5038
/www/nrich/html/content/id/5038/
1
2011-02-01T00:00:01
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Here is a grid:<br></br>
<br></br>
<mdo:image width="519" height="405" alt="grid with x from 0-4 and y from 0-3" src="Coord1.gif"></mdo:image><br></br>
<br></br>
Can you position these ten letters in their correct places
according to the eight clues below? <br></br>
<br></br>
<mdo:image width="363" height="78" alt="letters A, B, C, D, E, P, S, X, Y, Z" src="Coord2.gif"></mdo:image><br></br>
<mdo:image width="363" height="77" alt="letters P, S, X, Y, Z" src="Coord3.gif"></mdo:image><br></br>
<br></br>
<br></br>
Clues: <br></br>
<br></br>
<div>The letters at $(1, 1),$ $(1, 2)$ and $(1, 3)$ are all
symmetrical about a vertical line.</div>
<br></br>
<div>The letter at $(4, 2)$ is not symmetrical in any way.</div>
<br></br>
<div>The letters at $(1, 1),$ $(2, 1)$ and $(3, 1)$ are symmetrical
about a horizontal line.</div>
<br></br>
<div>The letters at $(0, 2),$ $(2, 0)$ have rotational
symmetry.</div>
<br></br>
<div>The letter at $(3, 1)$ consists of just straight lines.</div>
<br></br>
<div>The letters at $(3, 3)$ and $(2, 0)$ consist of just curved
lines.</div>
<br></br>
<div>The letters at $(3, 3),$ $(3, 2)$ and $(3, 1)$ are consecutive
in the alphabet.</div>
<br></br>
<div>The letters at $(0, 2)$ and $(1, 2)$ are at the two ends of
the alphabet.</div>
<br></br>
<div>You could use this interactivity to try out your ideas.</div>
<br></br>
<a href="/content/id/5038/Co-Ord3.swf">Full Screen Version</a>
<br></br>
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<p class="editorial">We had lots of solutions to this problem but
not many of you told us how you got your solutions. Luis and
Freddie, who go to Newton Prep, sent in well-explained solutions.
Here is what Luis wrote:</p>
<br></br>
<br></br>
The first thing I worked out is that the letter at (4,2) is not
symmetrical in any way, the only letter that isn't symmetrical in
any way is the letter P. <br></br>
<div style="clear: both;">Then I worked out that at (0,2) and (1,2)
are the ends of the alphabet and had to be A and Z but I needed to
know which one went where so I looked at the clues. One said that
(0,2) and (2,0) have rotational symmetry which means that A is on
(1,2) and Z is on (0,2).<br></br>
<div style="clear: both;">Then I saw the clue that said that (3,3),
(3,2) and (3,1) are all consecutive in the alphabet, because the A
and the Z are already used, it has to be C, D and E because it says
that (3,3) is made of only curved lines.<br></br>
<div style="clear: both;">Then I saw that one of the clues said
that (2,0) is only made of curved lines, so (2,0) is the S.<br></br>
(4,2) is not symmetrical in any way so (4,2) is a P.<br></br>
<div style="clear: both;">The first clue says that (1,1) and (1,3)
have a vertical line of symmetry, another clue says that (1,1) also
has a vertical line of symmetry, which means that (1,1) is a X and
(1,3) is a Y, which leaves the B which is at (2,1) because a clue
says that at (2,1) there is a letter with a horizontal line of
symmetry.</div>
<div style="clear: both;"></div>
<p style="clear: both;" class="editorial">Many children from
Stradbroke Primary School sent an image of their solution. Here is
George's:</p>
<div style="clear: both;"></div>
<div style="clear: both;"></div>
</div>
</div>
</div>
<div><mdo:image width="210" height="166" alt="solution" src="coordssol.gif"></mdo:image></div>
<br></br>
<p class="editorial">Well done!</p>
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<h2>Coordinate Challenge</h2>
<br></br>
Here is a grid:<br></br>
<br></br>
<mdo:image alt="grid with x from 0-4 and y from 0-3" height="405" src="Coord1.gif" width="519"></mdo:image><br></br>
<br></br>
Can you position these ten letters in their correct places according to the eight clues below?<br></br>
<br></br>
<mdo:image alt="letters A, B, C, D, E, P, S, X, Y, Z" height="78" src="Coord2.gif" width="363"></mdo:image><br></br>
<mdo:image alt="letters P, S, X, Y, Z" height="77" src="Coord3.gif" width="363"></mdo:image><br></br>
<br></br>
<br></br>
Clues:<br></br>
<br></br>
<div>The letters at $(1, 1),$ $(1, 2)$ and $(1, 3)$ are all symmetrical about a vertical line.</div>
<br></br>
<div>The letter at $(4, 2)$ is not symmetrical in any way.</div>
<br></br>
<div>The letters at $(1, 1),$ $(2, 1)$ and $(3, 1)$ are symmetrical about a horizontal line.</div>
<br></br>
<div>The letters at $(0, 2),$ $(2, 0)$ have rotational symmetry.</div>
<br></br>
<div>The letter at $(3, 1)$ consists of just straight lines.</div>
<br></br>
<div>The letters at $(3, 3)$ and $(2, 0)$ consist of just curved lines.</div>
<br></br>
<div>The letters at $(3, 3),$ $(3, 2)$ and $(3, 1)$ are consecutive in the alphabet.</div>
<br></br>
<div>The letters at $(0, 2)$ and $(1, 2)$ are at the two ends of the alphabet.</div>
<br></br>
<div>You could use this interactivity to try out your ideas.</div>
<br></br>
<a href="/content/id/5038/Co-Ord3.swf">Full Screen Version</a><br></br>
<mdo:flash height="475" id="/content/id/5038/Co-Ord3.swf" width="450"><param name="allowfullscreen" value="true"></param><param name="movie" value="/content/id/5038/Co-Ord3.swf"></param>
<param name="flashplayerversion" value="7"></param>
<param name="height" value="475"></param>
<param name="width" value="450"></param>
</mdo:flash><br></br>
</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
There are two main parts to <a href="http://nrich.maths.org/public/viewer.php?obj_id=5038&part=index">this problem</a> . Learners need to identify the symmetries of various capital letters and to practise reading coordinates. It needs to be approached systematically because not all the clues are straightforward.<br></br>
<h3>Possible approach</h3>
<div>It would be good to have the problem displayed on an interactive whiteboard or data projector so that you can use the interactivity throughout the lesson. Start by arranging some of the letters on the grid and asking the children a few questions about where they are placed which will remind them about symmetry and coordinates. For example, you could ask "Where is the letter that has two
lines of symmetry?"; "What do all the letters at ... have in common?".</div>
<br></br>
<div>After this learners could work in pairs on the problem so that they are able to talk through their ideas with a partner. They could either use the interactivity at computers or <a href="/content/id/5038/5038.pdf">this sheet</a> which has letters that can be cut out and moved around. Alternatively, the problem could be copied on to squared paper and the letters drawn on in pencil so that they
can be altered easily.</div>
<br></br>
<div>If pupils do not have much experience of dealing with clues and prioritising their order as this problem requires, it might be appropriate to read each clue in turn as a whole group before working on the task in pairs. This way, you can reassure the learners that it doesn't matter if you cannot immediately use a clue to place a letter definitively. You can also demonstrate that having the
letters to move around, or writing them on the grid in pencil, means that the solution can be refined as more information is revealed.</div>
<br></br>
<h3>Key questions</h3>
<div>What kind of symmetry does this letter have?</div>
<div>Where could this letter go on the grid? How do you know?</div>
<div>Which letters fit this clue?</div>
<br></br>
<h3>Possible extension</h3>
Learners could explore and list the symmetries of all the other letters of the alphabet.<br></br>
<h3>Possible support</h3>
Children could start by identifying the symmetries of the letters, listing whether they have vertical or horizontal line symmetry, rotational symmetry or no symmetry at all. They can then refer to their list as they try to place letters on the grid.<br></br>
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<div>It might help to read through each clue in turn and make a
note of which letters could be at those positions.</div>
<div>If you're not able to use the interactivity, how about
printing out <a href="/content/id/5038/5038.pdf">this sheet</a> and
then cutting out the letters so you can move them around the
grid?</div>
<div>What different types of symmetry do you know about? Which
letter is the only one not to have any symmetry at all? How does
that help you?</div>
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Coordinate Challenge
Use the clues about the symmetrical properties of these letters to
place them on the grid.
Using, Applying and Reasoning about Mathematics
Working systematically
Transformations and their Properties
Symmetry
Transformations and their Properties
Rotations
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