MATHEMATICS 4 TEXTBOOK – 1 Name: ___________________________________ Section: ________________ Roll No.: _________ School: __________________________________
Preface ClassKlap partners with schools, supporting them with learning materials and processes that are all crafted to work together as an interconnected system to drive learning. Our books strive to ensure inclusiveness in terms of gender and diversity in representation, catering to the heterogeneous Indian classroom. ClassKlap presents the Traveller series, designed specifically to meet the requirements of the new curriculum released in November 2016 by the Council for the Indian School Certificate Examinations (CISCE). Guiding principles: The 2016 CISCE curriculum states the following as a few of its guiding principles for Mathematics teaching: D evelop mathematical thinking and problem-solving skills and apply these skills to formulate and solve problems. A cquire the necessary mathematical concepts and skills for everyday life and for continuous learning in Mathematics and related disciplines. R ecognise and use connections among mathematical ideas and between Mathematics and other disciplines. R eason logically, communicate mathematically and learn cooperatively and independently. Each of these principles resonates with the spirit in which the ClassKlap textbooks, workbooks and teacher companion books have been designed. The ClassKlap team of pedagogy experts has carried out an intensive mapping exercise to create a framework based on the CISCE curriculum document. Key features of ClassKlap Traveller series: Theme-based content that holistically addresses all the learning outcomes specified by the CISCE curriculum. T he textbooks and workbooks are structured as per Bloom’s taxonomy to help organise the learning process according to the different levels involved. Student engagement through simple, age-appropriate content with detailed explanation of steps. Learning is supported through visually appealing images, especially for Grades 1 and 2. Increasing difficulty level in sub-questions for every question. Multiplication tables provided as per CISCE requirement. All in all, the Traveller Mathematics books aim to develop problem-solving and reasoning skills in the learners’ everyday lives while becoming adept at mathematical skills as appropriate to the primary level. – The Authors
Textbook Features I Will Learn About I Think Contains the list of concepts to be covered Arouses the student’s in the chapter along with the learning curiosity before objectives introducing the concept I Recall I RUenmdeermsbtearndand Pin-Up-Note Recapitulates the Elucidates the basic Highlights the key points or prerequisite knowledge for elements that form the definitions the concept learnt previously basis of the concept ? Train My Brain I Apply I Explore(H.O.T.S.) C hecks for learning to gauge Connects the concept E ncourages the student to the understanding level of the to real-life situations by extend the concept learnt student providing an opportunity to more complex scenarios to apply what the student has learnt Maths Munchies Connect the Dots Drill Time Aims at improving speed of Aims at integrating Revises the concepts with calculation and problem Mathematical concepts practice questions at the solving with interesting facts, with other subjects end of the chapter tips or tricks A Note to Parent E ngages the parent in the out-of- classroom learning of their child
Contents 1 Shapes 1.1 Introduction to Geometry������������������������������������������������������������������������������������������ 1 1.2 Perspective of Objects����������������������������������������������������������������������������������������������� 8 1.3 Reflection and Symmetry����������������������������������������������������������������������������������������� 13 2 Patterns 2.1 Growing and Reducing Patterns������������������������������������������������������������������������������ 22 3 Numbers 3.1 Introduction to Roman Numerals���������������������������������������������������������������������������� 30 3.2 Count 6-digit Numbers in Indian System���������������������������������������������������������������� 34 4 Addition and Subtraction 4.1 Add and Subtract 4-digit and 5-digit Numbers������������������������������������������������������ 43 5 Multiplication 5.1 Multiplication of 2-digit Numbers and 3-digit Numbers ��������������������������������������� 52 5.2 Multiply Using Lattice Algorithm����������������������������������������������������������������������������� 59 5.3 Mental Maths Techniques: Multiplication �������������������������������������������������������������� 63 6 Division 6.1 Divide 3-digit Numbers by 1-digit Numbers���������������������������������������������������������� 66
Shapes1Chapter I Will Learn About 1.1 • tangram shapes. • point, line, line segment and ray. • centre, radius and diameter. • perspectives of shapes. • reflection and symmetry. • tessellation. Introduction to Geometry I Think Surbhi drew around the inner edge of a bangle on a sheet of paper. She got a circle. She cut the circle and folded it twice carefully in such a way that each of the folds passes over the other. She was excited to show it to her teacher. What do you think the lines formed on the folds are? I Recall We have learnt about 2-dimensional figures. We also know about tangrams. Let us recall them. 1
Identify the number of triangles and squares in each of the following tangrams. a) b) c) I Remember and Understand Geometrical terms Figures are made of lines. Let us see what lines are made of. Point: A point is a dot. It has no shape or thickness. A point is denoted by a capital letter of the English alphabet. For example, A, X, Y, P and M shown below are points. X Y A point has no A M dimensions. P Line: Many points when placed close to each other without any gap between them form a straight line. It has no thickness and breadth. A line has only length. So, it is called a one-dimensional figure. A straight line has no ends. It extends on both the sides, as shown by the arrows pointing out in the given figure. A B We name two points, say A and B, on a line and write it as AB . We read it as line AB. AB is same as BA . Line segment: A line segment is a part of a line. It has two end points. A line segment has an exact length. 2
We name the end points as A and B and write A B the line segment AB as AB . We read it as segment AB. Either A or B can be taken as the starting point of the line segment AB . The length of the line segment is the distance between its end points A and B. So, segment AB is the same as segment BA . Ray: A ray is a straight line, which has a starting point but no A end point. It extends only on one side. B We name the end point as A and any other point on it as B and write it as ray AB as AB . We read it as ray AB. Parallel lines: Parallel lines are lines that do not meet or have a common point. They are always at the same distance apart. Intersecting lines: Two or more lines that meet at a common point are called intersecting lines. Parallel lines Intersecting lines Circles We know that a circle is a simple closed 2D figure. A circle is formed by joining many points that are at the same distance from the same fixed point. A bangle, a coin, a bottle lid, a tyre and a ring are a few examples of objects whose outlines give a circle. Parts of a circle Let us now understand the different parts of a circle through an activity. What we need: A paper sheet, a bangle, a pencil or pen, a pair of scissors Shapes 3
Step 1: T ake a white sheet of paper and draw a circle on it using a bangle. Step 2: Take a pair of scissors and cut along the circle drawn on the sheet. Step 3: S eparate the circle from the sheet of paper. Pencil Bangle Scissors White Paper Fold the circle into two halves and four quarters as shown. Half Quarter When we unfold the circle, two lines appear on it. These lines cross each other at a point. A Let us now define the parts of a circle. D Centre: The fixed point ‘O’ of a circle is called its centre. This point is at o C the same distance from any point on the edge of the circle. B Radius: The (fixed distance) line segment drawn from the centre ‘O’ to r or the edge of the circle is called its radius. The plural of radius is radii. We AB can draw any number of radii in a circle. All radii of a circle are of the same length. A radius of a circle is denoted as ‘r’. In the figure, AO and BO are two radii. Chord: A chord is a line segment that joins any two points on a circle. In C D the figure, AB and CD are two chords. o Diameter: A line segment drawn from one point on a circle to another A B and passing through the centre is known as its diameter. B The diameter is the longest chord of a circle. We can draw any A oC number of diameters in a circle. All the diameters of a circle are F of the same length. A diameter of a circle is denoted as ‘d’. In the D figure, AD, BE and CF are three diameters. From the figure, we observe that d = 2 × r or r = d ÷ 2. E 4
Semicircle: The diameter of a circle divides the circle into two halves. Each half is called a semicircle. r o cCairlcleudmtfheerecnircceu: mThfeerelenncgethoof tfhaeccirircclele.is o Circumference Let us summarise the parts of a circle from the figure: EC O = Centre of the circle o A OA = Radius D F BC = Diameter DE = Chord B BFC = Semicircle Try this! Draw circles using a bangle and a cap of a bottle. Show the radii, centres and diameters of these circles. Drawing a circle using a compass Let us now learn to draw a circle using a compass. In your geometry box or compass box, there are instruments such as a ruler, a divider, a compass, a protractor, a pair of set squares, a pencil and an eraser. Look at the picture of the compass. The needle of the compass: It is kept on a sheet of paper while drawing a circle. It should not be moved from its position while drawing a circle. It marks the centre of the circle on the sheet of paper. Hinge: It is used to tighten the compass to control the movement of its two arms. Pencil holder: It holds the pencil used to draw the circle. How to use a compass Step 1: Insert a well-sharpened small pencil in the pencil holder. Tighten the screw of the pencil holder till the pencil is fixed firmly. Step 2: Align the pencil with the needle of the compass. Step 3: P ress down the needle on a sheet of paper. The point where the needle touches the paper is the centre of the circle. Turn the arm having the pencil holder to the right or left till the Shapes 5
pencil returns to the starting point. The curve drawn is the required circle. The distance between the needle and the pencil tip is the radius of the circle. To draw a circle of a given radius, say 3 cm, follow these steps. Step 1: F ix the pencil in the pencil holder. Align it with the tip of the needle by placing the compass on a flat surface. Step 2: Adjust the pencil holder to get some distance between the needle and the tip of the pencil. Step 3: Place the needle of the compass at 0 cm on the ruler. Adjust the pencil holder such that the pencil is at 3 cm on 3 cm the ruler. The distance between the needle and the pencil is the radius, which is 3 cm. Step 4: P lace the needle of the compass on a sheet of paper. Without moving this arm, hold the hinge. Move the pencil holder arm right or left, till the pencil returns to the point where it started. Step 5: Remove the compass from the paper. Mark the needle point as ‘O’, the centre of the circle. Using a ruler, draw a line segment from O to a point A on the circle. This line segment OA is the radius of the circle, which is 3 cm long. Thus, the circle formed is the required circle. Example 1: Draw a circle of radius 2 cm. Solution: r = 2 cm P O ? Train My Brain Define the following: a) Chord b) Radius c) Centre 6
I Apply Let us see some examples using the concept of radius and diameter. We know that the diameter of a circle is two times its radius. So, the radius of a circle is half its diameter. That is, d = 2 × r and r = d ÷ 2. Example 2: Sonu has a circular disc having a diameter of 6 cm. What is its radius? Solution: We know that radius, r = d ÷ 2. Diameter of the disc = 6 cm Therefore, radius = 6 cm ÷ 2 = 3 cm. Example 3: The cap of a water bottle is 2 cm in radius. What is its diameter? Solution: Radius of a bottle cap = 2 cm We know that diameter, d = radius × 2. So, the diameter of the cap of the water bottle, d = 2 cm × 2 = 4 cm. I Explore (H.O.T.S.) Let us now see some figures drawn using circles. Can you guess how these figures are drawn? a) b) c) a) We observe that all the circles in this figure have the same centre but different radii. Such circles are called concentric circles. Now, try guessing how the figures b) and c) are drawn. Example 4: Draw a figure that has only circles. Solution: The sign of the Olympic games has only circles. Example 5: How many circles can be drawn with the same point as the centre? Solution: We can draw any number of circles with the same point as the centre. Shapes 7
1.2 Perspective of Objects I Think Surbhi’s teacher asked the class to draw a Surbhi’s drawing car. Surbhi and Pushkar, both drew a green Pushkar’s drawing car but the cars did not look the same. Who do you think has drawn correctly? I Recall We know that we get different shapes when we draw the outlines of the bases of different objects. Draw the outlines of the bases of the following objects. Object Outline of the base I Remember and Understand We see that the railway tracks appear to be wider at our end, but appear to be narrower at the other end. Similarly, roads and bridges too appear to be broader at our end and narrower at the other end. 8
Such a view is known as the perspective view. It is widely used in art and architecture. Objects look differently when viewed from different sides. A cube looks the Observe this cube. same from the front, side and top views. This is also true for a sphere. Front view: Top view: Side view: A cube looks like a square when we view it from the top, front or from the side. Now, observe this cuboid. Front view: Top view: Side view: The top and the front views of the cuboid look like rectangles whereas,the side view looks like a square. Consider an example to see what some objects look like in different views. Example 6: Label the top view and side view of the objects whose front views are given. One has been done for you. S.No. Front view Top or side view a) Top view Side view b) Shapes 9
S.No Front view Top or side view c) Example 7: Draw the top, front and side views of the given solids. a) b) c) Solution: 3D Object Top view Front view Side view S.No. a) b) Train My Brain c) ? Train My Brain Draw the top, front and side views of the given solid. 10
I Apply Let us see some more examples based on the different views of a few solid objects. Example 8: Draw the shape of the given objects from different views: a) b) c) Solution: Front view Top view Side view Object a) b) c) Example 9: Circle the correct solid based on the given views. One has been done for you. Front view Top view Side view Front view Top view Side view a) b) Shapes 11
Front view Top view Side view Front view Top view Side view c) d) Example 10: Write the number of cubes in the given solids: a) b) c) Solution: a) 7 b) 8 c) 7 I Explore (H.O.T.S.) Let us now see some solids made of unit cubes and identify their top, front and side views. Example 11: Draw the top, front and side views of the given solid. Solution: The top, front and side views of the given solid are as follows: Top view Front view Side view 12
Example 12: Draw the top, front and side views of the given solids. a) b) c) Solution: Object Top view Front view Side view a) b) c) 1.3 Reflection and Symmetry I Think Surbhi was standing near a lake. She was excited to see the images of the trees in the water. Do you know what such images are called? I Recall We have learnt various 2-dimensional shapes. They are triangle, circle, oval, square, rectangle and so on. Shapes 13
Name the given 2D shapes. ____________ __________ ________ __________ _______________ I Remember and Understand Reflections Reflection can be seen in mirrors, water, When an object is placed in front of a mirror, we see its image in oil, shiny surfaces and so on. the mirror. This image is called the reflection of the object. E E w wConsider these examples. The mirror image or reflection of an object looks exactly the same as the object. The dotted line is the mirror or the line of reflection. The image formed by the vertical line of reflection is also called the mirror image. The image formed by a horizontal line of reflection is called the water image. Example 13: Draw the reflections of the given figures with the dotted line as the line of reflection. a) b) c) d) 14
Solution: a) b) c) d) Symmetry When a line divides a shape into two parts where both parts are reflections of each other, the line is called the line of symmetry. Shapes with one or more lines of symmetry are called symmetrical shapes. Shapes with no line of symmetry are called asymmetrical shapes. Example 14: Draw the lines of symmetry of the given shapes. a) b) c) d) M Solution: a) b) c) d) M ? Train My Brain Answer the following: a) How many lines of symmetry does a square have? b) has ________ (number) lines of symmetry. c) Draw the lines of symmetry of the given triangle. Shapes 15
I Apply Symmetry can also be seen in tiling. We know that a tiling pattern is formed by repeating a basic tile. A tiling pattern that has a repeating pattern is called a periodic tiling. A tiling pattern that does not have a repeating pattern is called a non-periodic or aperiodic tiling. Using reflection of tiles about a vertical or a horizontal line results in different designs to the tile. Arranging such tiles in different ways, we can create decorative patterns on floors, walls, roofs, pavements and so on. Tessellation is a tiling pattern made of ceramic or cement hexagons or squares. Tessellations are found on floors, pavements, roofs of historical monuments, quilting and so on. The arrays of hexagonal cells in honeycombs are a classic example of tessellation in nature. Example 15: Create a few tessellations using the given basic tile. Solution: A few of the tessellations using the given tile are: Example 16: Find the basic shape in each of these tessellations. a) b) Solution: Basic shapes are – a) b) 16
I Explore (H.O.T.S.) We can identify if a given shape can tessellate or not by placing/drawing the shape such that there are no overlaps or gaps. Let us now identify the shapes that tessellate and those that do not. Example 17: Which of these shapes tessellate? a) b) c) d) Solution: a) Forms patterns but does not tessellate b) Forms patterns and also tessellates Try the remaining shapes. Example 18: Make a tessellation using the given shape. Shapes 17
Solution: Try this! a) Use two 4-sided figures and a triangle to create a tessellation. b) Use a 4-sided figure and a triangle to create a tessellation. c) Make a tessellation using any three shapes. Maths Munchies Today, tessellations are improved to the level of visual illusion. Do you know what a visual illusion is? Visual illusion makes the viewer believe that he or she is seeing something that isn’t present. Look at the tessellation shown here. Do you see some black phantom dots? When you look directly at those dots, do they disappear? Grey shades seem to appear at the intersections of the white lines. But actually, these dots are not present. Connect the Dots Social Studies Fun A great circle is defined as the largest possible circle that can be drawn around the sphere. The equator and all the meridians are the circles of the Earth. The equator is the largest circle around the Earth. 18
English Fun Can you think of a word that rhymes with ‘circle’? Drill Time 1.1 Introduction to Geometry 1) Draw circles with the given measures. a) diameter = 8 cm b) radius = 6 cm c) radius = 7 cm d) radius = 5 cm e) radius = 1 cm 2) Word problems a) Reena has a bangle of radius 2 cm. What is its diameter? b) The lid of a soft drink can is 8 cm in diameter. What is its radius? 1.2 Perspective of Objects 3) Label the top view and side view of the objects whose front views are given. Front view Top or side view a) b) Shapes 19
4) Draw the front, top and side views of the given solids. Solid Front view Top view Side view 5) Draw the shapes of the given objects from different views. Object Front view Top view Side view 20
1.3 Reflection and Symmetry 6) Draw reflections of the following figures about the given lines. a) b) c) d) e) 7) Draw lines of symmetry for the symmetrical letters of the English alphabet. 8) Find the basic shape in each of the following tessellations. b) a) A Note to Parent Show your child different circular objects such as a clock, a plate, a bottle cap and so on in your house. Ask him or her to measure their diameter and calculate the radii of a few of them. Shapes 21
Patterns2Chapter I Will Learn About • growing and reducing patterns. • the rule to extend a growing/ reducing pattern. 2.1 Growing and Reducing Patterns I Think Surbhi observed the following pictures: She is reminded of a mathematical concept that she learnt in her previous class. What do you think the concept is? 22
I Recall An arrangement of figures or designs in a certain way is called a pattern. We see patterns everywhere. Patterns can be natural or artificial. Natural patterns We observe natural patterns on stones, leaves of plants, spots and stripes on animals and so on. Artificial patterns Artificial patterns are made by humans. We create these patterns using a general rule. We place all the items in the pattern according to that rule. Patterns in lines and shapes Patterns in lines and shapes are created using repetitive basic lines or shapes. We can also find patterns in numbers, language, music and so on. Number patterns A sequence of numbers following a specific rule is called a number pattern. We observe that multiplication tables have a pattern too. 1) Odd numbers: 1, 3, 5, 7, 9, 11, ……….. (Beginning with 1 and increasing by 2.) Patterns 23
2) Even numbers: 2, 4, 6, 8, 10, 12, ………. (Beginning with 2 and increasing by 2.) Once we know the rule, we can continue a pattern any number of times or endlessly. Patterns that end after a few terms are called finite patterns and those that do not end are called infinite patterns. I Remember and Understand A pattern always follows a particular rule. Now, let us see growing and reducing patterns. Growing and reducing patterns in numbers Observe the following patterns. a) 1283, 1290, 1297, 1304, 1311, 1318 b) 632, 652, 672, 692, 712, 732 c) 3142, 4143, 5144, 6145, 7146, 8147 In pattern a), 1290 – 1283 = 7 and 1297 – 1290 = 7. So, the terms increase by 7. Similarly, in b) the terms increase by 20 and in c) they increase by 1001. As the terms in these patterns increase by a certain number than the previous term, they are called growing patterns. Now, observe these patterns. A pattern can be made by a) 5748, 5736, 5724, 5712, 5700, 5688 increasing or decreasing b) 7237, 6235, 5233, 4231, 3229, 2227 the numbers by 1. Such c) 914, 908, 902, 896, 890, 884 numbers are called consecutive numbers In pattern a), 5748 – 5736 = 12 and 5736 – 5724 = 12. which follow each other So, the terms decrease by 12. without skipping any numbers in between. Similarly, in b) the terms decrease by 1002 and in c) they decrease by 6. As the terms in these patterns decrease by a certain number than the previous term, they are called reducing patterns. Growing and reducing patterns in shapes Growing and reducing patterns can also be formed using shapes. Let us see a few examples. 24
Example 1: Complete the following growing patterns. One has been done for you. a) b) c) Example 2: Complete the following reducing patterns. One has been done for you. a) b) c) Patterns 25
? Train My Brain Complete the following patterns. a) 4735, 4741, 4747, 4753, _______, _______, _______ b) 854, 843, 832, 821, _______, _______, _______ c) 681, 732, 783, 834, _______, _______, _______ I Apply Let us now see how to draw a pattern when its rule is given. Example 3: Create a pattern starting with the number 219 and with its terms increasing by 27. Find the next three terms. Solution: The first term = 219 The terms increase by 27. The second term = 219 + 27 = 246 The third term = 246 + 27 = 273 The fourth term = 273 + 27 = 300 Therefore, the pattern is 219, 246, 273, 300, ... Example 4: Create a pattern starting with the number 6537 and with its terms decreasing by 104. Find the next three terms. Solution: The first term = 6537 The terms decrease by 104 The second term = 6537 – 104 = 6433 The third term = 6433 – 104 = 6329 The fourth term = 6329 – 104 = 6225 Therefore, the pattern is 6537, 6433, 6329, 6225... Example 5: Create a pattern starting with a triangle and with its terms increasing by 2 triangles. Find the next three terms. Solution: The first term is and the terms increase by . The second term is ; 26
The third term is ; The fourth term is ; Therefore, the pattern is: Growing or reducing patterns are not always formed by adding or subtracting a number. They can also be formed by multiplying or dividing a number. Let us see some examples. Example 6: Create a pattern starting with 4 and multiplying each term by 3. Solution: The first term = 4 Given that the terms increase as their product by 3, the second number is 4 × 3 = 12 The third term is 12 × 3 = 36 The fourth term is 36 × 3 = 108 Therefore, the pattern is 4, 12, 36, 108, ... Example 7: Create a pattern starting with the number 64 and dividing each term by 2. Solution: The first term = 64 Given that the terms decrease as they are divided by 2, the second term is 64 ÷ 2 = 32. The third term is 32 ÷ 2 = 16 The fourth term is 16 ÷ 2 = 8 Therefore, the pattern is 64, 32, 16, 8, ... I Explore (H.O.T.S.) Let us look at some other patterns. Example 8: Complete the following patterns by writing their next two terms. a) 3, 7, 6, 11, 9, 15, ... b) 5, 2, 10, 5, 15, 8, ... c) 2, 9, 4, 11, 6, 13, ... Patterns 27
Solution: a) The given series is a mixed series. 3, 6, 9, ... form a series and 7, 11, 15, ... form another series. In the series 3, 6, 9, ..., the terms increase by 3. Therefore, the next term in this series will be 12. In the series 7, 11, 15, ...,the terms increase by 4. Therefore, the next term in this series will be 19. Therefore, the series can be written as 3, 7, 6, 11, 9, 15, 12, 19. In similar way, the patterns of b) and c) are: b) 5, 2, 10, 5, 15, 8, 20, 11. c) 2, 9, 4, 11, 6, 13, 8, 15. Maths Munchies Fibonacci series: A series of numbers in which each number (Fibonacci number) is the sum of the two preceding numbers. The simplest is the series 1, 1, 2, 3, 5, 8, 13, 21, ... Connect the Dots Science Fun Growing and reducing patterns can be seen in nature also. For example, spider’s web, fern leaves and so on. Social Studies Fun As the Moon orbits around the Earth, the portion of the Moon that faces the Sun will light up. The different shapes of the lit portion of the Moon that can be seen from Earth are known as the phases of the Moon. These phases repeat once every 15 days. 28
Drill Time 2.1 Growing and Reducing Patterns 1) Complete the following patterns. a) 254, 279, 304, 329, _________, _________, _________. b) 6842, 6826, 6810, 6794, _________, _________, _________. c) 1200, 1000, 800, 600, _________, _________, _________. d) e) f) 2) Write the next four terms of the patterns with the given rules. a) Pattern starting with 74 and increasing by 35 b) Pattern starting with 406 and decreasing by 22 c) Pattern starting with 351 and increasing by 55 d) Pattern starting with 758 and decreasing by 15 e) Pattern starting with 1 and each term increasing by multiplying 2 f) Pattern starting with 80 and each term decreasing by dividing 2 A Note to Parent Ask your child to observe the growing patterns in multiplication tables that he or she has already learnt. Ask him or her to observe the surroundings and find a reducing pattern. Patterns 29
Numbers3Chapter I Will Learn About • Roman numerals up to 39. • reading and writing 5-digit and 6-digit numbers using the Indian system. • using place values to write numbers in its expanded forms and vice versa. • comparing and ordering 5-digit and 6-digit numbers. • forming the greatest and the smallest numbers using the given digits. 3.1 Introduction to Roman Numerals I Think Surbhi bought a watch but found it difficult to read the time as she was not familiar with the numbers on it. Have you ever seen such numbers? Do you know what these numbers are? I Recall We have already learnt numbers from 1 to 40 and their number names. Let us recall the same through the following. Write the numbers for the given number names. a) Thirty-nine = __________________ b) Twenty-seven = _______________ 30
c) Thirty-one = ___________________ d) Ten = _________________________ e) Thirteen = _____________________ This system of writing numbers is called the Hindu-Arabic numeration. Apart from this, there is another system called the Roman numeral system. Let us learn about it. I Remember and Understand The numerals that we use in our day-to-day life are 1, 2, 3, ..., and so on. These numbers are called the Hindu-Arabic numerals as they were developed in ancient India. They were spread to the other parts of the world by the Arab traders. The Roman numeral system was followed in the The Roman numeral system ancient city of Rome. Nowadays, Roman numerals has seven letters of the English are mainly used because of their historical alphabet with the help of importance. which all other numbers are written. The Roman numerals for the first 10 counting numbers are given in this table. A few of the Roman numbers are - I, V and X. Hindu-Arabic numerals (values) 1 2 3 4 5 6 7 8 9 10 Roman numerals (symbols) I II III IV V VI VII VIII IX X We follow certain rules to read and write numbers in the Roman numeral system. Rule Description Examples I I = 1 + 1 = 2 1) A symbol can be repeated to a XX = 10 + 10 = 20 maximum of three times. Repetition of numbers means addition. XV = 10 + 5 = 15 XII = 10 + 1 + 1 = 12 2) If a symbol is placed after a symbol of greater value, their values are added. 3) If a symbol is placed before a symbol of IV = (5 – 1) = 4 greater value, the smaller value is IX = (10 – 1) = 9 subtracted from the greater one. 4) The symbol 'I' can be subtracted from IV = 4, IX = 9 the symbols 'V' and 'X' only. Note: The symbols 'V' and 'L' can never be repeated. Using these rules, we can find the equivalent Hindu-Arabic numerals for the given Roman numerals and vice versa. Numbers 31
Example 1: Write the Hindu-Arabic numerals for the given Roman numerals: a) XIX b) XXVII c) XXXIX d) VI e) VIII e) 5 Solution: a) XIX = 10 + (10 – 1) = 19 b) XXVII = 10 + 10 + 5 + 1 + 1 = 27 c) XXXIX = 10 + 10 + 10 + (10 – 1) = 39 d) VI = 5 + 1 = 6 e) VIII = 5 + 1 + 1 + 1 = 8 Example 2: Write the Roman numerals for the given numbers: d) 36 a) 10 b) 29 c) 14 Solution: a) 10 = X b) 29 = 10 + 10 + (10 – 1) = XXIX c) 14 = 10 + (5 – 1) = XIV d) 36 = 10 + 10 + 10 + (5 + 1) = XXXVI e) 5 = V ? Train My Brain Write the Roman numerals for the given numbers: a) 12 b) 27 c) 17 I Apply Let us see some real-life examples where we apply the knowledge of Roman numerals. Example 3: Rahul is studying in class XI and Rohan is studying in class IX. Who is senior between the two students? Solution: Rahul is studying in class XI. The Hindu-Arabic numeral for XI is 10 + 1 = 11. Rohan is studying in class IX. The Hindu-Arabic numeral for class IX is 10 – 1 = 9. As 11 > 9, Rahul is senior to Rohan. 32
Example 4: Read the following clocks and write the time they are showing using Hindu-Arabic numbers. a) b) Solution: a) T he short (hour) hand has crossed IV. The Hindu-Arabic numeral for IV is 4. The long (minute) hand is on ‘V’ which is 5. So, it shows 25 minutes. Therefore, the time is 4:25. b) The short (hour) hand is at ‘II’. The Hindu-Arabic numeral for II is 2. The long (minute) hand is on ‘III’ which is 3. So, it shows 15 minutes. Therefore, the time is 2:15. I Explore (H.O.T.S.) Let us learn to add and subtract Roman numerals. Example 5: Add XX and V. Write the sum as a Roman numeral. Solution: The Hindu-Arabic numerals for XX and V are 20 and 5 respectively. So, XX + V = 20 + 5 = 25. The Roman numeral of 25 is XXV. Therefore, XX + V = XXV. Example 6: Subtract IV from IX. Solution: The Hindu-Arabic numerals for IV and IX are 4 and 9 respectively. So, IX – IV = 9 – 4 = 5. The Roman numeral for 5 is V. Therefore, IX – IV = V. Numbers 33
3.2 Count 6-digit Numbers in Indian System I Think Surbhi’s father bought a TV for ` 55,515. She read its cost as five thousand five hundred and fifty-one and one more five. Her father told her that she read it wrong and asked her to learn the correct way of reading 5-digit and 6-digit numbers. Can you read such big numbers? I Recall We know that 10 ones make a ten, 10 tens make a hundred and 10 hundreds make a thousand. Counting by 10s: 10, 20, 30, 40, 50, 60, 70, 80, 90 Counting by 100s: 100, 200, 300, 400, 500, 600, 700, 800, 900 Counting by 1000s: 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000 Write the numbers for the following number names: a) Eighty = _____________________ b) Eight hundred = _____________ c) Eight thousand = _____________ d) Eight hundred eighty-eight = ______________ Let us recall the smallest and the largest 2-digit, 3-digit and 4-digit numbers and name them. Number of Digits Smallest Largest 2 10 (Ten) 99 (Ninety-nine) 3 100 (Hundred) 999 (Nine hundred ninety-nine) 4 1000 (Thousand) 9999 (Nine thousand nine hundred ninety-nine) There are numbers greater than 9999. Let us learn about them. 34
I Remember and Understand We know that the smallest 4-digit number comes after the greatest 3-digit number. 999 + 1 = 1000. T Th Th H T O Similarly, the smallest 5-digit number comes just after + 111 (successor of) the largest 4-digit number. When we add 1 to the largest 4-digit number, we get a new place in the place 9999 value chart. It is called the ten thousands place. In short, we 1 write it as T Th. 1 0000 Similarly, we can find the smallest 6-digit number by adding 1 to the largest 5-digit number. The new place in the place value chart is called the lakhs place. In short, we write it as L. L T Th Th H T O The numbers starting from 1 1 111 are called natural numbers. 9 9999 The natural numbers along with ‘0’ are called whole +1 numbers. 1 0 0000 Place value and face value Place value: Every digit in a number occupies a place in the place value chart. Each digit gets its value from the place it occupies. This value is called its place value. Face value: The face value of a number is the number itself. It does not depend on its position in the place value chart. Let us write the place value of '4' in each of the following numbers: Numbers Place value of '4' 36346 4 is in the tens place. So, its place value is forty. 81423 4 is in the hundreds place. So, its place value is four hundred. 647211 4 is in the ten thousands place. So, its place value is forty thousand. 431311 4 is in the lakhs place. So, its place value is four lakh. From the above, we observe that the value of 4 changes according to its place in a number. The face value of 4 in each of the above numbers is 4. Let us understand the place value chart for such 5-digit and 6-digit numbers. Numbers 35
In the place value chart, as we move left from the ones place, the place value becomes 10 times more than the current place value. Let us now name a few numbers. Number Lakhs Place Value Chart Hundreds Tens Ones a) 25436 L Ten thousands Thousands H T O b) 382741 6 3 T Th Th 4 3 25 1 74 82 We can write the number names of the given numbers as: a) 25436: 2 ten thousands = 20000; 5 thousands = 5000; 4 hundreds = 400; 3 tens = 30; 6 ones = 6 Therefore, 25436 = 20000 + 5000 + 400 + 30 + 6. We read it as twenty-five thousand four hundred thirty-six. b) 382741: 3 lakhs = 300000; 8 ten thousands = 80,000; 2 thousands = 2000; 7 hundreds = 700; 4 tens = 40; 1 ones = 1 Therefore, 382741 = 300000 + 80000 + 2000 + 700 + 40 + 1. We read it as three lakh eighty-two thousand seven hundred forty-one. S.No. Lakhs Ten thousands Thousands Hundreds Tens Ones a) 43 6 3 46 b) Four lakh thirty-six thousand three hundred forty-six c) 8 1 4 23 d) Eighty-one thousand four hundred twenty-three 86 4 7 21 Eight lakh sixty-four thousand seven hundred twenty-one 4 1 3 11 Forty-one thousand three hundred eleven Writing numbers using periods We can write large numbers using periods. This helps in writing number names of large numbers. 36
We can also show 6-digit numbers in a place value chart, by dividing it into three parts called periods. The three periods are ones (O, T, H), thousands (Th, T Th) and lakhs (L). Let us write 65274 and 192658 in different periods of the place value chart. To show the periods, separate the digits using Lakhs Thousands Ones commas. L T Th Th H TO 274 So, 65,274 is sixty-five thousand two hundred 65 seventy-four. 658 1 92 Similarly, 1,92,658 is one lakh ninety-two thousand six hundred fifty-eight. Example 7: Place the commas at the appropriate places and write the number names of these numbers: a) 82558 b) 266756 Solution: a) 82558: 82,558; Eighty-two thousand five hundred fifty-eight b) 266756: 2,66,756; Two lakh sixty-six thousand seven hundred fifty-six Expanded form Once we know the place values of numbers, we can write its expanded forms. A number is said to be written in its expanded form when it is expressed as a sum of the place values of its digits. The place of the digit 0 is ignored. Example 8: Expand the number 53842. Solution: First, we find the place value of each digit by T Th Th H T O writing the number in the place value chart. 5 3842 5 is in the ten thousands place, so its place value is 50000. 3 is in the thousands place. So, its place value is 3000. 8 is in the hundreds place. So its place value is 800. 4 is in the tens place. So its place value is 40. 2 is in the ones place. So its place value is 2. Therefore, the expanded form of 53842 is 50000 + 3000 + 800 + 40 + 2. Example 9: Find the number and its number name of the given expanded form: 900000 + 60000 + 200 + 50 + 7 Solution: Place the appropriate digits in the place L T Th Th H T O value chart as shown. 9 6 0257 Therefore, the number is 9,60,257. Its number name is nine lakh sixty thousand two hundred fifty-seven. Numbers 37
? Train My Brain Say the number names of the following: a) 10024 b) 420010 c) 60600 I Apply We use the concept of place value to: 1) compare numbers. 2) arrange numbers in the ascending and the descending orders. 3) form the greatest and the smallest numbers. Compare numbers To compare two large numbers, follow these steps: Step 1: Write the numbers in the place value chart. Step 2: Check if the number of digits of the two numbers is the same. If yes, then proceed to step 3. Else, write the number with the fewer number of digits as the smaller one. Step 3: C ompare the digits in each place of the two numbers. The number with the smaller digit in the same place of the given numbers is the smaller number. Note: Always start comparing the digits from the extreme left. Example 10: Compare and put the correct symbol >, < or = in the blank. a) 9,48,137 ________ 8,48,137 b) 41,141 ________ 4,41,141 Solution: a) W rite the numbers in the place value chart. L T Th Th H T O In the lakhs place, 9 > 8. 9 4 8137 Therefore, 9,48,137 > 8,48,137. 8 4 8137 b) Write the numbers in the place value chart. There are 5 digits in 41,141 and 6 digits in L T Th Th H T O 4,41,141. 4 1 141 Therefore, 41,141 < 4,41,141. 4 4 1 141 38
Arrange numbers in the ascending and the descending orders Ascending order: The arrangement of numbers from the smallest to the largest is known as their ascending order. Descending order: The arrangement of numbers from the largest to the smallest is known as their descending order. Example 11: Arrange the given numbers in the ascending and descending orders. 348975; 27818; 424721; 187542 Solution: Write the numbers in the place value chart as shown: There are 5 digits in 27818 while all the other numbers have 6 digits. So, 27818 is the smallest number. L T Th Th H T O In the lakhs place, 1 < 3 < 4. 3 4 8 975 27818 < 187542 < 348975 < 424721 2 7 818 Therefore, the required ascending order is 4 2 4 721 27818, 187542, 348975 and 424721. 1 8 7 542 The descending order of numbers is just the reverse of their ascending order. Thus, 424721 > 348975 > 187542 > 27818. Therefore, the required descending order is 424721, 348975, 187542 and 27818. Form the greatest and the smallest 6-digit numbers Example 12: Form the greatest and the smallest 6-digit numbers using the digits, 4, 6, 5, 9, 7 and 8 without repeating them. Solution: The given digits are 4, 6, 5, 9, 7 and 8. The steps to find the greatest 6-digit number are as follows: Step 1: Arrange the digits in descending order; 9 > 8 > 7 > 6 > 5 > 4. Step 2: Place the digits in the place value chart from left to right. So, the greatest 6-digit number formed is L T Th Th H T O 9,87,654. 9 8 7 654 The steps to find the smallest 6-digit number are as follows: Step 1: Arrange the digits in ascending order; 4 < 5 < 6 < 7 < 8 < 9. Step 2: Place the digits in the place value chart from L T Th Th H T O left to right. 4 5 6 789 So, the smallest 6-digit number formed is 456789. Numbers 39
I Explore (H.O.T.S.) Let us see some more examples using 5-digit numbers. Example 13: Find the difference between the face value and place value of the digits in bold, in the following numbers: a) 50572 b) 84395 Solution: a) 50572: Face value = 5; Place value = 500; Difference = 500 – 5 = 495 b) 84395: Face value = 8; Place value = 80000; Difference = 80000 – 8 = 79992 Example 14: Write the number from the clues given. a) The only digit in 867891 with the same place value and face value. b) A few 6-digit numbers which have the same digit in all the six places. Solution: a) 1 b) 9,99,999; 1,11,111; 6,66,666; 4,44,444 and so on. Maths Munchies Apart from I, V and X, the rest of the Roman numerals are L, C, D and M. The Hindu-Arabic numerals for these Roman numerals are 50, 100, 500 and 1000 respectively. Connect the Dots Social Studies Fun Different languages are spoken in different states of our country. These languages have different scripts. Look at the numbers in the Gujarati script. Now, write the following numbers in Gujarati. One has been done for you. a) 23457 = 23457 b) 45786 = __________ c) 69 = ___________ d) 86 = __________ Science Fun The Challenger Deep is the deepest point known in the Earth’s oceans. Its depth is about 10,900 m approximately. 40
Drill Time 3.1 Introduction to Roman Numerals 1) Write the Hindu-Arabic numerals for the given Roman numerals: a) IV b) XX c) XXIV d) XXXII e) I 2) Read the clocks and write the time in Hindu-Arabic numerals. a) b) c) d) 3.2 Count 6-digit Numbers in Indian System 3) Write the following numbers in the place value chart. a) 871301 b) 49130 c) 84019 d) 591045 e) 18938 4) Write the place value and face value of the digits given in the following numbers: a) 4 in 41351 b) 8 in 491829 c) 6 in 76193 d) 3 in 124131 e) 0 in 40139 5) Write the following numbers using periods. a) 859251 b) 52048 c) 104506 d) 98204 e) 75920 6) Write the expanded forms of the following numbers: a) 410494 b) 58104 c) 956408 d) 65930 e) 10482 7) Form numbers and use periods for the following: a) 6 in the lakhs place, 8 in the ten thousands place, 4 in the thousands place, 1 in the hundreds place, 0 in the tens place and 7 in the ones place b) 4 in the ten thousands place, 1 in the thousands place, 0 in the hundreds place, 3 in the tens place and 8 in the ones place Numbers 41
c) 7 in the ten thousands place, 9 in the thousands place, 6 in the hundreds place, 5 in the tens place and 3 in the ones place d) 9 in the lakhs place, 6 in the ten thousands place, 4 in the thousands place, 5 in the hundreds place, 6 in the tens place and 2 in the ones place e) 1 in the ten thousands place, 5 in the thousands place, 7 in the hundreds place, 9 in the tens place and 6 in the ones place 8) Compare the numbers using the symbols <, > or = in each of the following pairs: a) 85704, 459101 b) 5814, 41049 c) 750314, 518403 d) 158131, 62104 e) 395206, 395260 9) Arrange the numbers in the ascending and descending orders. a) 51058, 58104, 58401 and 58041 b) 98765, 87659, 76598 and 65987 c) 776541, 776532, 776513 and 776520 d) 65807, 26806, 96905 and 14068 e) 581042, 671041, 710486 and 403285 10) Form the largest and the smallest 6-digit numbers using the given digits: a) 5, 2, 6, 1, 0, 8 b) 9, 6, 1, 5, 3, 7 c ) 7, 4, 1, 8, 5, 6 d) 1, 5, 2, 3, 8, 4 e) 6, 9, 1, 5, 0, 8 11) Word problem The savings (in rupees) of Rohan and his friends are as given. Use this information to answer the questions that follow: Salim: ` 9,80,231 Pooja: ` 7,99,501 Soham: ` 2,98,651 Mona: ` 1,70,123 Joe: ` 4,00,002 Varun: ` 8,44,976 Farah: ` 5,22,012 Varsha: ` 3,24,523 Meera: ` 6,50,905 a) How much money did Meera save? Write it in words. b) Who has the most savings and who has the least savings? Write their savings in words. c) Who has more savings between Pooja or Varun? A Note to Parent Write the first five digits and the last five digits of your mobile number on a piece of paper. Ask your child to read the numbers. Give some more such numbers for his or her practice. 42
4Chapter SAdudbittrioanctainond I Will Learn About • adding and subtracting 4-digit and 5-digit numbers. • applying two operations addition and subtraction in solving real-life situations. • framing word problems based on mathematical statements. 4.1 Add and Subtract 4-digit and 5-digit Numbers I Think Surbhi tried to log into her father’s computer. Unfortunately, it was locked and she did not know the password. She saw the hint as given below: 6738 − 2345 = _______ Can you solve it? I Recall We have learnt the addition and subtraction of 4-digit numbers without regrouping. 43
Let us recall the steps followed. Step 1: Arrange the numbers one below the other according to their places. For subtraction, ensure that the smaller number is placed below the larger number. Step 2: Start adding or subtracting the digits from the ones place. Step 3: Write the answer. Let us use these steps to solve the following: a) Th H T O b) Th H T O c) Th H T O 4216 1335 5985 +1251 +1234 +2003 d) Th H T O e) Th H T O f) Th H T O 7453 4472 6134 −1322 −1322 −1021 I Remember and Understand Let us see the examples of addition and subtraction of While subtracting numbers 4-digit and 5-digit numbers with regrouping. by the vertical method, always write the smaller Add 4-digit numbers number below the bigger number. We regroup the sum when it is equal to or more than 10. Example 1: Add 1456 and 1546. Solution: Arrange the numbers one below the other. Add and regroup if necessary. Solved Step 1: Add the ones. Step 2: Add the tens. Th H T O Th H T O 1 11 1456 1456 +1 5 4 6 +1 5 4 6 2 02 44
Solved Step 3: Add the hundreds. Step 4: Add the thousands. Th H T O Th H T O 111 111 1456 1456 +1 5 4 6 +1 5 4 6 002 3002 Th H T O Solve these Th H T O Th H T O 1758 4592 +5662 2678 +1456 +1332 Subtract 4-digit numbers In the subtraction of 4-digit numbers, we can regroup the digits in thousands, hundreds and tens places. Let us see an example. Example 2: Subtract: 4868 from 7437 Solution: Write the smaller number below the larger number. Steps Solved Solve these Step 1: Subtract the ones. But, 7 – 8 is not possible as 7 < 8. So, Th H T O Th H T O regroup the tens digit, 3. 2 17 3 tens = 2 tens + 1 ten. Borrow 1 ten to 1654 the ones place and add it (1 ten = 10 7 4 \\3 \\7 –1246 ones) to the ones place. Reduce the –4868 tens digit by 1. Then subtract the ones and write the answer. 9 So, 17 – 8 = 9. Addition and Subtraction 45
Steps Solved Solve these Step 2: Subtract the tens. But, 2 – 6 is Th H T O not possible as 2 < 6. So, regroup the Th H T O hundreds digit, 4. 12 5674 4 hundreds = 3 hundreds + 1 hundred. –2382 Borrow 1 hundred to the tens place and 3 \\2 17 add it (1 hundred = 10 tens) to the tens 7 4\\ \\3 7\\ Th H T O place. Reduce the hundreds digit by –4868 1. Then subtract the tens and write the 7468 answer. 69 –4837 So, 12 – 6 = 6. Th H T O Th H T O Step 3: Subtract the hundreds. But, 13 12 3 – 8 is not possible as 3 < 8. So, regroup 9276 the thousands digit, 7. 7 thousands = 6 \\3 2\\ 17 –5147 6 thousands + 1 thousand. Borrow 1 \\7 \\4 3\\ \\7 thousand to the hundreds place and –4868 add it (1 thousand = 10 hundreds) to the hundreds place. Reduce the thousands 569 digit by 1. Then subtract the hundreds and write the answer. Th H T O 13 12 So, 13 – 8 = 5. Step 4: Subtract the thousands and write 6 \\3 \\2 17 the answer. \\7 4\\ \\3 \\7 –4868 6–4=2 2569 Therefore, 7437 - 4868 = 2569. Add 5-digit numbers Example 3: Add: 48415 and 20098 Solution: Arrange the numbers in columns, one below the other. Steps Solved Solve these T Th Th H T O Step 1: Add the ones. Write T Th Th H T O the sum under the ones. 1 5 7383 Regroup if needed. + 3 1347 4 8415 + 2 0098 3 46
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