Hallway Borders
Problem
The hallway floor is rectangular in shape.
Its length and breadth are whole numbers of feet.
The hallway 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.
Now take a look here at possible follow-up extension questions.
Getting Started
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Student Solutions
Carla, Michael and Andrew from Smithdon HS, Hunstanton sent in a correct solution.
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.
Then the solution is based on solving:
$$2x +2y - 4 = \frac{1}{2} xy$$
which after a clever rearrangement looks like
$$ y = 4 + \frac{8}{(x - 4)} $$
so we know $x - 4$ is a factor of $8$.
Ignoring negative solutions leaves the positive solutions of $x=5,\,6,\,8,\,12$ and therefore $y=12,\,8,\,6,\,5$ correspondingly.
Hence the hallway is either $5$ by $12$ feet or $6$ by $8$ feet.
Did you expect to find two possible answers? Well done if you got both!
Teachers' Resources
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.
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.
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.
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.
"Hints" [A] shows a related spreadsheet - it mentions a single border as later on we will look at wider borders.
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; ].
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!
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".
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.
