Verifying Algebraic identity(II)

Today I am sharing with U the 2nd cubic algebraic activity which can be done

using the unit cubes.

•To prove the algebraic identity
(a-b)^3 = a^3 -3a^2b+3ab^2-b^3
using unit cubes.
Take any suitable value for a and b.
Let a=3 and b=1.
To represent (a-b)^3 make a cube of dimension

(a-b) x (a-b) x (a-b) i.e. 2x2x2 cubic units as shown below.

To represent (a)^3 make a cube of dimension a x a x a
i.e. 3x3x3 cubic units as shown below.

To represent 3ab^2 make 3 cuboids of dimension

a x b x b i.e. 3x1x1 cubic units as shown below.

To represent a^3 + 3ab^2 , join the cube and the cuboids

formed in steps 2 and 3 as shown below.

To represent a^3 + 3ab^2- 3a^2b extract from the shape

formed in the previous step 3 cuboids of dimension

3x3x1 to get the shape shown below.

To represent a^3 + 3ab^2- 3a^2b-b^3 extract from the

shape formed in the previous step 1 cube of dimension

1x1x1 .The shape shown below will be obtained.

Arrange the unit cubes left to make a cube of dimension
2x2x2 cubic units.

Observe the following
•The number of unit cubes in a^3 = …27…..
•The number of unit cubes in 3ab^2 =…9……
•The number of unit cubes in 3a^2b=…27……
•The number of unit cubes in b^3 =…1……
•The number of unit cubes in
a^3 - 3a^2b + 3ab^2- b^3 = …8…..
•The number of unit cubes in (a-b)^3 =…8…
It is observed that the number of unit cubes
in (a-b)^3 is equal to the number of unit cubes
in a^3 -3a^2b+3ab^2-b^3 .

Using the same strategy we can verify the other cubic identities.

Verifying Algebraic identity(I)

Today I will share with U how to verify cubic algebraic identities using unit cubes by activity method.

Proving the algebraic identity

(a+b)^3=a^3+3a^2b+3ab^2+b^3

Let a=3 and b=1.
To represent a^3 make a cube of dimension axaxa
i.e. 3x3x3 cubic units.

To represent 3a^2b make 3 cuboids of dimension
axaxb i.e. 3x3x1 cubic units.

To represent 3ab^2 make 3 cuboids of dimension
axbxb i.e. 3x1x1 cubic units.

To represent b^3 make a cube of dimension bxbxb
i.e. 1x1x1 cubic units.

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 a^3 = ……..
The number of unit cubes in 3a^2b =………
The number of unit cubes in 3ab^2 =………
The number of unit cubes in b^3 =………
The number of unit cubes in a^3 + 3a^2b + 3ab^2 + b^3
= ……..
The number of unit cubes in (a+b)^3 =……

It is observed that the number of unit cubes
in (a+b)^3 is equal to the number of unit cubes
in a^3 +3a^2b+3ab^2+b^3 .

Students in my class find this activity ineresting and they enjoy playing with unit cubes and learning algebraic formulae by this method.

In my next post I will share the 2nd algebraic formula.

Bye..

Using unit cubes for learning math

We at "Planet Infinity" are using Unit cubes for making our teaching/learning of cubic algebraic identities in a different way.The first question which comes in the mind is what is a unit cube?
A unit cube is nothing but a cube all of whose sides are 1 unit long as shown below.

• We say its dimension is 1 X 1 X 1.
• The volume of a 3-dimensional unit cube is 1 cubic unit.
• By joining unit cubes we can form cubes and cuboids of varied dimension.

In our “Planet infinity” we have designed the activities using the unit cubes. We have 500 wooden unit cubes in our laboratory which were prepared by the help of the carpenter in our school.It is necessary to arrange the unit cubes first.You can purchase plastic unit cubes from a toy shop also. I believe that "Where there is a will there is a way".

We have designed the following activities for the students of grade 8 and grade 9.
1) the formation of cubes and cuboids using unit cubes.
2) Verification of algebraic identities
i) (a+b)^3 = a^3 +3a^2b +3ab^2 +b^3
ii) (a-b)^3 = a^3 -3a^2b +3ab^2 -b^3
iii) a^3+b^3 = (a+b)(a^2-ab+b^2)
iv) )a^3-b^3 = (a-b)(a^2+ab+b^2)
Firstly, I will discuss the formation of cubes and cuboids using unit cubes.
(a)To make a cuboid of dimension 2 x 1 x 1 ,we need 2 unit cubes.Place the two unit cubes adjacent to each other.You will observe that the shape which is obtained by joining 2 unit cubes is not a cube,but a cuboid whose length is 2 units ,breadth is 1 unit and height is 1 unit as shown below.

Children love to do this activity because of two reasons ,one,they are able to do this on their own which motivates them for further learning and second,they are able to visualise the concept of 3-dimension in mathematics.
(b)To make a cube of dimension 3 x 3 x 3,we need 27 unit cubes.Firstly,try to get the length by joining 3 unit cubes,then try to get the breadth 3 units by adding more unit cubes.By doing this a base cuboid of dimension 3 x 3 x 1 will be formed.Now ask the students to add unit cubes to get the heigth 3 units.Thus a cube which is obtained is of dimension 3 x 3 x 3 .

Like this you can get cubes and cuboids of varied dimensions using unit cubes.
In my next post I will share with U "how we are verifying the cubic algebraic identities using the unit cubes"?