7282
/www/nrich/html/content/id/7282/
1
2011-05-26T13:06:28
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Charlie has been investigating square numbers. He decided to
organise his work in a table:<br></br>
<br></br>
<div style="text-align: center;"><mdo:image height="415" width="311" src="pythag.jpg" alt="table of square numbers"></mdo:image></div>
<br></br>
<br></br>
Charlie noticed some special relationships between certain square
numbers:<br></br>
<br></br>
$$3^2+4^2=5^2$$ $$5^2+12^2=13^2$$<br></br>
<br></br>
Sets of integers like ${3,4,5}$ and ${5,12,13}$ are called
<span style="font-weight: bold;">Pythagorean Triples</span>,
because they could be the lengths of the sides of a right-angled
triangle.<br></br>
<br></br>
He wondered whether he could find any more...<br></br>
<br></br>
Can you extend Charlie's table to find any more sets of Pythagorean
Triples where the hypotenuse is 1 unit longer than one of the other
sides?<br></br>
Do you notice any patterns?<br></br>
Can you make any predictions?<br></br>
<br></br>
<span style="font-weight: bold;">Can you find a formula that
generates Pythagorean Triples like Charlie's?</span><br></br>
<span style="font-weight: bold;">Can you prove that your formula
works?</span><br></br>
<br></br>
<br></br>
Alison has been working on Pythagorean Triples where the
hypotenuse is 2 units longer than one of the other sides.<br></br>
So far, she has found these:<br></br>
$$4^2 + 3^2 = 5^2$$ $$6^2+8^2=10^2$$ $$8^2+15^2=17^2$$<br></br>
<br></br>
Some of these are just scaled-up versions of Charlie's
triples, but some of them are new and can't be divided by a common
factor (these are called <span style="font-weight: bold;">primitive
triples</span>).<br></br>
<br></br>
Can you find more Pythagorean Triples like Alison's?<br></br>
<br></br>
<span style="font-weight: bold;">Can you find a formula for
generating Pythagorean Triples like Alison's?</span><br></br>
<span style="font-weight: bold;">Can you prove that your formula
works?</span><br></br>
<br></br>
Here are some follow-up questions you might like to consider:<br></br>
<br></br>
<ul>
<li>Can you find Triples where the hypotenuse is 3 units longer
than one of the other sides? Or 4 units longer? Or...?</li>
<li>Can you say anything about when such triples will be primitive
triples?</li>
</ul>
<br></br>
<span style="font-style: italic;">For a challenging extension
investigation, why not take a look at</span> <a style="font-style: italic;" href="http://nrich.maths.org/7542&part=">Few and Far
Between?</a> <br></br>
<a href="http://nrich.maths.org/7542&part="></a><br></br>
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<p><span class="editorial">Elliott from Wilson's school used a</span> <a href="/content/id/7282/Elliott.xls">spreadsheet</a> <span class="editorial">to find sets of triples where the hypotenuse is 1, 2, 3 and 4 units bigger than one of the other sides. He explains his method:</span></p>
<p>If you square each number, subtract one square from the square greater than it, then square root this number, you can find Pythagorean Triples. If the result is a whole number, the two numbers and the square rooted number make up a Pythagorean Triple.</p>
<p>For example, $24^2 = 576$, and $25^2 = 625$. $625-576 = 49 = 7^2$, so $(7, 24, 25)$ make up a Pythagorean Triple. </p>
<p>The first square, when the hypotenuse is one greater than the other side, is always odd, increasing by two each time.</p>
<p>$9^2 + 40^2 = 41^2$<br></br>
$11^2 + 60^2 = 61^2$<br></br>
$13^2 + 84^2 = 85^2$</p>
<p>If, instead of subtracting the square from the square one greater than it, you subtract it from the square two greater, you can work out the Pythagorean Triples similar to Alison's.</p>
<p>For example, $15^2 = 225$ and $17^2 = 289$. $289 - 225 = 64 = 8^2$ so $(8, 15, 17)$ makes up a Pythagorean Triple.</p>
<p>The first square number is always even and increases by two each time.</p>
<p>$10^2 + 24^2 = 26^2$<br></br>
$12^2 + 35^2 = 37^2$<br></br>
$14^2 + 48^2 = 50^2$<br></br>
</p>
<p>If the hypotenuse is three longer than a side, the shortest side increases by 6 each time (starting on 3) e.g. 3, 15, 21, 27 etc. If the hypotenuse is four units longer, the shortest side increases by 4 each time e.g. 4, 8, 12, 16 etc.</p>
<p><span class="editorial">Michael from St Patrick's in Australia used the following method:</span></p>
<p>All of Charlie's triples have an even square in them. Using that, the triples for Charlie are <br></br>
$1^2 + 0^2 = 1^2$<br></br>
$3^2 + 4^2 = 5^2$<br></br>
$5^2 + 12^2 = 13^2$<br></br>
$7^2 + 24^2 = 25^2$<br></br>
$9^2 + 40^2 = 41^2$<br></br>
$11^2 + 60^2 = 61^2$<br></br>
$13^2 + 84^2 = 85^2$<br></br>
There is a pattern. The smaller, odd, square number will always rise by two. The larger, even, square number will go up by multiples of four in order, 4,8,12,16,20,24 etc. The answer will always be one square number more than the largest number in the equation.<br></br>
The next equation in the pattern will be $15^2+112^2=113^2$. $15^2=225, 112^2=12544,$ and $113^2=12769$. $12544+225=12769$.</p>
<p><span class="editorial">Thomas, also from St Patrick's sent us his work on Alison's Pythagorean triples:</span></p>
<p>The hypotenuse is two units longer than one of the other sides. Can you find a formula for generating Pythagorean Triples like Alison's? Can you prove that your formula works?</p>
<p>The three examples given are: 4²+3²=5² 6²+8²=10² 8²+15²=17² There is a pattern here (from left to right) the first number goes up by 2. The second number goes up by 5 then by 7 so it only makes sense that in the next equation the second number would be (15+9)=24. The third number again goes up by 5 then 7 so the third number should be (17+9) =26 Using this info I can make the accurate
assumption that the next equation in line will be; 10²+24²=26². Now to prove this, 10²=100 and 24²=576 so 100+576=676. 26²=676. The next two equations in line will be 12²+35²=37² 14²+48²=50²</p>
<p><span class="editorial">I wonder if you can explain why Michael's and Thomas's patterns generate Pythagorean triples?</span></p>
<p><span class="editorial">Nathan from Rushmore School observed:</span></p>
<p>If $x+x+1=y^2$, $y,x,x+1$ make a Pythagorean triple.</p>
<p>Get two consecutive whole numbers that add up to a square number. For example, $4$ and $5$ - $4+5=9$ and $\sqrt9=3$ 3,4 and 5 make a Pythagorean triple. $12$ and $13$ - $12+13=25$ $\sqrt{25}=5$ 5,12 and 13 make a Pythagorean triple.<br></br>
The inverse of this is get any odd number ($x$) and square it. Halve the square number and add/take away half to make $y$ and $z$. $x,y$ and $z$ will be Pythagorean triples.</p>
<p>Once you have these numbers you can multiply the triples by $n$ to make more Pythagorean triples.</p>
<p><span class="editorial">Lucy from Belgium sent us her solution:</span></p>
<p>The formula is (2n+1)² + (2n²+2n)² = (2n²+2n+1)²</p>
<p>I tried it a few times :<br></br>
If n=1 => 3² + 4² = 5²<br></br>
If n=5 => 11² + 60² = 61²</p>
<p>Finding that you need an odd square number as the difference between the two other numbers wasn't so difficult. And then I found that between 4 and 12 there were 8 units, between 12 and 24 there are 12 units, and then 16 units and 20 units and so on... Which meant that the n was multiplied each time with the previous number+2<br></br>
n=1 was multiplied by 4 to get to 4<br></br>
n=2 was multiplied by 6 to get to<br></br>
12 n=3 was multiplied by 8 to get to 24 and so on again... in other words n was multiplied by the odd square number plus one => n(2n+2) => (2n²+2n) the last number is just one unit more so that is (2n²+2n+1)</p>
<p><span class="editorial">Well done Lucy! Can you show algebraically that the expression will always give a Pythagorean triple?</span></p>
<p><span class="editorial">We also received solutions from</span> <a href="/content/id/7282/Niharika.pdf">Niharika</a>, <a href="/content/id/7282/Chuyi.pdf">Chuyi</a> and <a href="/content/id/7282/Rajeev.pdf">Rajeev</a>.</p>
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<h3>Why do this problem?</h3>
<p>This problem connects different areas of mathematics. Students are required to apply their knowledge of Pythagoras' theorem and properties of numbers, while using their algebra skills to make generalisations and prove their conjectures.</p>
<h3>Possible approach</h3>
<div>"Look at this right-angled triangle:"</div>
<div style="text-align: center;"> <mdo:image alt="right-angled triangle" height="216" src="pythag3.jpg" width="305"></mdo:image></div>
<div> </div>
<div>"We know that the three sides have to satisfy the relationship $a^2 + b^2 = c^2$.</div>
<div>Can you find examples where $a, b$ and $c$ are whole numbers? Here is a table of square numbers you might find useful."</div>
<div style="text-align: center;"><mdo:image alt="List of square numbers" height="395" src="pythag2.jpg" width="211"></mdo:image></div>
<div>Give students a few minutes to seek sets of 3 numbers. Then bring the class together again and collect the examples they found on the board. <span style="font-style: italic;">If ${3, 4, 5}$ and ${6, 8, 10}$ have both been suggested, it may be worth taking the time to discuss whether we should be surprised that doubling a, b and c still maintained the Pythagorean relationship.</span></div>
<div> </div>
<div>For students who are wondering why the difference between consecutive square numbers is always odd, <a href="http://nrich.maths.org/content/id/2275/SumOddIntegers.htmlf">this representation</a> might be useful.</div>
<div> </div>
<div> </div>
<div>Now introduce the name <span style="font-weight: bold;">Pythagorean triple</span> to describe a set of three whole numbers that satisfy $a^2 + b^2 = c^2$.</div>
<div> </div>
<div>"<span style="font-weight: bold;">To start with, let's focus on Pythagorean Triples where b and c differ by just 1.</span> The annotations on this table might give you some ideas on how to find some more."</div>
<div style="text-align: center;"><mdo:image alt="" height="415" src="pythag.jpg" width="311"></mdo:image></div>
<p>Set students to work in pairs looking for more Pythagorean triples. "At the end of the lesson, I want you to be able to explain a method for generating more Pythagorean triples where $b$ and $c$ differ by just 1, and to be able to explain how you know your method will always work."<br></br>
<br></br>
For some students, it may be appropriate to nudge them towards algebraic representations to justify their findings. This could start with writing the relationship as $a^2 + b^2 = (b+1)^2$<br></br>
to explain why $a$ needs to be an odd number.<br></br>
<br></br>
Then students could move on to the more sophisticated idea of writing $a^2$ as $(2n+1)^2$ and using their results to find expressions for $b$ and $c$ in terms of $n$.<br></br>
<br></br>
If more lesson time is available, students can go on to use their insights from investigating the case where $b$ and $c$ differ by 1 to explore triples where they differ by more than 1. There are some suggestions for follow-up questions in the problem.</p>
<h3>Key questions</h3>
<div>Look at the table:</div>
<div><mdo:image alt="" height="93" src="pythag4.jpg" width="301"></mdo:image></div>
<div>Where is the $5^2$?</div>
<div>Where is the $12^2$?</div>
<div>Where is the $13^2$?</div>
<div> </div>
<div>Where might we find a similar set of related square numbers?</div>
<div> </div>
<h3>Possible extension</h3>
<div><a href="http://nrich.maths.org/7542&part=">Few and Far Between?</a> challenges students to find triples where $a$ and $b$ differ by just 1.</div>
<div> </div>
<h3>Possible support</h3>
<p><a href="http://nrich.maths.org/742&part=">What's Possible?</a> offers a rich exploration of properties of square numbers that could be a good starting point for working on this problem.</p>
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<mdoxml version="1.0">Here is an extract from Charlie's table:<br></br>
<mdo:image height="93" width="301" src="pythag4.jpg" alt=""></mdo:image><br></br>
Where is the $5^2$? <br></br>
Where is the $12^2$? <br></br>
Where is the $13^2$? <br></br>
<br></br>
Where might we find a similar set of related square
numbers? <br></br></mdoxml>
<mdoxml version="1.0">link to pythag triple articles<br></br></mdoxml>
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Generating Triples
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Numbers and the Number System
Square numbers
Numbers and the Number System
Properties of numbers
Algebra
Creating expressions/formulae
Using, Applying and Reasoning about Mathematics
Generalising
2D Geometry, Shape and Space
Pythagorean triples