Question

how to make a model of algebraic identity on (a+b)^3

Answer 1

Aim : To prove the algebraic identity (a+b)3 = a3 +3a2b+3ab2+b3 using unit cubes
Material required: Unit Cubes.
Take any suitable value for a and b. 
Let a=3 and b=1
Step 1. To represent a3 make a cube of dimension a x a x a i.e. 3x3x3 cubic units
Step 2. To represent 3a2b make 3 cuboids of dimension a x a x b i.e. 3x3x1 cubic units.
Step 3. To represent 3ab2 make 3 cuboids of dimension a x b x b i.e. 3x1x1 cubic units.
Step 4. To represent b3 make a cube of dimension a x a x a i.e. 1x1x1 cubic units.
Step 5. Join all the cubes and cuboids formed in the previous steps to make a cube of dimension (a +b) x ( a +b) x ( a +b) i.e. 4x4x4 cubic units. 
Observe the following  The number of unit cubes in a3 = ..27…..  The number of unit cubes in 3a2b =…27…  The number of unit cubes in 3ab2 =…9……  The number of unit cubes in b3 =…1……  The number of unit cubes in a3 + 3a2b + 3ab2 + b3 = ..64….  The number of unit cubes in (a+b)3 =…64…
It is observed that the number of unit cubes in (a+b)3 is equal to the number of unit cubes in a3 +3a2b+3ab2+b3 . It is observed that the number of unit cubes in (a+b)3 is equal to the number of unit cubes in a3 +3a2b+3ab2+b3 .