#### Topics

##### Integers

- Concept for Natural Numbers
- Concept for Whole Numbers
- Concept of Negative Numbers
- Concept of Integers
- Representation of Integers on the Number Line
- Concept for Ordering of Integers
- Addition of Integers
- Addition of Integers on Number line
- Subtraction of Integers
- Properties of Addition and Subtraction of Integers
- Multiplication of a Positive and a Negative Integers
- Multiplication of Two Negative Integers
- Product of Three Or More Negative Integers
- Closure Property of Multiplication of Integers
- Commutative Property of Multiplication of Integers
- Associative Property of Multiplication of Integers
- Distributive Property of Multiplication of Integers
- Multiplication of Integers with Zero
- Multiplicative Identity of Integers
- Making Multiplication Easier of Integers
- Division of Integers
- Properties of Division of Integers

##### Fractions and Decimals

- Concept of Fractions
- Types of Fraction
- Concept of Proper Fractions.
- Improper Fraction and Mixed Fraction
- Concept for Equivalent Fractions
- Like and Unlike Fraction
- Comparing Fractions
- Addition of Fraction
- Subtraction of Fraction
- Multiplication of a Fraction by a Whole Number
- Fraction as an Operator 'Of'
- Multiplication of a Fraction by a Fraction
- Division of Fractions
- Concept for Reciprocal of a Fraction
- Concept of Decimal Numbers
- Multiplication of Decimal Numbers
- Multiplication of Decimal Numbers by 10, 100 and 1000
- Division of Decimal Numbers by 10, 100 and 1000
- Division of a Decimal Number by a Whole Number
- Division of a Decimal Number by Another Decimal Number

##### Data Handling

##### Simple Equations

##### Lines and Angles

- Concept of Points
- Concept of Line
- Concept of Line Segment
- Concept of Intersecting Lines
- Concept of Angle - Arms, Vertex, Interior and Exterior Region
- Complementary Angles
- Supplementary Angles
- Adjacent Angles
- Concept of Linear Pair
- Concept of Vertically Opposite Angles
- Concept of Intersecting Lines
- Concept of Parallel Lines
- Pairs of Lines - Transversal
- Pairs of Lines - Angles Made by a Transversal
- Pairs of Lines - Transversal of Parallel Lines
- Checking Parallel Lines

##### The Triangle and Its Properties

- Concept of Triangles - Sides, Angles, Vertices, Interior and Exterior of Triangle
- Classification of Triangles (On the Basis of Sides, and of Angles)
- Equilateral Triangle
- Isosceles Triangles
- Scalene Triangle
- Acute Angled Triangle
- Obtuse Angled Triangle
- Right Angled Triangle
- Median of a Triangle
- Altitudes of a Triangle
- Exterior Angle of a Triangle and Its Property
- Angle Sum Property of a Triangle
- Some Special Types of Triangles - Equilateral and Isosceles Triangles
- Sum of the Lengths of Two Sides of a Triangle
- Right-angled Triangles and Pythagoras Property

##### Congruence of Triangles

##### Comparing Quantities

- Concept of Ratio
- Concept of Equivalent Ratios
- Concept of Proportion
- Concept of Unitary Method
- Concept of Percent and Percentage
- Converting Fractional Numbers to Percentage
- Converting Decimals to Percentage
- Converting Percentages to Fractions
- Converting Percentages to Decimals
- Estimation in Percentages
- Interpreting Percentages
- Converting Percentages to “How Many”
- Ratios to Percents
- Increase Or Decrease as Percent
- Concepts of Cost Price, Selling Price, Total Cost Price, and Profit and Loss, Discount, Overhead Expenses and GST
- Profit or Loss as a Percentage
- Concept of Principal, Interest, Amount, and Simple Interest

##### Rational Numbers

- Concept of Rational Numbers
- Equivalent Rational Number
- Positive and Negative Rational Numbers
- Rational Numbers on a Number Line
- Rational Numbers in Standard Form
- Comparison of Rational Numbers
- Rational Numbers Between Two Rational Numbers
- Addition of Rational Number
- Subtraction of Rational Number
- Multiplication of Rational Numbers
- Division of Rational Numbers

##### Practical Geometry

- Construction of a Line Parallel to a Given Line, Through a Point Not on the Line
- Construction of Triangles
- Constructing a Triangle When the Length of Its Three Sides Are Known (SSS Criterion)
- Constructing a Triangle When the Lengths of Two Sides and the Measure of the Angle Between Them Are Known. (SAS Criterion)
- Constructing a Triangle When the Measures of Two of Its Angles and the Length of the Side Included Between Them is Given. (ASA Criterion)
- Constructing a Right-angled Triangle When the Length of One Leg and Its Hypotenuse Are Given (RHS Criterion)

##### Perimeter and Area

- Mensuration
- Concept of Perimeter
- Perimeter of a Rectangle
- Perimeter of Squares
- Perimeter of Triangles
- Perimeter of Polygon
- Concept of Area
- Area of Square
- Area of Rectangle
- Triangles as Parts of Rectangles and Square
- Generalising for Other Congruent Parts of Rectangles
- Area of a Triangle
- Area of a Parallelogram
- Circumference of a Circle
- Area of Circle
- Conversion of Units
- Problems based on Perimeter and Area
- Problems based on Perimeter and Area

##### Algebraic Expressions

- Algebraic Expressions
- Terms, Factors and Coefficients of Expression
- Like and Unlike Terms
- Types of Algebraic Expressions as Monomials, Binomials, Trinomials, and Polynomials
- Addition of Algebraic Expressions
- Subtraction of Algebraic Expressions
- Evaluation of Algebraic Expressions by Substituting a Value for the Variable.
- Use of Variables in Common Rules

##### Exponents and Powers

- Concept of Exponents
- Multiplying Powers with the Same Base
- Dividing Powers with the Same Base
- Taking Power of a Power
- Multiplying Powers with Different Base and Same Exponents
- Dividing Powers with Different Base and Same Exponents
- Numbers with Exponent Zero, One, Negative Exponents
- Miscellaneous Examples Using the Laws of Exponents
- Decimal Number System Using Exponents and Powers
- Expressing Large Numbers in the Standard Form

##### Symmetry

##### Visualizing Solid Shapes

- Plane Figures and Solid Shapes
- Faces, Edges and Vertices
- Nets for Building 3-d Shapes - Cube, Cuboids, Cylinders, Cones, Pyramid, and Prism
- Drawing Solids on a Flat Surface - Oblique Sketches
- Drawing Solids on a Flat Surface - Isometric Sketches
- Visualising Solid Objects
- Viewing Different Sections of a Solid

#### description

- Multiplication of a fraction by a fraction
- Value of the Products

#### notes

**Multiplication of a fraction by a fraction:**

Let us now see how to find the product of two fractions

To do this we first learn to find the products like `1/2 xx 1/3`.

We divide the whole into three equal parts. Each of the three parts represents `1/3` of the whole. Take one part of these three parts and shade it.

Divide this one-third `(1/3)` shaded part into two equal parts. Each of these two parts represents `1/2 "of" 1/3 "i.e.," 1/2 xx 1/3`.

Take out 1 part of these two and name it ‘A’. ‘A’ represents `1/2 xx 1/3`.

The whole was divided in 6 = 2 x 3 parts and 1 = 1 x 1 part was taken out of it.

Thus, `1/2 xx 1/3 = 1/6 = (1 xx 1)/(2 xx 3)`

or `1/2 xx 1/3 = (1 xx 1)/(2 xx 3)`

The value of `1/3 xx 1/2` can be found in a similar way. Divide the whole into two equal parts and then divide one of these parts into three equal parts. Take one of these parts. This will represent `1/3 xx 1/2 "i.e.," 1/6`.

Therefore, `1/3 xx 1/2 = 1/6 = (1 xx 1)/(3 xx 2)` as discussed earlier.

Hence, `1/2 xx 1/3 = 1/3 xx 1/2 = 1/6`.

Now,

Each of these five equal shapes are parts of five similar circles. Take one such shape. To obtain this shape we first divide a circle into three equal parts. Further, divide each of these three parts into two equal parts. One part out of it is the shape we considered

It will represent `1/2 xx 1/3 = 1/6`.

The total of such parts would be `5 xx 1/6 = 5/6`.

Two fractions are multiplied by multiplying their numerators and denominators separately and writing the product as

`"Product of numerators"/"Product of denominators"`

For example, `2/3 xx 5/7 = (2 xx 5)/(3 xx 7) = 10/21`.

**2. Value of the Products:**

You have seen that the product of two whole numbers is bigger than each of the two whole numbers.

For example, 3 × 4 = 12 and 12 > 4, 12 > 3.

**A) Product of two proper fractions:**

When two proper fractions are multiplied, the product is less than each of the fractions. Or, we say the value of the product of two proper fractions is smaller than each of the two fractions.

`2/3 xx 4/5 = 8/15` |
`8/15 < 2/3, 8/15 < 4/5` |
Product is less than each of the fractions |

**B) Product of two improper fractions:**

The product of two improper fractions is greater than each of the two fractions. Or, the value of the product of two improper fractions is more than each of the two fractions.

`7/3 xx 5/2 = 35/6` | `35/6 > 7/3, 35/6 > 5/2` | Product is greater than each of the fractions |

**C)** **Product of improper fractions and proper fractions:**

The product obtained is less than the improper fraction and greater than the proper fraction involved in the multiplication.

`2/3 xx 7/5 = 14/15` | `14/15 < 7/5 and 14/15 > 2/3` | Product is less than the improper fraction and greater than the proper fraction. |

#### Example

Sulochanabai owns 42 acres of farmland. If she planted wheat on `2/7` of the land, on how many acres has she planted wheat?

We must find out 2 7 of 42 acres

∴ `42/1 xx 2/7 = (42 xx 2)/(1xx 7) = (6 xx 7 xx 2)/7 × × = 12`

Sulochanabai has planted wheat on 12 acres of land.