Math Borders Investigation
Figure below shows a dark cross-shape that has been surrounded by
white squares to create a bigger cross-shape;
The bigger cross-shape consists of 25 small squares in total.
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The next cross-shape is always made by surrounding the previous
cross-shape with small squares.
Part 1- Investigate to see how many squares would be needed to make
any cross-shape built in this way.
Part 2- Extend your investigation to 3 dimensions.
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Introduction -
I am doing an investigation to see how many squares would be needed to
make any cross-shape built up in this way. Each cross-shape is made by
using the previous cross-shape and adding another layer of white
squares, making all the inner squares black. The first cross-shape in
the sequence is a single black square.
To start my investigation I must draw the first 7 cross-shapes. This
will enable me to see a pattern in the shapes so I can make a table
and record how many black and white squares there are in each
cross-shape I have drawn. From my table I must use the results to work
out formulae for black, white and total number of squares.
After this I will test the formulae on a pattern I have already drawn
and on one I have not already drawn.
I will be working systematically in my investigation because if I work
in a
particular order it will be easier for to see a pattern and links in
the sequences.
Finally I will be looking at different ways of getting the formulae
and also extending my investigation into 3 dimensions.
Part 1
Drawing the cross-shapes
Pattern 1 Pattern 2 Pattern 3 Pattern 4
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Pattern 5 Pattern 6 Pattern 7
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Here is my table of results telling me how many black, white and total
numbers of squares there are in patterns 1 up to 7.
6x6x6 cube and see if I can find a pattern. When I have found a
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I am going to begin by investigating a square with a side length of 10
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