Mind your \Ps and \Qs

6382 /www/nrich/html/content/id/6382/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Here are 16 propositions involving a real number $x$: <table style="border-spacing:1px;center" border="1"> <tbody> <tr> <td style="" width="150">$x\int^x_0 ydy &lt; 0$</td> <td style="" width="150">$x&gt; 1$</td> <td style="" width="150">$0&lt; x&lt; 1 $</td> <td style="" width="150">$x^2+4x+4 =0$</td> </tr> <tr> <td>$x=0 $</td> <td>$\cos(x/2)&gt; \sin(x/2)$</td> <td>$x&gt; 2$</td> <td>$x=1$</td> </tr> <tr> <td>$2\int^{x^2}_0ydy&gt; x^2 $</td> <td>$x&lt; 0 $</td> <td>$x^2+x-2=0$</td> <td>$x=-2 $</td> </tr> <tr> <td>$x^3&gt; 1$</td> <td>$|x|&gt; 1$</td> <td>$x&gt; 4$</td> <td>$\int^x_0 \cos y dy =0$</td> </tr> </tbody> </table> [Note: the trig functions are measured in radians] By choosing $p$ and $q$ from this list, how many correct mathematical statements of the form $p\Rightarrow q$ or $p\Leftrightarrow q$ can you make? It is possible to rearrange the statements into four statements $p\Rightarrow q$ and four statements $p\Leftrightarrow q$. Can you work out how to do this? <div class="framework"> NOTES AND BACKGROUND Logical thinking is at the heart of higher mathematics: In order to construct clear, correct arguments in ever more complicated situations mathematicians rely on clarity of language and logic. Logic is also at the heart of computer programming and circuitry. To find out more, look at the ideas surrounding the <a href="http://nrich.maths.org/public/viewer.php?obj_id=5931&part=">Adding Machine</a> problem and related set of activities.</div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Leo, Lily, Pippa and Lewis from Fakenham College sent us this entertaining solution We transferred the equations onto cards so we could move them about easily. Good idea! These are the ones we got to match, then we coloured them in and added glitter.Glittery maths? Excellent idea . $ x< 0 \Leftrightarrow x\int^x_0 ydy < 0$ $x=0 \Rightarrow \int^x_0 \cos y dy =0$ $0< x< 1\Rightarrow \cos(x/2)> \sin(x/2) $ $x> 2\Rightarrow x^3> 1 $ $x> 4\Rightarrow 2\int^{x^2}_0ydy> x^2$ $x=-2\Rightarrow |x|> 1$ $x=1\Rightarrow x^2+x-2=0 $ We didn't manage to get four of < => , but were pleased we got this :) You should be pleased! This was good logical thinking. These seven were all correct, but there was a mistake in an eighth one you included: $x^2+4x+4 =0 \Rightarrow x> 1$. Can you see why? The full solution that we obtained was $x^3> 1 \Leftrightarrow x> 1$ $2\int^{x^2}_0ydy> x^2 \Leftrightarrow |x|> 1$ $x=-2 \Leftrightarrow x^2+4x+4 =0$ $x=1\Rightarrow x^2+x-2=0$ $0< x< 1 \Rightarrow \cos(x/2)> \sin(x/2)$ $x> 4\Rightarrow x> 2$ $x< 0 \Leftrightarrow x\int^x_0 ydy < 0$ $x=0 \Rightarrow \int^x_0 \cos y dy =0$ </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <h3>Why do this problem</h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=6382&part=">This problem</a> will help to train students in the art of careful, logical, pure thinking which will help to develop their general mathematical skill. It will require students to address issues surrounding integration, use of functions, and inequalities, without needing to go into any particular detail with calculation of integrals. <h3>Possible approach</h3> <div style="font-style: italic;">Note: This problem might work best once students have tried <a href="http://nrich.maths.org/public/viewer.php?obj_id=6331&part=">Iffy logic</a> as a starter at a previous time.</div> <div>Give the students Mind Your Ps and Qs and let them read it carefully and think about what it is asking.</div> <div>Suggest that students discuss in pairs what they think that the arrow symbols mean. Then, as a group discuss, for example, why</div> <div>$$x> 4 \Rightarrow x> 3\mbox{ and } x=-2 \Leftrightarrow x^3=-8$$</div> <div>are correct but</div> <div>$$|x|> 2 \Rightarrow x> 1 \mbox{ and }x^2=4\Leftrightarrow x=2$$</div> <div>are incorrect.</div> <div>The next step is to ensure that everyone can construct their own individual examples of correct mathematical statements using propositions from the list. Once students have a couple of examples of such statements they should share them with the class and explain their reasoning. Do others agree or disagree? TALKING about such results will quickly highlight woolly or fallacious thinking and is an important part of the mathematical process.</div> <div>Once the group has a feel for constructing the implications, they need to concentrate on using all of the statements to construct a complete set of 8 statements. Encourage students to consider their reasoning clearly in each case. Can the class complete the task with a clear explanation in each case?</div> <h3>Key questions</h3> <ul> <li>What do the arrow symbols mean?</li> <li>Have you read the question carefully?</li> <li>Are there certain statementswhich look likely to go together in a pair?</li> <li>If an integral is positive or zero, what can we say about the area enclosed?</li> <li>Do you remember what the graphs of $\cos$ and $\sin$ look like?</li> </ul> <h3>Possible support</h3> It is rather helpful to draw diagrams and number lines when thinking about inequalities. Shade the parts of the number line which apply to a particular inequality to help see which way round the logic flows. If possible, start off with <a href="http://nrich.maths.org/public/viewer.php?obj_id=6331&part=">Iffy logic</a> and the support materal suggested there. <h3>Possible extension</h3> Are there multiple solutions? Can students make up a similar set of questions to give to each other to try? Can they write down really clear explanations of why, for example, $x> 4\Rightarrow x> 2$? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> To do this question you need to be really sure what the two symbols $\Rightarrow$ and $\Leftrightarrow$ mean. $p\Rightarrow q$ essentially means that IF $p$ is True THEN $q$ is true. $p\Leftrightarrow q$ means that $p$ is true if and only if $q$ is true. Alternatively, they are either both true or both false. To get started, pick one of the simpler looking statements for $p$. Go through the other statements in turn and see which of them work as $q$ in $p\Rightarrow q$. You will soon be able to pair-off possible partners. Finally, don't forget to read the details of the question carefully! </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> $x^3> 1 \Leftrightarrow x> 1$ $2\int^{x^2}_0ydy> x^2 \Leftrightarrow |x|> 1$ $x=-2 \Leftrightarrow x^2+4x+4 =0$ $x=1\Rightarrow x^2+x-2=0$ $0< x< 1 \Rightarrow \cos(x/2)> \sin(x/2)$ $x> 4\Rightarrow x> 2$ $x< 0 \Leftrightarrow x\int^x_0 ydy < 0$ $x=0 \Rightarrow \int^x_0 \cos y dy =0$ </mdoxml> 2 4 0 0 0 0 1 Mind your \Ps and \Qs Sort these mathematical propositions into a series of 8 correct statements. Pre-Calculus and Calculus Integration Decision Mathematics and Combinatorics Logic Using, Applying and Reasoning about Mathematics Mathematical reasoning & proof Admin Stage 5 - Core Mapping