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Crossing the Town Square

Age 7 to 11 Challenge Level:

Crossing the Town Square


Imagine the central square in a big city and its paved with large square tiles. It may be rectangular rather than square! You are going to go straight from one corner, diagonally across to the other corner. You may be walking, cycling, skateboarding or using roller blades. Whichever way you travel you will need to go absolutely straight from corner to corner.

You see in the picture above that shows a very, very small example (a 4 by 3 rectangle). The blue line of travel goes through six of the square tiles. Maybe there are other small rectangles other than this one that crosses 6 tiles.

Your challenge is to find what different sizes of rectangles would mean you travelled across 10 tiles. 

Your extra challenge is to find a set of answers for 12 tiles being crossed. There are probably more than you first discover!

Can you find a generalization, a pattern, a  pattern or a system from the ones you've done that would enable you to find solutions more easy for other numbers of tiles being crossed?



Why do this problem?

This activity challenges the pupils in their abilities to link the numerical with the spatial. Many pupils will find this a fascinating idea and evoke their curiosity. Their perseverance will be challenged as they try different examples in order to find a generalization/pattern/system so that they can predict answers for other sizes of the town square. The "Teacher Support" at the bottom of this page is recommended in regard to the curiosity aspect of this task.


Possible approach

As this is intended for older/more experienced pupils I would, first of all, suggest printing out the activity and discussing together, then let them produce their solutions and their generalizations/patterns/systems.


Key questions

Tell me about how you arrive at these solutions.
So what relationships do you think are involved here?
What further questions can you ask?

Possible Extension

What happens if all the tiles are not square but just rectangular but the same as each other?

Teacher Support

This task was created to help in the pursuance of curiosity within the Mathematics lessons.
Help may be found in the realm of curiosity in watching parts of these excellent videos.
Firstly "The Rise & Fall of Curiosity", particularly the extract [23.50 - 37.15] on "adult encouragement answering and teacher behaviour."
Secondly, "The Hungry Mind: The Origins of Curiosity", particularly the extract [8.22 - 12.29]  on "Children asking questions"


First can also be found at -
Second can also be found at