Topics
Rational Numbers
- Rational Numbers
- Closure Property of Rational Numbers
- Commutative Property of Rational Numbers
- Associative Property of Rational Numbers
- Distributive Property of Multiplication Over Addition for Rational Numbers
- Identity of Addition and Multiplication of Rational Numbers
- Negative Or Additive Inverse of Rational Numbers
- Concept of Reciprocal or Multiplicative Inverse
- Rational Numbers on a Number Line
- Rational Numbers Between Two Rational Numbers
Linear Equations in One Variable
- Variable of Equation
- Concept of Equation
- Expressions with Variables
- Balancing an Equation
- The Solution of an Equation
- Linear Equation in One Variable
- Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
- Some Applications Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
- Solving Equations Having the Variable on Both Sides
- Some More Applications on the Basis of Solving Equations Having the Variable on Both Sides
- Reducing Equations to Simpler Form
- Equations Reducible to the Linear Form
Understanding Quadrilaterals
- Concept of Curves
- Different Types of Curves - Closed Curve, Open Curve, Simple Curve.
- Concept of Polygons
- Classification of Polygons
- Properties of a Quadrilateral
- Interior Angles of a Polygon
- Exterior Angles of a Polygon and Its Property
- Concept of Quadrilaterals
- Properties of Trapezium
- Properties of Kite
- Properties of a Parallelogram
- Properties of Rhombus
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Property: The adjacent angles in a parallelogram are supplementary.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Property: The diagonals of a rhombus are perpendicular bisectors of one another.
- Property: The Diagonals of a Rectangle Are of Equal Length.
- Properties of Rectangle
- Properties of a Square
- Property: The diagonals of a square are perpendicular bisectors of each other.
Practical Geometry
- Introduction to Geometric Tool
- Constructing a Quadrilateral When the Lengths of Four Sides and a Diagonal Are Given
- Constructing a Quadrilateral When Two Diagonals and Three Sides Are Given
- Constructing a Quadrilateral When Two Adjacent Sides and Three Angles Are Known
- Constructing a Quadrilateral When Three Sides and Two Included Angles Are Given
- Some Special Cases
Data Handling
- Concept of Data Handling
- Interpretation of a Pictograph
- Interpretation of Bar Graphs
- Drawing a Bar Graph
- Interpretation of a Double Bar Graph
- Drawing a Double Bar Graph
- Organisation of Data
- Frequency Distribution Table
- Graphical Representation of Data as Histograms
- Concept of Pie Graph (Or a Circle-graph)
- Interpretation of Pie Diagram
- Chance and Probability - Chance
- Basic Ideas of Probability
Squares and Square Roots
- Concept of Square Number
- Properties of Square Numbers
- Some More Interesting Patterns of Square Number
- Finding the Square of a Number
- Concept of Square Roots
- Finding Square Root Through Repeated Subtraction
- Finding Square Root Through Prime Factorisation
- Finding Square Root by Division Method
- Square Root of Decimal Numbers
- Estimating Square Root
Cubes and Cube Roots
Comparing Quantities
- Concept of Ratio
- Basic Concept of Percentage
- Increase Or Decrease as Percent
- Concept of Discount
- Estimation in Percentages
- Basic Concepts of Profit and Loss
- Sales Tax, Value Added Tax, and Good and Services Tax
- Calculation of Interest
- Concept of Compound Interest
- Deducing a Formula for Compound Interest
- Rate Compounded Annually Or Half Yearly (Semi Annually)
- Applications of Compound Interest Formula
Algebraic Expressions and Identities
- Algebraic Expressions
- Terms, Factors and Coefficients of Expression
- Types of Algebraic Expressions as Monomials, Binomials, Trinomials, and Polynomials
- Like and Unlike Terms
- Addition of Algebraic Expressions
- Subtraction of Algebraic Expressions
- Multiplication of Algebraic Expressions
- Multiplying Monomial by Monomials
- Multiplying a Monomial by a Binomial
- Multiplying a Monomial by a Trinomial
- Multiplying a Binomial by a Binomial
- Multiplying a Binomial by a Trinomial
- Concept of Identity
- Expansion of (a + b)2 = a2 + 2ab + b2
- Expansion of (a - b)2 = a2 - 2ab + b2
- Expansion of (a + b)(a - b) = a2-b2
- Expansion of (x + a)(x + b)
Mensuration
Visualizing Solid Shapes
Exponents and Powers
Direct and Inverse Proportions
Factorization
- Factors and Multiples
- Factorising Algebraic Expressions
- Factorisation by Taking Out Common Factors
- Factorisation by Regrouping Terms
- Factorisation Using Identities
- Factors of the Form (x + a)(x + b)
- Dividing a Monomial by a Monomial
- Dividing a Polynomial by a Monomial
- Dividing a Polynomial by a Polynomial
- Concept of Find the Error
Introduction to Graphs
- Concept of Bar Graph
- Interpretation of Bar Graphs
- Drawing a Bar Graph
- Concept of Double Bar Graph
- Interpretation of a Double Bar Graph
- Drawing a Double Bar Graph
- Concept of Pie Graph (Or a Circle-graph)
- Graphical Representation of Data as Histograms
- Concept of a Line Graph
- Linear Graphs
- Some Application of Linear Graphs
Playing with Numbers
- Other Patterns in Squares
- Pythagorean Triplets
Notes
Finding the Square of a Number:
A number which produces a specified number when multiplied by itself is known as square numbers.
If a natural number m can be expressed as n2, where n is also a natural number, then m is a square number.
Example, 4, 9, 16, 25....etc.
Finding square of a number using identity:
Find the square of the following numbers without actual multiplication.
1) 392
= (30 + 9)2
= 30(30 + 9) + 9(30 + 9)
= 302 + 30 × 9 + 9 × 30 + 92
= 900 + 270 + 270 + 81
= 1521.
2) 422
= (40 + 2)2
= 40(40 + 2) + 2(40 + 2)
= 402 + 40 × 2 + 2 × 40 + 22
= 1600 + 80 + 80 + 4
= 1764.
Other patterns in squares:
1) 752 = 5625 = (7 × 8) hundreds + 25
2) 1252 = 15625 = (12 × 13) hundreds + 25
3) (a5)2 = (10a + 5)2 = 10a(10a + 5) + 5(10a + 5)
= 100a2 + 50a + 50a + 25
= 100a(a + 1) + 25
= a(a + 1) hundred + 25.
Pythagorean triplets:
For any natural number m > 1, we have (2m)2 + (m2 – 1)2 = (m2 + 1)2. So, 2m, m2 – 1 and m2 + 1 forms a Pythagorean triplet.
32 + 42 = 9 + 16 = 25 = 52
The collection of numbers 3, 4, and 5 is known as a Pythagorean triplet.
6, 8, 10 is also a Pythagorean triplet, since 62 + 82 = 36 + 64 = 100 = 102.
Example
Write a Pythagorean triplet whose smallest member is 8.
We can get Pythagorean triplets by using general form 2m, m2 - 1, m2 + 1.
m2 - 1 = 8
m2 = 8 + 1 = 9
m = 3
2m = 6 and m2 + 1 = 10
The triplet is thus 6, 8, 10. But 8 is not the smallest member of this.
2m = 8
m = 4
m2 – 1 = 16 – 1 = 15
m2 + 1 = 16 + 1 = 17
The triplet is 8, 15, 17 with 8 as the smallest member.
Example
Find a Pythagorean triplet in which one member is 12.
m2 - 1 = 12
m2 = 12 + 1 = 13.
Then the value of m will not be an integer.
2m = 12
m = 6
Thus, m2 - 1 = 36 - 1 = 35 and m2 + 1 = 36 + 1 = 37.
Therefore, the required triplet is 12, 35, 37.
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