triplet</a> 8² + 6²
= 10² with {<i>a</i>,
<i>b</i>} = {8, 6}. The question seems to be taken from some
Kangaroo mathematics competition.<span style="color: #3366ff;"><o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="color: #3366ff;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html">Heuristics
Used</a><o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H05. Work backwards<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H08. Make suppositions<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H09. Restate the problem in
another way<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H13* Use Equation / write a
Mathematical Sentence<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">Suitable Levels<o:p></o:p></span></b></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">Lower Secondary Mathematics competition<o:p></o:p></span></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">GCE ‘O’ Level “Elementary” Mathematics (challenge)<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses that involve whole
numbers and Pythagorean triplets</span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* any precocious or independent learner who
loves a challenge<o:p></o:p></span></div>
lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-79293027877874340052016-01-15T23:28:00.003+08:002016-01-15T23:37:11.644+08:00[AM_20160115SEAV] Simultaneous equations? Absolutely!<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Problem / Question</span></b></div>
<div class="MsoNormal" style="text-align: center;">
<span style="font-size: 12.0pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJWfcJWuxEglEoLPj2o6ujAkoPpobfiy3dakdrhnof8Q-osyVYSb6oCZd_6K6LEmE5Pc7EgxvO3bspAJQpiJUqJGJz1lLUchL5tX34HLk7xUPEk1ydsSzwj-_5lfZu4lMRPBi8LEka-Jhk/s1600/abs+simult+-+q.png" imageanchor="1"><img border="0" height="142" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJWfcJWuxEglEoLPj2o6ujAkoPpobfiy3dakdrhnof8Q-osyVYSb6oCZd_6K6LEmE5Pc7EgxvO3bspAJQpiJUqJGJz1lLUchL5tX34HLk7xUPEk1ydsSzwj-_5lfZu4lMRPBi8LEka-Jhk/s640/abs+simult+-+q.png" width="640" /></a></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"><br /></span></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Introduction<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"> This
question was from a Facebook group not dedicated to “<a href="http://genuinesingaporemaths.blogspot.sg/2012/01/about-this-blog.html" target="_blank">Singapore math</a>”. The thing is, mathematics is really
an international experience, especially with modern social media.</span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;">
Simultaneous equations can pose a challenge to students, but this one is
absolutely more challenging, because of the <i>absolute
value</i>.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">The Absolute Value<o:p></o:p></span></b></div>
<div class="MsoNormal" style="text-align: center;">
<b><span style="font-size: 12.0pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4dI63jn1DMCUMCmsvj1KeqYFym0zT-LG4Gl3AT9TiU33JBV5FsSrMpOM-BZx1aqc1yJgQAztHC54DQNa2cFEDkXJFjmFPRDjQklxigdg16h8wSGXomqgkIU99HPC5y1LXoVn3YNeFA5AD/s1600/abs+simult+-+r.png" imageanchor="1"><img border="0" height="358" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj4dI63jn1DMCUMCmsvj1KeqYFym0zT-LG4Gl3AT9TiU33JBV5FsSrMpOM-BZx1aqc1yJgQAztHC54DQNa2cFEDkXJFjmFPRDjQklxigdg16h8wSGXomqgkIU99HPC5y1LXoVn3YNeFA5AD/s640/abs+simult+-+r.png" width="640" /></a></span></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Strategy<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> The problem
is: we do not know whether each of <i>x</i>
and <i>y</i> is negative or otherwise. That potentially raises complications. There seems to be 4 cases to check. However, by making assumptions <span style="color: #3366ff;">[H08]</span> </span><span style="font-size: 12pt;">separately</span><span style="font-size: 12pt;"> </span><span style="font-size: 12pt;">and checking for contradictions </span><span style="color: #3366ff; font-size: 12pt;">[H07]</span><span style="font-size: 12pt;">, we can narrow down the possibilities. </span><span style="color: #3366ff; font-size: 12pt;">[H10]</span><span style="font-size: 12pt;"> Finally, we can simplify the problem to a
regular pair of simultaneous linear equations </span><span style="color: #3366ff; font-size: 12pt;">[H10]</span><span style="font-size: 12pt;"> and solve it by the method of elimination. </span><span style="color: #3366ff; font-size: 12pt;">[H10, H11]</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12pt;">Solution</span></b></div>
<div class="MsoNormal">
<b><span style="font-size: 12pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-EsDMkr8tkpNgTPC8R9wgiD7QLe6L_axPZ_T2vsKiQuJssHYRy9dlfPCf89vwYDGj3OT1pyrYxlcn7vBwgCVsb6H_Nk6Nu6ph4asBGG7WmBJbJHNCNXTt2JZaDGGcFrJdI5kHYbb3gLlZ/s1600/abs+simult+-+s.png" imageanchor="1"><img border="0" height="362" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-EsDMkr8tkpNgTPC8R9wgiD7QLe6L_axPZ_T2vsKiQuJssHYRy9dlfPCf89vwYDGj3OT1pyrYxlcn7vBwgCVsb6H_Nk6Nu6ph4asBGG7WmBJbJHNCNXTt2JZaDGGcFrJdI5kHYbb3gLlZ/s640/abs+simult+-+s.png" width="640" /></a></span></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="color: #3366ff;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html">Heuristics
Used</a><o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H04. Look for pattern(s) [deciding what to eliminate]<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H07. Use guess and check [is <i>x</i>
negative? is <i>y</i>
negative?]<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H08. Make suppositions [is
<i>x</i> negative? is <i>y</i> negative?]<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H10. Simplify the problem<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H11. Solve part of the problem<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H13* Use Equation / write a
Mathematical Sentence<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;"><br /></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">Suitable Levels<o:p></o:p></span></b></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">Lower Secondary Olympiad<o:p></o:p></span></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">GCE ‘O’ Level Additional Mathematics, “IP Mathematics”
(challenge)<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses that involve simultaneous
equations and absolute values<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* any precocious or independent learner who
wants his/her mind tickled<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<br /></div>
lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-35319451087148076732016-01-09T14:34:00.001+08:002016-01-09T21:12:05.206+08:00[OlymLSec_20160109CBDR] Derangement of Cars in a Roundabout<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Problem / Question</span></b></div>
<div align="center" class="MsoNormal" style="tab-stops: 36.0pt right 432.0pt; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV8it7lIZuR5YbhbVrSAi6nrfaKjS27VGoBpC-WITWQaqhwgq6CkJit_FfhhFAbUvC9Y6N2-OrjLRqYIPYsxk5UACYzVflZXezpPJZXMsZHE38sL5VyjrrNEXXcAODgzc6vrd5J2OGKIm2/s1600/kangaroo+Q30+-+q1.png" imageanchor="1"><img border="0" height="284" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjV8it7lIZuR5YbhbVrSAi6nrfaKjS27VGoBpC-WITWQaqhwgq6CkJit_FfhhFAbUvC9Y6N2-OrjLRqYIPYsxk5UACYzVflZXezpPJZXMsZHE38sL5VyjrrNEXXcAODgzc6vrd5J2OGKIm2/s640/kangaroo+Q30+-+q1.png" width="640" /></a></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;"><br /></span></b></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Introduction<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> Some
enthusiastic student posted this “deranged” question on Facebook. It is taken from the Singapore Math Kangaroo Contest. The question involves <i>derangements</i>, which does not appear in any syllabus in <st1:country-region w:st="on"><st1:place w:st="on">Singapore</st1:place></st1:country-region> before
university level.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> If the
problem involved a <i>circular permutation</i>,
the correct answer would have been (5 –
1)! = 4 × 3 × 2 × 1 = 24. But this is
not a circular permutation, because the roads are distinguished by their different
positions. If one rotates the roundabout,
this would be considered a different pattern. If it were a <i>permutation</i>, the answer would have been 5! = 5 × 4 × 3 × 2 × 1 = 120. However, it is not a permutation, because the
cars cannot go back to their original road in the opposite direction. That is what the Kangaroo phrase “drives less
than one round” mean.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> A <i>derangement</i> is a some rearrangement in
which things are not allowed to go back to their original position. If one has not learned the formula for derangements,
how could this be solved?<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Strategy<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> We can consider simpler cases <span style="color: #3366ff;">[H10]</span> of the problem and build up the answer from
there. We may also split the problem
into two cases. <span style="color: #3366ff;">[H11]<o:p></o:p></span></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Solution<o:p></o:p></span></b></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> Let !<i>n</i> denote the number of derangements if there
are <i>n</i> cars and
<i>n</i> road branches in the roundabout. [This is called the subfactorial of <i>n</i>.]<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> For <i>n</i> =
1, !<i>n</i>
= 0 because the car has no way but to go
back on the original road.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> For <i>n</i> =
2, !<i>n</i>
= 1 because the only way is for the two
cars to swap.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> For <i>n</i> =
3: Car #1 has
two choices: either road #2 or road #3. But
it cannot end up swapping roads with any car, else the remaining car would have
to go back to its original road, which is not allowed. So it is either #1 </span><span style="font-family: Symbol; font-size: 12.0pt; mso-ascii-font-family: "Times New Roman"; mso-char-type: symbol; mso-hansi-font-family: "Times New Roman"; mso-symbol-font-family: Symbol;">®</span><span style="font-size: 12.0pt;"> #2 </span><span style="font-family: Symbol; font-size: 12.0pt; mso-ascii-font-family: "Times New Roman"; mso-char-type: symbol; mso-hansi-font-family: "Times New Roman"; mso-symbol-font-family: Symbol;">®</span><span style="font-size: 12.0pt;"> #3 </span><span style="font-family: Symbol; font-size: 12.0pt; mso-ascii-font-family: "Times New Roman"; mso-char-type: symbol; mso-hansi-font-family: "Times New Roman"; mso-symbol-font-family: Symbol;">®</span><span style="font-size: 12.0pt;"> #1 or #1 </span><span style="font-family: Symbol; font-size: 12.0pt; mso-ascii-font-family: "Times New Roman"; mso-char-type: symbol; mso-hansi-font-family: "Times New Roman"; mso-symbol-font-family: Symbol;">®</span><span style="font-size: 12.0pt;"> #3 </span><span style="font-family: Symbol; font-size: 12.0pt; mso-ascii-font-family: "Times New Roman"; mso-char-type: symbol; mso-hansi-font-family: "Times New Roman"; mso-symbol-font-family: Symbol;">®</span><span style="font-size: 12.0pt;"> #2 </span><span style="font-family: Symbol; font-size: 12.0pt; mso-ascii-font-family: "Times New Roman"; mso-char-type: symbol; mso-hansi-font-family: "Times New Roman"; mso-symbol-font-family: Symbol;">®</span><span style="font-size: 12.0pt;"> #1. Therefore
!3 = 2.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> For <i>n</i> =
4: Car #1 has 3
choices. For each of these choices,
either it ends up (1) swapping roads with the other car, or (2) it does not swap. <o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;">Case (1): If it
swaps with the other car, then the remaining
2 cars will have !2 = 1 way.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;">Case (2): If car #1 goes
to some road but not swapping roads with it,
then the other 3 cars will have !3 = 2
ways to choose roads different from their original roads.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;">Therefore !4 =
3 × (!3 + !2) = 3 × (2 + 1) = 9.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> For <i>n</i> =
5: Car #1 has 4
choices.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;">Case (1): It swaps
with the other car. The remaining 3 cars
will have !3 = 2 ways.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;">Case (2): It does
not swap with any car. The other 4 cars
will have !4 = 9 ways.<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;">Therefore !5 = 4
× (!4 + !3) = 4 × (9 + 2) = 44.<o:p></o:p></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Ans:</span></b><span style="font-size: 12.0pt;"> <b>(B)</b> 44.<o:p></o:p></span></div>
<br />
<div class="MsoNormal">
<b><span style="font-size: 12.0pt;">Remarks<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12.0pt;"> For
more information regarding derangements, please refer to <a href="https://en.wikipedia.org/wiki/Derangement" target="_blank">here</a> and <a href="http://mathworld.wolfram.com/Derangement.html" target="_blank">here</a>.<o:p></o:p></span></div>
<br />
<div class="MsoNormal">
<b><span style="color: #3366ff;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html">Heuristics
Used</a><o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H02. Use a diagram / model (can be used for small number cases)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H03. Make a systematic list (can be used for small number cases)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H04. Look for pattern(s)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H05. Work backwards<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H09. Restate the problem in
another way<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H10. Simplify the problem<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H11. Solve part of the problem<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">Suitable Levels<o:p></o:p></span></b></div>
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">Lower Secondary Mathematics Competition<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* University / College Combinatorics<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses that involve derangements</span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; tab-stops: 253.5pt;">
<span style="font-size: 8.0pt;">* any precocious
or interested learner who is interested<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; tab-stops: 253.5pt;">
<span style="font-size: 8.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; tab-stops: 253.5pt;">
<span style="font-size: 8.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; tab-stops: 253.5pt;">
<span style="font-size: 8.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; tab-stops: 253.5pt;">
<span style="font-size: 8.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; tab-stops: 253.5pt;">
<span style="font-size: 8.0pt;"><br /></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none; tab-stops: 253.5pt;">
<span style="font-size: 8.0pt;"><br /></span></div>
lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-7113573952636431782016-01-02T17:50:00.000+08:002016-01-02T17:50:32.999+08:00[Maths_Education] Is Learning Mathematics a "Performance"?<b>Article</b><br />
<a href="http://www.theatlantic.com/education/archive/2015/12/math-class-performing/421710/" target="_blank">The Math-Class Paradox</a><br />
<br />
<br />
<b>My Comments</b><br />
This is an excellent article addressing the issue of mathematics education in schools today. <a href="http://genuinesingaporemaths.blogspot.sg/2012/01/mathed000001-proposed-new-framework-for.html" target="_blank">School "mathematics" barely touches even the tip of my iceberg</a>. Is the learning of mathematics merely a matter of performing under time pressure on tests and exams (and that puts many students off)? What about exploration, creativity, asking questions and discovering connections ... etc?<br /> The thought provoking article is written by Jo Boaler, one of my favorite professors of mathematics education. It is definitely worth your while reading it!<br />
<br />
<br />lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-9810018161262087842015-12-26T22:25:00.002+08:002015-12-26T22:27:34.536+08:00[AM_20151226EIQR] Looking for a Pea among Quadratic Roots?<div class="MsoNormal">
<b><span style="font-size: 12pt;">Question<o:p></o:p></span></b></div>
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<v:stroke joinstyle="miter"/>
<v:formulas>
<v:f eqn="if lineDrawn pixelLineWidth 0"/>
<v:f eqn="sum @0 1 0"/>
<v:f eqn="sum 0 0 @1"/>
<v:f eqn="prod @2 1 2"/>
<v:f eqn="prod @3 21600 pixelWidth"/>
<v:f eqn="prod @3 21600 pixelHeight"/>
<v:f eqn="sum @0 0 1"/>
<v:f eqn="prod @6 1 2"/>
<v:f eqn="prod @7 21600 pixelWidth"/>
<v:f eqn="sum @8 21600 0"/>
<v:f eqn="prod @7 21600 pixelHeight"/>
<v:f eqn="sum @10 21600 0"/>
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<span style="font-size: 12pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9bILM5swU2-vatwxOMBjsveLF6iwq2uqdIKgXl6dPYVeCqdh6hmHZ4wRrseaNKXPG4SHzjC0qE6BAX1BVlqH0nx-mxGiGrkX8BilYAtW0sKkEZQ2HZAR7FpSCkVfl611G_YYvme13hPyS/s1600/quadratic+-+q.png" imageanchor="1"><img border="0" height="64" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9bILM5swU2-vatwxOMBjsveLF6iwq2uqdIKgXl6dPYVeCqdh6hmHZ4wRrseaNKXPG4SHzjC0qE6BAX1BVlqH0nx-mxGiGrkX8BilYAtW0sKkEZQ2HZAR7FpSCkVfl611G_YYvme13hPyS/s640/quadratic+-+q.png" width="640" /></a></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12pt;">Introduction<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="background: white; color: #141823; font-size: 12pt;"> </span><span style="font-size: 12pt;">This question is about
finding the parameter <i>p</i>, and not about solving for the
“unknown” <i>x</i>. It is heavy on algebra, one
has to be patient, careful and meticulous.
Please refer to this article for a recapitulation of (Vieta’s) <a href="http://genuinesingaporemaths.blogspot.sg/2015/11/am20151105qfer-new-quadratic-equation.html">theory
of Quadratic Roots</a>.<span style="background: white; color: #141823;"><o:p></o:p></span></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="font-size: 12pt;">Solution<o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="font-size: 12pt;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFO5YO4jzOza_fCkMTnTv2ZxDqbu3Z84ntvdAaFM7YxXkNdHKO8Wpyf4-j4czN7hX3ysG7nB_EfUjgqEIiaGnN28FDNCa93IOvCm-yKHa-GxYcjaRdDpIc5VOkRAWFVGe8f66N3fUlhsWH/s1600/quadratic+-+s.png" imageanchor="1"><img border="0" height="430" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjFO5YO4jzOza_fCkMTnTv2ZxDqbu3Z84ntvdAaFM7YxXkNdHKO8Wpyf4-j4czN7hX3ysG7nB_EfUjgqEIiaGnN28FDNCa93IOvCm-yKHa-GxYcjaRdDpIc5VOkRAWFVGe8f66N3fUlhsWH/s640/quadratic+-+s.png" width="640" /></a></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b><span style="color: #3366ff; font-size: 12pt;"><a href="http://genuinesingaporemaths.blogspot.sg/2015/03/math-ed-heuristics-in-mathematics.html" target="_blank"><span style="font-size: 10.0pt;">Heuristics Used</span></a></span></b><b><span style="color: #3366ff;"><o:p></o:p></span></b></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H04.
Look for pattern(s)<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H05.
Work backwards<o:p></o:p></span></div>
<div class="MsoNormal">
<span style="color: #3366ff;">H13*
Use Equation / write a Mathematical Sentence<o:p></o:p></span></div>
<br />
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">Suitable Levels</span></b></div>
<div class="MsoHeader">
<div class="MsoHeader">
<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">GCE ‘O’ Level
Additional Mathematics<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses that involve quadratic
roots<br />*<span style="font-size: 10.6667px;"> any learner who is interested</span></span><br />
<span style="font-size: 10.6667px;"><br /></span>
<span style="font-size: 10.6667px;"><br /></span>
<br /></div>
</div>
lefouquehttp://www.blogger.com/profile/12528621898369349519noreply@blogger.com0tag:blogger.com,1999:blog-7033336220707251714.post-32849906924172713952015-12-26T21:28:00.004+08:002015-12-26T21:29:54.321+08:00[S1_20151226NPSW] Finding the General Term of a Sequence (2)<div class="MsoNormal">
<b>Problem</b></div>
<div align="center" class="MsoNormal" style="text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhas090Vp2-4HC7jAfRUZ2WcyIXiHwQCk9yTBS45F33Mh_vDBtbkja_Xs-AJ3NVq0dTLj9-i30TnJd7GcLvekvVsTzbAAEM3xCmGnPHNMOlvBzTPMrwgZEPyX2tyq-7RUCICwMUgTK9QOka/s1600/Number+pattern+-+q.png" imageanchor="1"><img border="0" height="232" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhas090Vp2-4HC7jAfRUZ2WcyIXiHwQCk9yTBS45F33Mh_vDBtbkja_Xs-AJ3NVq0dTLj9-i30TnJd7GcLvekvVsTzbAAEM3xCmGnPHNMOlvBzTPMrwgZEPyX2tyq-7RUCICwMUgTK9QOka/s320/Number+pattern+-+q.png" width="320" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>Introduction</b></div>
<div class="MsoNormal">
The above was
discussed in this <a href="http://genuinesingaporemaths.blogspot.sg/2015/12/s120151226npgt-finding-general-term-of.html" target="_blank">previous article</a>. The earlier parts of the
problem are easy. The major sticking
point is finding the formula for <i>S<sub>n</sub></i>.
We solved that using factorisation and
observation, which I feel is the best way.
But what if you cannot do that and you are desperate (for example, in an
exam or test)?</div>
<div class="MsoNormal">
This article introduces <st1:city w:st="on">Newton</st1:city>’s Method, which can be used as a
back-up method, even though it is not in the regular syllabus.</div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>Solution </b>(<st1:place w:st="on">Newton</st1:place>’s Method)</div>
<div class="MsoNormal">
<o:p> </o:p><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDSTHlbvxNvrIfHdNlnj11uHys-SWMUBqS9iFU8oaHl9OFACUaPD9VQ7yR5DufShWTDwJ6X-fQUfsE7l9V0IDQHcnMVRp3ekuvE8NHViFiyOt7Jo01n_RTk35Y1cE-X8-9eF-uVtScvtKV/s1600/newtons+method+-+s.png" imageanchor="1"><img border="0" height="340" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDSTHlbvxNvrIfHdNlnj11uHys-SWMUBqS9iFU8oaHl9OFACUaPD9VQ7yR5DufShWTDwJ6X-fQUfsE7l9V0IDQHcnMVRp3ekuvE8NHViFiyOt7Jo01n_RTk35Y1cE-X8-9eF-uVtScvtKV/s640/newtons+method+-+s.png" width="640" /></a></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b>Remark<o:p></o:p></b></div>
<div class="MsoNormal">
Note that number sequences in “IQ tests” (with
no problem contexts) have been <a href="http://genuinesingaporemaths.blogspot.sg/2015/11/mathedn-20151118-abuse-of-equal-sign.html" target="_blank">debunked</a>. In our case here, the numbers do have a
certain regularity arising from the pattern of dots. In fact this is an arithmetic
progression. What we are calculating is
the sum of an arithmetic progression. However,
<st1:city w:st="on">Newton</st1:city>’s Method
extends beyond arithmetic progressions.</div>
<div class="MsoNormal">
<span style="color: #3366ff;"><br /></span></div>
<div class="MsoNormal">
<b><span style="font-size: 8pt;">Suitable Levels<o:p></o:p></span></b></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<b><span style="font-size: 8pt;">* </span></b><st1:place w:st="on"><st1:placename w:st="on"><span style="font-size: 8pt;">Lower</span></st1:placename><span style="font-size: 8pt;"> <st1:placetype w:st="on">School</st1:placetype></span></st1:place><span style="font-size: 8pt;"> Mathematics<o:p></o:p></span></div>
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<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">GCE ‘O’ Level “Elementary” Mathematics (revision)<o:p></o:p></span></div>
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<b><span style="font-size: 8pt;">* </span></b><span style="font-size: 8pt;">GCE ‘A’ Levels H2 Mathematics (revision)<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* other syllabuses that involve number
patterns and sequences<o:p></o:p></span></div>
<div style="layout-grid-mode: char; margin-bottom: .0001pt; margin: 0cm; mso-layout-grid-align: none;">
<span style="font-size: 8.0pt;">* any precocious or independent learner who
is interested<o:p></o:p></span></div>
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