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Hallway Borders

Age 11 to 14 Challenge Level:

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".
L
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!

The results of exploring these "L" shaped hallways is shown in "Hints" B.

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; ].

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.
Z3
In this case again you have to decide how to measure the length

The results of exploring these "L" shaped hallways is shown in "Hints" C.

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].

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 !

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.
LZW
E/ Just to finish off we need to perhaps consider other questions of the order "I wonder what woud happen if. . . . .?"
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;
Border2
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.