Topics
Integers
- Concept for Natural Numbers
- Concept for Whole Numbers
- Negative and Positive 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
- 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
- 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
notes
Use of Variables in Common Rules:
1. Using Variables To Find The Rule Of Perimeter:
- The perimeter of a square = `4l, "where" l` = the length of the side of the square.
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The perimeter of an equilateral triangle = `3l, "where" l` = length of the side of the equilateral triangle.
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Perimeter of a regular pentagon = `5l, "where" l` = the length of the side of the pentagon.
2. Using Variables To Find The Rule Of Area:
- If we denote the length of a square by l, then the area of the square = l2.
- If we denote the length of a rectangle by l and its breadth by b, then the area of the rectangle = l × b = lb.
- Similarly, If b stands for the base and h for the height of the triangle, then the area of the triangle = `1/2 × b × h`.
3. Rules for number patterns:
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If a natural number is denoted by n, its successor is (n + 1). We can check this for any natural number. For example, if n = 10, its successor is n + 1=11, which is known.
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If a natural number is denoted by n, 2n is an even number and (2n + 1) an odd number. Let us check it for any number, say, 15; 2n = 2 × n = 2 × 15 = 30 is indeed an even number and 2n + 1 = 2 × 15 + 1 = 30 + 1 = 31 is indeed an odd number.
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The term which occurs at the nth position is given by the expression 3n. You can easily find the term which occurs in the 10th position (which is 3 × 10 = 30); 100th position (which is 3 × 100 = 300) and so on.
4. Pattern in geometry:
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Algebraic expressions are used in writing patterns followed by geometrical figures.
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The number of diagonals that we can draw from one vertex of any polygon is (n – 3) where n is the number of sides of the polygon.
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Example,
Square = n – 3 = 4 – 3 = 1 diagonal from one vertex.
Pentagon = n – 3 = 5 – 3 = 2 diagonals from one vertex.
Hexagon = n – 3 = 6 – 3 = 3 diagonals from one vertex.
5. Rules from arithmetic:
(i) Commutativity of addition of two numbers:
We know that 4 + 3 = 7 and 3 + 4 = 7
i.e. 4 + 3 = 3 + 4
This property of numbers isknown as the commutativity of addition of numbers.
Let a and b be two variables which can take any number value. Then, a + b = b + a.
(ii) Commutativity of multiplication of two numbers:
4 × 3 = 12, 3 × 4 = 12
Hence, 4 × 3 = 3 × 4
This property of numbers is known as commutativity of multiplication of numbers.
Let a and b be two variables which can take any number value.
We can express the commutativity of multiplication of two numbers as a × b = b × a
(iii) Distributivity of numbers:
7 × 38 = 7 × (30 + 8) = 7 × 30 + 7 × 8 = 210 + 56 = 266.
This property is known as distributivity of multiplication over addition of numbers.
Let a, b and c be three variables, each of which can take any number. Then, a × (b + c) = a × b + a × c
(iv) Associativity Of Addition Of Numbers:
Rule For Associativity Of Addition Of Numbers i.e., (a + b) + c = a + (b + c).
Shaalaa.com | Using Variables To Find The Rule Of Perimeter Of A Square
Series: Use of Variables In Common Rule
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Related QuestionsVIEW ALL [4]
Use the given algebraic expression to complete the table of number patterns.
S. No |
Expression |
Terms | |||||||||
1st | 2nd | 3rd | 4th | 5th | ... | 10th | ... | 100th | ... | ||
1 | 2n - 1 | 1 | 3 | 5 | 7 | 9 | - | 19 | - | - | - |
2 | 3n + 2 | 5 | 8 | 11 | 14 | - | - | - | - | - | - |
3 | 4n + 1 | 5 | 9 | 13 | 17 | - | - | - | - | - | - |
4 | 7n + 20 | 27 | 34 | 41 | 48 | - | - | - | - | - | - |
5 | n2 + 1 | 2 | 5 | 10 | 17 | - | - | - | - | 10001 | - |