#### 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
- Reciprocal Or Multiplicative Inverse of Rational Numbers
- Rational Numbers on a Number Line
- Rational Numbers Between Two Rational Numbers

##### Linear Equations in One Variable

- The Idea of a Variable
- Expressions with Variables
- Concept of Equation
- 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 - Side, Vertex, Adjacent Sides, Adjacent Vertices and Diagonal
- Classification of Polygons
- Angle Sum Property of a Quadrilateral
- Interior Angles of a Polygon
- Exterior Angles of a Polygon and Its Property
- Concept of Quadrilaterals - Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles
- 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 Practical Geometry
- 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
- Concept of Percent and Percentage
- Increase Or Decrease as Percent
- Concept of Discount
- Estimation in Percentages
- Concepts of Cost Price, Selling Price, Total Cost Price, and Profit and Loss, Discount, Overhead Expenses and GST
- Sales Tax, Value Added Tax, and Good and Services Tax
- Concept of Principal, Interest, Amount, and Simple 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)
- Expansion of (x + a)(x + b)

##### Visualizing Solid Shapes

##### Mensuration

##### 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
- Linear Graphs
- Some Application of Linear Graphs

##### Playing with Numbers

## Notes

**Some More Interesting Patterns of Square Number:**

**1. Adding triangular numbers:**

If we combine two consecutive triangular numbers, we get a square number, like

**2. Numbers between square numbers:**

Between 1^{2}(=1) and 2^{2}(= 4), there are two (i.e., 2 × 1) non-square numbers 2, 3.

Between 2^{2}(= 4) and 3^{2}(= 9) there are four (i.e., 2 × 2) non-square numbers 5, 6, 7, 8.

Now, 3^{2} = 9, 4^{2} = 16.

Therefore, 4^{2} – 3^{2} = 16 – 9 = 7.

Between 9(= 3^{2}) and 16(= 4^{2}), the numbers are 10, 11, 12, 13, 14, 15 that is, six non-square numbers which is 1 less than the difference of two squares.

We have 4^{2} = 16 and 5^{2} = 25.

Therefore, 5^{2} – 4^{2} = 9.

Between 16 (= 4^{2}) and 25(= 5^{2}), the numbers are 17, 18,..., 24 that is, eight non-square numbers which is 1 less than the difference of two squares.

If we think of any natural number n and (n + 1), then,

(n + 1)^{2} – n^{2} = (n^{2} + 2n + 1) – n^{2} = 2n + 1.

We find that between n^{2} and (n + 1)^{2} there are 2n numbers which is 1 less than the difference of two squares.

Thus, in general, we can say that there are 2n non-perfect square numbers between the squares of the numbers n and (n + 1), where n is any natural number.

**3. Adding odd numbers:**

The sum of first n odd natural numbers is n^{2}.

1 [one odd number] = 1 = 1^{2}

1 + 3 [sum of first two odd numbers] = 4 = 2^{2}

1 + 3 + 5 [sum of first three odd numbers] = 9 = 3^{2}

If a natural number cannot be expressed as a sum of successive odd natural numbers starting with 1, then it is not a perfect square.

25 = 1 + 3 + 5 + 7 + 9. Also, 25 is a perfect square.

**4. A sum of consecutive natural numbers:**

We can express the square of any odd number as the sum of two consecutive positive integers `(n^2 – 1)/2 and (n^2 + 1)/2`.

For Example,

11^{2} = 121 = 60 + 61 = `(11^2 – 1)/2 + (11^2 + 1)/2`.

15^{2} = 225 = 112 + 113 = `(15^2 – 1)/2 + (15^2 + 1)/2`.

**5. Product of two consecutive even or odd natural numbers:**

Product of two consecutive even or odd natural numbers are calculated as (a + 1) × (a – 1) = a^{2} – 1 where a is a natural number and (a + 1), (a – 1) are two consecutive even or odd natural numbers.

For example,

29 × 31 = (30 – 1) × (30 + 1) = 30^{2} – 1.

44 × 46 = (45 – 1) × (45 + 1) = 45^{2} – 1.

**6. Some more patterns in square numbers:**

Observe the squares of numbers; 1, 11, 111,... etc. They give a beautiful pattern:

Another interesting pattern.

#### Shaalaa.com | Pattern of Square Numbers - Part 1

##### Series: Some More Interesting Patterns of Square Number

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