tag:blogger.com,1999:blog-50156005678588837432024-03-06T13:09:00.098+08:00Maths Is Interesting!Understanding principles | Appreciating concepts |
Maths is all about playing with mathematical symbols.EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.comBlogger410125tag:blogger.com,1999:blog-5015600567858883743.post-44649603145668036172013-03-22T16:46:00.002+08:002013-03-22T16:47:09.041+08:00Tips on Estimation<span style="font-family: Arial, Helvetica, sans-serif;">In maths, we do come across topics on estimation.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">This topics relate to our daily living very closely and is truly practical. </span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">Example is when we go shopping and starts to count the expenses to be paid. </span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">But mathematical estimation has one key concern.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">To what extent or how accurate does one wish to?</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">There is no right or wrong to an answer when dealing with estimation. After all it is an ESTIMATED answer.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">The basic requirement is thus to get as close to the true answer as possible.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">Let's start with one simple example to demonstrate the concept.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">Example:</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">y = 0.501 + square root(3.89)</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">What is y without using calculator ?</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">Answer 1:</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">y = 1 + square root (4)</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">y = 1 + 2 = 3</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">Answer 2:</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">y = 0.5 + square root (4)</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">y = 0.5 + 2 = 2.5</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">You can now see that both answers is close to the actual answer of 2.4733.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">However, it is the gap or extent of the <span style="color: #cc0000;">difference</span> you wish for.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">If possible, <span style="color: #cc0000;">LOOK carefully</span> at the numbers and give a best estimation closest to ability to compute the answer.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">Here, from the above, you will notice the first term of 0.501 decides the outcome.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">Estimate this 0.501 to what numeric value?</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">Think further and you will find that 0.501 to 1 will give you a bigger difference compared to 0.501 to 0.5.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">If you can mentally handle 0.5 as the estimated value for computation, go for it since this should be closer to the final outcome.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">Thus in conclusion, do look a bit closer to the numbers presented in your problem and simply do a brief calculation of the differences between estimated and raw data. Then you are one step closer to getting a good estimation.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">Estimation, finally, boils down to how far you are to the actual answer. Nothing difficult.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">Interesting? Any more suggestions?</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;">.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>
<span style="font-family: Arial, Helvetica, sans-serif;"><br /></span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com0tag:blogger.com,1999:blog-5015600567858883743.post-1365123229686770862013-03-09T22:28:00.001+08:002013-03-09T22:34:22.559+08:00Building Up Presentation Skills With MathsMaths is not about just doing and computing mathematical challenges. It is also not just about planning strategies to solve a problem. It involves more than mentioned.<br />
<br />
It includes skill seemingly not related to maths.<br />
<br />
What is it then?<br />
It is about presenting the solution and steps in approaching the maths problems.<br />
And presenting them well and in an understandable way.<br />
<br />
Example :<br />
<br />
3x - 6 = x. Find x.<br />
<br />
Solution A:<br />
x = 6 / 2 = 3<br />
<br />
Solution B:<br />
3x - x = 6<br />
2x = 6<br />
x = 6 / 2 =3<br />
<br />
In your view, which solution has a better presentation?<br />
Which shows the approach better?<br />
<br />
My view is Solution B is better. Why?<br />
It showed the thinking that goes on in the learner's mind.<br />
<br />
In real life, we need to make clear our thoughts in getting and convincing partners and team-mates to work with us. It is an important skill to present our ideas to others.<br />
<br />
By doing and presenting our maths solution, we are actually aligning ourselves to real working life.<br />
Thus performing good presentation in our maths solving helps.<br />
<br />
Agree?<br />
<br />
Maths is interesting.<br />
<br />
.<br />
<br />EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com0tag:blogger.com,1999:blog-5015600567858883743.post-23895954191500483942012-02-17T17:42:00.000+08:002012-02-17T17:42:13.563+08:00Fear Over Maths<span style="font-family: Arial, Helvetica, sans-serif;">Maths is one of the crucial and necessary subject we have to learn in school. Everyone has to go through it.</span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">Some like it, while some fear it. Have you wonder why?</span><br />
<br />
<span style="font-family: Arial;">Actually this fear is not only towards maths. The underlying reasons applies to anything we do.</span><br />
<span style="font-family: Arial;">It could be literature, Chinese language, dancing, or even driving and riding a bicycle.</span><br />
<br />
<span style="font-family: Arial;">What we have to do to reduce maths anxiety is to know why the fear occurs.</span><br />
<br />
<span style="font-family: Arial;">One of the issue links to the confidence level. This affects the comfort level. With things we are not comfortable, we tend to avoid. Avoiding make us do less of the subject.</span><br />
<br />
<span style="font-family: Arial;">We thus lack practices.</span><br />
<br />
<span style="font-family: Arial;">Learning maths involves 2 parts, namely:-</span><br />
<span style="font-family: Arial;">1) <strong><em>Procedural skills</em></strong>, and</span><br />
<span style="font-family: Arial;">2) <em><strong>Conceptual understanding</strong></em>.</span><br />
<br />
<span style="font-family: Arial;">Different stages / levels in the learning journey entails different focus of these 2 stages.</span><br />
<br />
<span style="font-family: Arial;">To have a better understanding, you may go to this <a href="http://www.mathsmap.com/why-fear-learning-mathematics"><strong>maths site</strong></a> to read more. </span><br />
<span style="font-family: Arial;">It is rather enlightening.</span><br />
<br />
<span style="font-family: Arial;">Learning anything needs a plan. (Procedure ==> Conceptual ==> Application)</span><br />
<span style="font-family: Arial;">If fear is the barrier, and causes obtrusion to learning, seek out why.</span><br />
<span style="font-family: Arial;">We grow as a result of this process.</span><br />
<br />
<span style="font-family: Arial;">Maths is one target in life that we can use to challenge yourself, besides the tools and techniques picked up to solve mathematical problems.</span><br />
<span style="font-family: Arial;">It involves character development as well, if you border to analyse the learning process.</span><br />
<br />
<span style="font-family: Arial;">Finally, to motive any maths learners,</span><br />
<br />
<span style="font-family: Arial;">just know that "Maths Is Interesting!".</span><br />
<br />
<span style="font-family: Arial;">You will then like maths, and anything you set forth to master.</span><br />
<br />
<span style="font-family: Arial;">Cheers! :-D.</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com3tag:blogger.com,1999:blog-5015600567858883743.post-3980562613748880212011-12-21T22:52:00.002+08:002011-12-23T10:17:10.889+08:00Caution on Mixed Number<span style="font-family: Arial, Helvetica, sans-serif;">'</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">Fractions are a necessary part of maths.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">They come in many forms; improper, proper and mixed number.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;"><br />
</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">Though improper and proper forms are direct in its presentation and interpretation, mixed number form may pose a potential mistake for young learners.</span><br />
<br />
<span style="font-family: Arial;">Example:</span><br />
<a href="http://www.codecogs.com/eqnedit.php?latex=\small 2\frac{3}{4}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\small 2\frac{3}{4}" title="\small 2\frac{3}{4}" /></a> <br />
<span style="font-family: Arial, Helvetica, sans-serif;">Is this <strong>2 + (3/4)</strong> or <strong>2 x (3/4)</strong> ?</span><br />
<br />
<span style="font-family: Arial;">Caution has to be taken to stress it as 2 + (3/4).</span><br />
<br />
<span style="font-family: Arial;">Some students have taken it to mean 2 pieces of (3/4) !</span><br />
<span style="font-family: Arial;">Dangerous isn't it.</span><br />
<br />
<span style="font-family: Arial;">But rest assure. </span><br />
<span style="font-family: Arial;">If you understand the language of maths and its "grammar", all will be well and interesting.</span><br />
<br />
<span style="font-family: Arial;">:-) </span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com2tag:blogger.com,1999:blog-5015600567858883743.post-16874892180805107462011-11-18T17:42:00.002+08:002011-12-07T18:04:54.947+08:00Tips On Using Substitution<span style="font-family: Arial, Helvetica, sans-serif;">Maths entails the usage of our brain juice in solving problems. It is a good platform for stretching our imagination and creativity by using simple concepts learned to handle seemingly complex maths questions.</span><br />
<br />
<span style="font-family: Arial;">Let's look at a "complex" simultaneous equations maths problem, and its way of solving (suggested).</span><br />
<br />
<span style="font-family: Arial;">Question:</span><br />
<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=\dpi{120} \fn_cs \frac{1}{y}@plus;\frac{1}{x}= \frac{81}{8}" target="_blank"><img src="http://latex.codecogs.com/png.latex?\dpi{120} \fn_cs \frac{1}{y}+\frac{1}{x}= \frac{81}{8}" title="\dpi{120} \fn_cs \frac{1}{y}+\frac{1}{x}= \frac{81}{8}" /></a> ---- (A)<br />
<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=\dpi{120} \fn_cs \frac{1}{2y}@plus;\frac{2}{x}= \frac{21}{4}" target="_blank"><img src="http://latex.codecogs.com/png.latex?\dpi{120} \fn_cs \frac{1}{2y}+\frac{2}{x}= \frac{21}{4}" title="\dpi{120} \fn_cs \frac{1}{2y}+\frac{2}{x}= \frac{21}{4}" /></a> --- (B)<br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">Solve for y and x.</span><br />
<br />
<span style="font-family: Arial;">How do you go about it?</span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">Look scary, right?</span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">But like what I said, looks can be deceiving. Use the brain to go around the issue!</span><br />
<br />
<span style="font-family: Arial;">Tips: The structure of the simultaneous equations looks similar to the conventional type.</span><br />
<span style="font-family: Arial;">(Conventional type:-</span><br />
<span style="font-family: Arial;">Ax + By = nn</span><br />
<span style="font-family: Arial;">Cx + Dy = kk )</span><br />
<br />
<span style="font-family: Arial;">So what we have to do can be to simply substitute </span><a href="http://www.codecogs.com/eqnedit.php?latex=\dpi{120} \fn_cs \frac{1}{y}" target="_blank"><img src="http://latex.codecogs.com/png.latex?\dpi{120} \fn_cs \frac{1}{y}" title="\dpi{120} \fn_cs \frac{1}{y}" /></a> <span style="font-family: Arial, Helvetica, sans-serif;">by <strong>m</strong>, and </span><a href="http://www.codecogs.com/eqnedit.php?latex=\dpi{120} \fn_cs \frac{1}{x}" target="_blank"><img src="http://latex.codecogs.com/png.latex?\dpi{120} \fn_cs \frac{1}{x}" title="\dpi{120} \fn_cs \frac{1}{x}" /></a> <span style="font-family: Arial, Helvetica, sans-serif;">by <strong>h</strong> (or any variable name).</span><br />
<br />
<span style="font-family: Arial;">What we thus convert to is:</span><br />
<a href="http://www.codecogs.com/eqnedit.php?latex=\dpi{120} \fn_cs m @plus; h = \frac{81}{8}" target="_blank"><img src="http://latex.codecogs.com/png.latex?\dpi{120} \fn_cs m + h = \frac{81}{8}" title="\dpi{120} \fn_cs m + h = \frac{81}{8}" /></a> ---- (A)<br />
<br />
<a href="http://www.codecogs.com/eqnedit.php?latex=\dpi{120} \fn_cs \frac{m}{2} @plus; 2h = \frac{21}{4}" target="_blank"><img src="http://latex.codecogs.com/png.latex?\dpi{120} \fn_cs \frac{m}{2} + 2h = \frac{21}{4}" title="\dpi{120} \fn_cs \frac{m}{2} + 2h = \frac{21}{4}" /></a> --- (B)<br />
<br />
<span style="font-family: Arial;">Will this <a href="http://mathsisinteresting.blogspot.com/2010/04/simultaneous-equations-re-write.html">simultaneous equations</a> be more comfortable to solve?</span><br />
<br />
<span style="font-family: Arial;">Hence, a simple twist to the former mathematical questions can result in a totally familiar situations where we have solve many a times.</span><br />
<br />
<span style="font-family: Arial;">Thus, the technique and usefulness of substitution cannot be under-estimated.</span><br />
<br />
<span style="font-family: Arial;">It can be powerful at times to reveal a beautiful mathematical expression for user to resolve.</span><br />
<br />
<span style="font-family: Arial;">Maths Is Interesting!</span><br />
<br />
<span style="font-family: Arial;">Treasure our brain and our thinking. </span><br />
<br />
<span style="font-family: Arial;">:-)</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com0tag:blogger.com,1999:blog-5015600567858883743.post-69560559667453349562011-10-30T23:23:00.001+08:002011-10-30T23:24:34.347+08:00Zippy Graphical Maths<span style="font-family: Arial, Helvetica, sans-serif;">Trigonometry is a fun topic in maths.</span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">It generates curves more than many other topics.</span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">By combining various trigonometrical functions, you can get interesting patterns on a graph.</span><br />
<br />
<span style="font-family: Arial;">Putting these functions on an algebraic expression produces even exciting diagram.</span><br />
<br />
<span style="font-family: Arial;">Below is one I created and an array of zips appears.</span><br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-WvL-1DbznoPxJXYFtFyoIfdRMRIWFUuSnYhtNSwKNHOtABmJRtXQvXCCrczYU3nMraNzs4q-HPnjRyUdkxquxEdZOpy0mrVf9DxCW3l7wrKl_sh6JSEwW8o396Cilw1D6-9k1bxKNlk_/s1600/Zippy+Graphical+Maths.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="363" ida="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-WvL-1DbznoPxJXYFtFyoIfdRMRIWFUuSnYhtNSwKNHOtABmJRtXQvXCCrczYU3nMraNzs4q-HPnjRyUdkxquxEdZOpy0mrVf9DxCW3l7wrKl_sh6JSEwW8o396Cilw1D6-9k1bxKNlk_/s640/Zippy+Graphical+Maths.jpg" width="640" /></a></div><br />
<span style="font-family: Arial, Helvetica, sans-serif;">Enjoy maths.</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">maths is interesting!</span><br />
<br />
<span style="font-family: Arial;">.</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com3tag:blogger.com,1999:blog-5015600567858883743.post-60089790725184278782011-09-09T23:47:00.000+08:002011-09-09T23:47:54.227+08:00Math Challenge 24<span style="font-family: Arial, Helvetica, sans-serif;">Math does not purely involve writing mathematical expression .</span><br />
<span style="font-family: Arial;">Sometime what you need is some logically deduction base on, of course, some mathematical principles.</span><br />
<br />
<span style="font-family: Arial;">Below is one good example of "deduction" type of math solving.</span><br />
<br />
<span style="font-family: Arial;">Let start the challenge, and have some fun!</span><br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjljvFuO633obaAnlVis3_SObeU5Bhux5FqZaK0QU_TgJiWbxYK_T194SmTMvvRXa1D6rGWxz2jQB4llG2KXFhn8HwzymC_-jMZP9jJbtm9vsJBbkQdzYUcXF9upU_cP6RMxgruZx_vswTF/s1600/Squares.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="138" nba="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjljvFuO633obaAnlVis3_SObeU5Bhux5FqZaK0QU_TgJiWbxYK_T194SmTMvvRXa1D6rGWxz2jQB4llG2KXFhn8HwzymC_-jMZP9jJbtm9vsJBbkQdzYUcXF9upU_cP6RMxgruZx_vswTF/s320/Squares.jpg" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"><br />
</div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;">Above you will find 3 squares. </span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;">Do note that the 2 yellows are of the same area and 1 blue of area bigger than the yellow ones.</span></div><div class="separator" style="clear: both; text-align: left;"><br />
</div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: Arial;">If the total area of the 3 squares are 57 sq cm, determine the area of the bigger blue square.</span></div><div class="separator" style="clear: both; text-align: left;"><br />
</div><div class="separator" style="clear: both; text-align: left;"><br />
</div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: Arial;">I believe you will enjoy this math question.</span></div><div class="separator" style="clear: both; text-align: left;"><span style="font-family: Arial;">.</span></div>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com15tag:blogger.com,1999:blog-5015600567858883743.post-33236343080603785872011-05-25T23:55:00.000+08:002011-05-25T23:55:12.680+08:00Using Units to Deduce Maths Formula<span style="font-family: Arial, Helvetica, sans-serif;">There are times when we cannot remember some simple formula for a maths application.</span><br />
<br />
<span style="font-family: Arial;">Or we have doubts to the some maths working especially when many parameters got involved.</span><br />
<br />
<span style="font-family: Arial;">I ave a simple tip.</span><br />
<br />
<span style="font-family: Arial;">Look at the units for the numerical item.</span><br />
<br />
<span style="font-family: Arial;">Example:</span><br />
<br />
<span style="font-family: Arial;">To calculate distance travelled by a vehicle, given the speed it goes and time taken, </span><br />
<br />
<span style="font-family: Arial;">we look at the speed's units.</span><br />
<br />
<span style="font-family: Arial;">Unit: m / s</span><br />
<br />
<span style="font-family: Arial;">What does it tell?</span><br />
<br />
<span style="font-family: Arial;">Yes, it gave an indirect answer that speed = distance / time.</span><br />
<br />
<span style="font-family: Arial;">Thus if time is given, we are able to know that we just need to multiple speed by time in order to retain only the distance.</span><br />
<br />
<span style="font-family: Arial;">(m / s) x s = m (only) ==> Distance</span><br />
<br />
<span style="font-family: Arial;">The above allow us to use units to deduce the working (and formula).</span><br />
<br />
<span style="font-family: Arial;">Hence, we should not overlook the power of knowing units.</span><br />
<br />
<span style="font-family: Arial;">It is simply disappointing to sometimes see people missing out on writing the units for certain parameters. Maths loses its value simply by ignoring this step.</span><br />
<br />
<span style="font-family: Arial;">Therefore treasure this little but powerful "units".</span><br />
<br />
<span style="font-family: Arial;">:-)</span><br />
<span style="font-family: Arial;">Maths is interesting!</span><br />
<br />
<span style="font-family: Arial;">.</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com5tag:blogger.com,1999:blog-5015600567858883743.post-70347961642056835212011-03-14T00:07:00.000+08:002011-03-14T00:07:52.120+08:00Watery Art using Maths Expression<span style="font-family: Arial, Helvetica, sans-serif;">Graphs are wonderful thing in the learning of maths.</span><br />
<br />
<span style="font-family: Arial;">Not only does it reflects visual symptom or trend in data collected, it displays, if allowed, beautiful images.</span><br />
<br />
<span style="font-family: Arial;">This is possible if you allow you maths juice to go free and create mathematical expressions to your fancy and view them on a graph.</span><br />
<br />
<span style="font-family: Arial;">Below I have created one. I visual it as water rippling through a surface (on the top view).</span><br />
<br />
<span style="font-family: Arial;">Hope you like this maths art of mine.</span><br />
<br />
<span style="font-family: Arial;">NOTE: It is created using trigonometry of circulatory expression.</span><br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_V5M0XQL-nMe-goGpWo80TLMUvz0_ExtHeYqAMlGOfPmop30HL3rDDR8x0E6ueQmASRCnQNN4JK8NdF6uSIRc6EngVR3Hz63jhov5YB14r_VsYdMn7O2oQWu6PrtPDQV2DpDO0DjByhy8/s1600/WaterRippling.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="296" q6="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_V5M0XQL-nMe-goGpWo80TLMUvz0_ExtHeYqAMlGOfPmop30HL3rDDR8x0E6ueQmASRCnQNN4JK8NdF6uSIRc6EngVR3Hz63jhov5YB14r_VsYdMn7O2oQWu6PrtPDQV2DpDO0DjByhy8/s400/WaterRippling.jpg" width="400" /></a></div><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">Here I view a water droplet going down into the centre. It then produces ripples or waves spreading outwards in a circular manner.</span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">Imagination ....</span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">Maths expressing ......</span><br />
<br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">:-)</span><br />
<br />
<br />
<br />
<br />
<br />
.EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com4tag:blogger.com,1999:blog-5015600567858883743.post-48366911354947006172011-03-05T23:42:00.001+08:002011-03-05T23:44:58.201+08:00Explanation of the Elimination Method<span style="font-family: Arial, Helvetica, sans-serif;">Solving of Simultaneous equations may require one common technique called "Elimination" method.</span><br />
<br />
<span style="font-family: Arial;">From the name, we know that it has to eliminate or remove something from the equations.</span><br />
<br />
<span style="font-family: Arial;">The target is one selected variable or unknown in the mathematical equations.</span><br />
<br />
<span style="font-family: Arial;">However, when approaching this method, you noticed that it involved the subtraction (or addition) of equations.</span><br />
<br />
<span style="font-family: Arial;">The question is <strong>"Can equations be subtracted?".</strong></span><br />
<span style="font-family: Arial;">And <strong>"What is the <span style="color: red;">real </span>meaning of subtracting equations?"</strong></span><br />
<br />
<span style="font-family: Arial;">My answers:-</span><br />
<br />
<span style="font-family: Arial;">Yes, equations can of course be subtracted. Equations are like other items, e.g. apples, chairs.</span><br />
<br />
<span style="font-family: Arial;">The real meaning of subtracting equations is not that apparent.</span><br />
<br />
<span style="font-family: Arial;">The true and desired wish to subtract equations boils down to commonising a certain coefficient of a variable.</span><br />
<span style="font-family: Arial;">With this common coefficient, it will then be able to remove this mathematical unknown.</span><br />
<br />
<span style="font-family: Arial;">(It is not really the direct processing of equations, and the magical removal of variable as a result!) </span><br />
<br />
<span style="font-family: Arial;">We commonise the coefficient of the selected variable first before subtracting the equations in order that same items are eliminated.</span><br />
<br />
<span style="font-family: Arial;">Hope this clarify some doubts of new learners to simultaneous equations solvers.</span><br />
<br />
<span style="font-family: Arial;">Concepts have to be learned upfront without pending questions for complete understanding and smooth follow-up learning in the later stage. Seek to clarify any doubts as far as possible.</span><br />
<br />
<span style="font-family: Arial;">It will reduce maths anxiety and allow you to enjoy maths as a result. The reward of clearing any doubts cannot be spelled out in words but through actual working and practice with proper analysis.</span><br />
<br />
<span style="font-family: Arial;">I believe you support this notion.</span><br />
<br />
<span style="font-family: Arial;">Cheers to maths.</span><br />
<br />
<span style="font-family: Arial;">.</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com0tag:blogger.com,1999:blog-5015600567858883743.post-6618900066288894962011-01-16T00:01:00.000+08:002011-01-16T00:01:20.582+08:00Simultaneous Equations - Decimal Numbered<span style="font-family: Arial, Helvetica, sans-serif;">In the learning of maths, questions grow challenging as one progress upwards.</span><br />
<br />
<span style="font-family: Arial;">One such example is the solving of simultaneous equations.</span><br />
<br />
<span style="font-family: Arial;">The easy type:</span><br />
<br />
<span style="font-family: Arial;">Solve for x and y using elimination method.</span><br />
<span style="font-family: Arial;">4x + 3y = 10 ---- (1)</span><br />
<span style="font-family: Arial;">3x + 4y = 11 ---- (2)</span><br />
<br />
<span style="font-family: Arial;">For the above problem can be solved easily by selecting a coefficient to be commonised.</span><br />
<span style="font-family: Arial;">The post of <a href="http://mathsisinteresting.blogspot.com/2008/07/elimination-method-simultaneous.html">elimination method</a> is reference here for review.</span><br />
<br />
<span style="font-family: Arial;">But moving on (higher) with more challenging maths question ... we may get the below.</span><br />
<br />
<span style="font-family: Arial;">0.4x + 0.3y = 1 ---- (A)</span><br />
<span style="font-family: Arial;">3x + 4y = 11 ---(B)</span><br />
<br />
<span style="font-family: Arial;">What should we do next?</span><br />
<br />
<span style="font-family: Arial;">Equation (A) may seems unusual. It is in decimal form!</span><br />
<br />
<span style="font-family: Arial;">But as the blog title claims "<strong>Maths Is Interesting!</strong>", we should not be worried.</span><br />
<br />
<span style="font-family: Arial;">This type of question is actually not new in concept or tricky as it seems.</span><br />
<span style="font-family: Arial;">It is there to test you understanding by being "different".</span><br />
<br />
<span style="font-family: Arial;">We have to remove the "catch", which is to change the decimated number to integer.</span><br />
<span style="font-family: Arial;">How we do it here is simply multiplying the coefficients by 10.</span><br />
<br />
<span style="font-family: Arial;">This makes equation (A) to be 4x + 3y = 10 (back to the original first set of simultaneous equations at the start of this post.</span><br />
<br />
<span style="font-family: Arial;">The above example serves to illustrate the simplicity of changing numbers to suit the condition for easy solving. (Other questions may be multiply by another decimal number, or integer).</span><br />
<br />
<span style="font-family: Arial;">Just have a <strong>clear mind</strong> and a <strong>confidence attitude</strong> will be enough to allow you to solve most of the maths questions.</span><br />
<br />
<span style="font-family: Arial;">Try it and you will believe what I say (or write).</span><br />
<br />
<span style="font-family: Arial;">Maths is interesting!</span><br />
<br />
<span style="font-family: Arial;">.</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com1tag:blogger.com,1999:blog-5015600567858883743.post-73162009289466704872010-12-08T23:48:00.000+08:002010-12-08T23:48:46.497+08:00Decimal Number Simplification.<br />
Algebraic expressions and equations normally come in integer or fraction form.<br />
<br />
Examples:<br />
1) 4x - 3 = x<br />
2) (3/4)x - 3x = 1/(3x)<br />
<br />
Simplification of the above examples will not pose much of a problem except maybe in the challenge of bringing the numbers and unknowns over the "equal" sign.<br />
<br />
But algebraic equations can come in decimal form too.<br />
<br />
Example:<br />
0.3(0.2x - 1) = 0.1x<br />
<br />
How do we go about solving the above "decimated" algebraic equation easily?<br />
<br />
A simple trick that I can think of (or maybe too simply a technique to call it 'trick").<br />
<br />
What I would do is to <span style="color: red;">multiply</span> the expression on both sides by 10.<br />
The idea is to bring the decimal number into the integer range.<br />
<br />
BUT do note that the expression on the left side has two decimal numbers.<br />
<br />
As such I would have to "x 10" <strong><span style="color: red;">twice</span></strong>.<br />
This means that there is a "x 100" on the <u>left </u>and <u>right</u> side.<br />
<br />
The new equation will thus be:<br />
<br />
3 (2x - 10) = 10 x<br />
==> 6x - 30 = 10x<br />
==> -30 = 10x - 6x = 4x<br />
==> x = -30 / 4 = -7.5<br />
<br />
Conclusion:<br />
Decimal can be seen to be intimidating when in the decimal form. However, it can be elevated to the familiar integer form through simple multiplication.<br />
However, do take note of how many decimal number has been multiplied.<br />
Left and right sides of the equation has to have the same number of multiplication (or division) to stay equal and valid.<br />
<br />
Maths is not that frightening. <br />
It can be interesting, if the method to "attack" it is properly done.<br />
<br />
:-)EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com0tag:blogger.com,1999:blog-5015600567858883743.post-26038857704648669832010-11-22T23:26:00.000+08:002010-11-22T23:26:50.238+08:00Basic Decimal Conversion`<br />
<span style="font-family: Arial, Helvetica, sans-serif;">Every thing falls back to basic. </span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">If the fundamentals are weak, any maths learners will have a hard time moving forward in their maths learning journey.</span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">Let me quote an example.</span><br />
<br />
<span style="font-family: Arial;">How do we change 4.75 to fraction.</span><br />
<br />
<span style="font-family: Arial;">We can use 475 / 100 and reduce it through long division. This will give 4 and 3/4.</span><br />
<br />
<span style="font-family: Arial;">However, if we know that 4.75 is actually consisting of 4 add to 0.75, the conversion will be simpler.</span><br />
<br />
<span style="font-family: Arial;">4 and 0.75 means 4 + (3/4) which leads directly to 4 whole and 3/4. Same as answer of above.</span><br />
<span style="font-family: Arial;">(NOTE: 0.75 is a quarter which equates to 3/4).</span><br />
<br />
<span style="font-family: Arial;">Simple isn't it?</span><br />
<br />
<span style="font-family: Arial;">Maths is interesting.</span><br />
<br />
<span style="font-family: Arial;">:-)</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com0tag:blogger.com,1999:blog-5015600567858883743.post-41851263509968915002010-10-03T00:59:00.001+08:002010-10-03T01:01:13.757+08:00Pointers in Teaching, Learning Speed.<br />
<span style="font-family: Arial, Helvetica, sans-serif;">At elementary level in maths education, speed is always a challenging topic for learners.</span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">It caught my attention and I started wondering why?</span><br />
<br />
<span style="font-family: Arial;">Many mistakes can be made when dealing with these types of questions.</span><br />
<br />
<span style="font-family: Arial;">After studying the various mistakes made by learners, I came to a few conclusion that I like to share here.</span><br />
<br />
<span style="font-family: Arial;"><strong>How to avoid confusion in doing Speed questions in maths</strong>:-</span><br />
<br />
<span style="font-family: Arial;">1) <strong>Speed involves two parameters, namely, distance and time.</strong></span><br />
<span style="font-family: Arial;">This is the key issue. Dealing with one parameter is already a challenge, and dealng with two is always a "headache".</span><br />
<span style="font-family: Arial;">The concept, has thus to be clearly addressed upon, before the ratio of distance and time leading to speed can be fully understood.</span><br />
<span style="color: blue; font-family: Arial;"><strong>What is distance?</strong></span><br />
<span style="color: blue; font-family: Arial;"><strong>What is time?</strong></span><br />
<span style="font-family: Arial;">These 2 items are variable in nature. They change in value. </span><br />
<span style="font-family: Arial;">They causes confusion when lumped together!</span><br />
<br />
<span style="font-family: Arial;">Examples of daily activities will help in this case.</span><br />
<br />
<span style="font-family: Arial;">Quote cases like running in a race, where the champion came back in the shortest time covering the same distance as all others.</span><br />
<br />
<span style="font-family: Arial;">Get the concept of distance versus time into them.</span><br />
<span style="font-family: Arial;">Also FAST and SLOW relation to speed.</span><br />
<br />
<span style="font-family: Arial;">2) <strong>Error in units</strong>:-</span><br />
<span style="font-family: Arial;">Break up the tasks of calculating km, m or cm and sec, hours, minutes separately.</span><br />
<span style="font-family: Arial;">In other words,deal with one item at a time.</span><br />
<span style="font-family: Arial;">Use basic unit if possible to reduce chances of making costly errors.</span><br />
<br />
<span style="font-family: Arial;">The learners have to handle the logical part of the question, and also the mechanical part of unit manipulation in speed problems.</span><br />
<span style="font-family: Arial;">Tell them to find one thing at a time, and the need for doing that. Be patience is the message.</span><br />
<br />
<span style="font-family: Arial;">3) <strong>Draw out a pictorial image of the question</strong>.</span><br />
<span style="font-family: Arial;">This method will help some kids to visualise the real issue.</span><br />
<br />
<span style="font-family: Arial;">By having drawn the length for distance to be covered (or covered), they will have a better idea of what distance is about in the maths question. They will not have to "keep" this disatnce in their mind together with the problematic "time" condition.</span><br />
<br />
<span style="font-family: Arial;">Use the seeing method helps them clear any doubts and can also reduce mistakes in interpreting the question.</span><br />
<br />
<span style="font-family: Arial;">There will definitely be more pointers to be added to my three above.</span><br />
<span style="font-family: Arial;">But with these 3 basic issues settled, most of the queries about speed and its maths problems should be clearer.</span><br />
<br />
<span style="font-family: Arial;">If you have any other pointers, you may share in the comment space.</span><br />
<span style="font-family: Arial;">Cheers :-)</span><br />
<br />
<span style="font-family: Arial;"><strong><span style="color: blue;">Maths is interesting</span></strong>, I suppose you cannot agree more.</span><br />
<br />
<span style="font-family: Arial;">.</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com0tag:blogger.com,1999:blog-5015600567858883743.post-5715435540238824012010-06-13T21:56:00.002+08:002010-10-03T00:23:18.015+08:00Simple Logarithm Tip.<br />
Maths expression may at times look challenging, but a bit of a thought may make it otherwise.<br />
<br />
Logarithm is always an exciting topics to new learners.<br />
<br />
With the "log" coming into the maths expression, one will be confused.<br />
Definitely!<br />
<br />
But do rest assure, as the tip below shows.<br />
<br />
<u><b>Maths Tip</b></u><br />
<br />
<span style="font-size: large;"><strong>e<sup>ln y</sup> = y</strong></span><br />
<br />
Proving this:<br />
"Natural log" both sides will give <strong><span style="color: red; font-size: large;">ln e<sup>ln y</sup> = ln y</span></strong><br />
<br />
Applying the law that ln a<sup>n</sup> = n ln a, and that ln e = 1, you will notice that the above mathematical expressions are true and equal.<br />
<br />
<strong>NOTE:</strong><br />
This tip applies to "log" too.<br />
<br />
10<sup>log y</sup> = y<br />
<br />
<br />
Hope this helps.<br />
<br />
:-)EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com3tag:blogger.com,1999:blog-5015600567858883743.post-58947992492139031442010-06-02T00:01:00.001+08:002010-06-02T00:02:56.396+08:00Model versus Variable Technique'<br />
In using Model method of solving maths question, we are using visual blocks to scope our thinking. This is followed by analysis through the models.<br />
<br />
Models become a link to our thinking process.<br />
<br />
The demerit is when we did not create the Model properly, or miss out some details that cause the model to be represented wrongly.<br />
<br />
The merit is that it can be simple and straight forward when drawn properly. It reflects outright the relationship between many unknowns.<br />
Less workings is thus needed, as visual que sets in.<br />
<br />
For algebraic variable technique, the unknowns are pre-defined and booked as "letters". A space, mentally, has been reserved for the answer. <br />
<br />
The working is just simply to accept that the answer is already there but only not numerical. Following through the working steps will ultimately reveal the letter of its numerical data which is what we want.<br />
<br />
Variable as letter is good in the sense that we need less analysis, but just mechanically following the rules and steps leading to the final step, of course with some logic and mathematical strategy.<br />
<br />
Each has its own advantages and weakness. It is up to us to make use of them in the correct way. <br />
Experience is the only way to overcome the proper selection of which technique.<br />
<br />
Thus practice to gain experience in maths is one good way to master maths.<br />
Skiving is a no-no.<br />
<br />
Through practice, you will sooner or later find that maths is interesting.<br />
<br />
:-)EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com0tag:blogger.com,1999:blog-5015600567858883743.post-48003055630182503382010-05-27T23:52:00.000+08:002010-05-27T23:52:33.081+08:00Purpose of Variables in Algebra'<br />
Unknowns are literally unknowns.<br />
<br />
In maths, these unknowns are a cuause formaths anxiety.<br />
When you are in unfamiliar territory, you will naturally be uncomfortable and unease.<br />
<br />
This is the same feelingwhen dealing with unknowns in maths.<br />
<br />
Algebra came to the rescue for this problems.<br />
<br />
Here, you will find unknowns named as "variables".<br />
<br />
They served as "parking lots" for the final answers or unknowns.<br />
<br />
In this algebra, you replace the variables for final numbers and work with them as though you already know them.<br />
<br />
You simply go through the motion of solving the question with any given condition and numbers / data.<br />
<br />
Upon finally reaching the last step, the numerical answers for the problem will be revealed.<br />
<br />
This is the power of the variables in algebra.<br />
<br />
Just simply work along, and not be fearful of the unknowns.<br />
The steps will align you to the final answers.<br />
<br />
Cheers!<br />
Maths is interesting!<br />
<br />
.EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com0tag:blogger.com,1999:blog-5015600567858883743.post-41847274901995999012010-04-24T18:21:00.001+08:002010-04-24T18:22:02.393+08:00A Mathematical Waterfall'<br />
Mathematics equation can be fun. <br />
It is not only used as a problem-solving tool, it can be used to create visual image simulating scene.<br />
<br />
By trying a few equations, anyone with patience and basic maths knowledge can do it.<br />
Simply create an expression or equation in a graph and tweet it to form any image.<br />
<br />
Here you will see an image formed up to look like a waterfall.<br />
<br />
Enjoy yourself.<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6AgkahK3Koe0crEJwLp_12AXTgl6AFXGlN6crzlj0RT1uoVslDtKQ0HWBAUlxW7vEFusXpDpzS7SA6SDJb2FCI0bWU427diZqe-o6jMeoKbRf-hg0v7rdHwYCxgMlYvu6ORibdOxRVpKR/s1600/maths_waterfall.GIF" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="297" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6AgkahK3Koe0crEJwLp_12AXTgl6AFXGlN6crzlj0RT1uoVslDtKQ0HWBAUlxW7vEFusXpDpzS7SA6SDJb2FCI0bWU427diZqe-o6jMeoKbRf-hg0v7rdHwYCxgMlYvu6ORibdOxRVpKR/s400/maths_waterfall.GIF" tt="true" width="400" /></a></div><br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
This was done with both logarithm and trigonometry functions.<br />
<br />
:-)EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com1tag:blogger.com,1999:blog-5015600567858883743.post-81300930053250123072010-04-14T23:30:00.000+08:002010-04-14T23:30:31.131+08:00Maths Symbol in Our Applications.<br />
<span style="font-family: Arial, Helvetica, sans-serif;">There are many symbols in maths. </span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">To learn and understand maths, we need to know the meaning of the symbols.</span><br />
<br />
<span style="font-family: Arial;">This is very much like talking to a foreigner. Without understanding each other's language, no communication can be carried out (other than the international body language!)</span><br />
<br />
<span style="font-family: Arial;">Hence, knowing the usage of the symbols in a mathematical expression helps.</span><br />
<br />
<span style="font-family: Arial;">But is it really so?</span><br />
<br />
<span style="font-family: Arial;">Partially.</span><br />
<br />
<span style="font-family: Arial;">Why do I say that?</span><br />
<br />
<span style="font-family: Arial;">Yes, you may know the symbol while doing maths, but if the same symbol is used elsewhere, do you still understand?</span><br />
<br />
<span style="font-family: Arial;">One example is:</span><br />
<br />
<span style="color: red; font-family: Arial;"><strong>y = x + 1</strong></span><br />
<span style="font-family: Arial;">This means x is added by one and their total is represented by the variable "y".</span><br />
<span style="font-family: Arial;">This is for the maths operator "<span style="color: red;"><strong>+</strong></span>".</span><br />
<br />
<span style="font-family: Arial;">But what about the expression <strong><span style="color: red;">x++</span></strong> ?</span><br />
<span style="font-family: Arial;">This looks odd, isn't it?</span><br />
<br />
<span style="font-family: Arial;">To the maths learner, this may be a typo error, or something is missing.</span><br />
<br />
<span style="font-family: Arial;">"x++" is actually commonly used in C programming.</span><br />
<br />
<span style="font-family: Arial;">What it means is <strong><span style="color: #741b47;">x = x + 1</span></strong>.</span><br />
<span style="font-family: Arial;">It is a short-cut way of writing the addition of x and replacing it by the same variable "x".</span><br />
<br />
<span style="font-family: Arial;">Thus this example showed the use of "+" in another application.</span><br />
<br />
<span style="font-family: Arial;">It is still maths in some sense, but written in another form.</span><br />
<br />
<span style="font-family: Arial;">Maths is therefore always around us. It is a matter of us applying them and understanding them.</span><br />
<br />
<span style="font-family: Arial;">Only by learning their "language", can we communicate with them.</span><br />
<br />
<span style="font-family: Arial;">Interesting? I bet you agree!</span><br />
<br />
<span style="font-family: Arial;">Other applications can be " += ", " :-) " and " x>>4 ". </span><br />
<span style="font-family: Arial;">Can you find their meaning?</span><br />
<br />
<span style="font-family: Arial;">:D</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com4tag:blogger.com,1999:blog-5015600567858883743.post-50177028294655914952010-04-10T15:41:00.000+08:002010-04-10T15:41:10.047+08:00Number of Answers | Common mistake<span style="font-family: Arial, Helvetica, sans-serif;">Maths can be tricky when you are not careful.</span><br />
<span style="font-family: Arial;">This is not to frighten you, though.</span><br />
<br />
<span style="font-family: Arial;">This post is just to remind you of the wonderful aspect of maths in covering all areas.</span><br />
<br />
<span style="font-family: Arial;">Below is an example of what I meant.</span><br />
<br />
<span style="font-family: Arial;">Let's take the quadratic eqaution solving as a starting point</span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">x<sup>2</sup> = 5x </span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">x = 5x / x = 5 (Answer)</span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">At first, this looks pretty fine. The answer, when substituted back, produces match of equation.</span><br />
<br />
<span style="font-family: Arial;">But this is actually not complete.</span><br />
<br />
<span style="font-family: Arial;">Those doing quadratic equation will know 2nd order (x<su>2</sup>) equation evaluates to 2 answsers.</span><br />
<span style="font-family: Arial;">The answers may be the same though.</span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">Now, if we approach it using another method, let's see the different.</span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">x<sup>2</sup> - 5x = 0</span><br />
<br />
<span style="font-family: Arial, Helvetica, sans-serif;">==> x (x - 5) = 0 , after factorising</span><br />
<br />
<span style="font-family: Arial;">==> x = 0 and (x - 5) = 0</span><br />
<span style="font-family: Arial;">==> x = 0 and x = 5</span><br />
<br />
<span style="font-family: Arial;">There are two answers now.</span><br />
<br />
<span style="font-family: Arial;">We had the x = 5 initially, but what about this new x = 0.</span><br />
<br />
<span style="font-family: Arial;">We have missed out on the x = 0 with the first mehtod. It looks OK then.</span><br />
<br />
<span style="font-family: Arial;">What happen?</span><br />
<span style="font-family: Arial;">It may be due to lack of experience handling this form of maths question.</span><br />
<br />
<span style="font-family: Arial;">The concept in solving quadratic equation is actually not limited to second order.</span><br />
<span style="font-family: Arial;">The hidden message is depending on the order, the number of answers will follow suit.</span><br />
<br />
<span style="font-family: Arial;">What I meant is :</span><br />
<span style="font-family: Arial;">2nd order gives 2 answers,</span><br />
<span style="font-family: Arial;">3rd order gives 3 answers,</span><br />
<span style="font-family: Arial;">4th order gives 4 answers, etc.</span><br />
<br />
<span style="font-family: Arial;">It is this verry message that maths learner should capture. Otherwise you will be tricked to give only one answer which leads you to "mistakes" of being incomplete.</span><br />
<br />
<span style="font-family: Arial;">I agree that this is tricky, but within reasonable argument.</span><br />
<span style="font-family: Arial;">If a student practice hard (and smart), he will not fall prey to this type of simple math problem.</span><br />
<br />
<span style="font-family: Arial;">Do not get con again.</span><br />
<br />
<span style="font-family: Arial;">Enjoy maths. It's fun and interesting.</span><br />
<br />
<span style="font-family: Arial;">:D</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com1tag:blogger.com,1999:blog-5015600567858883743.post-33144671410382929492010-04-05T23:34:00.001+08:002010-04-05T23:35:48.094+08:00Simultaneous Equations | Re-write equations*<br />
<span style="font-family: Arial, Helvetica, sans-serif;">Simple simultaneous equation problem comes as 2 straight forward mathematical expressions.</span><br />
<br />
<span style="font-family: Arial;">Example 1:</span><br />
<br />
<span style="font-family: Arial;">3x + y = 4</span><br />
<span style="font-family: Arial;">x + 2y = 3</span><br />
<br />
<span style="font-family: Arial;">But some may come in odd expressions (since life is always the case, which makes learning maths more exciting!)</span><br />
<br />
<span style="font-family: Arial;">Example 2:</span><br />
<br />
<span style="font-family: Arial;">(12 - x)(1 + y) = 15</span><br />
<span style="font-family: Arial;">(8 - x) (1 + y) = -15</span><br />
<br />
<span style="font-family: Arial;">Here you will notice that the unknowns are biased towards one side.</span><br />
<br />
<span style="font-family: Arial;">Approach 1:</span><br />
<span style="font-family: Arial;">Multiply the 2 factors to get something like example 1.</span><br />
<span style="font-family: Arial;">Using elimination method, remove one of the unknown.</span><br />
<span style="font-family: Arial;">Solve for the only one unknown left.</span><br />
<span style="font-family: Arial;">Using the result found, compute the other unknown.</span><br />
<br />
<span style="font-family: Arial;">Approach 2: (The focus of this post)</span><br />
<span style="font-family: Arial;"><strong><span style="color: red;">Re-write</span></strong> the expression to make it look <span style="color: red;"><strong>simpler</strong></span>.</span><br />
<br />
<span style="font-family: Arial;">The example 2 can be re-written into below <strong><span style="color: red;">simpler form</span></strong>:</span><br />
<span style="font-family: Arial;">(12 - x) = 15 / (1 + y) ===> (A)</span><br />
<span style="font-family: Arial;">(8 - x) = -15 / (1 + y) ===> (B)</span><br />
<br />
<span style="font-family: Arial;">Equation (B) can then be seen to be the negative of equation (A).</span><br />
<br />
<span style="font-family: Arial;">With the re-writing, we will be visually aided to see another form, a simpler one, of the simultaneous equations.</span><br />
<br />
<span style="font-family: Arial;">Moving forward with the solution...</span><br />
<br />
<span style="font-family: Arial;">12 - x = -(8 - x) = -8 + x</span><br />
<span style="font-family: Arial;">12 + 8 = 2x</span><br />
<span style="font-family: Arial;">x = 20 / 2 = 10 (ANSWER)</span><br />
<span style="font-family: Arial;">Putting x = 10 back into either equation (A) or (B),</span><br />
<br />
<span style="font-family: Arial;">We will get (12 - 10) (1 + y) = 15, if we select equation (A)</span><br />
<span style="font-family: Arial;">1 + y = 15 / 2 = 7.5</span><br />
<span style="font-family: Arial;">y = 7.5 - 1 = 6.5 (ANSWER)</span><br />
<br />
<span style="font-family: Arial;">The solution is not the issue in this post.</span><br />
<br />
<span style="font-family: Arial;">The key message here is the technique of "re-writing" the equations to reveal the simplicity of the question.</span><br />
<br />
<span style="font-family: Arial;">Maths is not that difficult if you look and think to make it easy.</span><br />
<br />
<span style="font-family: Arial;">Cheers!</span><br />
<span style="font-family: Arial;">.</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com0tag:blogger.com,1999:blog-5015600567858883743.post-77486532292780924542010-04-02T22:01:00.000+08:002010-04-02T22:01:35.154+08:00Tips on Avoiding Mistakes (Unit writing).<br />
<span style="font-family: Arial, Helvetica, sans-serif;">Maths involves many traps.</span><br />
<br />
<span style="font-family: Arial;">Any one of this traps will make the solution looks odd or even to the extent of wrong answer.</span><br />
<br />
<span style="font-family: Arial;">What are this traps ?</span><br />
<br />
<span style="font-family: Arial;">Mathematical operators, symbols, units, transferring of numbers, size of the written symbols, decimal points are some of the examples of traps contributing to the error.</span><br />
<br />
<span style="font-family: Arial;">Here I would like to mention about "unit".</span><br />
<br />
<span style="font-family: Arial;">In maths, calculation of items are aplenty. One of them is the study of <strong><span style="color: #cc0000;">speed</span></strong>.</span><br />
<br />
<span style="font-family: Arial;">In the topic of speed, students are dealing with three basic elements.</span><br />
<span style="font-family: Arial;">They are the distance, time and their ratio (speed).</span><br />
<br />
<span style="font-family: Arial;">All these three elements have different units all to themselve.</span><br />
<br />
<span style="font-family: Arial;">Distance == metre</span><br />
<span style="font-family: Arial;">Time == second</span><br />
<span style="font-family: Arial;">Speed == metre / sec</span><br />
<br />
<span style="font-family: Arial;">There are variations of the above.</span><br />
<span style="font-family: Arial;">km, mintues, hours, km / h, m / min, etc</span><br />
<br />
<span style="font-family: Arial;">Do you now see the danger?</span><br />
<br />
<span style="font-family: Arial;">If you are dealing with so many units in one maths question, what are the chance of making mistakes?</span><br />
<span style="font-family: Arial;">If you are careful, the chance is low, but it does not mean zero.</span><br />
<br />
<span style="font-family: Arial;">You still have to be careful.</span><br />
<br />
<span style="font-family: Arial;">How to avoid having mistakes due to this undesired slip?</span><br />
<br />
<span style="font-family: Arial;">One tip is to write down the units in the working steps.</span><br />
<span style="font-family: Arial;">Do not leave the numerical answer (in the working) without any unit indicated.</span><br />
<br />
<span style="font-family: Arial;"><span style="color: red;">Make clear the item of interest</span>, whether it is distance or time <span style="color: red;">by reflecting the unit</span> besides the number.</span><br />
<br />
<span style="font-family: Arial;">Example: 5 km, 40 sec.</span><br />
<br />
<span style="font-family: Arial;">A complete maths example will push the message across, thus ....</span><br />
<br />
<span style="font-family: Arial;">Example :</span><br />
<span style="font-family: Arial;">Alan travelled at a speed of 60 km / h for 2 h. After that, he slowed down by 20 km / h and travelled the last quarter of the journey at this new speed. How long did he take to travel?</span><br />
<br />
<span style="font-family: Arial;">Working:</span><br />
<span style="font-family: Arial;">60 x20 = 120</span><br />
<span style="font-family: Arial;">120 / 3 = 40</span><br />
<span style="font-family: Arial;">60 - 20 = 40</span><br />
<span style="font-family: Arial;">40 / 40 = 1</span><br />
<span style="font-family: Arial;">2 + 1 = 3</span><br />
<span style="font-family: Arial;">Answer: 3 hrs.</span><br />
<br />
<span style="font-family: Arial;">What is your comment on the working?</span><br />
<br />
<span style="font-family: Arial;">I personally feel uncomfortable. What about you?</span><br />
<br />
<span style="font-family: Arial;">The danger in that sort of working is the lack of showing the actual item in the calculation.</span><br />
<span style="font-family: Arial;">It does not allow a good way for checking after completing the worksheet (if many maths problems are within).</span><br />
<br />
<span style="font-family: Arial;">Clearly writing the units will, at least, make checking later an easier task.</span><br />
<br />
<span style="font-family: Arial;">It also allows the marker (teacher) a clearer picture instead of guessing what you intend to show.</span><br />
<br />
<span style="font-family: Arial;">Along the way, during the working, you will also have a lesser chance of getting confuse as the items are listed with the proper message (through the units).</span><br />
<br />
<span style="font-family: Arial;">So are you convince proper unit presentation is worth the while?</span><br />
<br />
<span style="font-family: Arial;">A pointer for your thoughts.....</span><br />
<br />
<span style="font-family: Arial;">Cheers :-D</span><br />
<br />
<span style="font-family: Arial;">.</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com0tag:blogger.com,1999:blog-5015600567858883743.post-41530839221137808352010-03-30T23:04:00.001+08:002010-03-31T16:29:33.433+08:00Using Equation To Create A Square Graphically'<br />
<span style="font-family: Arial, Helvetica, sans-serif;">While studying maths, I have been exposed to equation that forms a circle.</span><br />
<br />
<span style="font-family: Arial;">We know that x^2 + y^2 = 1 creates a circle.</span><br />
<br />
<span style="font-family: Arial;">But I have been wondering what is an equation to form a square.</span><br />
<br />
<span style="font-family: Arial;">I had tried a few mathematical expressions till today.</span><br />
<br />
<span style="font-family: Arial;">And finally I found the interesting and mysteries equation. </span><br />
<span style="font-family: Arial, Helvetica, sans-serif;">It utilises the same concept as the circle except that hyperbolic trigonometry is applied.</span><br />
<br />
<span style="font-family: Arial;">Below is a graph plotted with that equation.</span><br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbX2fR4IEdI9SdUgumUbuPHdHeaHQXacN0snrOkW-yWAJUNthlEGWogciQFlRX_vANfin-X9jsizjUr-cWqLklOIJqFetcJsglg4g9CiHzd7XtLMtK2f1o7B1xsNG7S8I_pqjzW3U1oZsE/s1600/Square_graph.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="475" nt="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbX2fR4IEdI9SdUgumUbuPHdHeaHQXacN0snrOkW-yWAJUNthlEGWogciQFlRX_vANfin-X9jsizjUr-cWqLklOIJqFetcJsglg4g9CiHzd7XtLMtK2f1o7B1xsNG7S8I_pqjzW3U1oZsE/s640/Square_graph.jpg" width="640" /></a></div><br />
<span style="font-family: Arial;">The corners are rounded though. Any one has any try with a more sharper corner?</span><br />
<br />
<span style="font-family: Arial;">Graph is a wonderful tool as it can present results visually with one view.</span><br />
<span style="font-family: Arial;">Appreciating maths and using it appropriately can reduce many complex problems.</span><br />
<br />
<span style="font-family: Arial;">Maths is interesting.</span><br />
<br />
.EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com1tag:blogger.com,1999:blog-5015600567858883743.post-53250982255583863922010-03-27T13:25:00.001+08:002010-03-27T13:27:25.201+08:00Area Displacement Theory.<br />
<span style="font-family: Arial, Helvetica, sans-serif;">Maths is not all about calculation. </span><br />
<br />
<span style="font-family: Arial;">There are always more to it than meet the eyes.</span><br />
<br />
<span style="font-family: Arial;">This is especially true when you are doing geometrical questions where you are involved with area, perimeter and so on.</span><br />
<br />
<span style="font-family: Arial;">Displacement theory or its equivalent is always done without the knowledge of many people.</span><br />
<br />
<span style="font-family: Arial;">What is this theory about?</span><br />
<br />
<span style="font-family: Arial;">Let's look at one example below.</span><br />
<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMfsTuxQPzylw3skz8S_4AkRPF7dP6jcIkkhATMgXHkdJOVtrYjCQP6cW3-_a-oB1bwWut-C11eeFQeZxAG04E5TbjURhMLm75eRxqu-YHfyRb3GH2jQ7mDCJ_s7Jk7ws-lFsJUYd9I3L-/s1600/path+maths.jpg" imageanchor="1" style="clear: left; cssfloat: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="170" nt="true" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMfsTuxQPzylw3skz8S_4AkRPF7dP6jcIkkhATMgXHkdJOVtrYjCQP6cW3-_a-oB1bwWut-C11eeFQeZxAG04E5TbjURhMLm75eRxqu-YHfyRb3GH2jQ7mDCJ_s7Jk7ws-lFsJUYd9I3L-/s320/path+maths.jpg" width="320" /></a></div><br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
<span style="font-family: Arial;">In the diagram above, you will see a path (white coloured) going across a blue platform.</span><br />
<span style="font-family: Arial;">If you are asked to find the area of this path, what can you do to obtain this area?</span><br />
<br />
<span style="font-family: Arial;">If no data of dimension is given, it is definitely not possible.</span><br />
<br />
<span style="font-family: Arial;">Now if the width of the path and the vertical length of the blue platform is given, can you compute the answer?</span><br />
<br />
<span style="font-family: Arial;">Again , this need a bit of thinking.</span><br />
<br />
<span style="font-family: Arial;">Displacement theory kicks in here. Look at the diagram on the right.</span><br />
<br />
<span style="font-family: Arial;">It is the displaced or closed up portion of the blue platform that does the trick.</span><br />
<span style="font-family: Arial;">Here you will notice the dashed line forming a white rectangluar area on the right-most side of the white blue platform.</span><br />
<br />
<span style="font-family: Arial;">Are you able to find the area of this white rectangular piece?</span><br />
<span style="font-family: Arial;">The width of this rectangle piece is ACTUAL the width of the white path!</span><br />
<br />
<span style="font-family: Arial;">You should now be able to calculate the area of this rectangular piece since the path width and length of the rectangular block is known or deduced now.</span><br />
<br />
<span style="font-family: Arial;">How this is possibe is through the "hidden" clue or step of closing up the path revealing the simpler rectangular area that any decent maths student can calculate.</span><br />
<br />
<span style="font-family: Arial;">Hence, maths is wonderful in that it tests you not only about applcations of maths tools, but your other "intelligence".</span><br />
<br />
<span style="font-family: Arial;">Having known displacement theory here, I believe you are really for the <a href="http://mathsisinteresting.blogspot.com/2010/03/math-challenge-23.html">Math Challenge 23</a>.</span><br />
<span style="font-family: Arial;">Go there and answer the question, and <strong><span style="color: #cc0000;">be quick</span></strong> before others grap the position one ... </span><br />
<br />
<span style="font-family: Arial;">:-D</span><br />
<span style="font-family: Arial;">.</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com0tag:blogger.com,1999:blog-5015600567858883743.post-29994410900732689512010-03-24T15:57:00.002+08:002010-03-28T08:22:02.441+08:00Percentage Increase in Area.<br />
<span style="font-family: Arial, Helvetica, sans-serif;">Is there any formula or maths expression showing the ncrease in area when its length and its breadth are increase by m% ?</span><br />
<br />
<span style="font-family: Arial;">If you cannot find one, it does not matter. You can easily derive one!</span><br />
<br />
<span style="font-family: Arial;">Let us work on this and show the others how simple maths can help us solve daily issue.</span><br />
<br />
<span style="font-family: Arial;">Let the length be x and breadth be y.</span><br />
<br />
<span style="font-family: Arial;">If x and y increase by m%, </span><br />
<span style="font-family: Arial;">length becomes x + x(m/100), and breadth becomes y + y(m/100).</span><br />
<br />
<span style="font-family: Arial;">Area is length x breadth.</span><br />
<br />
<span style="font-family: Arial;">Thus new area becomes [ x + x(m/100)] [ y + y(m/100)]</span><br />
<br />
<span style="font-family: Arial;">This gives us an area of xy + (m/100)xy + (m/100)xy + (m/100)(m/100)xy.</span><br />
<br />
<span style="font-family: Arial;">From the above maths expression, we can deduce that increase in area is:</span><br />
<span style="font-family: Arial;"><strong>2 x m% + (m% x m%)/100</strong></span><br />
<br />
<span style="font-family: Arial;">Example with numbers will convince readers better, therefore ......</span><br />
<br />
<span style="font-family: Arial;"><u>Example</u> </span><br />
<br />
<span style="font-family: Arial;">If the increase in perimeter is 10%, what is the increase in area?</span><br />
<br />
<span style="font-family: Arial;">Answer is 2 (10%) + (10% x 10%) /100 = 20% + 1% = 21%</span><br />
<br />
<span style="font-family: Arial;">Easy isn't it? </span><br />
<br />
<span style="font-family: Arial;">For other post related to this concept in percentage increase, see the post <a href="http://mathsisinteresting.blogspot.com/2010/03/percentage-increase-in-perimeter.html">Percentage Increase in Perimeter.</a></span><br />
<br />
<span style="font-family: Arial;">Maths is interesting. </span><br />
<br />
<span style="font-family: Arial;">;-D</span><br />
<br />
<span style="font-family: Arial;">.</span>EeHaihttp://www.blogger.com/profile/02489802104300809304noreply@blogger.com1