IB Maths HL & SL / Analysis: Last blog entries
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InThinking IB Maths HL & SL / Analysis: www.thinkib.net/mathhlsl2019 InThinking Educational Consultants. All rights reserved.Probability - Complement Approach
https://www.thinkib.net/mathhlsl/blog/29825/probability-complement-approach
Sun, 02 Jun 2019 00:00:00 +0000<p> <img alt="" src="files/mathhlsl/files/Blog%20entries/dice-in-hand_img.jpg" style="float: right; width: 330px; height: 188px;" /></p><div class="blueBg"><p><u><strong>Question</strong></u>: Two fair six-sided dice are rolled. What is the probability that at least one of the dice shows a five facing up?</p></div><p>Three different solutions are given below. Let <em>X</em> be the random variable representing the # of fives.</p><p><img alt="" height="234" src="files/mathhlsl/files/Blog%20entries/table_img.jpg" style="float: left;" width="229" /><br /></p><p><u><strong>Method 1</strong></u>: <em>Analysing the sample space</em></p><p>“At least one five” means one or more fives – or specifically for this question – one or two fives. From the table (left) depicting the sample space, it can be seen that there are ten outcomes with one five and one outcome with two fives (shown in red). Therefore, the probability of getting at least one five when rolling a die twice is <span class="math-tex">\({\rm{P}}\left( {X \ge 1} \right) = \frac{{11}}{{36}}\)</span>.</p><p><u><strong>Method 2</strong></u>: <em>Adding probabilities of all the outcomes that make up the event</em></p><p>The ‘event’ of obtaining at least one five consists of three different outcomes: (1) a 5 on the 1st die, and not a 5 on the 2nd die, (2) not a 5 on the 1st die, and a 5 on the 2nd die, and (3) a 5 on both dice [note: 5´ means “not 5”]</p><p><span class="math-tex">\({\rm{P}}\left( {5,5'} \right) = \frac{1}{6} \cdot \frac{5}{6} = \frac{5}{{36}}\)</span> <span class="math-tex">\({\rm{P}}\left( {5',5} \right) = \frac{5}{6} \cdot \frac{1}{6} = \frac{5}{{36}}\)</span> <span class="math-tex">\({\rm{P}}\left( {5,5} \right) = \frac{1}{6} \cdot \frac{1}{6} = \frac{1}{{36}}\)</span></p><p>From these three probabilities, it follows that <span class="math-tex">\({\rm{P}}\left( {X \ge 1} \right) = \frac{5}{{36}} + \frac{5}{{36}} + \frac{1}{{36}} = \frac{{11}}{{36}}\)</span></p><p><u><strong>Method 3</strong></u>: <em>Complement approach</em></p><p><span class="math-tex">\({\rm{P}}\left( {X \ge 1} \right) + {\rm{P}}\left( {X = 0} \right) = 1\;\;\;\; \Rightarrow \;\;\;\;{\rm{P}}\left( {X \ge 1} \right) = 1 - {\rm{P}}\left( {X = 0} \right)\)</span></p><p><span class="math-tex">\({\rm{P}}\left( {X = 0} \right)\)</span> is the probability of not getting a 5 on either of the dice.</p><p>Therefore, <span class="math-tex">\({\rm{P}}\left( {X \ge 1} \right) = 1 - \frac{5}{6} \cdot \frac{5}{6} = 1 - \frac{{25}}{{36}} = \frac{{11}}{{36}}\)</span></p><p>Clearly, the complement approach is the most efficient way of computing this particular probability.</p><p>Students should be presented with several examples that illustrate the usefulness of a <strong>complement approach</strong> in a variety of probability questions. See the page <a href="mathhlsl/page/19805/the-birthday-problem-more"><strong>The Birthday Problem & More</strong></a> in my <strong>Exploration (IA) Ideas</strong> section.</p>https://www.thinkib.net/mathhlsl/blog/29825/probability-complement-approach#1559433600Worked Solutions - HL Paper 1 Exam
https://www.thinkib.net/mathhlsl/blog/29780/worked-solutions-hl-paper-1-exam
Wed, 22 May 2019 00:00:00 +0000<p><img alt="" src="files/mathhlsl/files/Blog%20entries/wrkd-solns-img0.jpg" style="float: right; width: 300px; height: 170px;" /><br />The May 2019 IB examination session is now finished. Teachers are eager to scrutinize an exam after it has been made available 24 hours after students have taken it. This usually leads to some interesting discussion amongst teachers about specific exam questions and their solutions. Some years I manage to summon enough time and energy to write my own set of complete <strong>worked solutions</strong> for one or more of the HL or SL exams. Although having discussions with other teachers about the exam questions is often instructive and worthwhile, if I do compose a set of worked solutions, I do it for primarily two reasons: (1) To satisfy my own curiosity. In my opinion, the only way to get a genuine ‘feel’ for an exam is to carefully work through all the questions yourself. (2) To assemble a high-quality assignment for my 1<sup>st</sup> year students. I am usually teaching integral calculus to my 1<sup>st</sup> year Maths HL or SL students about this time of the school year; and the only syllabus topic that has not yet been covered to any extent is Probability & Statistics. Usually, the <strong>Paper 1 exam</strong> will contain fewer questions on probability and/or statistics. Thus, most (if not all) of the questions on the Paper 1 exam can be attempted by my 1<sup>st</sup> year students. After removing any questions or parts of questions on the Paper 1 exam that are on syllabus items that have not been covered yet, I distribute the set of Paper 1 questions to my 1<sup>st</sup> year students and give them a few days to work on them. I then give them my worked solutions and we discuss. It’s a nice review assignment for students not long before the summer holiday break between the 1<sup>st</sup> and 2<sup>nd</sup> years. Also, it’s the first time that I share a complete IB maths exams with my 1<sup>st</sup> year students, so this assignment provides some useful insights into what an IB maths exam is really like.</p><p>You can view – and download – my worked solutions for the <span >HL Paper 1 exam (2019 TZ2) </span><a href="https://www.thinkib.net/mathhlsl/page/29871/worked-solutions-2019-hl-p1-exam-tz2" target="_blank"><strong><span >here</span></strong></a><span >.</span> Any feedback is appreciated.</p>https://www.thinkib.net/mathhlsl/blog/29780/worked-solutions-hl-paper-1-exam#1558483200Euler's method on the TI-Nspire
https://www.thinkib.net/mathhlsl/blog/28881/eulers-method-on-the-ti-nspire
Sun, 17 Mar 2019 00:00:00 +0000]]>Euler's method on the TI-Nspirehttps://www.thinkib.net/cache/blog-thumbs/11/28881-1552864135-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/28881/eulers-method-on-the-ti-nspire
<p>As they say, a picture (or a video, in this case) is worth a thousand words. The video here shows one way to use the spreadsheet on the TI-Nspire to perform Euler's method to find the approximate solution to a differential equation.<img src="https://www.thinkib.net/cache/blog-thumbs/11/28881-1552864135-thinkib.jpg" alt="Euler's method on the TI-Nspire" /><br /><br /></p><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="335" src="https://www.youtube.com/embed/Ptu6WiKRsW8?rel=0" width="595"></iframe>https://www.thinkib.net/mathhlsl/blog/28881/eulers-method-on-the-ti-nspire#1552780800what is path of midpoint of a falling ladder?
https://www.thinkib.net/mathhlsl/blog/28092/what-is-path-of-midpoint-of-a-falling-ladder
Sat, 24 Nov 2018 00:00:00 +0000]]>what is path of midpoint of a falling ladder?https://www.thinkib.net/cache/blog-thumbs/11/28092-1543065158-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/28092/what-is-path-of-midpoint-of-a-falling-ladder
<h3><span >Visualizing ... in your head</span></h3><p><img src="https://www.thinkib.net/cache/blog-thumbs/11/28092-1543065158-thinkib.jpg" alt="what is path of midpoint of a falling ladder?" /><br /><br /></p></section></div><p>The <strong>Geogebra applet</strong> below can be used to convince students of the correct answer. Thiis is best done with trace turned on for the midpoint. With the trace turned off, you can also use the applet to present the question. Even when seeing the movement of the line segment some students will still struggle to correctly describe the path of the midpoint.</p><p><iframe height="480px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/zb9t57vj/width/666/height/480/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="path of midpoint of falling ladder" width="666px"></iframe></p>https://www.thinkib.net/mathhlsl/blog/28092/what-is-path-of-midpoint-of-a-falling-ladder#1543017600Effective GDC Use #3
https://www.thinkib.net/mathhlsl/blog/28008/effective-gdc-use-3
Sat, 17 Nov 2018 00:00:00 +0000]]>Effective GDC Use #3https://www.thinkib.net/cache/blog-thumbs/11/28008-1542499022-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/28008/effective-gdc-use-3
<p>Concerning effective GDC use, I try to avoid giving my students the impression that mindlessly pushing buttons on a GDC will allow them to successfully answer an exam question. Although there may be exam questions where this occurs (more on this in future blog posts), I endeavour to give my students questions which combine smart use of their GDC with some clear thinking.</p><p>Early in Maths SL or HL, students will be expected to identify the domain and range of a given function. Although not inherently difficult, it’s possible to compose questions about the domain and range of a function in such a way that wise use of a GDC is necessary but not sufficient to obtain the correct answers. A student will also need to apply some number sense and knowledge about certain functions. One of the habits I try to engrain upon my students with such questions is to make an educated guess about the domain and range of a function <strong><em>before</em></strong> graphing the function on a GDC, and then make use of a GDC to confirm / modify the determination of the function’s domain and range. Although not really exam-like, I like these kind of questions because they can be used early in either Maths SL or HL to promote thoughtful application of a GDC. Here are two examples.</p><div class="blueBg"><p><u>Question</u>: Determine the domain and range of each function.</p><p>(a) <span class="math-tex">\(f\left( x \right) = \frac{1}{{{x^2} + 3x - 10}}\)</span> (b) <span class="math-tex">\(g\left( x \right) = \sqrt {\frac{{8x - 4}}{{x - 3}}} \)</span></p></div><div class="yellowBg"><p><u>Solution</u>:</p><p>(a) A student should immediately factorise the denominator to identify the vertical asymptotes for the graph of the function.</p><p><span class="math-tex">\(f\left( x \right) = \frac{1}{{{x^2} + 3x - 10}} = \frac{1}{{\left( {x + 5} \right)\left( {x - 2} \right)}}\)</span></p><p>Thus, the graph of <em>f</em> has vertical asymptotes (and discontinuities) at <span class="math-tex">\(x = - 5\)</span> and at <span class="math-tex">\(x = 2\)</span>. The domain is certain to be <span class="math-tex">\(x \in \mathbb{R},\;x \ne - 5,\;x \ne 2\)</span>. Determining the range is much less clear. One can figure out that the graph of <em>f</em> will have a horizontal asymptote <span class="math-tex">\(y = 0\)</span> (<em>x</em>-axis) because as <em>x</em> becomes either a very large positive or very large negative number the <span class="math-tex">\({x^2}\)</span> term will dominate and cause the denominator <span class="math-tex">\({x^2} + 3x - 10\)</span> to become a large positive number. Thus, as <span class="math-tex">\(x \to + \,\infty \)</span> or as <span class="math-tex">\(x \to - \,\infty \)</span> the value of <span class="math-tex">\(\frac{1}{{{x^2} + 3x - 10}}\)</span> approaches zero from positive values meaning that the graph of <em>f</em> will be asymptotic to the <em>x</em>-axis from above (i.e. the <em>end behaviour</em> of <em>f</em>). So, an educated guess is that the range is <span class="math-tex">\(y \in \mathbb{R},\;y \ne 0\)</span>.</p><p>Graphing on a GDC (see images below) confirms the domain as stated, but the ‘gap’ in <em>y</em> values is more than just zero. The downward-opening branch of the graph of <em>f</em> appears to have a maximum value less than zero. Analyzing the graph on the GDC reveals that this downward-opening branch of the graph has a maximum at <span class="math-tex">\(\left( { - 1.5, - \,0.08163} \right)\)</span>. It’s clear that the <em>x</em>-coordinate of this local maximum point is exactly <span class="math-tex">\( - 1.5\)</span> but what about the <em>y</em>-coordinate? Finding it exactly requires some effort. Somewhat surprisingly, it’s more of an effort on the TI-Nspire than the TI-84.</p><p><img src="https://www.thinkib.net/cache/blog-thumbs/11/28008-1542499022-thinkib.jpg" alt="Effective GDC Use #3" /><br /><br />A graph of <em>g</em> confirms the conjecture for the domain but appears to indicate there is a horizontal asymptote with an equation somewhere around <span class="math-tex">\(y = 3\)</span>. As with the function in part (a), considering the end behaviour of the function will lead to an exact equation for the horizontal asymptote. When <span class="math-tex">\(x \to + \,\infty \)</span> or <span class="math-tex">\(x \to - \,\infty \)</span>, the <em>x</em> terms in the numerator and denominator of <span class="math-tex">\(\frac{{8x - 4}}{{x - 3}}\)</span> will dominate so that the value of <span class="math-tex">\(\frac{{8x - 4}}{{x - 3}}\)</span> will come ever closer to <span class="math-tex">\(\frac{8}{1}\)</span>. Thus, the equation of the horizontal asymptote is <span class="math-tex">\(y = \sqrt 8 = 2\sqrt 2 \)</span>; also indicating where there will be a ‘gap’ in the range. Therefore, the correct range for function <em>g</em> is <span class="math-tex">\(0 \le y < 2\sqrt 2 ,\;y > 2\sqrt 2 \)</span>.</p></div>https://www.thinkib.net/mathhlsl/blog/28008/effective-gdc-use-3#1542412800P.o.t.D.- 301 problems
https://www.thinkib.net/mathhlsl/blog/27480/potd-301-problems
Mon, 01 Oct 2018 00:00:00 +0000<p><img alt="" src="files/mathhlsl/files/Blog%20entries/eqn_solver_img1.jpg" style="float: right; width: 227px; height: 119px;" />This is just a short entry to highlight the fact that I have now managed to compose more than <strong>300 problems</strong> in my <a href="https://www.thinkib.net/mathhlsl/page/23399/problem-of-the-day" target="_blank"><strong><span >Problem of the Day</span></strong></a> (P.o.t.D.) lists - one list for an <a href="mathhlsl/page/23398/sl-problem-of-the-day" target="_blank"><strong>SL Problem of the Day</strong></a>, and another list for an <a href="https://www.thinkib.net/mathhlsl/page/23396/hl-problem-of-the-day" target="_blank"><strong>HL Problem of the Day</strong></a>.</p><p>I am trying to think up problems that will continue to be relevant for the new IB mathematics courses which will start being taught in August 2019 - <strong>Analysis & Approaches</strong> and <strong>Applications & Interpretation</strong>. The Applications & Interpretation course (which I refer to as the 'Applications' course) will have a greater emphasis on the integration of technology in the learning and 'doing' of mathematics. One fact that underlines this is that all of the exams for the Applications course (2 exams for SL, 3 exams for HL) will allow (require) a student to have a graphic display calculator (GDC).</p><p>So, for my 301st problem (problem #151 in the SL list) I decided to compose a problem that would feature effective use of a GDC; nothing complicated, just making use of one of the basic tools on any GDC - but one that many students do not fully utilize. Here is the problem.</p><hr /><div class="blueBg"><h4>Problem of the Day #151 (SL)</h4><p>For all of the adults in the Netherlands, the ratio of the number of people who speak more than two languages to the number of people who speak two or fewer languages is <em>m</em> to 7 (or <span class="math-tex">\(\frac{m}{7}\)</span>), where <span class="math-tex">\(m \in {\mathbb{Z}^ + }\)</span>. It is known that if eight Dutch adults are chosen at random, then the probability that at least one of them speaks more than two languages is approximately 0.9424. Find the value of <em>m</em>.</p></div><div class="yellowBg"><p><u>Solution</u>:</p><p>There are eight independent ‘trials’ where only one of two results can be obtained – i.e. a person either speaks more than two languages or does <u>not</u> speak more than two languages. If <em>X</em> represents the random variable for the “the number of adults who speak more than two languages”, then <em>X</em> has a <strong>binomial distribution</strong> (assuming each selection of a Dutch adult is an independent event).</p><p>The fraction of Dutch adults that speak two or more languages is <span class="math-tex">\(\frac{m}{{m + 7}}\)</span>. Thus, the probability that a randomly selected adult speaks more than two languages is <span class="math-tex">\(\frac{m}{{m + 7}}\)</span>, and the probability that a randomly selected adult does <strong>not</strong> speak more than two languages is <span class="math-tex">\(1 - \frac{m}{{m + 7}} = \frac{{m + 7}}{{m + 7}} - \frac{m}{{m + 7}} = \frac{7}{{m + 7}}\)</span>.</p><p>It is given that <span class="math-tex">\({\rm{P}}\left( {X \ge 1} \right) = 0.9424\)</span>. </p><p>Either there is at least one person that speaks more than two languages or there is no one that speaks more than two languages. Hence, <span class="math-tex">\({\rm{P}}\left( {X \ge 1} \right) + {\rm{P}}\left( {X = 0} \right) = 1\;\;\;\; \Rightarrow \;\;\;\;{\rm{P}}\left( {X \ge 1} \right) = 1 - {\rm{P}}\left( {X = 0} \right)\)</span>.</p><p>Thus, <span class="math-tex">\({\rm{P}}\left( {X \ge 1} \right) = 1 - {\rm{P}}\left( {X = 0} \right) = 0.9424\)</span>.</p><p>Need to solve for <em>m</em> in the equation <span class="math-tex">\(1 - {\left( {\frac{7}{{m + 7}}} \right)^8} = 0.9424\)</span></p><p>This can be done with the equation solver on a GDC.</p><p><img alt="" src="files/mathhlsl/files/Blog%20entries/eqn_solver_img3.jpg" style="width: 360px; height: 65px;" /><br /></p><p>Therefore, <span class="math-tex">\(m = 3\)</span>.</p></div>https://www.thinkib.net/mathhlsl/blog/27480/potd-301-problems#1538352000Effective GDC Use #2
https://www.thinkib.net/mathhlsl/blog/27409/effective-gdc-use-2
Thu, 20 Sep 2018 00:00:00 +0000<h3><span >Checking tool</span></h3><p>Continuing my discussion on promoting effective and wise use of a calculator (GDC) by our students, I wish to highlight what I consider maybe the first and simplest piece of GDC advice to students: “When possible, use your GDC as a <strong>checking tool</strong>.” When a Paper 2 question (GDC allowed) requires an <strong>exact</strong> answer, it usually means that the solution needs to be carried out manually / analytically – and a GDC will not be helpful in obtaining the answer. However, in such a situation a GDC can often be used to quickly and easily check whether the exact answer obtained is correct.</p><p>For example, consider the following exam-like (Paper 2) question.</p><div class="blueBg" contenteditable="false"><div contenteditable="true"><p>Consider the region R bounded by the graph of <span class="math-tex">\(y = x - \frac{1}{x}\)</span>, the <em>x</em>-axis and the line <img height="19" src="file:///C:/Users/TIMGAR~1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.png" width="37" />.</p><p>(a) Find the <strong>exact</strong> area of the region R.</p><p>(b) A solid of revolution is formed by rotating the region R about the <em>x</em>-axis. Find the <strong>exact</strong> volume of this solid.</p></div></div><div class="pinkBg" contenteditable="false"><div contenteditable="true"><p><strong>Solution</strong>:<img alt="" height="209" src="files/mathhlsl/files/Blog%20entries/gdc2_img1.jpg" style="float: right;" width="311" /></p><p>(a) Region R is shaded in the diagram at right.</p><p>area of R<span class="math-tex">\( = \int_1^2 {\left( {x - \frac{1}{x}} \right)dx = \left. {\frac{{{x^2}}}{2} - \ln x} \right]} _1^2\)</span></p><p><span class="math-tex">\( = \left( {\frac{{{2^2}}}{2} - \ln 2} \right) - \left( {\frac{{{1^2}}}{2} - \ln 1} \right) = \frac{3}{2} - \ln 2\)</span></p><p>(b) volume</p><p><span class="math-tex">\( = {\rm{\pi }}\int_1^2 {\left( {{{\left( {x - \frac{1}{x}} \right)}^2}} \right)dx = {\rm{\pi }}} \int_1^2 {\left( {{x^2} + \frac{1}{{{x^2}}} - 2} \right)dx = {\rm{\pi }}\left. {\left( {\frac{{{x^3}}}{3} - \frac{1}{x} - 2x} \right)} \right]} _1^2\)</span></p><p><span class="math-tex">\( = {\rm{\pi }}\left[ {\left( {\frac{{{2^3}}}{3} - \frac{1}{2} - 2 \cdot 2} \right) - \left( {\frac{{{1^3}}}{3} - \frac{1}{1} - 2 \cdot 1} \right)} \right] = {\rm{\pi }}\left[ { - \frac{{11}}{6} - \left( { - \frac{8}{3}} \right)} \right] = \frac{5}{6}{\rm{\pi unit}}{{\rm{s}}^3}\)</span></p></div></div><p>With a GDC allowed on this question, it’s certainly a good idea to check the answers for (a) and (b) on a GDC, as shown below.</p><p><img alt="" src="files/mathhlsl/files/Blog%20entries/gdc2_img2.jpg" style="width: 225px; height: 154px; margin-left: 10px; margin-right: 10px;" /> <img alt="" src="files/mathhlsl/files/Blog%20entries/gdc2_img3.jpg" style="width: 225px; height: 143px;" /><br /></p><p>One of the simplest and most powerful uses of a GDC is to use it to check the correctness, accuracy or reasonableness of an answer.</p>https://www.thinkib.net/mathhlsl/blog/27409/effective-gdc-use-2#1537401600Effective GDC Use #1
https://www.thinkib.net/mathhlsl/blog/27356/effective-gdc-use-1
Wed, 12 Sep 2018 00:00:00 +0000<h3><span >Approximate answers on Paper 2 exams</span></h3><h4><strong>HL Paper 2 exam-like question #4 </strong></h4><div class="blueBg" contenteditable="false"><div contenteditable="true"><p>The region <em>R</em> is enclosed by the graph of <span class="math-tex">\(y = {\rm{\pi }} - 3\arccos \left( x \right)\)</span>, the <em>y</em>-axis and the line <span class="math-tex">\(y = 2\)</span>.</p><p>(a) Write down a definite integral to represent the area of <em>R</em>. [4]</p><p>(b) Calculate the area of <em>R</em>. [2]</p></div></div><div class="pinkBg" contenteditable="false"><div contenteditable="true"><p><strong>Solution</strong>:</p><p>(a) area <span class="math-tex">\( = \int_0^{0.928...} {\left[ {2 - \left( {{\rm{\pi }} - 3\arccos x} \right)} \right]} \,dx\)</span></p><p>(b) area <span class="math-tex">\( \approx 1.89\)</span> <span class="math-tex">\({\rm{unit}}{{\rm{s}}^2}\)</span></p></div></div><p><img alt="" src="files/mathhlsl/images/_Site/Blogs/gdc-use/gdc1_img1b.jpg" style="float: left; width: 315px; height: 235px;" /><img alt="" src="files/mathhlsl/images/_Site/Blogs/gdc-use/gdc1_img2b.jpg" style="width: 315px; height: 235px;" /></p><hr /><br /><p>A question very similar to the one above was question #4 on the May 2017 HL Paper 2 exam (time zone 1). The comment for the question that appeared in the subsequent Maths HL Subject Report for that exam session stated the following:</p><p><em>Part (a) was attempted by most candidates taking various approaches. Limits were often incorrect and showed that many did not fully understand how to approach this question.</em></p><p><em>Part (b) was well done by those who had the correct expression in part (a); again, too many tried to integrate by hand instead of using their GDC.</em></p><p>It is worrying that many students did not do well on such a relatively straightforward question. Not doing well on part (a) reveals a lack of conceptual understanding; whereas, not doing well on part (b) – given part (a) was answered correctly – indicates a poor awareness on the appropriate and effective use of a graphical display calculator (GDC).</p><p>On a Paper 2 exam (and Paper 3 for HL), it is essential that a student continually ask themselves whether the use of a GDC could be beneficial. This is important for any mathematical work they are doing (homework, quizzes, unit tests) for which the use of a GDC is allowed. Students need to read questions carefully – and keep a special eye out for whether a question requires an exact answer or not. If an exact answer is not required, then an <strong>approximate answer</strong> – accurate to three significant figures unless stated otherwise – is perfectly acceptable.</p><p>For this question, part (b) did not ask for an exact answer, and it was only worth two marks. From the Subject Report comment, it is clear that a significant number of students attempted to integrate manually which would have involved finding <span class="math-tex">\(\int {\left[ {2 - \left( {{\rm{\pi }} - 3\arccos x} \right)} \right]} \,dx\)</span>. Although it is possible to find the anti-derivative of <span class="math-tex">\(\arccos x\)</span> by applying a combination of ‘integration by parts’ and <em>u</em>-substitution, this would require quite a bit of time and certainly a quantity of working that would represent far more than just two marks.</p><p>But, it’s not enough just to point out the poor judgement of students. It is likely that some students do not make effective use of their GDC when there is an opportunity to do so because they have not had adequate practice with questions that make them think hard about whether a GDC will be helpful or not. In the same Subject Report, in the Paper 2 section <em>Recommendations and guidance for the teaching of future candidates</em>, it states:</p><p><em>Candidates need to recognize when they need to use algebra and calculus to gain marks and when using their GDC is a more effective and errorless way.</em></p><p>Hence, all of us teachers of IB Maths HL and SL need to strive to provide our students with regular opportunities (on homework, quizzes & tests) to develop skills in recognizing when the use of a GDC is – or is not – appropriate, efficient and wise.</p>https://www.thinkib.net/mathhlsl/blog/27356/effective-gdc-use-1#1536710400P.o.t.D.- 250 Problems
https://www.thinkib.net/mathhlsl/blog/26392/potd-250-problems
Mon, 07 May 2018 00:00:00 +0000]]>P.o.t.D.- 250 Problemshttps://www.thinkib.net/cache/blog-thumbs/11/26392-1525730306-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/26392/potd-250-problems
<p>I started writing problems for my <a href="mathhlsl/page/23399/problem-of-the-day" target="_blank"><span ><strong>Problem of the Day</strong></span></a> (P.o.t.D.) section on this site 15 months ago - and there are now 250 problems available - equally distributed between HL and SL. The distinction between a problem that I post on the <a href="http://www.thinkib.net/mathhlsl/page/23396/hl-problem-of-the-day" target="_blank"><span >HL problems</span></a> page and one that I post on the <a href="http://www.thinkib.net/mathhlsl/page/23398/sl-problem-of-the-day" target="_blank"><span >SL problems</span></a> page is certainly not well-defined. I try to compose problems that require some application of problem solving skills. Problems that I think are suitable for HL students usually involve more sophistication with regard to figuring out a successful strategy (not always just one method that works), and often a higher degree of rigour with regard to mathematical techniques. In a nutshell, the key differences between the Maths HL and Maths SL course is the much higher level of <strong>sophistication </strong>and <strong>rigour</strong> that exists in HL.</p><p><img src="https://www.thinkib.net/cache/blog-thumbs/11/26392-1525730306-thinkib.jpg" alt="P.o.t.D.- 250 Problems" /><br /><br /></p><p>This led me to consider finding the minimum distance between the graphs of two differet equations ... how about a circle and a parabola. And I thought it would be a nice problem in which a student would need to consider appropriate use of technology to help them solve it. Not only using technology in the solution, but also in assisting them in coming up with a strategy. It's not a super difficult question (with a GDC) but there is an important insight required (at least in the solution method I used) that many students may not come up with unless they see the conditions of the problem illustrated dynamically. For my <a href="http://www.thinkib.net/mathhlsl/page/23399/problem-of-the-day" target="_blank">Problems of the Day</a> for which a GDC is allowed (always stipulated in the instructions for each problem wheter a GDC is allowed or not), I encourage my students to consider using technology other than their GDC - for example, <strong>Geogebra</strong>. I like my students to gain some experience with <strong>Geogebra </strong>during the course, so that they feel confident enough with it so they can consider using it with their Exploration (IA). Below is a <strong>Geogebra applet</strong> that dynamically illustrates <a href="/files/mathhlsl/files/Assessment/prob-of-day/126_hl_problem_5_may_2018.pdf" target="_blank">HL P.o.t.D. #126</a> - and can help a student gain insight into devising a solution strategy. Can you? (solution is given on 2nd page of problem)</p><p><iframe height="492px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/m65Ed9qN/width/718/height/492/border/888888/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="parabola circle min distance" width="718px"></iframe></p>https://www.thinkib.net/mathhlsl/blog/26392/potd-250-problems#1525651200Some fun with primes ... and Geogebra
https://www.thinkib.net/mathhlsl/blog/25623/some-fun-with-primes-and-geogebra
Thu, 18 Jan 2018 00:00:00 +0000]]>Some fun with primes ... and Geogebrahttps://www.thinkib.net/cache/blog-thumbs/11/25623-1516316997-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/25623/some-fun-with-primes-and-geogebra
<p>Prime numbers are fascinating, mysterious and useful. The topic of prime numbers can pop up in a maths lesson at many different ages and ability levels. The announcement two weeks ago of a ‘new’ <a href="http://www.mersenne.org/primes/press/M77232917.html" target="_blank"><span ><strong>largest known prime number</strong></span></a> <span class="math-tex">\(\left( {{2^{77,232,917}} - 1} \right)\)</span> led to some interesting discussions in a couple of my classes the past couple of weeks. I have a poster in my classroom that lists all the prime numbers between 1 and 10,000 (there are 1229 of them) – and it gives some impression about the distribution of prime numbers. The 'density' of primes certainly seems to decrease as we move to ever larger positive integers. Students are intrigued by this. It might be an opportunity to mention the <strong>Prime Number Theorem</strong> (good <em>Numberphile </em>video <a href="http://www.youtube.com/watch?v=l8ezziaEeNE" target="_blank"><span >here</span></a> on it), but I think it might be a bit too much for most students. If time allows, I find it fun to give students some way to actively explore something about prime numbers, and how they are - or how they are not - distributed is very accessible given some easy-to-use maths software such as <strong>Geogebra</strong>.<img src="https://www.thinkib.net/cache/blog-thumbs/11/25623-1516316997-thinkib.jpg" alt="Some fun with primes ... and Geogebra" /><br /><br /> </p><p>download PDF file of the <a href="/files/mathhlsl/images/_Site/Blogs/prime-freq-dist/freq-dist-plot-of-primes-on-ggb.pdf" target="_blank"><span ><strong>11-step set of instructions</strong></span></a></p>https://www.thinkib.net/mathhlsl/blog/25623/some-fun-with-primes-and-geogebra#1516233600primitive Pythagorean triples
https://www.thinkib.net/mathhlsl/blog/25433/primitive-pythagorean-triples
Sat, 30 Dec 2017 00:00:00 +0000]]>primitive Pythagorean tripleshttps://www.thinkib.net/cache/blog-thumbs/11/25433-1514670883-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/25433/primitive-pythagorean-triples
<p>Over the years, I’ve got the impression that students are as fascinated with <strong>Pythagorean triples </strong>as they are with prime numbers. Many mathematical papers, books and websites delve into the various intriguing aspects of Pythagorean triples. For example, see Dr Ron Knott’s impressive website <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html" target="_blank">Pythagorean Right-Angles Triangles</a> (maybe some <strong>Exploration </strong>ideas to be found). My interest in Pythagorean triples motivated me to make this a key component in the most recent additions to my <a href="mathhlsl/page/23399/problem-of-the-day" target="_blank"><span ><strong>HL and SL Problem-of-the-Day</strong></span></a> (there are now 200 in total !) – and also to construct a Geogebra applet for generating Pythagorean triples ... but, I’m getting a little ahead of myself. First, here is some background information before presenting the Geogebra applet (but you can certainly jump ahead to the <a href="#GGB">applet</a>, if you wish).</p><p>A <strong>Pythagorean triple</strong> is a set of three positive integers <span class="math-tex">\(\left( {a,\;b,\;c} \right)\)</span> satisfying the Pythagorean theorem <span class="math-tex">\({a^2} + {b^2} = {c^2}\)</span>. Knowledge of (and, perhaps, fascination with) Pythagorean triples has existed for thousands of years. The ancient Babylonian tablet known as <a href="http://www.math.ubc.ca/%7Ecass/courses/m446-03/pl322/pl322.html" target="_blank">Plimpton 322</a> contains a list of Pythagorean triples and is dated from around 1800 BCE. The Great Pyramid of Giza in Egypt conforms almost precisely to the ratios in the 3-4-5 <strong>Pythagorean triangle</strong>. A Pythagorean triangle is a triangle whose sides are a <strong>primitive</strong> <strong>Pythagorean triple</strong>. A Pythagorean triple <span class="math-tex">\(\left( {a,\;b,\;c} \right)\)</span> is “primitive” when the three numbers have no common factors (relatively prime, or coprime). <span class="math-tex">\(\left( {3,\;4,\;5} \right)\)</span> and <span class="math-tex">\(\left( {5,\;12,\;13} \right)\)</span> are two well-known primitive Pythagorean triples. <span class="math-tex">\(\left( {6,\;8,\;10} \right)\)</span> is an example of a Pythagorean triple that is not primitive. It’s obvious that there are an infinite number of Pythagorean triples because a multiple of any primitive Pythagorean triple is also a Pythagorean triple (just not primitive). For example, Pythagorean triples <span class="math-tex">\(\left( {6,\;8,\;10} \right)\)</span> and <span class="math-tex">\(\left( {9,\;12,\;15} \right)\)</span> are multiples of <span class="math-tex">\(\left( {3,\;4,\;5} \right)\)</span>. In his famous book, <em>Elements</em>, Euclid proved that there are an infinite number of primitive Pythagorean triples.</p><p>One of the interesting aspects of primitive Pythagorean triples is different methods for generating them. The most well-known ‘formula’ for generating primitive Pythagorean triples appeared in Euclid’s Elements, and can be described as follows.</p><p><img src="https://www.thinkib.net/cache/blog-thumbs/11/25433-1514670883-thinkib.jpg" alt="primitive Pythagorean triples" /><br /><br />If <em>m</em> and <em>n</em> are two positive integers such that one is odd and the other is even, have no common factors other than 1 (relatively prime), and <span class="math-tex">\(m > n\)</span>, then the expressions <span class="math-tex">\(a = {m^2} - {n^2},\;b = 2mn,\;c = {m^2} + {n^2}\)</span> will generate a primitive Pythagorean triple <span class="math-tex">\(\left( {a,\;b,\;c} \right)\)</span>. This method can generate any possible primitive Pythagorean, but it’s not the only method. </p><p>A slightly different ‘formula’ is: if <em>p</em> and <em>q</em> are two relatively prime positive odd integers where <span class="math-tex">\(p > q\)</span>, then the expressions <span class="math-tex">\(a = pq,\;b = \frac{{{p^2} - {q^2}}}{2},\;c = \frac{{{p^2} + {q^2}}}{2}\)</span> generate a primitive Pythagorean triple. I prefer this ‘formula’ (or parameterization) and used it in composing the Geogebra applet below. It is easier to check if both numbers are odd, and because <em>a </em>is the product of <em>p</em> and <em>q</em> it’s not difficult to construct a list of primitive Pythagorean triples where one of the ‘legs’ (<em>a</em> in this case) of the associated right triangle is the same; although only for values of <em>a</em> that are odd. One just needs to choose an odd number and determine pairs of factors for the number. For example, for the odd number 15 we have <span class="math-tex">\(15 = 5 \times 3\)</span> and <span class="math-tex">\(15 = 5 \times 1\)</span>. When <span class="math-tex">\(p = 5\)</span> and <span class="math-tex">\(q = 3\)</span>, the primitive Pythagorean triple <span class="math-tex">\(\left( {15,\;8,\;17} \right)\)</span> is generated; and when <span class="math-tex">\(p = 15\)</span> and <span class="math-tex">\(q = 1\)</span>, the primitive Pythagorean triple <span class="math-tex">\(\left( {15,\;112,\;113} \right)\)</span> is generated. It’s a bit trickier with the hypotenuse (i.e. the value of <em>c</em>) because <em>c</em> is half of the sum of two squares (or 2c is sum of two squares). However, it’s not too hard to show that 30 (<span class="math-tex">\(2 \times 15\)</span>) cannot be expressed as the sum of two squares. Therefore, the only primitive Pythagorean triples in which the number 15 appears are <span class="math-tex">\(\left( {15,\;8,\;17} \right)\)</span> and <span class="math-tex">\(\left( {15,\;112,\;113} \right)\)</span>. A similar argument proves that <span class="math-tex">\(\left( {9,\;40,\;41} \right)\)</span> is the only primitive Pythagorean triple in which the number 9 appears.</p><p><a id="GGB" name="GGB"></a>Use the Geogebra applet below – and have some fun generating primitive Pythagorean triples for various values of <em>p</em> and <em>q</em>. What is true for all primitive Pythagorean triples generated when <span class="math-tex">\(q = 1\)</span> ? when <span class="math-tex">\(p = q + 2\)</span> ? when <span class="math-tex">\(p = q + k,\;k \in {\mathbb{Z}^ + }\)</span> ?</p><iframe height="436px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/NUhcYDZw/width/644/height/436/border/888888/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="generating primitive Pythagorean triples" width="644px"></iframe>https://www.thinkib.net/mathhlsl/blog/25433/primitive-pythagorean-triples#1514592000Problem: 12 circles circling a circle
https://www.thinkib.net/mathhlsl/blog/24457/problem-12-circles-circling-a-circle
Mon, 28 Aug 2017 00:00:00 +0000<p><img alt="" src="files/mathhlsl/images/_Site/Blogs/circles_blog_img2.jpg" style="float: right; width: 200px; height: 140px;" /><br />Early in the school year, I like to offer my first-year IB maths students - especially HL students - some <strong>challenging problems</strong>. I like creating problems of my own that students can solve using mathematics that they would have studied previously. </p> <p>Here is my latest creation which is best presented with the dynamic Geogebra applet shown below. The 12 'outside' circles are all congruent with each being tangent to the larger 'inside' circle and also tangent to the 'outside' circles on either side.</p> <iframe height="548px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/Rmf9tyxB/width/580/height/548/border/888888/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/false/ctl/false" style="border:0px;" title="12 circles circling a circle" width="580px"></iframe> <p>You can challenge your students by asking them to find the ratio of the radius of one of the 'outside' circles, <span class="math-tex">\({r_o}\)</span> , to the radius of the 'inside' circle, <span class="math-tex">\({r_i}\)</span> . That is, find <span class="math-tex">\(\frac{{{r_o}}}{{{r_i}}}\)</span>. A student (or teacher) should first play with the dynamic Geogebra applet and become convinced that the ratio <span class="math-tex">\(\frac{{{r_o}}}{{{r_i}}}\)</span> is the same regardless the size of the 'inside' circle; in other words, <span class="math-tex">\(\frac{{{r_o}}}{{{r_i}}}\)</span> is a constant.</p> <p>You can ask for an <strong>approximate </strong>value of <span class="math-tex">\(\frac{{{r_o}}}{{{r_i}}}\)</span> (e.g. accurate to 3 significant figures) and/or for the <strong>exact </strong>value of <span class="math-tex">\(\frac{{{r_o}}}{{{r_i}}}\)</span>. Obtaining an exact value is very challenging and there are different ways to express the it. [<em>see below for access to answer and worked solution</em>]</p> <p>When I offer problems like this to students, I usually do not require that they solve it for homework. It is optional. But, I'm very curious to see which students attempt problems like this - and whether they successfully solve it. Again, I'm especially interested in how HL students react to problems like this. It often provides me with some valuable insight into a students' willingness and capability to tackle a genuine problem that is unfamiliar to them - and to get a sense of whether they enjoy it. I hope they do. I did.</p> <p>Here is a link for the Geogebra applet (above) that can be used to access it directly. <a href="http://ggbm.at/gWkrnxj2" target="_blank">https://ggbm.at/gWkrnxj2</a></p> <p><u><strong>Answer</strong></u>: click on 'eye' below to reveal approximate and exact value of <span class="math-tex">\(\frac{{{r_o}}}{{{r_i}}}\)</span>.</p> <section class="tib-hiddenbox"> <p><span class="math-tex">\(\frac{{{r_o}}}{{{r_i}}} = 3\sqrt 6 - 4\sqrt 3 - 5\sqrt 2 + 7 \approx 0.3491981862...\)</span></p> <p>As mentioned, the expression for the exact value can be written in different ways but, I believe, this is the 'simplest'</p> </section> <p>Here is a two-page PDF file with worked solution & notes: <a href="/files/mathhlsl/files/Blog%20entries/12-circles-circling-a-circle---solution.pdf" target="_blank">12 Circles Circling a Circle - Solution</a></p> <p><strong>Tags:</strong> <em>problem, circle</em></p>https://www.thinkib.net/mathhlsl/blog/24457/problem-12-circles-circling-a-circle#1503878400P.o.t.D. - 150 Problems !
https://www.thinkib.net/mathhlsl/blog/24007/potd-150-problems-
Sun, 30 Jul 2017 00:00:00 +0000<p>Over the past six months, I’ve toiled away at composing <em>problems</em> to add to my two <a href="mathhlsl/page/23399/problem-of-the-day" target="_blank" title="Assessment » Problem of the Day"><span >Problem of the Day</span></a> lists – one list of problems for <a href="http://mathhlsl/page/23398/sl-problem-of-the-day" target="_blank"><span >Maths SL</span></a> and a second list of problems for <a href="mathhlsl/page/23396/hl-problem-of-the-day" target="_blank"><span >Maths HL</span></a>. Last week I managed to produce my 150<sup>th</sup> Problem of the Day – and to mark that milestone I thought I would try and create a problem that was a true problem for myself. That is, a question for which I did not know the answer and not immediately aware of an effective solution strategy.<img alt="" src="files/mathhlsl/files/Blog%20entries/bread-img0.jpg" style="float: right; width: 220px; height: 182px;" /></p> <p>The problem that I devised was motivated by a problem that I found in a short article entitled <a href="http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/how-do-you-slice-the-bread" target="_blank">How Do You Slice the Bread?</a> The authors of the article model a slice of bread in two dimensions with a rectangle and a semi-ellipse (as shown in diagram at right). One of the questions they pose – and solve – is where to cut a slice of bread to produce two “triangular” pieces that are equal in area with the ‘cut’ starting from the lower left corner of the rectangular section.</p> <p><img alt="" src="files/mathhlsl/files/Blog%20entries/semicircle-img0.jpg" style="float: left; width: 240px; height: 129px;" /><br />This made me come up of with a similar question: <span ><strong>Where do you cut a semicircle – also starting from the lower left ‘corner’ – to create two regions of equal area?</strong></span> Although someone may have posed and solved this problem previously, I’m not aware of such and was surprised not to find anything after searching on the internet. My work on solving the problem required me to put on my problem-solving hat and led to some engaging mathematics and a solution with an interesting and unexpected property. I became so engrossed with the problem that I ended up solving it two different ways.</p> <p>I have posted my solutions and notes for this problem at: <a href="http://www.thinkib.net/mathhlsl/page/24083/semicircle-chord-cut-v2-solutions-notes" target="_blank" title="Assessment » Problem of the Day » Semicircle Chord Cut v2 - Solutions & Notes">Semicircle Chord Cut v2 - Solutions & Notes</a>. Check out my Geogebra applet, <a href="mathhlsl/page/24006/semicircle-chord-cut-v2-applet" target="_blank" title="Assessment » Problem of the Day » semicircle chord cut v2 (applet)">semicircle chord cut v2 (applet)</a>, which dynamically reveals the solution and illustrates the unexpected property of point P. [<u>Note</u>: This is v2 (version 2) because there is another semicircle dissection problem on the site, <a href="files/mathhlsl/files/Basics/Course%20Planning/hl1-start/problem---semicircle-chord-cut-v1.pdf" target="_blank">semicircle chord cut v1</a>, where one needs to find a chord <em>horizontal </em>to the diameter that cuts the semicircle into two regions of equal area]</p> <p><strong>Tags:</strong> <em>problem of the day, problem solving, semicircle dissection</em></p>https://www.thinkib.net/mathhlsl/blog/24007/potd-150-problems-#1501372800Challenge Problems (#10)
https://www.thinkib.net/mathhlsl/blog/20778/challenge-problems-10
Sun, 10 Apr 2016 00:00:00 +0000<p>To me - and to most people who enjoy mathematics - solving a problem that requires insight, perseverance and creative use of mathematical knowledge and techniques can be a very enjoyable and rewarding activity. Most of what students do in mathematics classes around the world is solve <em><strong>exercises</strong></em>, not <em><strong>problems</strong></em>. An 'exercise' is a question for which a student usually knows beforehand what strategy and technique(s) to apply. An exercise set for homework will contain questions that will have students practice some facts and/or techniques they were recently taught. For example, use the sine rule to solve for a missing side or angle in a triangle; or set up a definite integral to find the area between two curves. This is all necessary and valuable - but students also need (I believe) regular experiences in solving a <u>true</u> problem - that is, a <strong>challenging problem </strong>for which it is not obvious what mathematical knowledge or technique(s) to apply.</p> <p>There are many good books and websites that contain mathematical problems and puzzles. The tough task for a teacher interested in having their students tackle <strong>challenging problems</strong> is to find ones that are at a sufficient level of difficulty and involve mathematics that is appropriate and useful for the particular content that is being taught in their course.</p> <p>I've worked hard to come up with such problems suitable for IB Maths HL and SL students. Of course, generally speaking, Maths HL students are going to be more receptive and capable in tackling difficult problems - but, I believe, Maths SL students also benefit from practicing problem-solving skills in a genuine way.</p> <p><img alt="" src="files/mathhlsl/images/_Site/Blogs/half-the-area/half_area_img1.jpg" style="float: left; width: 250px; height: 167px;" />Today, I am adding my <strong>10th Challenge Problem</strong> - called <a href="http://www.thinkib.net/mathhlsl/page/20779/half-the-area" target="_blank"><em><strong>Half the Area</strong></em></a>. I strive to come up with new problems or a modification of a known problem. Devising a new problem is not easy, but I'm always on the outlook for interesting questions that can lead to a problem suitably challlenging for Maths HL & SL students.</p> <p>Recently in a Maths SL class, I asked students to quickly determine the exact area (so do not use a GDC) bounded by the <em>x</em>-axis and the graph of <span class="tib-mathml" contenteditable="false"><span class="tib-mathml"><math> <mrow> <mi>y</mi><mo>=</mo><mi>sin</mi><mi>x</mi></mrow></math></span> </span> on the interval <span class="tib-mathml" contenteditable="false"><span class="tib-mathml"><math> <mrow> <mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mtext>π</mtext></mrow></math></span> </span> . It's good practice in setting up an appropriate definite integral, carrying out some simple anti-differentiation, and then evaluating the definite integral. The result of exactly 2 square units is interesting; most students do not expect to get an exact integer value for this area.</p> <p><img alt="" src="files/mathhlsl/images/_Site/Blogs/half-the-area/half_area_img2.jpg" style="float: right; width: 260px; height: 162px;" />But then the question came up in class about how we could divide this region in half. After a brief discussion, we thought it best to divide it with a horizontal line, <span class="tib-mathml" contenteditable="false"><span class="tib-mathml"><math> <mrow> <mi>y</mi><mo>=</mo><mi>k</mi></mrow></math></span> </span> , so that the area of the region under <span class="tib-mathml" contenteditable="false"><span class="tib-mathml"><math> <mrow> <mi>y</mi><mo>=</mo><mi>sin</mi><mi>x</mi></mrow></math></span> </span> is divided into two regions each with an area of exactly 1 square unit.</p> <p>Often a useful first step in a problem like this is to make a good guess at the answer. This is something students often do not consider doing. Simple logic informs any thoughtful student that the value of <em>k</em> must be less than <span class="tib-mathml" contenteditable="false"><span class="tib-mathml"><math> <mrow> <mfrac bevelled="true"> <mn>1</mn> <mn>2</mn> </mfrac> </mrow></math></span> </span> .</p> <p>It's also fun to use technology to 'play' around to refine a good guess even better. See the video below showing an animation on my TI-Nspire CX that tries to answer this problem.<br /> <iframe align="left" allowfullscreen="" frameborder="0" height="315" scrolling="no" src="https://www.youtube.com/embed/84sbq_XceVo?rel=0" style="padding:10px;border:10px #000;" width="420"></iframe>Is<em> </em>it possible that the value of <em>k</em> is exactly 0.36 ?</p> <p>Of course, we know that GDCs can give inaccurate or false results in certain situations. So, it seems reasonable (and correct, in this case) to be suspicious that the equation of the horizontal line that divides the region in half is exactly <span class="tib-mathml" contenteditable="false"><span class="tib-mathml"><math> <mrow> <mi>y</mi><mo>=</mo><mn>0.36</mn></mrow></math></span></span> .</p> <p>Is there an exact value for <em>k</em> ? I don't know, although I believe there is none. I've calculated <em>k</em> (approximate to 12 significant figures) to be <strong>0.360034982809</strong></p> <p>As a result of my attempt to express <em>k</em> exactly, I did determine the following interesting result.</p> <p>The desired value of <em>k</em> satisfies the following equation:</p> <p><span class="tib-mathml" contenteditable="false"><span class="tib-mathml"><math> <mrow> <mi>arcsin</mi><mi>k</mi><mo>=</mo><mfrac> <mrow> <mtext>π</mtext><mtext> </mtext><mi>k</mi><mtext> </mtext><mo>+</mo><mtext> </mtext><mn>1</mn><mo>−</mo><mn>2</mn><msqrt> <mrow> <mn>1</mn><mo>−</mo><msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <mrow> <mn>2</mn><mi>k</mi></mrow> </mfrac> </mrow></math></span></span></p> <p><img alt="" src="files/mathhlsl/images/_Site/Blogs/half-the-area/half-area-vid-img2.jpg" style="margin-left: 10px; margin-right: 10px; float: right; width: 420px; height: 315px;" /><br />This interesting result - which makes use of a healthy dose of integral calculus, trigonometry & problem solving - led me to devise my 10th Challenge Problem. Take a look at this <a href="http://www.thinkib.net/mathhlsl/page/20779/half-the-area" target="_blank"><em><strong>Half the Area</strong></em></a> Challenge Problem - and the other Challenge Problems that I've posted thus far. There are more to come.</p> https://www.thinkib.net/mathhlsl/blog/20778/challenge-problems-10#1460246400Measuring the earth's radius 1000 years ago
https://www.thinkib.net/mathhlsl/blog/19923/measuring-the-earths-radius-1000-years-ago
Tue, 10 Nov 2015 00:00:00 +0000]]>Measuring the earth's radius 1000 years agohttps://www.thinkib.net/cache/blog-thumbs/11/19923-1447199823-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/19923/measuring-the-earths-radius-1000-years-ago
<p><img src="https://www.thinkib.net/cache/blog-thumbs/11/19923-1447199823-thinkib.jpg" alt="Measuring the earth's radius 1000 years ago" /><br /><br /> Al-Biruni computed the radius of the earth to be about 6339 km (converting from cubits – the distance measure that al-Biruni used) which gives the earth’s circumference to be about 39830 km which is more accurate that Eratosthenes’ calculation of about 39690 km. At the equator, the earth’s radius and circumference are 6378.1 km and 40075 km. The earth is not a perfect sphere (radius to the poles is shorter than to the equator). The mean radius and circumference are 6371.0 km and 40030 km. Eratosthenes had to deal with significant measurement inaccuracies – in measuring the angle of inclination of the sun and especially in measuring the distance between Alexandria and Syene. Although the one distance required for al-Biruni’s method was inherently more accurate it was difficult to measure angles with a great deal of accuracy. However, al-Biruni was carrying out his method a little more than 1000 years after Eratosthenes and the astrolabe that al-Biruni used would certainly have been more accurate than whatever angle measuring device that Eratosthenes had used. Historians have surmised that al-Biruni’s astrolabe was probably able to measure angles up to 1 minute of an arc which is 1/60 of a degree.</p> <p>See the student handout <em><a href="files/mathhlsl/files/Assessment/Challenge-Qs/Radius-of-the-Earth.pdf" target="_blank"><strong>Radius of the Earth</strong></a> </em>in the set of <strong>Challenge Questions</strong> on the site (in <strong>Assessment </strong>section). I gave this handout to my students to work through and then had a discussion with them about the significant progress achieved by Islamic mathematicians and scientists during a time when scientific thought was at a low point in medieval Europe. After completing the handout and watching the 10-minute portion in the documentary that covered al-Biruni’s method of measuring the radius of the earth, I asked my students what <strong>“tools”</strong> were critical to the development and execution of the method. Obvious answers included an astrolabe (or giant protractor) and counting paces to measure distance, but most students did not go beyond those two. It did not take much to convince them that <strong><em>algebra</em></strong> was also a necessary “tool” that was absolutely critical for finding the two formulas that were central to al-Biruni’s method. But then I suggested that there was another necessary “tool” they had overlooked. I pointed out that it was necessary to evaluate trigonometric functions in order to apply the two formulas. How would al-Biruni have evaluated a trigonometric function (e.g. evaluate <span class="tib-mathml"><math> <mrow> <mi>tan</mi><msup> <mrow> <mn>0.57</mn></mrow> <mrow> <mtext> </mtext><mo>∘</mo></mrow> </msup> </mrow></math></span> )? It’s now been more than 30 years since students routinely used tables (and interpolation when necessary) to evaluate trigonometric functions. Of course, al-Biruni would have valued his trigonometric tables just as much – if not more so – than his precisely crafted astrolabe.</p> https://www.thinkib.net/mathhlsl/blog/19923/measuring-the-earths-radius-1000-years-ago#1447113600Perplexing Pentagons (Exploration Idea #499)
https://www.thinkib.net/mathhlsl/blog/19600/perplexing-pentagons-exploration-idea-499
Sun, 13 Sep 2015 00:00:00 +0000<p><br /> <img alt="" src="files/mathhlsl/images/_Site/Blogs/Perplexing-Pentagons/pentagon-tile-img1.jpg" style="width: 250px; height: 250px; float: right;" />Recently, a Maths SL student asked for my thoughts about her writing an <em>Exploration (IA) </em>on tilings of the plane (Euclidean tessellations). I must admit that my initial thought was that it would be easy for a student to write about tessellations and only use fairly elementary mathematics. Criterion <strong>E: Use of Mathematics</strong> contributes the largest portion of the total marks (6 out of 20) for a student <em>Exploration </em>– and the IB’s document <strong>Additional Notes and Guidance on the Exploration</strong> informs us that, “<em>If the level of mathematics is not commensurate with the level of the course, a maximum of two marks can be awarded for this criterion.</em>” The resulting loss of 4 marks is 20% of the total possible marks – ouch. However, with some effort it is possible to get into some interesting mathematics with tessellations that would be at least commensurate with the level of mathematics in Maths SL.</p> <p>I wanted to point her in the right direction so that she would encounter some mathematics beyond just simple plane geometry. Tiling the plane by a single convex polygon turns out to be a very intriguing area of mathematics – and one where new discoveries are being made and where there are still unresolved questions. Which convex polygons – on its own – will tessellate (tile) the plane? Obviously any triangle will the tessellate the plane; and any quadrilateral will also. A regular hexagon certainly tessellates the plane; but, not <em>any </em>convex hexagon tessellates. It turns out that there are three different types of regular hexagons that can tessellate the plane. These are illustrated in the image below. Even better is this <a href="http://tube.geogebra.org/student/m155779" target="_blank"><strong>interactive Geogebra applet</strong></a> that illustrates the <strong>three types of hexagonal tilings</strong>.</p> <p><img alt="" src="files/mathhlsl/images/_Site/Blogs/Perplexing-Pentagons/hex-tilings-3-img.jpg" style="width: 575px; height: 354px;" /></p> <hr class="hidden" /> <p>It gets quite a bit more interesting with the pentagon. Although a regular pentagon has its own inherent beauty and the golden ratio <img align="middle" alt="phi" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#966;«/mi»«/math»" src="/ckeditor//plugins/wiris/integration/showimage.php?formula=f8cb28869e4f932c0d04182b39a89cbb.png" /> is embedded in its shape, it does not tessellate the plane. <img alt="" class="noborder" src="files/mathhlsl/images/_Site/Blogs/Perplexing-Pentagons/reg-pentagons.jpg" style="width: 100px; height: 100px;" /> But there are some convex pentagons that do tessellate. What about polygons beyond the hexagon? It has been mathematically proven that it is not possible to tile the plane with a convex polygon that has more than 6 sides.</p> <p>During the 20<sup>th</sup> century, mathematicians – including a self-taught amateur mathematician (interesting story of housewife <a href="http://sites.google.com/site/intriguingtessellations/home" target="_blank">Marjorie Rice</a>) determined that there were 14 different types of convex pentagons that tile the plane. In her 1978 award-winning article <a href="http://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1979/0025570x.di021103.02p0247f.pdf" target="_blank"><strong><em>Tiling the Plane with Congruent Pentagons</em></strong></a>, American mathematician Doris Schattschneider, gives a thorough mathematical treatment of the 13 different convex pentagonal tilings that were known at that time (in 1985, a German mathematics graduate student found a 14<sup>th</sup> type). Schattschneider is an American mathematician and retired professor of mathematics who is known for her work on tessellations and being the co-director of a project in the 1980s that lead to the development of the powerful mathematics software <a href="http://www.dynamicgeometry.com/" target="_blank">Geometer’s Sketchpad</a>. I borrowed the title for this blog post, <a href="http://britton.disted.camosun.bc.ca/jbperplex.htm" target="_blank"><strong><em>Perplexing Pentagons</em></strong></a>, from a short <a href="http://britton.disted.camosun.bc.ca/jbperplex.htm" target="_blank">article </a>written by Schattschneider in 1996 which she concludes by writing “<em>Are all the types of convex pentagons that tessellate now known? The tessellating pentagon problem remains unsolved</em>.”</p> <p>So, in my conversation with the Maths SL student I told her that tessellations involving convex pentagons is an active area of mathematics and recently featured in the news. It had been 30 years since the German student Rolf Stein found the 14<sup>th</sup> type of convex pentagon that tiles the plane but two months ago a 15<sup>th</sup> convex pentagonal tiling was found. Two mathematics professors and a student at the University of Washington Bothell found the new convex pentagon that tiles the plane with the aid of a specially designed computer algorithm. (this new tiling is shown in the first image above)</p> <p><strong><em><a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/aug/10/attack-on-the-pentagon-results-in-discovery-of-new-mathematical-tile" target="_blank">Attack on the pentagon results in discovery of new mathematical tile</a> </em></strong>(Guardian, 11 August 2015)</p> <p><a href="http://www.dailymail.co.uk/sciencetech/article-3201424/A-new-way-tile-floor-Scientists-discover-strange-new-shape-trying-solve-one-maths-complex-problems.html" target="_blank"><strong><em>Mathematicians find new 'perfect shape' that solves one of their most complex problems</em></strong></a> (Daily Mail, 17 August 2015)</p> <p>A great way to see all 15 of the pentagon tilings is with this <a href="http://demonstrations.wolfram.com/PentagonTilings/" target="_blank"><strong>Pentagon Tilings - Wolfram Demonstration</strong></a>. If you download Wolfram's free <a href="http://www.wolfram.com/cdf-player/" target="_blank">CDF Player</a>, you can interact with the demonstration - choosing the particular tiling and specific characteristics of the chosen tiling.</p> <p>These articles may very well stimulate some interest in a student motivating them to look deeper into the topic. The challenge is narrowing the topic enough where it's manageable for a 6-12 page report and also focusing on mathematics which is sufficiently advanced but not too advanced. The mathematical (and computer) work that contributed to the recent discovery of the 15th convex pentagonal tiling is far too advanced but a focused look on the geometric construction of a particular tessellating convex polygon might be suitable for a student <em>Exploration</em>.</p> <p>Schattschneider's aforementioned 16-page article <a href="http://www.maa.org/sites/default/files/pdf/upload_library/22/Allendoerfer/1979/0025570x.di021103.02p0247f.pdf" target="_blank"><em><strong>Tiling the Plane with Convex Polygons</strong></em></a> is a good resource - and is what I suggested that the student read. It has perhaps too much detail but near the end of the article (pages 12-14), Schattschneider provides a table with geometric details on how to construct the different pentagonal tilings (the 13 that were known at that time).</p> <p><img alt="" src="files/mathhlsl/images/_Site/Blogs/Perplexing-Pentagons/Cairo-pentagonal-tiling.jpg" style="width: 237px; height: 237px; float: left;" />Another direction to take is to focus on just one of the pentagonal tilings which has particularly interesting properties or aesthetic appeal. One of the pentagonal tilings that has received a lot of attention is the <a href="http://en.wikipedia.org/wiki/Cairo_pentagonal_tiling">Cairo pentagonal tiling </a>(shown left). It was given the name because of the claim (though not substantiated until fairly recently) that some streets and squares in Cairo have paving with this design. A Cairo-type of tiling can be constructed on <a href="http://www.mhhe.com/math/ltbmath/bennett_nelson/conceptual/instructor/grids/IsometricDotPap.pdf" target="_blank">isometric paper</a> but it will have two different pentagons so it's using more than one type of pentagon - hence, not a true pentagonal tiling (like the 15 that are known). It can be interesting to mathematically determine the angle measures of the single pentagon that does create a Cairo pentagonal tiling as shown left and below. What are the exact measures of the angles?</p> <p><img alt="" src="files/mathhlsl/images/_Site/Blogs/Perplexing-Pentagons/Cairo-tile.jpg" style="width: 160px; height: 150px; margin-left: 25px; margin-right: 25px;" /><img alt="" src="files/mathhlsl/images/_Site/Blogs/Perplexing-Pentagons/Cairo-pentagon-geo-cons-img.jpg" style="width: 225px; height: 181px;" /><br /></p> <p>And, lastly, there is a short 5-page article, <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.106.1383&rep=rep1&type=pdf" target="_blank"><em><strong>Tiling Problem of Convex Pentagon</strong></em></a>, written by two Japanese mathematicians in 2000 on convex pentagon tilings and it references Euler's formula in graph theory on the number of vertices, edges and faces. It could be a good resource for students considering a student <em>Exploration </em>involving pentagonal tessellations.</p> <p><strong>Tags:</strong> <em>tessellation, tiling, pentagon, exploration, internal assessment</em></p>https://www.thinkib.net/mathhlsl/blog/19600/perplexing-pentagons-exploration-idea-499#1442102400'Too hard' Scottish Higher Maths exam
https://www.thinkib.net/mathhlsl/blog/19440/too-hard-scottish-higher-maths-exam
Mon, 10 Aug 2015 00:00:00 +0000<p>Although I live in Scotland, I'm not directly involved with the curriculum for Scottish secondary mathematics (worth mentioning that the education system in Scotland is separate from the rest of the UK - as are several other institutions). Nevertheless, I am familiar with mathematics teaching in Scottish secondary schools and often find mathematics education materials produced here very useful. So, when reports on issues related to mathematics exams in Scotland recently appeared in news publications here they caught my attention. And, as a maths teacher that has worked in international schools in a few countries, I'm just naturally curious about how aspects of mathematics teaching and assessment compare from one country, or system, to another.</p> <p>The news stories were specifically about the perceived increased difficulty of new exams for <strong>Scottish Higher Mathematics</strong> (one of many exams operated by the Scottish Qualifications Authority, SQA). There is another secondary maths course in Scotland that is more advanced, with the appropriate course title of Advanced Higher Mathematics. The curriculum in Scottish secondary schools has been undergoing some changes the past few years - and one of the consequences of this was a 'new' exam for Higher Maths (the 'new' exam for Advanced Higher Maths first occurs in 2016). Not long after students sat the 'new' Higher Maths exam about 10 weeks ago, there were so may complaints about the exam being "too hard" that it became news, i.e. it had enough 'sensational' aspects so various media outlets deemed it worthy of reporting.</p> <p>In my opinion (and some may differ with this), Scottish Higher Maths falls somewhere very roughly between IB Maths SL and IB Maths HL in terms of content and difficulty level. The Higher Maths exam has two papers - Paper 1 does not allow a calculator, and Paper 2 does. As a maths teacher, one of the aspects of the news reporting on the 'difficulty' of the Higher Maths exam that I found very intriguing was one particular question from the exam that different online news stories decided to highlight. One of the news stories even referred to the question in its title: <a href="http://www.bbc.co.uk/news/uk-scotland-33778968" target="_blank"><strong>New Higher Maths exam - why did the crocodile cross the stream?</strong></a> The question is shown below (it was #8 on Paper 2). By clicking on the question, you can obtain the entire 2015 Scottish Higher Maths exam.</p> <p><a href="/files/mathhlsl/images/_Site/Blogs/SQA-higher-maths/Scottish_Higher_Maths_P1_and_P2_2015.pdf" target="_blank"><img alt="" src="files/mathhlsl/images/_Site/Blogs/SQA-higher-maths/higher_maths_Q8_img.jpg" style="width: 576px; height: 576px;" /></a></p> <p>To some this question may 'look' difficult - it has a 'messy' function. Remember, it is on Paper 2, so a student will have a graphing calculator. With that in mind, the question is really rather trivial. To answer part (a) (i) one simply needs to evaluate the time function when <em><strong>x</strong></em>=20 metres, that is <span class="tib-mathml"><math> <mrow> <mi>T</mi><mrow><mo>(</mo> <mrow> <mn>20</mn> </mrow> <mo>)</mo></mrow><mo>≈</mo><mn>104.4</mn> </mrow></math></span> - and since the units are tenths of a second this is approximately 10.4 seconds. And part (a) (ii) is answered by evaluating the time function when x=0 metres, that is <span class="tib-mathml"><math> <mrow> <mi>T</mi><mrow><mo>(</mo> <mn>0</mn> <mo>)</mo></mrow><mo>=</mo><mn>110</mn> </mrow></math></span> which is equivalent to 11 seconds. To find the minimum possible time - and the value of <em><strong>x</strong></em> that produces this minimum - a student can simply graph <span class="tib-mathml"><math> <mrow> <mi>T</mi><mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow> </mrow></math></span> on their GDC and use a standard built-in 'minimum' function to find the minimum point on the graph. The graph below on a TI-Nspire shows that the minimum time is 98 (this is 9.8 seconds) and that it occurs when <em><strong>x</strong></em>=8 metres.</p> <p><img alt="" src="/files/mathhlsl/images/_Site/Blogs/SQA-higher-maths/Nspire_graph_img.jpg" style="width: 575px; height: 384px;" /><br /></p> <p>If there is a 'difficult' aspect to this question, I think it is being very careful to correctly interpret the given information in the context of the time function, <span class="tib-mathml"><math> <mrow> <mi>T</mi><mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow> </mrow></math></span> , that was given. For example, it was important to understand that the units of the function's output is tenths of a second, not seconds - and, also that 'not traveling on land' (part (a) (i)) means that <em><strong>x</strong></em>=20 metres; and, likewise, that 'swimming shortest distance' means that <em><strong>x</strong></em>=0 metres. And, perhaps most importantly, a student needs to resist being intimidated by the question because it might appear difficult given the amount of text and the 'not-so-simple-looking' time function.</p> <p>It is my firm belief that one of the most important skills that a student in either Maths HL or Maths SL (more important in HL) needs to gain from the course (if they do not already have it) is to not be easily intimidated by questions that initially look more difficult than they actually are; not to become unsettled by questions that look unfamiliar. This is probably what the large number of Scottish students were actually complaining about. The recent 'new' Higher Maths exam had some questions which were unfamiliar to them - when, in fact, these questions (like the one above) were actually relatively straightforward. Too many students become easily intimidated by questions that look 'different'. As teachers, we need to continually present our students (again, especially HL students) with a wide variety of questions, and to design assignments and assessments so that students routinely confront unfamiliar questions. This makes them practice good fundamental problem solving skills and builds up their exam-taking confidence - and to successfully answer questions which some would claim they "have not seen before."</p> https://www.thinkib.net/mathhlsl/blog/19440/too-hard-scottish-higher-maths-exam#1439164800Exploration idea #498
https://www.thinkib.net/mathhlsl/blog/19372/exploration-idea-498
Fri, 31 Jul 2015 00:00:00 +0000<p><img alt="" src="files/mathhlsl/images/_Site/Blogs/parab-line/para_line_img2.jpg" style="float: right;" />I'm always on the look out for interesting mathematical 'nuggets' that might lead to something suitable for a Maths HL or SL student <strong>Exploration </strong>(IA task). Recently, I came across the following fact about parabolas that I was not aware of and thought was very intriguing.</p> <p><img alt="" src="files/mathhlsl/images/_Site/Blogs/parab-line/para_fact_img3.jpg" style="width: 500px; height: 98px;" /></p> <p>So, if the distance <strong><span ><em>h</em></span></strong> between the two points where the secant line intersects the parabola is kept constant for a particular parabola <span class="tib-mathml"><math> <mrow> <mi>y</mi><mo>=</mo><mi>a</mi><mo>⋅</mo><msup> <mi>x</mi> <mn>2</mn> </msup> </mrow></math></span> then the area of the region bounded by the parabola and the secant line should remain constant.</p> <p>I consider this an interesting mathematical 'nugget' that has the potential to expand into a topic that a student could explore. I thought it would be fun to do some exploring myself - and this fact about parabolas is very appropriate for confirming with a dynamic graph. Recently I've been playing around with <a href="http://www.desmos.com/" target="_blank"><strong>Desmos</strong></a>, the free online graphing calculator. Below is a short video (30 sec) showing my Desmos graph in motion - with the secant line moving but the horizontal distance between the intersection points is always 4 units. Since Desmos does not have a built-in function for computing a definite integral I entered an expression to add 500 rectangles between the graphs from the left intersection point to the right intersection point to compute the area of the shaded region.</p> <p><iframe allowfullscreen="" frameborder="0" height="380" src="https://www.youtube.com/embed/hUGzzK_e_B0?autohide=1;rel=0" width="630"></iframe></p> <p><img alt="" src="/files/mathhlsl/images/_Site/Blogs/parab-line/desmos_list_img1.jpg" style="width: 200px; height: 329px; float: right;" /></p> <p>The expressions list for this Desmos graph (the user-defined items on the left side that control various aspects of the graph) is shown at right. The parabola used in this demonstration is <span class="tib-mathml"><math> <mrow> <mi>f</mi><mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow><mo>=</mo><mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow></math></span> and <span class="tib-mathml"><math> <mrow> <mi>g</mi><mrow><mo>(</mo> <mi>x</mi> <mo>)</mo></mrow> </mrow></math></span> is the equation for the secant line where <em><strong><span >p</span></strong></em> is the <span ><em><strong>x</strong></em></span>-coordinate of the left intersection point and <em><strong><span >h</span></strong></em> is the horizontal distance to the right intersection point. For this graph the value of <em><strong><span >p</span></strong></em> is -1 and <em><span >h</span></em> is 4.</p> <p>One of the nicest features of constructing a graph with Desmos is that it is so easy to make it dynamic. As soon as a constant is entered the program asks if you wish to make a slider so that you can easily change the value of the constant. So all of the constants contained in this graph - <em><strong><span >a</span></strong></em>, <em><strong><span >p</span></strong></em>, <em><strong><span >h</span></strong></em> and <em><strong><span >n</span></strong></em> - can be changed manually by dragging the slider or changed automatically by pressing the 'animate' button <img alt="" class="noborder" src="/files/mathhlsl/images/_Site/Blogs/parab-line/animate-button-img.jpg" style="width: 20px; height: 22px;" /> - and then the graph will be animated according to the changing parameter values. The video above is a recording of the animation created after pushing the animate button for <em><strong><span >p</span></strong></em>.</p> <p>Remember, according to the statement with which I started this blog post, the area of the bounded region is supposed to be dependent <u>only</u> on the value of <em><strong><span >a</span></strong></em> and <em><strong><span >h</span></strong></em>. By playing around with the values for <em><strong><span >a</span></strong></em> and <em><strong><span >h</span></strong></em> and making note of the area <em><strong><span >A</span></strong></em>, one may be able to conjecture a formula for <em><strong><span >A</span></strong></em> in terms of <em><strong><span >a</span></strong></em> and <em><strong><span >h</span></strong></em> (not easy).</p> <p><img alt="" src="/files/mathhlsl/images/_Site/Blogs/parab-line/desmos_list_img2.jpg" style="width: 325px; height: 83px; float: right;" /><br />Coming up with the correct formula and proving it would be a good start for an Exploration but would probably not be enough for an Exploration that would earn high marks. In order to have a good chance at writing a strong Exploration, I would encourage a student to devise their own further extensions to this initial investigation. For example, one could consider other parabolas that are not in the form <span class="tib-mathml"><math> <mrow> <mi>y</mi><mo>=</mo><mi>a</mi><mo>⋅</mo><msup> <mi>x</mi> <mn>2</mn> </msup> </mrow></math></span> - for example, <span class="tib-mathml"><math> <mrow> <mi>y</mi><mo>=</mo><mi>a</mi><msup> <mi>x</mi> <mn>2</mn> </msup> <mo>+</mo><mi></mi><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi> </mrow></math></span> or <span class="tib-mathml"><math> <mrow> <mi>y</mi><mo>=</mo><mi>a</mi><msup> <mrow> <mrow><mo>(</mo> <mrow> <mi></mi><mi>x</mi><mo>−</mo><mi>h</mi> </mrow> <mo>)</mo></mrow> </mrow> <mn>2</mn> </msup> <mo>+</mo><mi>k</mi> </mrow></math></span> . And perhaps consider if a similar property exists for other polynomial functions (e.g. cubic, quartic), or consider whether a similar property occurs in three dimensions.</p> <p><strong>Tags:</strong> <em>exploration, parabola, Desmos, dynamic graph, slider</em></p>https://www.thinkib.net/mathhlsl/blog/19372/exploration-idea-498#1438300800'Amazing' Calculus
https://www.thinkib.net/mathhlsl/blog/18980/amazing-calculus
Sun, 10 May 2015 00:00:00 +0000]]>'Amazing' Calculushttps://www.thinkib.net/cache/blog-thumbs/11/18980-1431273413-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/18980/amazing-calculus
<p><a href="http://www.youtube.com/watch?v=YX7csqVBzQE" target="_blank"><img src="https://www.thinkib.net/cache/blog-thumbs/11/18980-1431273413-thinkib.jpg" alt="'Amazing' Calculus" /><br /><br /></a>Even from the narrow view of just considering the syllabus contents for Maths HL & Maths SL, it is clear that the study of <strong>calculus </strong>is a very important component of both courses - it is the 'largest' syllabus topic in both courses in terms of recommended teaching hours. But I think it's beneficial to all students to gain a wider view of the importance of the calculus - both as a feat of human ingenuity and as a very effective tool for solving a wide range of problems.</p> <p>Recently, I stumbled across a <a href="http://www.youtube.com/watch?v=YX7csqVBzQE" target="_blank">YouTube video</a> (below) showing part of an interview with the very successful British novelist <a href="http://en.wikipedia.org/wiki/Ian_McEwan" target="_blank">Ian McEwan</a> in which he was asked about the difference between the humanities and the sciences - i.e. what he thought about the status of the "two cultures" debate that has been going on ever since C.P. Snow's famous 1959 lecture about the growing gulf between intellectuals in the humanities and scientists. I came across the video as I was preparing for a <strong>Theory of Knowledge </strong>lesson in which I wanted students to engage in a "two cultures" class discussion.</p> <p><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/YX7csqVBzQE" width="560"></iframe></p> <p>It was very interesting to hear McEwan admit to some 'intellectual shame' that, in his opinion, studying a humanities subject is not nearly as difficult or challenging as studying mathematics or a science. He was being interviewed after having taught literature for a few months at the California University of Technology (CalTech) which is one of the premiere universities in the USA for maths and the sciences.</p> <p>McEwan felt that university students in the humanities are missing out by not being required to get their "mind around a subject like calculus which is quite tough - but it's an amazing adventure." As a maths teacher, it was very reassuring to hear a person who has gained fame and riches in the arts proclaim that calculus is an "<strong>amazing adventure</strong>", and that it was fascinating that "a guy in a wig in the 18th century thought this up on how you'd describe the changing state of something ... amazing."</p> <p>Great to hear a 'non-maths' person support the idea that someone who pursues an expertise in the humanities should still make an effort - and enjoy and benefit from - studying some mathematics. As McEwan said - why not; experts in maths and the sciences are often very knowledgeable and interested in the arts - so, why not the other way around.</p> https://www.thinkib.net/mathhlsl/blog/18980/amazing-calculus#1431216000one week to go until exams
https://www.thinkib.net/mathhlsl/blog/18918/one-week-to-go-until-exams
Mon, 27 Apr 2015 00:00:00 +0000<p><img alt="" height="199" src="files/mathhlsl/images/_Site/Blogs/exam-revision/exam_rev_img1.png" style="float: left;" width="199" /><br />Here is an activity that I have found to be very effective revision for Maths HL exams.</p> <p>I have written <strong>10 exam questions for section A of a Paper 1 practice (mock) exam</strong> and and another <strong>10 questions for section A of a Paper 2 practice exam</strong>. All the questions are <strong>'original' questions</strong> - although some of them are similar to a past IB exam question, none of them are actually past exam questions. I have written full <strong>worked solutions</strong> for all the questions.</p> <p>What I find very effective to do with students - especially at this time with only a week to go before the May exam session begins - is to have them sit for about 70 min in a classroom with me. We spend about 10 minutes reading through the 10 questions - making mental notes (similar to what they should do during the 5-min 'reading time' they get for each exam); but we do this as a group activity. Then I write a time interval on the board for each question indicating a target "schedule" for doing each question with the # of minutes corresponding to the # of marks. After they start working I will indicate when each time period for each question has expired. Of course, they do not have to stick precisely to this schedule - but it does help them realize if their pace is about right or too slow - and whether they will be forced to skip any questions.</p> <p>After the students have finished I give them the worked solutions and they go away to review the work / answers that they were able to do in the 60 minutes. I invite them to discuss any questions they may have after they read through the worked solutions.</p> <p>Clcik <a href="mathhlsl/page/18684/sets-of-hl-review-exercises" target="_blank"><strong><u>here</u> </strong></a>to go to <a href="http://www.thinkib.net/mathhlsl/page/18684/sets-of-hl-review-exercises" target="_blank">Sets of HL Review Exercises</a> page (Assessment > Exams > Review Materials > Sets of HL Review Exercises) where you will find the <strong>set of 10 Paper 1 Section A questions</strong> and the set of 10 Paper 2 Section A questions - and <strong>Worked Solutions</strong> for both. Although they are not full mock exams (no Section B) i have referred to them as "Mock B". There is a full "Mock A" exam for HL and SL (both Paper 1 and Paper 2 for both) in the Review Materials section of Assessment > Exams.</p> https://www.thinkib.net/mathhlsl/blog/18918/one-week-to-go-until-exams#1430092800Geometry problem in work of art
https://www.thinkib.net/mathhlsl/blog/18864/geometry-problem-in-work-of-art
Sun, 12 Apr 2015 00:00:00 +0000<p>During the years 1795 to 1805 the famous English poet, printmaker and painter <strong>William Blake </strong>(1757-1827) created a color print entitled <strong><em>Newton</em></strong>, shown below.</p> <p><img alt="" height="278" src="files/mathhlsl/images/_Site/Blogs/geo-prob-in-art/Blake_Newton_sqr_img0.jpg" style="float: left;" width="278" /></p> <p>The work of art depicts a naked Isaac Newton manipulating a geometer’s compass on a scroll of paper. Upon closer inspection of the scroll, one can clearly see a diagram that has what appears to be an equilateral triangle with an arc that intersects two of the triangle’s vertices. It also appears that a side of the triangle is tangent to the arc at each of the two lower vertices. This diagram (given the assumptions just made) poses an interesting geometry problem – which can be stated as follows (with some labeled points and shading added to the diagram):</p> <h4 class="yellowBg"><u>Question</u>: What fraction is the area of the shaded region (segment of a circle) of the area of the equilateral triangle DAB? (see diagram below right)</h4> <p><img alt="" src="/files/mathhlsl/images/_Site/Blogs/geo-prob-in-art/Blake_Newton_geo_diag1.jpg" style="width: 206px; height: 300px; float: right;" /><img alt="" height="155" src="/files/mathhlsl/images/_Site/Blogs/geo-prob-in-art/Blake_Newton_lower_rt_img2.jpg" style="float: right;" width="178" /><br /><br /> click on file name below to download this <strong>question </strong>and a full <strong>worked solution</strong></p> <p><strong><a href="/files/mathhlsl/files/Blog%20entries/Geometry_problem_in_work_of_art.pdf" target="_blank">Geometry_problem_in_work_of_art</a></strong></p> <p>The astrophysicist <strong>Mario Livi</strong> wrote an interesting article about this print called <a href="http://www.huffingtonpost.com/mario-livio/on-william-blakes-newton_b_6036258.html" target="_blank"><em><strong>On Blake's 'Newton'</strong></em></a> which discusses Blake's resistance to the growing prominence of science and empiricism during the Enlightenment. It turns out that Blake was not trying to compliment Newton and all that he represented in terms of science's progress in explaining natural phenomena - but, in fact, he was trying to do the contrary.</p> <p><strong>Tags:</strong> <em>challenge, geometry</em></p>https://www.thinkib.net/mathhlsl/blog/18864/geometry-problem-in-work-of-art#1428796800pi day 2015
https://www.thinkib.net/mathhlsl/blog/18685/pi-day-2015
Sat, 14 Mar 2015 00:00:00 +0000]]>pi day 2015https://www.thinkib.net/cache/blog-thumbs/11/18685-1426355681-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/18685/pi-day-2015
<p><img src="https://www.thinkib.net/cache/blog-thumbs/11/18685-1426355681-thinkib.jpg" alt="pi day 2015" /><br /><br />. Ask your students what famous mathematicians and scientists were alive at that time. Of course, there were no Americans around to write the date in a way which allows us to celebrate a very special number.</p> <p><strong>Tags:</strong> <em>pi, pi day</em></p>https://www.thinkib.net/mathhlsl/blog/18685/pi-day-2015#1426291200A Galton Board
https://www.thinkib.net/mathhlsl/blog/18669/a-galton-board
Tue, 10 Mar 2015 00:00:00 +0000]]>A Galton Boardhttps://www.thinkib.net/cache/blog-thumbs/11/18669-1425947029-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/18669/a-galton-board
<p>Some day I'm going to build my very own <strong><a href="http://mathworld.wolfram.com/GaltonBoard.html" target="_blank">Galton Board</a></strong> (also known as a quincunx). It's a fantastic device for experimentally demonstrating a binomial distribution or binomical coefficients. Without a real one in my hands, I use 'electronic' ones that I can run on my computer for class demonstrations.</p> <p><a href="http://www.mathsisfun.com/data/quincunx.html" target="_blank"><img src="https://www.thinkib.net/cache/blog-thumbs/11/18669-1425947029-thinkib.jpg" alt="A Galton Board" /><br /><br /></a>My favorite is the Galton Board on the <a href="http://www.mathsisfun.com/" target="_blank">Math is Fun</a> website - click <a href="http://www.mathsisfun.com/data/quincunx.html" target="_blank"><strong>here</strong> </a>or on the image to open a page showing it. I've used it often in class - and it never fails to get students' attention. It has some nice features - including: (1) easily increase or decrease the number of rows of 'pegs' that the balls bounce off of; (2) possible to change the probability of bouncing left (obviously also affecting probability of bouncing right); (3) fast forward button is a very nice feature because it allows one to quickly get a result for a very large number of trials (# of balls). The image at left shows the result for 5000 balls having bounced through the triangular arrangement of 'pegs'.</p> <p>There are some decent youtube videos showing physical Galton Boards made from various materials (I <u>will</u> make one some day - or maybe I'll get a student to make one as part of their Exploration (IA) ...now there's <em><strong>personal engagement</strong></em>). Below is a short (19 sec) youtube video of a Galton Board in action. The device appears to have been constructed with a fairly simple design (yeah, I could definitely make that).</p> <p ><iframe align="middle" allowfullscreen="" frameborder="0" height="315" scrolling="no" src="https://www.youtube.com/embed/oPCcOtQKU8M" width="420"></iframe></p> <p>And, finally, I have come across two or three digital animations of Galton Boards that have been very cleverly made with Geogebra. The one below, called <a href="http://ggbtu.be/m33736" target="_blank"><strong>Bino Stat</strong></a> by its creator, is my favorite with Geogebra.</p> <p ><iframe height="408px" scrolling="no" src="https://tube.geogebra.org/material/iframe/id/33736/width/595/height/408/border/888888/rc/false/ai/true/sdz/true/smb/false/stb/true/stbh/true/ld/false/sri/true/at/auto" style="border:0px;" width="595px"></iframe></p> <p><strong>Tags:</strong> <em>probability, binomial coefficients, quincunx, galton, binomial distribution</em></p>https://www.thinkib.net/mathhlsl/blog/18669/a-galton-board#1425945600Exploration idea #497
https://www.thinkib.net/mathhlsl/blog/18618/exploration-idea-497
Sun, 01 Mar 2015 06:30:00 +0000]]>Exploration idea #497https://www.thinkib.net/cache/blog-thumbs/11/18618-1425236818-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/18618/exploration-idea-497
<p><img src="https://www.thinkib.net/cache/blog-thumbs/11/18618-1425236818-thinkib.jpg" alt="Exploration idea #497" /><br /><br />. Results for this 3rd set of perpendicular tangent lines is shown at left.</p> <p>So, if the parabola property I mentioned at the start of this blog entry was unknown then someone would hopefully conjecture the property from these three results.</p> <p>What about this being a nice topic for a Maths HL or Maths SL student Exploration (IA task)? I think it could be a suitable topic. I think a key aspect that would make it a strong Exploration is for a student to prove this property. This would require some thoughtful input from the student on setting up parameters and variables - and then organizing a clear algebraic proof annotated with clearly written text and supported by clearly labeled diagrams. I think it is also conducive for a good student to think of some nice extensions. For example, do any other conic sections (ellipse, hyperbola, etc) have a similar property? Do other conic sections have a 'directrix'? If so, what role does it play in constructing the particular conic section as a locus of points? Is there a way to illustrate the property of parabolas shown above using another procedure?</p> <p>And, finally, would it not be nice if students could submit their Exploration electronically so that it could include dynamic images. For this particular Exploration a very nice addition would be an animation showing how by dragging one point on the graph of the parabola would automatically show the two perpendicular tangent lines and their intersection point - and how a dynamic 'capture' of these intersection points would 'trace out' the directrix.</p> <p><strong>Tags:</strong> <em>exploration, internal assessment, directrix, locus</em></p>https://www.thinkib.net/mathhlsl/blog/18618/exploration-idea-497#1425191400Exploration idea #496
https://www.thinkib.net/mathhlsl/blog/18569/exploration-idea-496
Wed, 18 Feb 2015 06:30:00 +0000]]>Exploration idea #496https://www.thinkib.net/cache/blog-thumbs/11/18569-1424219489-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/18569/exploration-idea-496
<p><img src="https://www.thinkib.net/cache/blog-thumbs/11/18569-1424219489-thinkib.jpg" alt="Exploration idea #496" /><br /><br /> and the ratio of the volumes of the 'lower' truncated pyramid to the 'smaller' (or 'upper') pyramid.</p> <p>And after school was finished for the day and I thought about it some more, it came to me that another direction (and a good one for a Maths HL student Exploration) is to invert the pyramid (upside down) and consider water being added into this solid at a constant rate. What would be the rate of change of the height of the water (height of the 'smaller' pyramid) at the point at which the solid is half full? And how does the rate of change - at this height where the volume is one-half of the entire pyramid - change for pyramids with different bases (e.g. square, regular pentagon, regular hexagon, etc)? </p> <p>Certainly there are some elements of an Exploration (more than one?) lurking amongst these questions.</p> <p>p.s. why idea #<strong>496</strong>? wanted to come up with an 'interesting' three digit number - and 496 is the third <em><strong>perfect number</strong></em></p> <p><strong>Tags:</strong> <em>internal assessment, exploration, geometry, pyramid</em></p>https://www.thinkib.net/mathhlsl/blog/18569/exploration-idea-496#1424241000maths competition
https://www.thinkib.net/mathhlsl/blog/18498/maths-competition
Mon, 02 Feb 2015 06:30:00 +0000]]>maths competitionhttps://www.thinkib.net/cache/blog-thumbs/11/18498-1422921278-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/18498/maths-competition
<p><a href="http://www.ismtf.org" target="_blank"><img src="https://www.thinkib.net/cache/blog-thumbs/11/18498-1422921278-thinkib.jpg" alt="maths competition" /><br /><br /> below question to see answer ]</p> <h5><span >The date for 1 February 2015 can be written in an 8-digit format as 01-02-2015. In what month is the next soonest date which can be written in the same format using 8 different digits (i.e. no digits can repeat)?</span></h5> <section class="tib-hiddenbox"> <p>next soonest date is in month of <strong>June </strong>; and the date is <strong>17-06-2345</strong></p> </section> https://www.thinkib.net/mathhlsl/blog/18498/maths-competition#1422858600distribution of primes on calculator
https://www.thinkib.net/mathhlsl/blog/18454/distribution-of-primes-on-calculator
Wed, 28 Jan 2015 06:30:00 +0000]]>distribution of primes on calculatorhttps://www.thinkib.net/cache/blog-thumbs/11/18454-1422449605-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/18454/distribution-of-primes-on-calculator
<p>It all started out of a conversation between students during a short break in one of my maths classes. The focus of their conversation was a poster I have in my classroom which <strong>lists all the prime numbers below 10,000</strong>. One was commenting to the other that there seemed to be more prime numbers between 1 and 100 (25 primes) than there are between 100 and 200 (21 primes), and so they started counting. Then they counted that there only 16 primes between 200 and 300. They started wondering whether this <strong>decline in the distribution of primes</strong> continued - but then counted that there were 17 primes between 400 and 500 (more than between 200 and 300). But they thought there was something to their first hunch when they went to the end of the list and counted that there were only 9 primes in the last group of hundred - between 9,900 and 10,000. I entered the conversation around this time and suggested they do something to investigate further - but more than just searching around the internet. So they decided to write a <strong>computer program</strong> which motivated me to write a <strong>program on my TI-Nspire</strong> to count the number of primes in intervals of my choosing between a pair of integers.</p> <p>The image below left shows the <strong>scatter plot on my TI-Nspire</strong> created using the results from my program (shown right) which counted the <strong>number of primes between 1 to 10,000 in intervals of 100</strong>. That is, it counted the number of primes between 1 to 100, 100 to 200, 200 to 300, etc, up to between 9,900 to 10,000.</p> <p><img src="https://www.thinkib.net/cache/blog-thumbs/11/18454-1422449605-thinkib.jpg" alt="distribution of primes on calculator" /><br /><br /></p> https://www.thinkib.net/mathhlsl/blog/18454/distribution-of-primes-on-calculator#1422426600CAS - why not?
https://www.thinkib.net/mathhlsl/blog/18186/cas-why-not
Mon, 22 Dec 2014 03:30:00 +0000]]>CAS - why not?https://www.thinkib.net/cache/blog-thumbs/11/18186-1419292400-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/18186/cas-why-not
<p><img src="https://www.thinkib.net/cache/blog-thumbs/11/18186-1419292400-thinkib.jpg" alt="CAS - why not?" /><br /><br />In the mathematical world, <strong>CAS </strong>stands for <strong>computer algebra system</strong>, not creativity, action & service.</p> <p>Any calculator (I prefer the word "device") with CAS is not allowed to be used on an IB exam. But, in my opinion, this should not mean that an IB maths teacher never uses CAS (handheld device or computer software) as a teaching tool - or does not consider guiding his/her students to use CAS when appropriate.</p> <p>Ah, but that begs the question "when is using CAS in an IB maths class appropriate given that it's not allowed on an IB exam?" I certainly accept that many (most?) teaching IB maths would answer that question with "never" - but I disagree.</p> <p>How about just one example to illustrate why I disagree.</p> https://www.thinkib.net/mathhlsl/blog/18186/cas-why-not#1419219000Is Reality a Mathematical Structure?
https://www.thinkib.net/mathhlsl/blog/18129/is-reality-a-mathematical-structure
Mon, 15 Dec 2014 03:30:00 +0000]]>Is Reality a Mathematical Structure?https://www.thinkib.net/cache/blog-thumbs/11/18129-1418680135-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/18129/is-reality-a-mathematical-structure
<p><img src="https://www.thinkib.net/cache/blog-thumbs/11/18129-1418680135-thinkib.jpg" alt="Is Reality a Mathematical Structure?" /><br /><br /></p> <p><strong>Tags:</strong> <em>TOK, theory of knowledge, cosmology, philosophy</em></p>https://www.thinkib.net/mathhlsl/blog/18129/is-reality-a-mathematical-structure#1418614200counting primes
https://www.thinkib.net/mathhlsl/blog/18045/counting-primes
Tue, 02 Dec 2014 03:30:00 +0000]]>counting primeshttps://www.thinkib.net/cache/blog-thumbs/11/18045-1417484333-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/18045/counting-primes
<p><img src="https://www.thinkib.net/cache/blog-thumbs/11/18045-1417484333-thinkib.jpg" alt="counting primes" /><br /><br /></p> <p><strong>Tags:</strong> <em>prime number, program, calculator, proof</em></p>https://www.thinkib.net/mathhlsl/blog/18045/counting-primes#1417491000volume of a donut
https://www.thinkib.net/mathhlsl/blog/17974/volume-of-a-donut
Wed, 19 Nov 2014 03:30:00 +0000]]>volume of a donuthttps://www.thinkib.net/cache/blog-thumbs/11/17974-1416420630-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/17974/volume-of-a-donut
<p>One of my students came up with - what I thought - is a good idea for an Exploration: finding the volume of a donut. Or, more precisely, deriving a general formula for the volume of a torus. The torus (donut) can be interpreted as a solid of revolution where the solid is created by revolving a circle of radius r around a line such that the center of the circle is a distance R from the line (axis of revolution).</p> <p>Here is an image of a torus graphed in 3D parametric mode on a TI-Nspire CX (using the software rather than the handheld device).</p> <p><img src="https://www.thinkib.net/cache/blog-thumbs/11/17974-1416420630-thinkib.jpg" alt="volume of a donut" /><br /><br /></p> https://www.thinkib.net/mathhlsl/blog/17974/volume-of-a-donut#1416367800Dealing with Infinity
https://www.thinkib.net/mathhlsl/blog/17837/dealing-with-infinity-
Sun, 02 Nov 2014 03:30:00 +0000]]>Dealing with Infinity https://www.thinkib.net/cache/blog-thumbs/11/17837-1414942232-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/17837/dealing-with-infinity-
<p><img src="https://www.thinkib.net/cache/blog-thumbs/11/17837-1414942232-thinkib.jpg" alt="Dealing with Infinity " /><br /><br /></p> <p><strong>Tags:</strong> <em>infinity, series, sequence, geometric series</em></p>https://www.thinkib.net/mathhlsl/blog/17837/dealing-with-infinity-#1414899000Remembering Martin Gardner
https://www.thinkib.net/mathhlsl/blog/17769/remembering-martin-gardner
Sat, 25 Oct 2014 03:30:00 +0000]]>Remembering Martin Gardnerhttps://www.thinkib.net/cache/blog-thumbs/11/17769-1414252665-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/17769/remembering-martin-gardner
<p><img src="https://www.thinkib.net/cache/blog-thumbs/11/17769-1414252665-thinkib.jpg" alt="Remembering Martin Gardner" /><br /><br /></p> <p>I highly recommend including both books in your collection of teaching resources - and in your personal library.</p> <p><strong>Tags:</strong> <em>recreational mathematics, puzzles, resources</em></p>https://www.thinkib.net/mathhlsl/blog/17769/remembering-martin-gardner#1414207800Is infinity a number?
https://www.thinkib.net/mathhlsl/blog/17766/is-infinity-a-number
Fri, 24 Oct 2014 03:30:00 +0000<p><img alt="" src="files/mathhlsl/images/_Site/Blogs/infinity-img2.jpg" style="width: 236px; height: 236px; border-width: 0px; border-style: solid; float: right;" />Today I wanted to quickly convince my Maths SL students that for any triangle the ratio of the length of a side to the sine of the angle opposite it is always equal to the ratio of another side of the same triangle to the side opposite it - what we know as the Law of Sines (or Sine Rule). I used a sketch on the dynamic geometry software Geometers Sketchpad to do this. So, in front of the class (projected onto my whiteboard) I dragged around one of the vertices of a triangle and we watched to see what happened - observing that the three ratios (side length to sine of opposite angle) were always equal (measurements were updated dynamically in the sketch). It's fun to play with a dynamic image. One of my students 'played' with it and some of her classmates encouraged her to see what would happen if she tried to move a vertex so that the three vertices were collinear - thus, making it no longer a triangle but just a line segment (two of the angles having a measure of 0 degrees). What would happen? Watch the video below to see. Is infinity <img align="middle" alt="infinity" class="Wirisformula" data-mathml="«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»§#8734;«/mo»«/math»" src="/ckeditor//plugins/wiris/integration/showimage.php?formula=82b86695a433d33a65325bbec67a8805.png" /> a number?</p> <p><iframe allowfullscreen="" frameborder="0" height="315" src="//www.youtube.com/embed/JCHCiP8MDTQ?rel=0" width="420"></iframe></p> <p><img alt="" src="files/mathhlsl/images/_Site/Blogs/law-of-sines-img1.jpg" style="width: 430px; height: 329px; border-width: 1px; border-style: solid;" /></p> <p><img alt="" src="files/mathhlsl/images/_Site/Blogs/law-of-sines-img2.jpg" style="width: 400px; height: 416px; border-width: 1px; border-style: solid;" /><br /> </p> <p>But, infinity is <strong>not </strong>a number, but a concept. Not easy to communicate to students. Click <a href="http://www.bbc.co.uk/programmes/p01mwtqf" target="_blank">here </a>to listen to a clip from the BBC radio program <a href="http://www.bbc.co.uk/programmes/p01mwtqf" target="_blank">The Infinite Monkey Cage</a> - and you'll hear why it's not so easy to describe.</p> <p><strong>Tags:</strong> <em>infinity, geometers sketchpad, law of sines</em></p>https://www.thinkib.net/mathhlsl/blog/17766/is-infinity-a-number#1414121400the invention(?) of logarithms 400 years ago
https://www.thinkib.net/mathhlsl/blog/17740/the-invention-of-logarithms-400-years-ago
Sun, 19 Oct 2014 03:30:00 +0000<p><img alt="" src="files/mathhlsl/images/_Site/Blogs/logarithms/John-Napier-img1.jpg" style="width: 220px; height: 220px; border-width: 0px; border-style: solid; float: left;" />In 1614, the Scottish nobleman, mathematician, scientist and theologian <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Napier.html" target="_blank">John Napier</a> (1550-1617) published <em>Mirifici Logarithmorum Canonis Descriptio </em>(A Description of the Wonderful Law of Logarithms). This Latin text explained (although it consisted mostly of tables of numbers) Napier's invention of logarithms. His motivation is made clear in the book's preface when he wrote (taken from 1616 English translation):</p> <p><em>Seeing there is nothing </em>(<em>right well-beloved Students of the Mathematics</em>)<em> that is so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.</em></p> <p><img alt="" src="files/mathhlsl/images/_Site/Blogs/logarithms/Napiers-table-img.jpg" style="width: 320px; height: 210px; border-width: 0px; border-style: solid; float: right;" />The logarithms that Napier invented are not the same as common logarithms (base 10) and natural logarithms (base e) with which all Maths HL & SL students are very familiar. However, Napier certainly introduced the central concept of logarithms where one manipulates the powers (exponents) of numbers to carry out a calculation (multiplication, division, roots, etc) instead of performing the tedious calculation directly - thus, turning difficult multiplications and divisions into simple additions and subtractions. The invention was incredibly beneficial to scientists - especially astronomers - who routinely had to perform operations with very large numbers. The French mathematician and astronomer <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Laplace.html" target="_blank">Pierre-Simon Laplace</a> (1749-1827) is reputed to have said that logarithms, "by shortening the labours, doubled the life of the astronomer."</p> <p><img alt="" src="files/mathhlsl/images/_Site/Blogs/logarithms/slide-rule-book.jpg" style="width: 190px; height: 280px; border-width: 0px; border-style: solid; float: left;" />The English mathematician <a href="http://www-history.mcs.st-and.ac.uk/Biographies/Briggs.html" target="_blank">Henry Briggs</a> (1561-1630) met with Napier and made improvements to Napier's invention - the first was changing logarithms to have a base of 10 (common logarithms). After the invention of logarithms came the invention of the logarithmic scale by <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Gunter.html" target="_blank">Edmund Gunter </a>(1581-1626). Gunter was able to perform calculations by using dividers (a device similar to a compass) to add and subtract distances along the logarithmic scale. A few years later, the English mathematician <a href="http://www-history.mcs.st-and.ac.uk/Mathematicians/Oughtred.html" target="_blank">William Oughtred</a> (1574-1660) - who invented the notation <strong>×</strong> for multiplication - put two logarithmic scales together (so that one could slide along the other) and inventing the first linear slide rule. In 1632 he published <em>Circles of Proportion and the Horizontal Instrument</em> which described slide rules and sundials. For over 300 years, the slide rule was the most effective handheld computing device.</p> <p><img alt="" src="files/mathhlsl/images/_Site/Blogs/logarithms/using-slide-rule.jpg" style="width: 243px; height: 212px; border-width: 0px; border-style: solid; float: left;" /><img alt="" height="198" src="/files/mathhlsl/images/_Site/Blogs/logarithms/slide-rule-img3.jpg" style="border-width: 0px; border-style: solid; float: left;" width="310" /><br /></p> <p>Up to this point in this blog entry some form of the word <strong>invent </strong>is used eight times. Virtually all references to John Napier state that he "invented" logarithms. Was it an 'invention' or a 'discovery'? Certainly, all would agree that Oughtred's introduction of the <strong>×</strong> symbol for multiplication is an invention because the symbol was new and had not been used for this purpose before, but Oughtred did not invent multiplication. Early in the 18th century the greek letter <strong><span class="texhtml">π</span></strong> (pi) started to be used to represent the ratio of a circle's circumference to its diameter. Again, most would agree that the use of this symbol was an invention but what about the number (i.e. ratio of circumference to diameter) that the symbol represents - was that invented? Most would say 'no'. Even if humans did not exist, wouldn't the ratio of the circumference of any circle to its diameter still be the same constant that we decided to represent with the greek letter <strong><span class="texhtml">π </span></strong>?</p> <p>No doubt Napier <em>invented </em>the name <strong>logarithm </strong>(or, in Latin, <em>logarithmus</em>) which combined the Greek words <em>logos </em>(reckoning or ratio) and <em>arithmos </em>(number). But did he <em>invent </em>the idea behind logarithms, or did he simply uncover and utilize a concept that already existed and was waiting to be discovered by someone who was sufficiently astute and motivated? A discussion of this question can be a very useful and interesting way to integrate some <strong>TOK thinking/activity</strong> into a maths lesson. It can also serve as the starting point of an activity that could be used in the <strong>TOK </strong>classroom itself.</p> <p>Consider the following passage from the <a href="http://www.stir.ac.uk/events/calendarofevents/400yearsoflogarithms1614-2014-thestoryof/" target="_blank">University of Stirling (Scotland)</a> announcing a lecture to commemorate what it refers to as the "400th anniversary of Napier's <strong>discovery </strong>of logarithms." (bold highlights added here and below)</p> <p><em>One of the most important <strong>discoveries </strong>in the history of mathematics was the <strong>invention </strong>in 1614 of logarithms as a calculating aid by the famous Scottish mathematician John Napier. This came at an opportune time for the great astronomers of the day who were making massive strides in their understanding of the movement of the planets. Those of a certain age will also remember using logarithms in school before the days of electronic calculators! To celebrate this quatercentenary we shall have a look at Napier's life and some of his other <strong>inventions </strong>as well as seeing why logarithms proved such an important breakthrough.</em></p> <h5>The question is whether mathematics is an invention (a creation of the human mind) or a discovery (something that exists independent of us).</h5> <p><strong>Intuitionists </strong>say that mathematics is a creation of the human mind; therefore, it is <strong>invented </strong>by humans. Any mathematical idea or object exists only in our mind and does not have an existence independent of us.</p> <p>Alternatively, <strong>Platonists </strong>claim that any mathematical idea or object exists and we can only "see" them through our mind. This means that any mathematical concept or construction exists independent of humans and can, therefore, be <strong>discovered</strong>.</p> https://www.thinkib.net/mathhlsl/blog/17740/the-invention-of-logarithms-400-years-ago#1413689400GDC - visual check ... solution?
https://www.thinkib.net/mathhlsl/blog/17679/gdc-visual-check-solution
Tue, 07 Oct 2014 03:30:00 +0000<p><img alt="" src="files/mathhlsl/images/_Site/Blogs/quad_funct_disc_graph.jpg" style="width: 265px; height: 266px; border-width: 1px; border-style: solid; float: left;" />Today I posed a fairly common type of 'quadratic functions' question to my first year Maths HL students. The question is also suitable for SL students although questions like this are less common on SL exams. It is a 'fair' question for both HL and SL students. The reason I'm highlighting it in a blog post came about because the way in which I and some of my students verified our answer to it brings up some interesting questions about how technology might blur the distinction between verification (checking answer) and proof (formal, analytic process of obtaining answer).</p> <p><img alt="crying" class="ico" height="21" src="/img/tib-icons/hex-tok-24.png" title="crying" width="24" /> The ensuing brief discussion in class was an opportunity to ask an interesting question about knowledge. Are results (knowledge) obtained by means of technology any more or less reliable than that obtained without the use of technology?</p> <p>OK, below is the question and its solution. But, like I said, what generated an interesting discussion was how some of us decided to use the dynamic features of our TI-Nspire CX handheld device (not 'calculator') to check the answer ... and whether this check could be considered a legitimate way to solve/prove.</p> <p ><img alt="" src="files/mathhlsl/images/_Site/Blogs/discriminant-question-solution.jpg" style="width: 577px; height: 232px; border-width: 0px; border-style: solid;" /><br /></p> <p ></p> <p>Below is a short video of what we did on a TI-Nspire CX to check the answer. If we skipped the analytic solution (shown above) and obtained the answer from the animated graph shown below, would that be considered a legitimate solution?</p> <p ><iframe allowfullscreen="" frameborder="0" height="315" src="//www.youtube.com/embed/etI5a2r48rY?rel=0" width="420"></iframe></p> https://www.thinkib.net/mathhlsl/blog/17679/gdc-visual-check-solution#1412652600free graphing technology-1
https://www.thinkib.net/mathhlsl/blog/17645/free-graphing-technology-1
Sun, 28 Sep 2014 03:30:00 +0000]]>free graphing technology-1https://www.thinkib.net/cache/blog-thumbs/11/17645-1411942465-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/17645/free-graphing-technology-1
<p><img src="https://www.thinkib.net/cache/blog-thumbs/11/17645-1411942465-thinkib.jpg" alt="free graphing technology-1" /><br /><br /> </p> https://www.thinkib.net/mathhlsl/blog/17645/free-graphing-technology-1#1411875000Yang Hui's triangle
https://www.thinkib.net/mathhlsl/blog/16609/yang-huis-triangle
Wed, 26 Mar 2014 10:22:00 +0000]]>Yang Hui's trianglehttps://www.thinkib.net/cache/blog-thumbs/11/16609-1395851760-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/16609/yang-huis-triangle
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/11/16609-1395851760-thinkib.jpg" alt="Yang Hui's triangle" /><br /><br />Perhaps the most well-known two-dimensional number pattern is the triangular array of binomial coefficients most commonly referred to as <strong>Pascal's triangle</strong>. However, the pattern had been known and studied well before <strong>Blaise Pascal (1623-1662)</strong>. In centuries before Pascal, mathematicians in India, Greece, Iran, China, Germany and Italy had studied it - although it is fair to say that Pascal conducted the most thorough and organized study of this fascinating array of numbers.</p> <p><strong>Tags:</strong> <em>binomial coefficients,combinations,binomial expansion,Pascal,Tartaglia</em></p>https://www.thinkib.net/mathhlsl/blog/16609/yang-huis-triangle#1395829320GDC solution times - Specimen P2 question
https://www.thinkib.net/mathhlsl/blog/16475/gdc-solution-times-specimen-p2-question
Sun, 02 Mar 2014 10:55:00 +0000]]>GDC solution times - Specimen P2 questionhttps://www.thinkib.net/cache/blog-thumbs/11/16475-1393780575-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/16475/gdc-solution-times-specimen-p2-question
<p > <img src="https://www.thinkib.net/cache/blog-thumbs/11/16475-1393780575-thinkib.jpg" alt="GDC solution times - Specimen P2 question" /><br /><br /></p> <p><strong>Tags:</strong> <em>TI-Nspire,GDC,calculator,tips</em></p>https://www.thinkib.net/mathhlsl/blog/16475/gdc-solution-times-specimen-p2-question#1393757700cosine of pi/5
https://www.thinkib.net/mathhlsl/blog/16384/cosine-of-pi5
Tue, 18 Feb 2014 07:26:00 +0000]]>cosine of pi/5https://www.thinkib.net/cache/blog-thumbs/11/16384-1392730348-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/16384/cosine-of-pi5
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/11/16384-1392730348-thinkib.jpg" alt="cosine of pi/5" /><br /><br /> is in the first quadrant.</p> <p><strong>Tags:</strong> <em>trigonometry,sine rule,trigonometric identities</em></p>https://www.thinkib.net/mathhlsl/blog/16384/cosine-of-pi5#1392708360Conditional Probability Brainteaser
https://www.thinkib.net/mathhlsl/blog/16326/conditional-probability-brainteaser
Tue, 04 Feb 2014 21:03:00 +0000]]>Conditional Probability Brainteaserhttps://www.thinkib.net/cache/blog-thumbs/11/16326-1391569663-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/16326/conditional-probability-brainteaser
<p> You have two coins in your pocket. One of them is double-headed and the other is a fair coin (head-tail). You randomly choose one of the coins from your pocket. You look at one side of it. Given that you see 'heads', what is the probability that you've chosen the fair coin?</p> <p > <img src="https://www.thinkib.net/cache/blog-thumbs/11/16326-1391569663-thinkib.jpg" alt="Conditional Probability Brainteaser" /><br /><br /></p><p><strong>Tags:</strong> <em>probability,conditional probability,puzzle,paradox,tree diagram</em></p>https://www.thinkib.net/mathhlsl/blog/16326/conditional-probability-brainteaser#1391547780Unfamiliar Questions
https://www.thinkib.net/mathhlsl/blog/16120/unfamiliar-questions
Sun, 01 Dec 2013 19:39:00 +0000]]>Unfamiliar Questionshttps://www.thinkib.net/cache/blog-thumbs/11/16120-1385949096-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/16120/unfamiliar-questions
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/11/16120-1385949096-thinkib.jpg" alt="Unfamiliar Questions" /><br /><br />A common comment from examiners that appears on Maths HL Subject Reports is that teachers are strongly encouraged to take measures to develop their students' problem solving skills and their ability to tackle unfamiliar exam questions. This is especially important for Maths Higher Level. Below are some excerpts from past Maths HL Subject Reports in the section entitled "Recommendations and guidance for the teaching of future students":</p>https://www.thinkib.net/mathhlsl/blog/16120/unfamiliar-questions#1385926740Counting by Twelves
https://www.thinkib.net/mathhlsl/blog/15597/counting-by-twelves
Mon, 22 Jul 2013 20:41:00 +0000]]>Counting by Twelveshttps://www.thinkib.net/cache/blog-thumbs/11/15597-1374547659-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/15597/counting-by-twelves
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/11/15597-1374547659-thinkib.jpg" alt="Counting by Twelves" /><br /><br /> Why is our number system based on the number 10 and not some other number?</p> <p> All very interesting ... but how do you deal with them during class?</p> <p><strong>Tags:</strong> <em>sexagesimal,history,counting</em></p>https://www.thinkib.net/mathhlsl/blog/15597/counting-by-twelves#1374525660Drawing a Line with Circles
https://www.thinkib.net/mathhlsl/blog/15405/drawing-a-line-with-circles
Mon, 27 May 2013 18:01:00 +0000]]>Drawing a Line with Circleshttps://www.thinkib.net/cache/blog-thumbs/11/15405-1369700632-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/15405/drawing-a-line-with-circles
<h4> Cycloids</h4> <p> I've always been fascinated with cycloids. It's a shape that is best illustrated not by a formula but by a particular type of motion. A cycloid is the shape that is traced out by a fixed point on a circle as the circle rolls (without slipping) along a line.</p> <p > <img src="https://www.thinkib.net/cache/blog-thumbs/11/15405-1369700632-thinkib.jpg" alt="Drawing a Line with Circles" /><br /><br /></p> <p> It seems that the number of cusps is equal to the ratio of the larger radius to the smaller radius provided that the ratio is an integer.</p> <p> What do you think happens when the radius of the larger circle is <strong>two </strong>times that of the smaller circle? Before you play the animation below and see the answer to this question - try and guess the answer given what you know about the 4-cusped hypocycloid and the 3-cusped hypocycloid shown above. Try to visualize the answer - and then play the animation. The radius of the larger black circle is two times the radius of the rolling blue circle.</p> <p> <iframe allowfullscreen="" frameborder="0" height="440" src="http://www.youtube.com/embed/Qfq1pTXntcU?HD=1;rel=0;showinfo=0;controls=0;modestbranding=1" width="440"></iframe></p> <p><strong>Tags:</strong> <em>cycloid,hypocycloid,geometry</em></p>https://www.thinkib.net/mathhlsl/blog/15405/drawing-a-line-with-circles#1369677660Do not completely trust your GDC
https://www.thinkib.net/mathhlsl/blog/15290/do-not-completely-trust-your-gdc
Sun, 21 Apr 2013 11:35:00 +0000]]>Do not completely trust your GDChttps://www.thinkib.net/cache/blog-thumbs/11/15290-1366566151-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/15290/do-not-completely-trust-your-gdc
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/11/15290-1366566151-thinkib.jpg" alt="Do not completely trust your GDC" /><br /><br /></p> <p><strong>Tags:</strong> <em>GDC,technology</em></p>https://www.thinkib.net/mathhlsl/blog/15290/do-not-completely-trust-your-gdc#1366544100What About Computer Algebra Systems (CAS)?
https://www.thinkib.net/mathhlsl/blog/15247/what-about-computer-algebra-systems-cas
Sat, 06 Apr 2013 19:41:00 +0000]]>What About Computer Algebra Systems (CAS)?https://www.thinkib.net/cache/blog-thumbs/11/15247-1365300556-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/15247/what-about-computer-algebra-systems-cas
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/11/15247-1365300556-thinkib.jpg" alt="What About Computer Algebra Systems (CAS)?" /><br /><br /></p> <p> In the nearly 30 years since then, computer algebra systems have been much more user friendly, more powerful, and far more accessible. It used to be that one who have to spend quite a chunk of money to buy a CAS software package such as Mathematica or Maple. One of the exceptions was the program Derive which was developed by the same company, Soft Warehouse, that produced muMATH. Derive was relatively inexpensive, did not take up an enormous amount of memory, and there were considerable number of mathematics educators actively writing materials and promoting its usefulness as a teaching and learning tool. The <a href="http://www.austromath.at/dug/" target="_blank" title="International Derive Users Group (DUG)">International Derive User Group (DUG)</a> was founded in 1991 and is still active. Derive was purchased by Texas Instruments and the CAS capability on the handheld calculator models TI-89 and TI-92 (Voyage 200) was based on Derive.</p> <p> During the latter half of the 1980s and the 1990s many teachers (me included) were convinced that CAS was going to have a major impact on how mathematics is taught. I have no research to back me up, but my sense is that the feeling that CAS can (and should?) change teaching practices is held by a significantly smaller portion of the mathematics education community than, say, 20 years ago. If so, why is that? I'm not sure, but there is no doubt that the accessibility of CAS - and, hence, the opportunity to use it in the classroom is so much easier nowadays. Take, for example, the computational search engine <strong><a href="http://www.wolframalpha.com/" target="_blank" title="Wolfram Alpha">Wolfram Alpha</a></strong> - developed by Wolfram Research and based on its mega-powerful (and expensive) mathematics software pacakage Mathematica. What do you think of the following widget created by <a href="http://www.andykemp.org.uk/" target="_blank" title="Andy Kemp">Andy Kemp</a> from Wolfram Alpha showing the power of CAS as a teaching and learning tool ?</p> <script type="text/javascript" id="WolframAlphaScripte951ccd95572a67138f4572c1c7d7ee8" src="http://www.wolframalpha.com/widget/widget.jsp?id=e951ccd95572a67138f4572c1c7d7ee8"></script><p><strong>Tags:</strong> <em>computer algebra,CAS,wolfram alpha</em></p>https://www.thinkib.net/mathhlsl/blog/15247/what-about-computer-algebra-systems-cas#1365277260quod erat demonstrandum
https://www.thinkib.net/mathhlsl/blog/15170/quod-erat-demonstrandum
Sat, 23 Feb 2013 11:51:00 +0000]]>quod erat demonstrandumhttps://www.thinkib.net/cache/blog-thumbs/11/15170-1361642487-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/15170/quod-erat-demonstrandum
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/11/15170-1361642487-thinkib.jpg" alt="quod erat demonstrandum" /><br /><br />There is a great little book entitled <strong><a href="http://www.woodenbooks.com/browse/qed/index.php" target="_blank" title="QED - Beauty in Mathematical Proof">QED - Beauty in Mathematical Proof</a></strong> that I keep on my desk at school - not as a resource for juicy example questions to use in class lectures, but as something to share with individual students when they ask me a question which reveals a genuine interest in how mathematics works. I've also used as a resource for discussions about the nature of mathematical proof in TOK classes. I've also used it in pre-IB classes to offer students a nice insight into some classic examples of an elegant proof in mathematics, with emphasis on 'elegant'. I've even used it to assist students writing an extended essay in mathematics - when they've given me the impression that they do not fully understand what a proof in mathematics entails. And, of course, I hold it up in class the first time that I write the acronymn Q.E.D. at the end of finishing a brief proof - or, more likely, after answering a "show that" question which commonly appear on IB mathematics exams.</p> <p> The simple act of writing Q.E.D. at the end of some work in a classroom lesson gives a teacher an opportunity to give a mini-lesson about the nature of proof mixed with a little history of mathematics. Any discussion has to start (hopefully initiated by a student asking what Q.E.D. stands for) with giving a translation of the phrase "quod erat demonstrandum". The most accepted translation into English is "that which was to be demonstrated (or proved)".</p> <p><strong>Tags:</strong> <em>QED,show that,proof</em></p>https://www.thinkib.net/mathhlsl/blog/15170/quod-erat-demonstrandum#1361620260The Monty Hall Problem
https://www.thinkib.net/mathhlsl/blog/15136/the-monty-hall-problem
Tue, 12 Feb 2013 20:26:00 +0000]]>The Monty Hall Problemhttps://www.thinkib.net/cache/blog-thumbs/11/15136-1360722801-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/15136/the-monty-hall-problem
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/11/15136-1360722801-thinkib.jpg" alt="The Monty Hall Problem" /><br /><br />Like much of mathematics, without much effort it is certainly possible to teach probability in a stale and uninteresting way. I find that engaging the students with some well known probability paradoxes can keep them interested and usually generates some very constructive classroom discussions - and the teaching of some basic probability skills and concepts are even reinforced. Certainly one of the most well known, and intriguing, probability paradoxes is the Monty Hall Problem - to which it is usually referred.</p> <p> <span >[ under constructiopn ]</span></p> <p><strong>Tags:</strong> <em>probability,paradox,monty hall,three doors</em></p>https://www.thinkib.net/mathhlsl/blog/15136/the-monty-hall-problem#1360700760Probability, Coincidences and Birthdays
https://www.thinkib.net/mathhlsl/blog/15088/probability-coincidences-and-birthdays
Thu, 31 Jan 2013 21:31:00 +0000]]>Probability, Coincidences and Birthdayshttps://www.thinkib.net/cache/blog-thumbs/11/15088-1359690739-thinkib.jpghttps://www.thinkib.net/mathhlsl/blog/15088/probability-coincidences-and-birthdays
<p> <img src="https://www.thinkib.net/cache/blog-thumbs/11/15088-1359690739-thinkib.jpg" alt="Probability, Coincidences and Birthdays" /><br /><br />On our past two family holidays, my family and I have experienced very unlikely (or at least what we felt to be very unlikely) coincidences - namely, having chance meetings with people we know well while travelling far from home. One of these "unlikely" coincidences occurred when we were visiting Washington DC last spring. On our first day in the city we decided to go to the popular Air & Space Museum. There were literally thousands of people visiting the museum the day we were there. While walking around the exhibits we literally ran into a teaching colleague of ours - someone who has taught all three of our children. All involved were astounded by this event. Three days previously we were teaching in the same school and then bump into each other in a very crowded museum thousands of miles and on the other side of an ocean. We took the required group photo and after returning to school the next week, we shared the story with many colleagues - who all expressed the same keen sense of our chance meeting being a very unlikely coincidence. But is it really that unlikely?</p> <p> <span ><strong>[under construction]</strong></span></p> <p><strong>Tags:</strong> <em>probability,birthday problem,coincidences,complement</em></p>https://www.thinkib.net/mathhlsl/blog/15088/probability-coincidences-and-birthdays#1359667860Snow Day
https://www.thinkib.net/mathhlsl/blog/15053/snow-day
Wed, 23 Jan 2013 00:00:00 +0000<p><img alt="" class="left noborder" height="200" src="files/mathhlsl/images/Blogs/snowflake_img0.jpg" title="snowflake" width="200" />There are plusses and minuses of living in any particular location – and one of the plusses of teaching in the north of Scotland is that during the winter months we sometimes get enough snow that school is cancelled. Not everyone is going to consider that a plus (and depends on what particular day it occurs) but when you live near a nice hill for sledding (or "sledging", if you’re British) – as I and my family do – then a day cancelled due to snow means some fun sliding down a hill on some piece of equipment that usually offers little in terms of directional control (but a very low coefficient of friction). Today our school was closed due to snow, so me and my 13-year old daughter spent some time on our local sledding hill. My inability to control the direction (and spin) of my sled is obvious in the video below. I was not thinking about school - just enjoying the snow - but once I got home I did have some thoughts about the association between snow and mathematics, and even an idea for a good student exploration that would fit into the new internal assessment program. See below the video.</p> <p><iframe allowfullscreen="" frameborder="0" height="315" src="http://www.youtube.com/embed/NQG67DknqT8?rel=0" width="560"><br /> </iframe></p> <h4>Snow and Mathematics</h4> <p>After returning home from the hill and warming up with some hot chocolate– and knowing that snow would be a topic of discussion at school – I thought of opportunities for mathematical exploration in connection to snow. It just so happens that I came across a wonderfully written short piece on associations between winter and mathematics in a recent edition of The New Yorker magazine submitted by Gregory Buck, a professor of mathematics at Saint Anselm College in Manchester, New Hampshire USA and called <strong><a href="http://www.newyorker.com/online/blogs/culture/2012/12/the-wondrous-mathematics-of-winter-and-snow.html" target="_blank" title="The Wondrous Mathematics of Winter">The Wondrous Mathematics of Winter</a></strong>. I may not have my Math HL students read it but I will definitely share it with my Theory of Knowledge classes. Reading it again got me to thinking of some exploration ideas that would investigate how to do some mathematical modelling (or "modeling", if you're American) connected to snow.</p> <p>We're told that each snowflake displays some innate symmetry and that each snowflake is unique. Apparently this is due to the conditions in which a snowflake is created. A snowflake is first 'born' when water vapor in a cloud freezes around a dust particle or a bit of pollen. As water molecules are added the snowflake 'grows' but the way in which this growth occurs is affected by temperature and humidity which will vary depending on the path the growing snowflake takes to descend from the cloud to the ground. Until recently this process has proven very difficult to model using mathematics. Nearly a year ago, a team of mathematicians based at universities in Germany and the UK succeeded in modelling snowflake growth on a computer using laws of physics and mathematics. See the article <strong><a href="http://www.scientificamerican.com/article.cfm?id=how-do-snowflakes-form" target="_blank" title="Snowflake Growth Successfully Modeled from Physical Laws">Snowflake Growth Successfully Modeled from Physical Laws</a></strong> in the March 16, 2012 issue of Scientific American. The mathematics used in the computer model involved partial differential equations. So this is a topic that is beyond Math HL, even the Calculus HL Option, so it is not a suitable topic for students to research with a possible exploration in mind to satisfy their internal assessment requirement. But I would suggest having students read about this example of mathematical modelling (e.g. the Scientific American article) to illustrate a good real-life example of using mathematics to model a natural process. I do have an idea that involves mathematical modelling and snow that would be suitable for a student exploration - probably best suited for a student in HL.</p> <p><span ><strong>Modelling Melting Snow</strong></span></p> <p><img alt="" class="right noborder" height="215" src="files/mathhlsl/images/Blogs/snowman1.jpg" title="snowman" width="178" />Anyone who lives in a location that gets enough snow to cancel school from time to time will have seen (or been involved in) the building of a snowman. Inevitably the snowman melts away - but at what rate does the snow melt? Of course, it depends on various factors - the most important of which is temperature. If it is known that a particular portion (e.g. half) of a snowman melts away during a certain amount of time (e.g. 12 hours), then can we determine how long it will take for the rest of the snowman to melt? Questions about the rate at which a snowman, or any particular formation of snow, melts is a ripe area for applying some mathematical modeling. And I think that it can serve as a good example of illustrating to students the characteristics of a mathematical model and some insights into how one goes about trying to construct a mathematical model - especially of a natural process, such as snow melting.</p> <p><span ><strong>Other resources</strong></span></p> <p>I found a nice Java applet that demonstrates dynamically how the differential calculus topic of <strong>related rates</strong> (only in HL syllabus) can provide a useful way to mathematically analyse how a snowball melts. See <strong><a href="http://www.calculusapplets.com/snowballproblem.html" target="_blank" title="Related Rates - Melting a Snowball">Related Rates - Melting a Snowball</a></strong>.</p> <p>If you have the free <a href="http://demonstrations.wolfram.com/download-cdf-player.html" target="_blank" title="Wolfram CDF Player">Wolfram (Mathematica) CDF Player</a> installed on your computer, you can also interact with a dynamic illustration (shown below) of solving the following <strong>related rates problem</strong>: If a snowball melts so that its surface area decreases at a rate of 1 <img align="middle" class="Wirisformula" data-mathml="«math style=¨font-size:12px¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«msup»«mtext»cm«/mtext»«mn»2«/mn»«/msup»«mo»/«/mo»«mtext»min«/mtext»«/math»" src="http://www.inthinking.co.ukhttp://www.inthinking.co.uk/ckeditor/plugins/ckeditor_wiris/integration/showimage.php?formula=61d356b0bc7061016e7770eabf68e78b.png" /><br /> find the rate at which the diameter decreases when the diameter is 10 cm.<script type='text/javascript' src='http://demonstrations.wolfram.com/javascript/embed.js' ></script></p> <p><script type='text/javascript'>var demoObj = new DEMOEMBED(); demoObj.run('ASnowballsRateOfChange', '', '525', '403');</script></p> <div id="DEMO_ASnowballsRateOfChange"><a class="demonstrationHyperlink" href="http://demonstrations.wolfram.com/ASnowballsRateOfChange/" target="_blank">A Snowball's Rate of Change</a> from the <a class="demonstrationHyperlink" href="http://demonstrations.wolfram.com/" target="_blank">Wolfram Demonstrations Project</a> by Cindy Piao</div> <p><strong>Tags:</strong> <em>modelling, modeling, exploration, nature, snow, chain rule, related rates, CDF player, Wolfram Demonstrations Project</em></p>https://www.thinkib.net/mathhlsl/blog/15053/snow-day#1358899200