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A working model for science is very rare.Anyway,you can try this: First of all make a potractor shaped model representing all the degrees in it.Then just attach a twin in it's top part as hanging way.Then we can adjust it in any way and get angles.While the base part of the potractor can be called as 'the line of sight' and can be used in applications for trigonometry.
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You can make a model to prove the identity .(a + b)²  This is easy. On a piece of paper, make a square of length (a + b) [Choose a and b as per your wish t be any two lengths].Its area represents (a + b)². Make another square of side a, and another of side b.   
Make a rectangle of side 2a and b or 2b and a, whichever is convenient.
Put the side a square, side b square and the rectangle on the bigger square of side (a + b). You can show that this fits perfectly, thus proving the identity.
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