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Tiling Into Slanted Rectangles

Age 7 to 11 Challenge Level:

Tiling into Slanted Rectangles

 This is seen as a possible follow-on from Tiles in the Garden.

This activity takes "Tiles in the Garden", much further. We can keep the main ideas the same - 

  • Square tiles
  • A corner of a tile at each corner of the rectangle
  • The ability to slice a tile into parts so as to use each part

So this one used $26$ and the slope was generated by going along 1 and up to 5.

 

This time let's put on a limit of using less than 100 tiles.

What sizes of rectangles could be filled obeying the three rules?

How many tiles for each rectangle you find?

Are there any numbers of tiles between 10 and 100 for which there cannot be a rectangle?

 


Why do this problem?

This activity acts as a further extension to Tiling the Garden. It's an activity that is intended to give opportunities for those pupils to explore more deeply using their intuition and flair in the areas of both spatial awareness and number relationships. Many pupils may find this whole idea sparks off their curiosity and through persevering will come up with some great new ideas. If you want to pursue curiosity more see the Teacher Support below.

Possible approach

As this is designed for the highest attaining, it might be presented as on the website or in a one-to-one situation, encouraging discussion between adult and pupil. The pupils may need access to a spreadsheet once many number results are being acquired.

Key questions

Tell me about what you have found?
Can you describe the ways that you arrived at these numbers?
How did you construct this on the spreadsheet you used?
 

Possible Extension

It would be good to handing over to the pupil and encouraging the curiosity question "I wonder what would happen if . . . ?"


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 - https://www.youtube.com/watch?v=X-0NOrIU67w
Second can also be found at https://www.youtube.com/watch?v=Wh4WAdw-oq8