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All Lesson Plans

Border Tiles Puzzle

Puzzle Board

polypad.org/OEQmCL7HXLIyg

The border tiles is a traditional problem in which you are asked to write an expression to calculate the number of tiles needed to border a given pool. The shape of the pool is usually a square or a rectangle, but any shape can be used as a pool too!

Hints

For younger students, starting with a square pool of 1 by 1 and enlarging the pool one square at a time can help to find the general pattern.

polypad.org/DfAAOQFThZxLFg

For the rectangular pool, the following questions may be helpful:

  • What are the dimensions of the pool? What is its area?
  • What are the dimensions of the area enclosed by the border tiles? Its area?
  • What does the difference between areas represent?
  • For an n x m pool, what are the dimensions of the area enclosed by the border tiles?

Solution

Square Pool

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The beauty of this problem is the endless ways of analyzing the shape. Here are a few different ways to look at a 5 by 5 square pool.

Rectangular Pool

Rectangular Pools are very similar to square pools. The general formula for an n x m rectangle is 2n+2m+42n+ 2m+ 4.

For a 12 x 16 pool, the number of red tiles will be 60.

More on the Border Tiles Puzzle

You may continue to study other shapes to solve the border problems. Here is an L shape pool example;

polypad.org/PLZqWIp9IeJgMQ

Visit the Visual Patterns Tasks Page to practice exploring and creating patterns.

mathigon.org/task/visual-patterns

Next

If you like this puzzle, try this one!

polypad.org/O2KD9Dg7qNRnA