Table of Contents
Integers
Operations on Integers and Their Properties
| Operation | Rule | Example |
|---|---|---|
| Addition | Same signs – add and keep sign; Different signs – subtract and take sign of larger number | \((-3) + (-5) = -8\), \((-6) + (4) = -2\) |
| Subtraction | Subtracting a number is same as adding its additive inverse | \(5 – (-3) = 5 + 3 = 8\) |
| Multiplication | Product of two integers with same sign is positive; with different signs is negative | \((-4) imes (-2) = 8\), \((-3) imes 2 = -6\) |
In this chapter, students learn How To perform arithmetic operations on integers, including addition, subtraction, multiplication, and division. The key is understanding how positive and negative numbers interact. For instance, subtracting a negative number is equivalent to adding a positive one, which simplifies expressions greatly.By mastering integer operations, students develop number sense, which is crucial in algebra, coordinate geometry, and data handling. The rules of integers also help in solving higher-level equations in future grades.
Fractions and Decimals
Representation and Operations on Fractions
| Operation | Rule | Example |
|---|---|---|
| Addition/Subtraction | Find LCM of denominators and then add or subtract numerators | \(\frac{2}{5} + \frac{3}{10} = \frac{7}{10}\) |
| Multiplication | Multiply numerators and denominators directly | \(\frac{2}{3} imes \frac{4}{5} = \frac{8}{15}\) |
| Division | Multiply the first fraction by reciprocal of the second | \(\frac{3}{4} \div \frac{2}{5} = frac{3}{4} imes frac{5}{2} = frac{15}{8}\) |
Fractions and decimals are essential for understanding ratios, percentages, and measurements. In this topic, students learn How To perform arithmetic operations systematically using denominators and reciprocals. Real-life examples include calculating discounts, distances, or cooking measurements.Regular practice with fractions improves accuracy in numerical problem-solving. It also provides a solid base for topics like rational numbers and algebraic simplification in higher classes.
Algebraic Expressions
Basic Terms and Operations in Algebra
| Concept | Explanation | Example |
|---|---|---|
| Monomial | Expression with one term | \(5x\), \(-3y^2\) |
| Binomial | Expression with two terms | \(x + 4\), \(2a – 5b\) |
| Polynomial | Expression with more than two terms | \(3x^2 + 2x + 1\) |
Algebra introduces the use of variables and constants to express mathematical relationships. Students learn to form and simplify expressions, evaluate identities, and substitute values. Understanding the structure of terms like monomials and polynomials helps in solving higher-level equations later.For example, simplifying \(3x + 5x = 8x\) demonstrates the combination of like terms. Algebra serves as a bridge between arithmetic and advanced mathematics, enhancing logical and abstract reasoning skills.
Practical Geometry
Construction Using Compass and Ruler
| Construction | Steps | Use |
|---|---|---|
| Perpendicular Bisector | Draw an arc from both ends of a line segment intersecting above and below; join intersection points | Used to divide a segment into equal halves |
| Angle Bisector | Draw an arc cutting both arms; from intersections draw another arc to intersect previous arc; join with vertex | Used in geometry and engineering drawing |
| Triangle Construction | Based on SSS, SAS, or ASA criteria | Used for geometrical proofs and architecture |
Practical Geometry helps students visualize and construct shapes accurately using geometrical tools like a compass and protractor. The chapter emphasizes step-by-step construction of triangles, perpendicular bisectors, and angle bisectors. Understanding these processes enhances spatial awareness and precision.Geometry develops a student’s ability to think logically and geometrically. The use of tools and the understanding of congruence rules such as SSS, SAS, and ASA are fundamental for advanced geometrical reasoning in higher grades.