In this portfolio, I am required to investigate the number of regions obtained by making cuts in one, two and three dimensional objects

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MATHS PORTFOLIO

HOW MANY PIECES?

INVESTIGATION QUESTIONS –

A line segment is a finite one-dimensional object. Find a rule which relates the maximum number of segments (S) obtained from n cuts. Comment on your results.

A circle is a finite two-dimensional object. Investigate the maximum number of regions (R) obtained when n chords are drawn.

A cuboid is a finite three-dimensional object. Investigate the maximum number of parts (P) that are obtained with n cuts.

If a finite four-dimensional object exists and the procedure is repeated what would you expect to find?

In this portfolio, I am required to investigate the number of regions obtained by making cuts in one, two and three dimensional objects. By finding a rule in all the three cases, I need to develop a rule for a four-dimensional object by searching for a definite pattern.

The data is as follows-

One – dimensional object

In the data table, we can see that in all the 5 cases, the number of segments formed is 1 more than the number of cuts made. After noting this simple pattern, I derived the recursive rule as follows –

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No of segments = No of cuts made + 1

   

Two – dimensional object

For a circle, it was hard to find the maximum number of regions because chords can cut different number of regions in the way they are drawn and because I needed maximum number of regions, I had to draw chords in a specific manner.

The diagrams are as follows –

 

After drawing all the sketches and finding the number of regions obtained, I proceeded to find the pattern of the sequence.

First I found the difference between ...

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