Class 08 Activity2 – Algebraic Expressions

Objective: 

To verify the identity (x-y)² = x² + y² – 2xy.

Materials Required: 

Some thick sheets of paper (cardboard), scissors, geometry box, sketch pen, pencil, etc.

Procedure: 

Let us verify the identity (x-y)² = x² + y² - 2xy by taking x = 6, y = 4.

1. On a thick sheet of paper (cardboard), draw a figure as shown below. This shape is a combination of two squares, one with side 6 cm and other with side 4 cm. Using scissors, cut it out. Area of this shape = (6² + 4²) cm²

2. From the above piece, cut out a rectangle of size 6 cm x 4 cm, marked as I in the figure. 

Area of piece I = (6 x 4) cm².

3. From the remaining shape, cut out a rectangle of size 6 cm x 4 cm, marked as II in the figure.

Area of piece II = (6 x 4) cm²

4.Finally, you are left with a square piece of side = (6 -4) cm = 2 cm Area of this shape = (6-4) x (6 – 4)= (6-4)² cm²

5. Area of the shape in fig. 1 = (6² +4²) cm² 

Area of the shape after cutting out I =  (6² + 4²) –  (6 x 4) 

Area of the shape after cutting out II  

= (6² + 4²) –  (6 x 4) - (6x4) 

= [6² + 4² – 2 x (6x4)] cm² 

But, after cutting out II, you are left with a square piece of

 area (6 - 4)² cm² 

Thus, (6-4)² = 6² + 4² – 2 x (6 x 4)

 (x-y)² = x² + y² – 2xy [Q x = 6, y = 4]

Do Yourself:

Verify the identity (x-y)² = x² + y² – 2xy for the following pairs of numbers:

(i) x = 3, y = 1 (ii) x = 5, y = 2

(iii) x = 7, y = 5 (iv) x = 8, y = 3

Class 08 Activity – Algebraic Expressions

 Activity – Algebraic Expressions

Objective: 

To verify the identity (x + y)² = x² + y² + 2xy.

Materials Required: 

Some thick sheets of paper (cardboard), scissors, geometry box, sketch pen, pencil, etc.

Procedure: 

Let us verify the identity (x + y)² = x² + y² + 2xy for x = 5, y = 3

On a thick sheet of paper, draw two squares one with side 5 cm and another with 4.side 3 cm. Using scissors, cut them out.

2. On another thick sheet of paper, draw two rectangles each with dimensions5 cm x 3 cm. Using scissors, cut them out.

Activity – Algebraic Expressions

3. Now, arrange the four pieces in such a way that the resulting figure becomes a square as shown in the figure.

4. We see that the resulting figure is a square of side (5+3)cm.

Area of the resulting square  = (5 + 3)² cm²

Also, area of piece I = 5 cm x 5 cm = 5² cm² 

area of piece II = 3 cm x 3 cm = 3² cm²

area of piece III = 5 cm x 3 cm = (5 x 3) cm² and 

area of piece IV = 5 cm x 3 cm = (5 x 3) cm² 

Thus, area of the resulting piece = area of the four pieces.

 (5 + 3)² = 5² + 3² + (5 x 3) + (5 x 3)

(5 + 3)² = 5² + 3² + 2 x (5 x 3)

 (x + y)² = x² + y² + 2xy {Q x = 5, y = 3]

Do Yourself:

Verify the identity (x + y)² = x² + y² + 2xy for following pair of numbers:

x = 4, y = 2 (ii) x = 3, y = 6

(iii) x = 5, y = 4 (iv) x = 7, y = 8

Class 08 Activity – Cube and Cube roots

 Based on CHAPTERs7. Cubes and cube roots16. Playing with numbers9.Algebraic Expression14.Factorization2.Linear Equation in one Variable

Activity – Cube and Cube roots

 Objective : 

To find the cube of a number by paper folding.

Materials Required:

 A chart paper and a pair of scissors.

Procedure :

Any rectangular piece of paper represents the base. The number of times the paper is folded represents the exponent of the base.

2. To find the cube of 2, take a chart paper and fold it into two equal parts along the length and then along width. Again, fold it into two parts along the length.

3. You have folded the chart paper three times that represents 2³. Unfold it and cut along the folds. You will get 8 pieces i.e., 2³.

4. To find the cube of 3, take another chart paper and fold it into three equal parts as given in the figure.

5. Now, fold along the width, dividing it further into three equal parts.

6. This time fold it along the length, dividing it further into three equal parts.

7. Now unfold the rectangular sheet and cut along the folds, you will get 27 equal pieces, i.e., 3³

In this way you can find the cube of any number by folding a paper.

Class 08 PUZZLES

 PUZZLES 

I have two digits. I am a square. I am also a cube. What number am I? 

2. I am a two digit number. I am the square of the sum of my digits. What number am I?

3. The sum of the squares of six consecutive whole numbers is 1111. Find the six whole numbers. 

4. Great Grandmother wouldn't tell when she was born. She did say that she was A years old in the year A? What year was she born? (Hint: A is between 40 and 50)

5. The difference of the squares of two consecutive even numbers is 20. What are these even numbers? 

I have two digits. I am a square. I am also a cube. What number am I? 

Solution : 

64 = 8² = 43

2. I am a two digit number. I am the square of the sum of my digits. What number am I?

Solution : 81 = (8+1)² = 9²

3. The sum of the squares of six consecutive whole numbers is 1111. Find the six whole numbers. 

11² + 12² + 13² + 14² + 15² + 16² = 121+144+169+196+225+256=

4. Great Grandmother wouldn't tell when she was born. She did say that she was A years old in the year A²? What year was she born? (Hint: A is between 40 and 50)

Solution: 49 ( because only square number 49 is between 40 &50)

5. The difference of the squares of two consecutive even numbers is 20. What are these even numbers? 

Solution: 6² - 4² =  36 – 16 = 20