## Topic outline

• • ### 01 Knowing our Numbers

Counting things is easy for us now. We can count objects in large numbers, for example, the number of students in the school, and represent them through numerals. We can also communicate large numbers using suitable number names. It is not as if we always knew how to convey large quantities in conversation or through symbols. Many thousands years ago, people knew only small numbers. Gradually, they learnt how to handle larger numbers. They also learnt how to express large numbers in symbols. All this came through collective efforts of human beings. Their path was not easy, they struggled all along the way. In fact, the development of whole of Mathematics can be understood this way. As human beings progressed, there was greater need for development of Mathematics and as a result Mathematics grew further and faster.We use numbers and know many things about them. Numbers help us count concrete objects. They help us to say which collection of objects is bigger and arrange them in order e.g., first, second, etc. Numbers are used in many different contexts and in many ways. Think about various situations where we use numbers. List five distinct situations in which numbers are used.

• ### 02 Whole Numbers

As we know, we use 1, 2, 3, 4,... when we begin to count. They come naturally when we start counting. Hence, mathematicians call the counting numbers as Natural numbers.

• ### 03 Playing with Numbers

The numbers other than 1 whose only factors are 1 and the number itself are called Prime numbers.

Try to find some more prime numbers other than these.

There are numbers having more than two factors like 4, 6, 8, 9, 10 and so on.

These numbers are composite numbers. Numbers having more than two factors are called Composite numbers.

• ### 04 Basic Geometrical Ideas

Basic geometrical ideas (2 -D):

Introduction to geometry. Its linkage with and reflection in everyday experience.

• Line, line segment, ray.

• Open and closed figures.

• Interior and exterior of closed figures.

• Curvilinear and linear boundaries

• Angle — Vertex, arm, interior

and exterior,

• Triangle — vertices, sides, angles, interior and exterior, altitude and median

• Quadrilateral — Sides, vertices, angles, diagonals, adjacent sides and opposite sides (only convex quadrilateral are to be discussed),

interior and exterior of a quadrilateral.

• Circle — Centre, radius, diameter, arc, sector, chord, segment, semicircle, circumference, interior and exterior.

(ii) Understanding Elementary Shapes (2-D and 3-D):

• Measure of Line segment

• Measure of angles

• Pair of lines

– Intersecting and perpendicular lines

– Parallel lines

• Types of angles- acute, obtuse, right, straight, reflex, complete and zero angle

• Classification of triangles (on the basis of sides, and of angles)

Trapezium, parallelogram, rectangle, square, rhombus.

• Simple polygons (introduction)

(Upto octagons regulars as well as non regular).

• Identification of 3-D shapes: Cubes,Cuboids, cylinder, sphere, cone, prism (triangular), pyramid (triangular and square)

Identification and locating in the surroundings

• Elements of 3-D figures. (Faces, Edges and vertices)

• Nets for cube, cuboids, cylinders, cones and tetrahedrons.

(iii) Symmetry: (reflection)

• Observation and identification of 2-D symmetrical objects for reflection symmetry

• Operation of reflection (taking mirror images) of simple 2-D objects

• Recognising reflection symmetry (identifying axes)

(iv) Constructions (using Straight edge Scale,protractor, compasses)

• Drawing of a line segment

• Construction of circle

• Perpendicular bisector

• Construction of angles (using protractor)

• Angle 60°, 120° (Using Compasses)

• Angle bisector- making angles of 30°, 45°, 90° etc. (using compasses)

• Angle equal to a given angle (using compass)

• Drawing a line perpendicular to a given line from a point a) on the line b) outside the line.

• ### 05 Understanding Elementary Shapes

All the shapes we see around us are formed using curves or lines. We can see corners, edges, planes, open curves and closed curves in our surroundings.We organise them into line segments, angles, triangles, polygons and circles.We find that they have different sizes and measures. Let us now try to develop tools to compare their sizes.

• ### 06 Integers

An integer (from the Latin integer meaning "whole")[note 1] is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5

1

/

2

, and √2 are not.

The set of integers consists of zero (0), the positive natural numbers (1, 2, 3, …), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, …). The set of integers is often denoted by a boldface Z ("Z") or blackboard bold {\displaystyle \mathbb {Z} } \mathbb {Z}  (Unicode U+2124 ℤ) standing for the German word Zahlen ([ˈtsaːlən], "numbers").

Z is a subset of the set of all rational numbers Q, in turn a subset of the real numbers R. Like the natural numbers, Z is countably infinite.

The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, the (rational) integers are the algebraic integers that are also rational numbers.

• ### 07 Fractions

In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number, corresponding to division of fractions. For example, and are complex fractions.

• ### 08 Decimals

The decimal numeral system (also called base-ten positional numeral system, and occasionally called denary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as Decimal notation.

• • ### 10 Mensuration

When we talk about some plane figures as shown below we think of their regions and their boundaries.

We need some measures to compare them. Interpret and analyse them. These provide accuracy in findings.

• ### 11 Algebra

Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians.

• ### 12 Ratio and Proportion

If two ratios are not equal, then we say that they are not in proportion. In a statement of proportion, the four quantities involved when taken in order are known as respective terms. First and fourth terms are known as extreme terms. Second and third terms are known as middle terms.

• ### 13 Symmetry

Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance.[a] In mathematics, "symmetry" has a more precise definition, that an object is invariant to any of various transformations; including reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are related, so in this article they are discussed together.

Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, theoretic models, language, music and even knowledge itself.

• ### 14 Practical Geometry

We see a number of shapes with which we are familiar. We also make a lot of pictures. These pictures include different shapes. We have learnt about some of these shapes in earlier chapters as well. Why don’t you list those shapes that you know about along with how they appear? In this chapter we shall learn to make these shapes. In making these shapes we need to use some tools. We shall begin with listing these tools, describing them and looking at how they are used.