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 Reciprocals or Multiplicative Inverses
- Rational Numbers on a Number Line
- Rational Numbers Between Two Rational Numbers
- Multiples and Common Multiples
Linear Equations in One Variable
- Constants and Variables in Mathematics
- Equation in Mathematics
- Expressions with Variables
- Word Problems on Linear Equations
- 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 Linear Equations
Understanding Quadrilaterals
- Concept of Curves
- Different Types of Curves - Closed Curve, Open Curve, Simple Curve.
- Basic Concept of Polygons
- Classification of Polygons
- Properties of Quadrilateral
- Sum of Interior Angles of a Polygon
- Sum of Exterior Angles of a Polygon
- 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.
Data Handling
Practical Geometry
- 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
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
- Ratio
- Increase Or Decrease as Percent
- Concept of Discount
- Estimation in Percentages
- Basic Concepts of Profit and Loss
- 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
- Classification of Terms in Algebra
- 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
Exponents and Powers
Visualizing Solid Shapes
Direct and Inverse Proportions
Factorization
- Factors and Common Factors
- 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
Playing with Numbers
Definition
Line graph: A line graph displays data that changes continuously over periods of time.
Notes
Concept of a Line Graph:
A line graph displays data that changes continuously over periods of time.
1. Consider the following data, given in tabular form.
|
Time |
6 a.m. |
10 a.m. |
2 p.m. |
6 p.m. |
| Temperature(°C) |
37 |
40 |
38 |
35 |
-
The horizontal line (usually called the x-axis) shows the timings at which the temperatures were recorded.
-
Temperature(°C) is labelled on the vertical line (usually called the y-axis).
-
By this line graph, we can easily understand the increase and decrease of temperature over the course of 12 hours from 6 AM to 6 PM.
Interpretation of line graph:
1) The given graph represents the total runs scored by two batsmen A and B, during each of the ten different matches in the year 2007. Study the graph and answer the following questions.
(i) What information is given on the two axes?
(ii) Which line shows the runs scored by batsman A?
(iii) Were the run scored by them the same in any match in 2007? If so, in which match?
(iv) Among the two batsmen, who is steadier? How do you judge it?
Solution:
(i) The horizontal axis (or the x-axis) indicates the matches played during the year 2007. The vertical axis (or the y-axis) shows the total runs scored in each match.
(ii) The dotted line shows the runs scored by Batsman A. (This is already indicated at the top of the graph).
(iii) During the 4th match, both have scored the same number of 60 runs. (This is indicated by the point at which both graphs meet).
(iv) Batsman A has one great “peak” but many deep “valleys”. He does not appear to be consistent. B, on the other hand, has never scored below a total of 40 runs, even though his highest score is only 100 in comparison to 1 15 of A. Also, A has scored a zero in two matches, and in a total of 5 matches, he has scored less than 40 runs. Since A has a lot of ups and downs, B is a more consistent and reliable batsman.
Example
The given graph describes the distances of a car from a city P at different times when it is travelling from City P to City Q, which are 350 km apart. Study the graph and answer the following:
(i) What information is given on the two axes?
(ii) From where and when did the car begin its journey?
(iii) How far did the car go in the first hour?
(iv) How far did the car go during (i) the 2nd hour? (ii) the 3rd hour?
(v) Was the speed same during the first three hours? How do you know it?
(vi) Did the car stop for some duration at any place? Justify your answer.
(vii) When did the car reach City Q?
(i) The horizontal (x) axis shows the time. The vertical (y) axis shows the distance of the car from City P.
(ii) The car started from City P at 8 a.m.
(iii) The car travelled 50 km during the first hour.
(iv) The distance covered by the car during
(a) the 2nd hour (i.e., from 9 am to 10 am) is 100 km, (150 – 50).
(b) the 3rd hour (i.e., from 10 am to 11 am) is 50 km (200 – 150).
(v) From the answers to questions (iii) and (iv), we find that the speed of the car was not the same all the time. (In fact, the graph illustrates how the speed varied).
(vi) We find that the car was 200 km away from city P when the time was 11 a.m. and
also at 12 noon. This shows that the car did not travel during the interval of 11 a.m. to 12 noon. The horizontal line segment representing “travel” during this period is illustrative of this fact.
(vii) The car reached City Q at 2 p.m.
Related QuestionsVIEW ALL [33]
Which graphs of the following represent the table below?
| Length of Side of a Square | 1 | 2 | 3 | 4 | 5 |
| Perimeter | 4 | 8 | 12 | 16 | 20 |
Use the tables below to draw linear graphs.
Population (in thousands) of men and women in a village in different years.
| Year | 2003 | 2004 | 2005 | 2006 | 2007 |
| Number of men | 12 | 12.5 | 13 | 13.2 | 13.5 |
| Number of women | 11.3 | 11.9 | 13 | 13.6 | 12.8 |
