Harmonic Triangle

4716 /www/nrich/html/content/id/4716/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> This is the start of the harmonic triangle: \begin{array}{ccccccccccc} & & & & &\frac{1}{1} & & & & & \\ & & & & \frac{1}{2} & & \frac{1}{2} & & & & \\ & & & \frac{1}{3} & &\frac{1}{6} & & \frac{1}{3} & & & \\ & & \frac{1}{4} & &\frac{1}{12} & & \frac{1}{12} & & \frac{1}{4} & & \\ & \frac{1}{5} & & \frac{1}{20} & & \frac{1}{30} & & \frac{1}{20} & & \frac{1}{5} & \\ \frac{1}{6} & & \frac{1}{30} & & \frac{1}{60} & & \frac{1}{60} & & \frac{1}{30} & & \frac{1}{6}\\ & & & & & \ldots& & & & & \end{array} Each fraction is equal to the sum of the two fractions below it. Look at the triangle above and check that the rule really does work. Can you work out the next two rows? The $n$th row starts with the fraction $\frac{1}{n}$. <div style="font-weight: bold;"> </div> <div>We can continue the first diagonal ($\frac{1}{1}$, $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, and so on) using this rule. Take a look at the second diagonal: ($\frac{1}{2}$, $\frac{1}{6}$, $\frac{1}{12}$, $\frac{1}{20}$, and so on). What do you notice about the numerators and denominators of these fractions?</div> Can you prove the pattern will continue? What about the third and fourth diagonals? <a href="http://nrich.maths.org/public/viewer.php?obj_id=4745&part=">Click here for a poster of this problem</a>. <div style="font-weight: bold;"> </div> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Congratulations to Alice who sent in a correct solution to this problem: <mdo:image width="648" height="602" alt="solution" src="harmonic triangle.GIF"></mdo:image> Congratulations to Ray who also worked on this problem and sent us the following result: <div><mdo:image width="600" height="519" alt="Ray's solution" src="harmonic2.gif"></mdo:image></div> <div></div> <div> </div> </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=4716">This problem</a> provides a fraction-based challenge for students who already possess a good understanding of fraction addition and subtraction, and it leads to algebraic manipulation of that same process. <h3>Possible approach</h3> Silently, and with the full class attention (!) begin writing the triangle on the board, slowly, row by row. Students can put up hands when they know what is coming next. Allow whispered explanations, until everyone seems to have some idea, then invite explanations from students. In pairs let students generate as much of the pyramid array as they can. Bring the group together and ask about what is easy/hard, and any short cuts/observations anyone has made. Suggest that students work on one diagonal at a time, and redefine their task asfinding, and trying to prove, general methods for calculating numbers in this table, (for example, can they establish the second number in the $46$th row? The nth row? What about the third numbers?). Note : How far this problem goes will depend on the confidence students have at using algebra to represent and explore generality. The general term in the second diagonal should be accessible to most students who can manage algebraic fractions and many who can't but who can reason generally based on the patterns in the numerical values. There is no rush to finalise a proof for any term in the array, the algebra involved isn't completely simple and the reasoning based on the algebra needs to be thorough. But this is an excellent context in which to sense generality while proof requires some care and imagination. <h3>Key questions</h3> <ul> <li>What's the reason why your pattern must continue? How sure are you?</li> </ul> <ul> <li>Try to find the first case where it doesn't work.</li> </ul> <ul> <li>If you think it will continue indefinitely, can that be proved?</li> </ul> <h3>Possible extension</h3> Press students to justify their conjectures using algebraic reasoning, extended gradually to cover all terms across a row. <h3>Possible support</h3> <div>Learners might find it useful to use <a href="/content/id/4716/Harmonic%20Cards.doc">these cards</a> to recreate the triangle as a group activity. There are a few fractions missing and the blank cards can be completed to fill in the gaps.</div> <div><a href="http://nrich.maths.org/public/viewer.php?obj_id=2281&part=">Number Pyramids</a> and <a href="http://nrich.maths.org/public/viewer.php?obj_id=2282&part=">More Number Pyramids</a> are useful less-demanding challenges, using mainly whole numbers, and based on a similar structure.</div> An alternative, easier task working with unit fractions is <a href="http://nrich.maths.org/public/viewer.php?obj_id=1173&part=">Egyptian Fractions</a> This could be a replacement for, or a preliminary to Harmonic Triangles. <blockquote>Gordon Davis, who teaches at Colyton Grammar School in Devon, UK said:</blockquote> <blockquote>"After introducing the structure of the triangle briefly, I gave groups sugar paper and slips of paper with all the fractions that they would need for the first seven rows of the triangle. There was a massive amount of mental calculation, as students organised and stuck down their fractions. It worked well as there was a lot of space on the sugar paper for students to note down any observations they had. I asked them to highlight any of these comments that they could prove to be true always. We then spent most of the lesson trying to establish the terms on the 100th row."</blockquote> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> $$\frac{1}{2} = \frac{1}{3} + \frac{1}{6}$$ $$\frac{1}{3} = \frac{1}{4} + \frac{1}{12}$$ $$\frac{1}{12} = \frac{1}{20} + \frac{1}{30}$$ Look at the diagonal lines running from the right down to the left. The fractions in the first one are $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and so on. If $\frac{1}{n}$ is at the end of the nth row, the fraction above it must be $\frac{1}{n-1}$ and the fraction below it must be $\frac{1}{n+1}$. Have a look at the second diagonal (the one formed by taking the second number in each row: it starts $\frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20}$. Can you find a pattern for these numbers so that you can work them out easily (without having to subtract fractions)? Can you explain why the pattern works? </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0">Next two rows: $$\frac{1}{7}, \frac{1}{42}, \frac{1}{105}, \frac{1}{140}, \frac{1}{105}, \frac{1}{42}, \frac{1}{7}$$ $$\frac{1}{8}, \frac{1}{56}, \frac{1}{168}, \frac{1}{280}, \frac{1}{280}, \frac{1}{168}, \frac{1}{56}, \frac{1}{8}$$ Pattern: the second number in the $n$th row is $\frac{1}{n(n-1)}$. We can always fill the second diagonal, because $$\frac{1}{n}+\frac{1}{n(n-1)}=\frac{[(n-1)+1]}{[n(n-1)}=\frac{1}{(n-1)}$$ And this is clearly a unit fraction. </mdoxml> 2 5 0 0 1 0 0 Harmonic Triangle Can you see how to build a harmonic triangle? Can you work out the next two rows? 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