7453
/www/nrich/html/content/id/7453/
1
2011-02-01T00:00:01
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<span style="font-weight: bold;">Charlie likes to go for walks around a square park.</span><br></br>
Here is a diagram of the journey he took one day:<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="vectors forming a square" height="143" src="vector1.jpg" width="144"></mdo:image></div>
He started his journey by walking along the black vector $\pmatrix{3\cr 1}$<br></br>
What vectors did he need to walk along to complete his journey?<br></br>
<br></br>
Draw some other square parks that Charlie could walk around, and find the vectors that would describe his journey.<br></br>
<br></br>
<span style="font-weight: bold;">Can you describe and explain any relationships between the vectors that determine Charlie's journey around any square park?</span><br></br>
<br></br>
Once you know the first vector of a journey, can you work out what the second, third and fourth vectors will be? Is there more than one possibility?<br></br>
<br></br>
<br></br>
<span style="font-weight: bold;">Alison likes to walk across parks diagonally.</span><br></br>
One day, she walked along the blue vector $\pmatrix{2\cr 4}$:<br></br>
<br></br>
<div style="text-align: center;"> <mdo:image alt="vector square with diagonal" height="143" src="vector2.jpg" width="144"></mdo:image></div>
<div style="text-align: left;">For the parks you drew above, find the vectors that would describe Alison's diagonal path.</div>
<ul>
<li style="text-align: left;">Can you describe any relationships between the vectors that determine Alison's and Charlie's journey, for any square park?<br></br>
</li>
<li style="text-align: left;">Given the vector that describes Alison's journey, how can you work out the first stage of Charlie's journey?<br></br>
</li>
<li style="text-align: left;">If all square parks have their vertices on points of a dotty grid, what can you say about the vectors that describe Alison's diagonal journey?</li>
</ul>
<span style="font-weight: bold;">Can you explain and justify your findings?</span><br></br>
<br></br>
<span style="font-style: italic;"> </span><br></br>
<br></br>
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<span class="editorial">James from Wilson's School told us
something he'd noticed about the second, third and fourth vectors
forming a square park:</span> <br></br>
<br></br>
The vector journey goes: $\pmatrix{3\cr 1} \pmatrix{-1\cr 3}
\pmatrix{-3\cr -1} \pmatrix{1\cr -3}$<br></br>
<br></br>
After finding the first vector both top and bottom numbers should
be positive, next vector the top number is made negative then next
vector both numbers are negative and finally the last vector the
bottom number is negative. The second and last vectors the
left/right number switches with the up/down number. <br></br>
<br></br>
<span class="editorial">Elliott, also from Wilson's, made some
observations:</span><br></br>
<br></br>
Charlie walks on vectors of $\pmatrix{3\cr 1} \pmatrix{-1\cr
3} \pmatrix{-3\cr -1} \pmatrix{1\cr -3}$<br></br>
<br></br>
Another square he could walk would have vectors of $\pmatrix{5\cr
2} \pmatrix{-2\cr 5} \pmatrix{-5\cr -2} \pmatrix{2\cr -5}$<br></br>
<br></br>
These vectors must only consist of four numbers: $x, y, -x$ and
$-y$.<br></br>
It can only be two numbers, and their negatives, so that all the
sides of the square are equal in length.<br></br>
<br></br>
After travelling along the first vector, you can then move left or
right. From there, you must do the opposite of your first move,
then the opposite of your second, to get back to your original
position.<br></br>
<br></br>
<span class="editorial">Niharika from Leicester High School
for Girls sent us</span> <a class="editorial" href="/content/id/7453/vector%20journeys.pdf">this
solution</a><span class="editorial">. Well done to you
all.</span><br></br>
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<h3>Why do this problem?</h3>
This problem offers a simple context for exploring vectors
that leads to some interesting generalisations that can be proved
with some vector algebra.<br></br>
<br></br>
<a href="http://nrich.maths.org/7472&part=">Here is an
article</a> that describes some of the background thinking that
informed the creation of this problem. <br></br>
<h3>Possible approach</h3>
This problem requires students to draw tilted squares reliably. <a href="http://nrich.maths.org/content/id/2293/squareAnimation3.swf">This
interactivity</a> might be helpful to demonstrate to students what
a tilted square looks like. Students could play <a href="http://nrich.maths.org/2526&part=">Square It</a> until
they can reliably spot tilted squares on a dotty grid.<br></br>
<br></br>
<span style="font-weight: bold;">A possible start which involves
the minimum of teacher input</span> is to draw the
vector $\pmatrix{3\cr 1}$ and say:<br></br>
"Imagine we are drawing squares using vectors with whole
numbers.<br></br>
This vector could be the side of a square, or the
diagonal of a square.<br></br>
<br></br>
Find the vectors that describe the journeys around the squares
that include this vector as either a side or a diagonal."<br></br>
<br></br>
This leads on to the challenge "In a while, I am going to ask you
to find the vectors that describe the journeys around squares that
could be drawn using a <span style="font-weight: bold;">different</span> vector as either the
side or the diagonal. The challenge will be to answer without doing
any drawing."<br></br>
<br></br>
<br></br>
<span style="font-weight: bold;">Alternatively, start</span> by
showing the picture of Charlie's walk.<br></br>
"If the black vector is $\pmatrix{3\cr 1}$ what are the other three
vectors?"<br></br>
Once everyone is confident with vector notation, ask students to
draw a square park of their own on dotty paper, making sure the
vertices are on lattice points, and to work out the vectors that
would describe Charlie's journey.<br></br>
<br></br>
On the board, draw a table to collect together some of the vector
journeys the students have devised. After the first few, can they
start predicting what the second, third and fourth vectors will be
once they know the first vector of a journey? Is there more than
one possibility?<br></br>
Give students some time to work on their own or in pairs to test
any conjectures they make.<br></br>
<br></br>
"Could we have worked out the vectors if we'd been given a diagonal
of the square instead of a side?"<br></br>
Show Alison's diagonal walk, and ask students to consider this
question with regard to the squares they drew earlier on. After a
short while, the diagonal vectors could be added to the information
already collected on the board.<br></br>
Then set students the three questions from the problem:<br></br>
<ul>
<li>Can they describe any relationships between the vectors that
determine Alison's and Charlie's journey, for any square park?
</li>
<li>Given the vector that describes Alison's journey, how can they
work out the first stage of Charlie's journey?</li>
<li>If all square parks have their vertices on points of a dotty
grid, what can they say about the vectors that describe Alison's
diagonal journey? </li>
</ul>
<div>Finally bring the class together to share their ideas and
justify their findings.</div>
<div> </div>
<div>One technique for testing ideas at the end is to set a
specific challenge, for example, to find the vectors describing
Charlie's route if Alison's diagonal route is given by the vector
$\pmatrix{35 \cr 15}$</div>
<div> </div>
<div>Use of dynamic geometry software such as the free-to-download
<a href="http://www.geogebra.org">GeoGebra</a> can help students to
develop insights into the structure of this problem. The example
below shows a construction which could be shared with students.
Alternatively, an extension activity might be to encourage students
to create their own constructions.</div>
<div> </div>
<div> </div>
<mdo:applet height="491" width="495" code="geogebra.GeoGebraApplet" archive="http://www.geogebra.org/webstart/geogebra.jar" datafile=""><param value="http://nrich.maths.org/content/id/7453/vertices.ggb" name="filename" ></param><param value="true" name="framePossible" ></param><param value="true" name="showResetIcon" ></param><param value="true" name="enableRightClick" ></param><param value="true" name="showMenuBar" ></param><param value="true" name="showToolBar" ></param><param value="true" name="showToolBarHelp" ></param><param value="true" name="showAlgebraInput" ></param></mdo:applet>
<div> </div>
<div> </div>
<br></br>
<h3>Key questions</h3>
<div>How can I use the first vector to work out the other three
vectors which describe a journey around a square?</div>
<div> </div>
<div>How can I use the diagonal vector to work out the four vectors
which describe a journey around a square? </div>
<div> </div>
<div>Is there a quick way to determine whether a given vector could
be the diagonal of a square with corners on the lattice points of a
square grid?</div>
<br></br>
<h3>Possible extension</h3>
<a href="http://nrich.maths.org/6572&part=">Vector Walk</a>
challenges students to explore relationships between vector algebra
and geometry, and to consider the points that can be reached on a
grid using a set of vectors. <br></br>
<h3>Possible support </h3>
<div>The interactivity in <a href="http://nrich.maths.org/2667&part=">Square Coordinates</a>
helps students to visualise tilted squares.</div>
<div><a href="http://nrich.maths.org/7381&part=">Opposite
Vertices</a> explores similar mathematical ideas but without
vector notation.</div>
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The interactivity in <a href="http://nrich.maths.org/2667&part=">Square Coordinates</a>
may help you to visualise tilted squares.<br></br>
You can challenge yourself by playing <a href="http://nrich.maths.org/2526&part=">Square It</a> against
the computer.<br></br>
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Vector journeys
Charlie likes to go for walks around a square park, while Alison
likes to cut across diagonally. Can you find relationships between
the vectors they walk along?
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