2292
/www/nrich/html/content/id/2292/
1
2011-02-01T00:00:01
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<mdoxml version="1.0"><p>Charlie and Alison have been drawing patterns on coordinate grids. You may want to choose just one to explore or you may like to try all three.</p>
<p> </p>
<h4>Charlie's Squares<br></br>
</h4>
<div class="toggle"><br></br>
Charlie has been drawing squares.<br></br>
<br></br>
<mdo:image src="SquarePatternSmall.png"></mdo:image><br></br>
<br></br>
<strong>What will the coordinates of the centre of square number 3 be?</strong><br></br>
How do you know?<br></br>
<br></br>
Charlie wants to know where the centre of square number 20 will be.<br></br>
Can you use the diagram above to help you to work this out?<br></br>
<br></br>
Can you suggest a quick and efficient strategy for working out the coordinates of the centre of any square?<br></br>
<br></br>
Would your strategy work if Charlie's sequence extended to the left? $$....-2, -1, 0, 1, 2, 3....$$<br></br>
Can you adapt your strategy to work out the coordinates of the corners of <strong>any</strong> square?</div>
<p> </p>
<h4>Alison's Triangles</h4>
<p> </p>
<div class="toggle"><br></br>
Alison has been drawing triangles.<br></br>
<br></br>
<mdo:image src="TrianglePattern.png"></mdo:image><br></br>
<br></br>
She wants to know where the vertices of triangle number 23 will be.<br></br>
<br></br>
Can you use the diagram to work it out?<br></br>
<br></br>
Can you suggest a quick and efficient strategy for working out the coordinates of the vertices of <strong>any</strong> triangle?<br></br>
<br></br>
Would your strategy work if Alison's sequence extended to the left? $$....-2, -1, 0, 1, 2, 3....$$<br></br>
</div>
<h4> </h4>
<h4>More Squares from Charlie</h4>
<p> </p>
<div class="toggle"><br></br>
Charlie has been drawing more squares.<br></br>
<br></br>
<mdo:image src="MoreSquarePattern.png"></mdo:image><br></br>
<br></br>
He wants to know what the coordinates of the centre of square 22b will be.<br></br>
<br></br>
Can you use the diagram to work it out?<br></br>
<br></br>
Can you suggest a quick and efficient strategy for working out the coordinates of the vertices of <strong>any</strong> square?<br></br>
</div>
<p> </p>
<p> </p>
<p> </p>
<div class="framework">The ideas for these problems originally came from the SMP11-16 booklets on Coordinate Patterns published by CUP.</div>
<p><br></br>
<br></br>
</p></mdoxml>
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<mdoxml version="1.0"><p><span class="editorial">Wow, we received loads of solutions here! Let's have a look at a few of them:</span><br></br>
</p>
<h4>Charlie's Squares</h4>
<p><span class="editorial">William and Chris, from Croftlands Junior School, tried to approach Charlie's Squares this way:</span></p>
<p>We found that the best way to organise our information was to draw a table. Here is the table we drew:</p>
<table style="width: 271px" border="1">
<tbody>
<tr>
<td style="text-align: center;">Square</td>
<td style="text-align: center;">x-coordinate</td>
<td style="text-align: center;">y-coordinate</td>
</tr>
<tr>
<td style="text-align: center;">1</td>
<td style="text-align: center;">2</td>
<td style="text-align: center;">2</td>
</tr>
<tr>
<td style="text-align: center;">2</td>
<td style="text-align: center;">5</td>
<td style="text-align: center;">3</td>
</tr>
<tr>
<td style="text-align: center;">3</td>
<td style="text-align: center;">8</td>
<td style="text-align: center;">4</td>
</tr>
<tr>
<td style="text-align: center;">4</td>
<td style="text-align: center;">11</td>
<td style="text-align: center;">5</td>
</tr>
<tr>
<td style="text-align: center;">5</td>
<td style="text-align: center;">14</td>
<td style="text-align: center;">6</td>
</tr>
</tbody>
</table>
<p>Using the table, we found out that the x-coordinate was going up 3 every time. We added a couple of extra examples in the table. We spotted that the y-coordinate was the number of the square plus 1. Then we tried spotting patterns by working with the y coordinates to find the x coordinates.</p>
<p><span class="editorial">A great way to start! They then went on to find the general formula correctly. Callum, Elys, Cerys, Elgan, Cullen, Ethan, Ifan and Twm, from Ysgol Llanegryn, jumped straight in with the following:</span></p>
<p>We first found a pattern: (2,2), (5,3), (8,4), and we then discussed how to find any centre square. We turned to algebra. The n<sup>th</sup> term in the pattern is (3n -1, n + 1) so the coordinates of the 20th centre point would be (60-1, 20+1) which is (59,21).</p>
<p><span class="editorial">Justin, from the John of Gaunt School, correctly noted:</span></p>
<p>In the case of going to the left, the process is repeated but instead of increasing the horizontal and vertical coordinates, the coordinates decrease by the same amounts.</p>
<p><span class="editorial">Sam, from Fern Avenue, gave the following answer to Alison's Triangles:</span></p>
<p>I started by comparing triangles 1 and 3, and realised that 'middle' vertices on nearby odd triangles were exactly 8 points horizontally apart. That, with the fact that all odd middle vertices have coordinates of the form (x,10), allows you to find the middle vertex of any odd numbered triangle.</p>
<p>The same applies for all even numbered triangles, except that the vertices are on coordinates that are 4 grid squares to the right of the odd triangles.</p>
<p>After noticing this, I realised that the even isosceles triangles continued to the middle vertex of the odd numbered triangles, and vice-versa. This meant that they either went down or up 5 squares, and right 2 squares.</p>
<p>I now realised that if I knew the coordinates of any triangle, then I knew the middle vertex of the next triangle.</p>
<p>Therefore, triangle 23 will have vertices at (88,5), (90,10), and (92,5).</p>
<p><span class="editorial">Joe and Jack, from Springfield Primary School, gave us their formula for the x-coordinate of the top/bottom vertex of the n<sup>th</sup> triangle, which was 4n-2. From this they plugged in n = 23, which gave them the correct x-coordinate, and noticed that the y-coordinate alternates between 10 (when n was odd) and 0 (when n was even). Great!</span><br></br>
</p>
<h4>More Squares from Charlie</h4>
<p><span class="editorial">Penny, Jacob, Patrick and Cameron, from Inter Lakes, submitted solutions to More Squares from Charlie. Cameron wrote:</span></p>
<p>The coordinates of the centre of square 22b are (44,-36). To work this out, first I made a chart for the x-coordinates and the y-coordinates. To get from one 'b' square to the next, add 2 to the x-coordinate and subtract 2 from the y-coordinate. My quick and efficient strategy is knowing the x and y axis rules.</p>
<p><span class="editorial">Brilliant!</span><br></br>
<br></br>
</p>
<h4>A recap of rules for the n<sup>th</sup> term</h4>
<span class="editorial">Like some of the students featured above, Ivan Ivanov from the 47th High School in Sofia, Bulgaria put his reasoning into algebraic form. He found formulas for the coordinates of the shapes using the nice notation below. Well done to everyone who found similar formulae.</span>
<h4><br></br>
Charlie's Squares</h4>
<ol>
<li>If $C(n)$ is the centre of square $n$,<br></br>
then the coordinates of $C(n)$ satisfy the equations: $x(n) = 3n - 1$, and $y(n) = n + 1$.</li>
<li>If $L(n)$ is the bottom left hand vertex of square $n$,<br></br>
then the coordinates of $L(n)$ satisfy the equations: $x(n) = 3n - 2$, and $y(n) = n - 1$.</li>
</ol>
These formulas both give the correct coordinates even when Charlie goes left with values $n = ....-2, -1, 0, 1, 2, 3....$.<br></br>
<br></br>
<h4>Alison's Triangles</h4>
<ol>
<li>If $C(n)$ is the top or bottom vertex of triangle $n$ (for odd and even $n$ respectively), then the coordinates of $C(n)$ satisfy the equations: $x(n) = 4n - 2$, $y(n) = 10$ when $n$ is odd and $y(n) = 0$ when $n$ is even.</li>
<li>If $L(n)$ is the left-most vertex of triangle $n$, then the coordinates of $L(n)$ satisfy the equations: $x(n) = 4n - 4$, $y(n) = 5$.</li>
<li>The right-most vertex of triangle $n$ is the same as the left-most vertex of triangle $n-1$.</li>
</ol>
Again, these formulas give the correct coordinates when Alison works left with values $n = ....-2, -1, 0, 1, 2, 3....$.<br></br>
<br></br>
<h4>More Squares from Charlie</h4>
<br></br>
If $B(n)$ is the centre of square $nb$, then the coordinates of $B(n)$ satisfy the equations: $x(n) = 2n$, and $y(n)B = -2n + 8$.<br></br>
<br></br>
<span class="editorial">You might like to think if you can find a formula for squares such as 19c or 199a. What about if Charlie continues the pattern and draws squares such as 1d and 1e, etc?</span><br></br>
<p><br></br>
<span class="editorial"><span class="editorial">Michael Sena from NSBH sent in a little computer program that could give the coordinates of any of Charlie's first set of squares, even producing a table of the first few. Well done! </span><br></br>
<br></br>
Thanks again to everyone for their solutions!</span></p></mdoxml>
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<mdoxml version="1.0"><br></br>
<h3>Why do this problem?</h3>
<p>This problem offers a good opportunity for students to discuss patterns and find convincing arguments for their solutions.</p>
<p>Reuben Hersh has written that:</p>
<p>"In the classroom, convincing is no problem. Students are too easily convinced. Two special cases will do it."</p>
<p>This problem offers an opportunity to ensure that students are justified in generalising from the particular cases that they have selected.</p>
<p> </p>
<h3>Possible approach</h3>
<div><br></br>
Show the first diagram. It's available as a <a href="/content/id/2292/Coordinate%20Patterns.ppt">PowerPoint</a>, or you could print out this <a class="pdflink" href="/content/id/2292/Coordinate%20Patterns.pdf">worksheet</a>.</div>
<div> </div>
<div>"Have a look at this image. Can you work out the coordinates of the centre of square number 3?"<br></br>
<br></br>
"I wonder if you can now work out the coordinates of the centre of square number 20 from the image, without working out the centres of the squares in between."</div>
<div> </div>
<div>"Spend a short while thinking about it on your own, then discuss it with your partner, and together develop a convincing explanation for your answer to share with the class."</div>
<div> </div>
<div>As students are working, if they get stuck you could offer the following hints:</div>
<div>"How do you move from one square to the next?</div>
<div>What do you notice about the x coordinates of the centres?</div>
<div>What do you notice about the y coordinates of the centres?"</div>
<div> </div>
<div>While pairs are talking, circulate and eavesdrop on discussions, drawing attention to mistakes and making a mental note of any students with clear explanations.<br></br>
Bring the class together and invite those students with interesting or elegant strategies to present their ideas to the rest of the class.</div>
<div> </div>
<div><strong>"In a while I'm going to choose a square and ask you to work out the coordinates of one of the vertices. Can you find a quick and elegant strategy?"</strong></div>
<div> </div>
<div>"Again, you may want to start by working on your own before discussing it with your partner."</div>
<div> </div>
<div>Finally bring the class together and challenge them with a few examples. Students could be asked to display their solutions on their mini-whiteboards. Allow some time for discussion of their strategies.</div>
<div> </div>
<div>Students can work on Alison's Triangles and More Squares from Charlie in a similar way.</div>
<div> </div>
<h3>Possible extension</h3>
<p>Find a general symbolic expression for the coordinates of the vertices of the $n$th square or triangle.</p>
<h3> </h3>
<h3>Possible support</h3>
<p>Before working on this problem students could develop fluency in using coordinates by working on <a href="/6288">Cops and Robbers</a> and fluency with linear sequences by taking a look at <a href="/6713">Shifting Times Tables</a>.<br></br>
</p></mdoxml>
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<mdoxml version="1.0"><p> </p>
<p>How do you move from one square to the next?<br></br>
What do you notice about the x coordinates of the centres?<br></br>
What do you notice about the y coordinates of the centres?</p>
<p> </p>
<p> </p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><p><strong>Squares</strong></p>
<p> </p>
<p>Coordinates of the centre of the 20th square: (59,21)</p>
<p>Coordinates of the bottom left hand vertex of the 34th square: (100,33)</p>
<p>Coordinates of the centre of the -15th square: (-46,-14)</p>
<p> </p>
<p><strong><strong>Triangles</strong></strong></p>
<p> </p>
<p>Coordinates of the top vertex of the 23rd triangle: (90,10)</p>
<p>Coordinates of the top left hand vertex of the 58th triangle: (228,5)</p>
<p> </p>
<p> </p>
<p><strong><strong>And more squares</strong></strong></p>
<p> </p>
<p>Coordinates of the centre of 22B: (44,-36)</p>
<p>Coordinates of the left hand vertex of 26G: (48,-46)</p>
<p> </p>
<br></br>
<p class="editorial">We received good solutions from Ivan Ivanov from the 47th High School in Sofia, Bulgaria and from Michael Sena from NSBH. Well done to you both.</p>
<p class="editorial">Ivan reasoned as follows:</p>
<p style="font-weight: bold;">SQUARES</p>
<p>1. If $C(n)$ is the centre of square $n$,</p>
<p>then the coordinates of $C(n)$ satisfy the equations: $x(n) = 3n - 1$, and $y(n) = n + 1$.</p>
<p>2. If $L(n)$ is the bottom left hand vertex of square $n$,</p>
<p>then the coordinates of $L(n)$ satisfy the equations: $x(n) = 3n - 2$, and $y(n) = n - 1$.</p>
<p>Given the above equations:</p>
<p>a. The coordinates of the centre of the $20$th square are $(59,21)$</p>
<p>b. The coordinates of the centre of the $-15$th square are $(-46,-14)$</p>
<p>c. The coordinates of the bottom left hand vertex of the $34$th square are $(100,33)$</p>
<p style="font-weight: bold;">TRIANGLES</p>
<p>1. If $C(n)$ is the vertex of triangle $n$,</p>
<p>then the coordinates of $C(n)$ satisfy the equations: $x(n) = 4n - 2$, $y(n) = 10$ when $n$ is odd and $y(n) = 0$ when $n$</p>
<p>$L(n)$ is the left top (bottom) vertex of triangle $n$,</p>
<p>then the coordinates of $L(n)$ satisfy the equations: $x(n) = 4n - 4$, $y(n) = 5$.</p>
<p>Given the above equations:</p>
<p>a. The coordinates of the vertex of the $23$rd triangle are $(90,10)$</p>
<p>b. The coordinates of the left top vertex of the $58$th triangle are $(228,5)$</p>
<p style="font-weight: bold;">AND MORE SQUARES</p>
<p>1. If $C(n)B$ is the centre of square $nB$,</p>
<p>then the coordinates of $C(n)B$ satisfy the equations: $x(n)B = 2n + 2$, and $y(n)B = (-2)n + 10$.</p>
<p>2. If $L(n)G$ is the left hand vertex of square $nG$,</p>
<p>then the coordinates of $C(n)G$ satisfy the equations: $x(n)G = 2n - 4$, and $y(n)G = (-2)n + 6$.</p>
<p>Given the above equations:</p>
<p>a. The coordinates of the centre of square $22B$ are $(46,-34)$</p>
<p>b. The coordinates of the left hand vertex of square $26G$ are $(48,-46)$</p>
<p class="editorial">Michael wrote a short program for each problem that produced the same resultsere is his solution to SQUARES:</p>
<p><mdo:image alt="Michael's program" height="707" src="squaresprog.png" width="255"></mdo:image></p></mdoxml>
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Coordinate Patterns
Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?
Sequences, Functions and Graphs
Arithmetic sequence
Coordinates and Coordinate Geometry
Coordinates - all quadrants
Using, Applying and Reasoning about Mathematics
Visualising
Admin
Workshop
Secondary Mapping Document
Patterns and sequences LS
Secondary Mapping Document
Coordinate geometry
Secondary Mapping Document
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