Topics
Number Systems
Number Systems
Polynomials
Algebra
Algebraic Expressions
Algebraic Identities
Coordinate Geometry
Linear Equations in Two Variables
Coordinate Geometry
Geometry
Area
Constructions
- Introduction of Constructions
- Geometric Constructions
- Some Constructions of Triangles
Introduction to Euclid’S Geometry
Mensuration
Statistics and Probability
Lines and Angles
- Introduction to Lines and Angles
- Basic Terms and Definitions
- Intersecting Lines and Non-intersecting Lines
- Parallel Lines
- Concept of Pairs of Angles
- Concept of Transversal Lines
- Basic Properties of a Triangle
Probability
Triangles
Quadrilaterals
- Properties of Quadrilateral
- Another Condition for a Quadrilateral to Be a Parallelogram
- Theorem of Midpoints of Two Sides of a Triangle
- Property: The Opposite Sides of a Parallelogram Are of Equal Length.
- Theorem: A Diagonal of a Parallelogram Divides It into Two Congruent Triangles.
- Theorem : If Each Pair of Opposite Sides of a Quadrilateral is Equal, Then It is a Parallelogram.
- Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
- Theorem: If in a Quadrilateral, Each Pair of Opposite Angles is Equal, Then It is a Parallelogram.
- Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
- Theorem : If the Diagonals of a Quadrilateral Bisect Each Other, Then It is a Parallelogram
Circles
Areas - Heron’S Formula
- Area of a Triangle by Heron's Formula
- Application of Heron’s Formula in Finding Areas of Quadrilaterals
- Geometric Interpretation of the Area of a Triangle
Surface Areas and Volumes
Statistics
- Introduction
- Types of Expressions
- Rules for Separating Terms
- Framing Algebraic Expressions and Formulas
- Real-Life Applications
- Key Points Summary
Introduction
Algebraic expressions consist of terms joined by +/– signs. Understanding term separation, expression types, and framing processes is essential.
An algebraic expression is like a grocery cart of items (terms) joined by + or – signs.
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Term: A single item (e.g., 5x, 7).
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Constant term: No letters (e.g., 7).
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Note: × and ÷ do not split terms.
Types of Expressions
| Type | Definition | Example |
|---|---|---|
| Monomial | 1 term | -8, xy |
| Binomial | 2 unlike terms | 5x + 2y |
| Trinomial | 3 unlike terms | ax² + bx + c |
| Multinomial | ≥ 2 terms | 7 + x - xy + y² |
| Polynomial | One or more unlike terms; whole-number powers | 3x + 2y - 7z + 8 |
Rules for Separating Terms
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Plus (+) and minus (−) signs separate terms.
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Multiplication (×) and division (÷) do not separate terms.
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Example: In 3p + 5z − 7y, there are three terms (3p, 5z, −7y).
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In 3p × 5z − 7y, there are only two terms (3p × 5z, −7y).
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Framing Algebraic Expressions and Formulas
Framing Algebraic Expressions:
- Converting a statement into mathematical terms (no equals sign).
- Example: The area of a rectangle is the product of length and breadth:
Expression: l × b
Framing Formulas:
- Converting a complete statement into an equation (with an equals sign).
- Examples: "The sum of x and y is 75" → x + y = 75
- The formula for time taken by a hiker is distance divided by speed:
Formula: t = `D/S`
Real-Life Applications
| Situation | Expression/Formula |
|---|---|
| Recipe ingredients | 2c + 3s (2 cups flour + 3 spoons sugar) |
| Movie ticket cost | T = 150n (150 rupees per person × n people) |
| Pizza sharing | P ÷ f (Pizza slices ÷ friends) |
| Savings goal | S = M × 12 (Monthly savings × 12 months) |
Key Points Summary
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Terms are separated by '+/–' only.
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Expression types depend on term count and whole-number exponents.
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Frame as an expression (no “=”) or a formula (with “=”).
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Polynomials need whole number powers only
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A constant term has no letters (like the number 7)
