545
/www/nrich/html/content/97/12/six6/
1
2011-02-01T00:00:01
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p>The hallway floor is rectangular in shape.</p>
<p>Its length and breadth are whole numbers of feet.</p>
<p>The hallway is tiled and each tile is one foot square.</p>
<p>Given that the number of tiles around the perimeter is EXACTLY
half the total number of tiles, find the possible dimensions of the
hallway.</p>
<p>Now have a look in <a href="http://nrich.maths.org/public/viewer.php?obj_id=545&part=note">
Notes</a> to see how this could be extended for work at Stages 2, 3
& 4.</p>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
<p><strong><em class="editorial" style="font-style: normal;">Carla, Michael and Andrew from Smithdon HS, Hunstanton sent in a correct solution.</em></strong></p>
<p>It is important to realise that if the hallway is $x$ by $y$ feet, then the perimeter involves $2x +2y - 4$ tiles, rather than $2x + 2y$. This is because the tiles in the four corners go along two edges each, so are counted twice when we calculate that the perimeter is $2x + 2y$ feet.<br></br>
<br></br>
Then the solution is based on solving:</p>
<p>$$2x +2y - 4 = \frac{1}{2} xy$$<br></br>
which after a clever rearrangement looks like</p>
<p>$$ y = 4 + \frac{8}{(x - 4)} $$<br></br>
so we know $x - 4$ is a factor of $8$.</p>
<p>Ignoring negative solutions leaves the positive solutions of $x=5,\,6,\,8,\,12$ and therefore $y=12,\,8,\,6,\,5$ correspondingly.</p>
<p>Hence the hallway is either $5$ by $12$ feet or $6$ by $8$ feet.<br></br>
<br></br>
<span class="editorial">Did you expect to find two possible answers? Well done if you got both!</span></p></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
Having got a solution for this problem let's have a look at some
ways of taking it much further which allows a lot of investigations
to take place.<br></br>
Having twice as many tiles in the total, compared to the number in
the perimeter, could be worded as "the ratio of Total Tiles to
those in the Border is 2", which is more helpful when exploring
further.<br></br>
<br></br>
A/ We could first of all look at the ratio being some other numbers
instead. When doing this you may notice that some hallways appear
as almost square and could be explored as a separate item.<br></br>
<br></br>
Some pupils might use arithmetic and geometric knowledge to pursue
it further, others might go for practical trial and error linked
with a calculator, others may be able to handle spreadsheets.<br></br>
<br></br>
"Hints" [A] shows a related spreadsheet - it mentions a single
border as later on we will look at wider borders.<br></br>
<br></br>
GENERAL IDEAS:-I suggest that Patterns and Relationships can be
explored among those results that generate the same Ratio, [ eg.
widths & lengths of 7 30; 8 18; 9 14; 10 12;] as well as going
between one Ratio and another [ eg. widths & lengths of 5 12; 7
30; 9 56; ].<br></br>
<br></br>
Also in this case the numbers that are present in "Those that are
Nearly a Square" could be explored OR could in fact just be
presented to pupils for exploration of a set of numbers!<br></br>
<br></br>
B/ Another way of extending this invesigation is to explore the
idea of a hallway of constant width but with a right angle turn in
it producing a plan view in the shape of an "L".<br></br>
<div style="text-align: center;"><mdo:image height="72" width="79" src="An%20%22L%22.jpg" alt="L"></mdo:image></div>
<div>You have to decide how the "Length" is measured, in the
picture anove I went for the length across the top added to the
length down the right hand side, but you choose!</div>
<br></br>
<div style="text-align: left;">
<div style="clear: both;">The results of exploring these "L" shaped
hallways is shown in "Hints" B.</div>
<div style="clear: both;"></div>
<div style="clear: both;"></div>
</div>
<div style="text-align: left;"></div>
<br></br>
<div>GENERAL IDEAS:- I suggest that Patterns and Relationships can
be explored among those that generate the same Ratio, [ eg. widths
& lengths of 7 27; 8 26; 9 23; 10 22;] as well as going between
one ratio and another [ eg. widths & lengths of 5 17; 7 37; 9
126; ].</div>
<br></br>
<div style="text-align: left;"></div>
C/ So, why not go on a step further and consider a "Z" shaped
hallway keeping the same width and with two right angle turns.
<div style="text-align: left;"></div>
<div style="text-align: center;"><mdo:image height="112" width="187" src="Z3.jpg" alt="Z3"></mdo:image></div>
<div style="text-align: center;"></div>
<div style="text-align: center;"></div>
<div>In this case again you have to decide how to measure the
length</div>
<br></br>
<div style="text-align: left;"></div>
<div style="text-align: left;"></div>
<div>The results of exploring these "L" shaped hallways is shown in
"Hints" C.</div>
<br></br>
<div style="text-align: left;"></div>
<div>GENERAL IDEAS:- I suggest that Patterns and Relationships can
be explored among those that generate the same Ratio, [eg. widths
& lengths of 9 56; 10 32; 11 24] as well as going between one
ratio and another [ widths & lengths of 7 30; 9 56; 11
90].</div>
<br></br>
<div style="text-align: left;"></div>
<div>D/ Like in many maths investigations when we come to the point
of having explored more then one variation of the original
challenge its good to compare the resluts of them all. So I've
shown a table in "Notes ",D which brings together some of the
measurements but does not deal with already discovered
relationships and patterns that YOU discovered, it may just be a
handy start !</div>
<br></br>
<div>You may like to look at them geometrically and so I gathered
them together as ones with right angled corners in them but all
having the same ratio.</div>
<div style="text-align: left;"><mdo:image height="169" width="688" src="LZW.jpg" alt="LZW"></mdo:image></div>
<div style="text-align: left;"></div>
<div style="text-align: left;">E/ Just to finish off we need to
perhaps consider other questions of the order "I wonder what woud
happen if. . . . .?"</div>
<div style="text-align: left;"></div>
<div style="text-align: left;">Pupils can be asked - if they have
not already suggested it about border tiles that are double width.
So we'd have something like this;</div>
<div style="text-align: center;"><mdo:image height="86" width="307" alt="Border2" src="Border2jpg.jpg"></mdo:image></div>
<div style="text-align: left;">More possibilities here particularly
for those with spreadsheet skills as the rectangles, "L" & "Z"
shapes increase and maybe the width of the border increases to 3, 4
or 5 etc.</div>
<div style="text-align: left;"></div>
<br></br>
<br></br>
<br></br>
<br></br>
<br></br></mdoxml>
<?xml version="1.0" encoding="UTF-8"?>
<mdoxml version="1.0"><br></br>
A/<br></br>
<mdo:image width="461" height="451" src="Picture%20A.jpg" alt="A"></mdo:image><br></br>
---------------------------------------------------<br></br>
B/<br></br>
<mdo:image width="425" height="379" alt="B2" src="B2.jpg"></mdo:image><br></br>
---------------------------------------------------<br></br>
<br></br>
C/<br></br>
<mdo:image width="225" height="449" src="Picture%20C.jpg" alt="C"></mdo:image><br></br>
----------------------------------------------<br></br>
D/<br></br>
<mdo:image width="783" height="247" alt="D1" src="D1.jpg"></mdo:image><br></br>
--------------------------------------------------<br></br>
<br></br>
E/<br></br>
<br></br>
<mdo:image width="369" height="425" src="FdiffBorders.jpg" alt="F-Diff Borders"></mdo:image><br></br>
<br></br></mdoxml>
2
5
0
0
1
0
0
Hallway Borders
A hallway floor is tiled and each tile is one foot square. Given
that the number of tiles around the perimeter is EXACTLY half the
total number of tiles, find the possible dimensions of the hallway.
Algebra
Creating expressions/formulae
Algebra
Diophantine equations
2D Geometry, Shape and Space
Rectangles
Measures and Mensuration
Area
Using, Applying and Reasoning about Mathematics
Trial and improvement