Topic outline

• • 01 Rational Numbers

Rational numbers are closed under the operations of addition, subtraction and multiplication. 2. The operations addition and multiplication are (i) commutative for rational numbers. (ii) associative for rational numbers. 3. The rational number 0 is the additive identity for rational numbers. 4. The rational number 1 is the multiplicative identity for rational numbers. 5. The additive inverse of the rational number a b is a b − and vice-versa. 6. The reciprocal or multiplicative inverse of the rational number a b is c d if 1 a c b d × = . 7. Distributivity of rational numbers: For all rational numbers a, b and c, a(b + c) = ab + ac and a(b – c) = ab – ac 8. Rational numbers can be represented on a number line.

• 02 Linear Equations in One Variable

An algebraic equation is an equality involving variables. It says that the value of the expression on one side of the equality sign is equal to the value of the expression on the other side. 2. The equations we study in Classes VI, VII and VIII are linear equations in one variable. In such equations, the expressions which form the equation contain only one variable. Further, the equations are linear, i.e., the highest power of the variable appearing in the equation is 1.

• Parallelogram: A quadrilateral with each pair of opposite sides parallel.

Rhombus: A parallelogram with sides of equal length.

Rectangle: A parallelogram with a right angle.

Square: A rectangle with sides of equal length.

Kite: A quadrilateral with exactly two pairs of equal consecutive sides

• 04 Practical Geometry

We require three measurements (of sides and angles) to draw a unique triangle. Since three measurements were enough to draw a triangle, a natural question arises whether four measurements would be sufficient to draw a unique four sided closed figure, namely, a quadrilateral.

• 05 Data Handling

The information collected in all such cases is called data. Data is usually collected in the context of a situation that we want to study. For example, a teacher may like to know the average height of students in her class. To find this, she will write the heights of all the students in her class, organise the data in a systematic manner and then interpret it accordingly. Sometimes, data is represented graphically to give a clear idea of what it represents.

• 06 Squares and Square Roots

If a natural number m can be expressed as n 2 , where n is also a natural number, then m is a square number. 2. All square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place. 3. Square numbers can only have even number of zeros at the end. 4. Square root is the inverse operation of square.

• 07 Cubes and Cube Roots

Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They can be expressed as sum of two cubes in two different ways. 2. Numbers obtained when a number is multiplied by itself three times are known as cube numbers. For example 1, 8, 27, ... etc. 3. If in the prime factorisation of any number each factor appears three times, then the number is a perfect cube.

• 08 Comparing Quantities

Discount is a reduction given on marked price. Discount = Marked Price – Sale Price. 2. Discount can be calculated when discount percentage is given. Discount = Discount % of Marked Price 3. Additional expenses made after buying an article are included in the cost price and are known as overhead expenses. CP = Buying price + Overhead expenses 4. Sales tax is charged on the sale of an item by the government and is added to the Bill Amount. Sales tax = Tax% of Bill Amount 5. Compound interest is the interest calculated on the previous year’s amount (A = P + I)

• 09 Algebraic Expressions and Identities

Expressions are formed from variables and constants. 2. Terms are added to form expressions. Terms themselves are formed as product of factors. 3. Expressions that contain exactly one, two and three terms are called monomials, binomials and trinomials respectively. In general, any expression containing one or more terms with non-zero coefficients (and with variables having non- negative exponents) is called a polynomial. 4. Like terms are formed from the same variables and the powers of these variables are the same, too. Coefficients of like terms need not be the same. 5. While adding (or subtracting) polynomials, first look for like terms and add (or subtract) them; then handle the unlike terms. 6. There are number of situations in which we need to multiply algebraic expressions: for example, in finding area of a rectangle, the sides of which are given as expressions.

• 10 Visualising Solid Shapes

Plane shapes have two measurements like length and breadth and therefore they are called two-dimensional shapes whereas a solid object has three measurements like length, breadth, height or depth. Hence, they are called three-dimensional shapes. Also, a solid object occupies some space. Two-dimensional and three-dimensional figures can also be briefly named as 2-D and 3- D figures. You may recall that triangle, rectangle, circle etc., are 2-D figures while cubes, cylinders, cones, spheres etc. are three-dimensional figures.

• 11 Mensuration

We have learnt that for a closed plane figure, the perimeter is the distance around its boundary and its area is the region covered by it. We found the area and perimeter of various plane figures such as triangles, rectangles, circles etc. We have also learnt to find the area of pathways or borders in rectangular shapes. In this chapter, we will try to solve problems related to perimeter and area of other plane closed figures like quadrilaterals. We will also learn about surface area and volume of solids such as cube, cuboid and cylinder.

• 12 Exponents and Powers

Very small numbers can be expressed in standard form using negative exponents.

• 13 Direct and Inverse Proportions

. Two quantities x and y are said to be in direct proportion if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant. That is if x k y = [k is a positive number], then x and y are said to vary directly.

Two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a manner that the product of their corresponding values remains constant.

• 14 Factorisation

When we factorise an expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions. 2. An irreducible factor is a factor which cannot be expressed further as a product of factors. 3. A systematic way of factorising an expression is the common factor method. It consists of three steps: (i) Write each term of the expression as a product of irreducible factors (ii) Look for and separate the common factors and (iii) Combine the remaining factors in each term in accordance with the distributive law. 4. Sometimes, all the terms in a given expression do not have a common factor; but the terms can be grouped in such a way that all the terms in each group have a common factor. When we do this, there emerges a common factor across all the groups leading to the required factorisation of the expression. This is the method of regrouping

• 15 Introduction to Graphs

The purpose of the graph is to show numerical facts in visual form so that they can be understood quickly, easily and clearly. Thus graphs are visual representations of data collected. Data can also be presented in the form of a table; however a graphical presentation is easier to understand. This is true in particular when there is a trend or comparison to be shown. We have already seen some types of graphs.

• 16 Playing with Numbers

In this chapter, we will explore numbers in more detail. These ideas help in justifying tests of divisibility.

• 