Page 1 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. 5.2 Measuring Line Segments We have drawn and seen so many line segments. A triangle is made of three, a quadrilateral of four line segments. A line segment is a fixed portion of a line. This makes it possible to measure a line segment. This measure of each line segment is a unique number called its “length”. We use this idea to compare line segments. T o compare any two line segments, we find a relation between their lengths. This can be done in several ways. (i) Comparison by observation: By just looking at them can you tell which one is longer? You can see that AB is longer. But you cannot always be sure about your usual judgment. For example, look at the adjoining segments : 5.1 Introduction Chapter 5 Chapter 5 Chapter 5 Chapter 5 Chapter 5 Understanding Understanding Understanding Understanding Understanding Elementary Elementary Elementary Elementary Elementary Shapes Shapes Shapes Shapes Shapes 2020-21 Page 2 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. 5.2 Measuring Line Segments We have drawn and seen so many line segments. A triangle is made of three, a quadrilateral of four line segments. A line segment is a fixed portion of a line. This makes it possible to measure a line segment. This measure of each line segment is a unique number called its “length”. We use this idea to compare line segments. T o compare any two line segments, we find a relation between their lengths. This can be done in several ways. (i) Comparison by observation: By just looking at them can you tell which one is longer? You can see that AB is longer. But you cannot always be sure about your usual judgment. For example, look at the adjoining segments : 5.1 Introduction Chapter 5 Chapter 5 Chapter 5 Chapter 5 Chapter 5 Understanding Understanding Understanding Understanding Understanding Elementary Elementary Elementary Elementary Elementary Shapes Shapes Shapes Shapes Shapes 2020-21 Fractions and Fractions and Fractions and Fractions and Fractions and Integers Integers Integers Integers Integers Fractions and Fractions and Fractions and Fractions and Fractions and Integers Integers Integers Integers Integers UNDERSTANDING ELEMEN TA RY SHAPES 87 The difference in lengths between these two may not be obvious. This makes other ways of comparing necessary. In this adjacent figure, AB and PQ have the same lengths. This is not quite obvious. So, we need better methods of comparing line segments. (ii) Comparison by Tracing To compare AB and CD , we use a tracing paper, trace CD and place the traced segment on AB . Can you decide now which one among AB and CD is longer? The method depends upon the accuracy in tracing the line segment. Moreover, if you want to compare with another length, you have to trace another line segment. This is difficult and you cannot trace the lengths everytime you want to compare them. (iii) Comparison using Ruler and a Divider Have you seen or can you recognise all the instruments in your instrument box? Among other things, you have a ruler and a divider. Note how the ruler is marked along one of its edges. It is divided into 15 parts. Each of these 15 parts is of length 1cm. Each centimetre is divided into 10subparts. Each subpart of the division of a cm is 1mm. How many millimetres make one centimetre? Since 1cm = 10 mm, how will we write 2 cm? 3mm? What do we mean by 7.7 cm? Place the zero mark of the ruler at A. Read the mark against B. This gives the length of AB . Suppose the length is 5.8 cm, we may write, Length AB = 5.8 cm or more simply as AB = 5.8 cm. There is room for errors even in this procedure. The thickness of the ruler may cause difficulties in reading off the marks on it. Ruler Divider 1 mm is 0.1 cm. 2 mm is 0.2 cm and so on . 2.3 cm will mean 2 cm and 3 mm. 2020-21 Page 3 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. 5.2 Measuring Line Segments We have drawn and seen so many line segments. A triangle is made of three, a quadrilateral of four line segments. A line segment is a fixed portion of a line. This makes it possible to measure a line segment. This measure of each line segment is a unique number called its “length”. We use this idea to compare line segments. T o compare any two line segments, we find a relation between their lengths. This can be done in several ways. (i) Comparison by observation: By just looking at them can you tell which one is longer? You can see that AB is longer. But you cannot always be sure about your usual judgment. For example, look at the adjoining segments : 5.1 Introduction Chapter 5 Chapter 5 Chapter 5 Chapter 5 Chapter 5 Understanding Understanding Understanding Understanding Understanding Elementary Elementary Elementary Elementary Elementary Shapes Shapes Shapes Shapes Shapes 2020-21 Fractions and Fractions and Fractions and Fractions and Fractions and Integers Integers Integers Integers Integers Fractions and Fractions and Fractions and Fractions and Fractions and Integers Integers Integers Integers Integers UNDERSTANDING ELEMEN TA RY SHAPES 87 The difference in lengths between these two may not be obvious. This makes other ways of comparing necessary. In this adjacent figure, AB and PQ have the same lengths. This is not quite obvious. So, we need better methods of comparing line segments. (ii) Comparison by Tracing To compare AB and CD , we use a tracing paper, trace CD and place the traced segment on AB . Can you decide now which one among AB and CD is longer? The method depends upon the accuracy in tracing the line segment. Moreover, if you want to compare with another length, you have to trace another line segment. This is difficult and you cannot trace the lengths everytime you want to compare them. (iii) Comparison using Ruler and a Divider Have you seen or can you recognise all the instruments in your instrument box? Among other things, you have a ruler and a divider. Note how the ruler is marked along one of its edges. It is divided into 15 parts. Each of these 15 parts is of length 1cm. Each centimetre is divided into 10subparts. Each subpart of the division of a cm is 1mm. How many millimetres make one centimetre? Since 1cm = 10 mm, how will we write 2 cm? 3mm? What do we mean by 7.7 cm? Place the zero mark of the ruler at A. Read the mark against B. This gives the length of AB . Suppose the length is 5.8 cm, we may write, Length AB = 5.8 cm or more simply as AB = 5.8 cm. There is room for errors even in this procedure. The thickness of the ruler may cause difficulties in reading off the marks on it. Ruler Divider 1 mm is 0.1 cm. 2 mm is 0.2 cm and so on . 2.3 cm will mean 2 cm and 3 mm. 2020-21 MATHEMATICS 88 Think, discuss and write 1. What other errors and difficulties might we face? 2. What kind of errors can occur if viewing the mark on the ruler is not proper? How can one avoid it? Positioning error To get correct measure, the eye should be correctly positioned, just vertically above the mark. Otherwise errors can happen due to angular viewing. Can we avoid this problem? Is there a better way? Let us use the divider to measure length. Open the divider. Place the end point of one of its arms at A and the end point of the second arm at B. T aking care that opening of the divider is not disturbed, lift the divider and place it on the ruler. Ensure that one end point is at the zero mark of the ruler. Now read the mark against the other end point. EXERCISE 5.1 1. What is the disadvantage in comparing line segments by mere observation? 2. Why is it better to use a divider than a ruler, while measuring the length of a line segment? 3. Draw any line segment, say AB . Take any point C lying in between A and B. Measure the lengths of AB, BC and AC. Is AB = AC + CB? [Note : If A,B,C are any three points on a line such that AC + CB = AB, then we can be sure that C lies between A and B.] 1. T ake any post card. Use the above technique to measure its two adjacent sides. 2. Select any three objects having a flat top. Measure all sides of the top using a divider and a ruler. 2020-21 Page 4 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. 5.2 Measuring Line Segments We have drawn and seen so many line segments. A triangle is made of three, a quadrilateral of four line segments. A line segment is a fixed portion of a line. This makes it possible to measure a line segment. This measure of each line segment is a unique number called its “length”. We use this idea to compare line segments. T o compare any two line segments, we find a relation between their lengths. This can be done in several ways. (i) Comparison by observation: By just looking at them can you tell which one is longer? You can see that AB is longer. But you cannot always be sure about your usual judgment. For example, look at the adjoining segments : 5.1 Introduction Chapter 5 Chapter 5 Chapter 5 Chapter 5 Chapter 5 Understanding Understanding Understanding Understanding Understanding Elementary Elementary Elementary Elementary Elementary Shapes Shapes Shapes Shapes Shapes 2020-21 Fractions and Fractions and Fractions and Fractions and Fractions and Integers Integers Integers Integers Integers Fractions and Fractions and Fractions and Fractions and Fractions and Integers Integers Integers Integers Integers UNDERSTANDING ELEMEN TA RY SHAPES 87 The difference in lengths between these two may not be obvious. This makes other ways of comparing necessary. In this adjacent figure, AB and PQ have the same lengths. This is not quite obvious. So, we need better methods of comparing line segments. (ii) Comparison by Tracing To compare AB and CD , we use a tracing paper, trace CD and place the traced segment on AB . Can you decide now which one among AB and CD is longer? The method depends upon the accuracy in tracing the line segment. Moreover, if you want to compare with another length, you have to trace another line segment. This is difficult and you cannot trace the lengths everytime you want to compare them. (iii) Comparison using Ruler and a Divider Have you seen or can you recognise all the instruments in your instrument box? Among other things, you have a ruler and a divider. Note how the ruler is marked along one of its edges. It is divided into 15 parts. Each of these 15 parts is of length 1cm. Each centimetre is divided into 10subparts. Each subpart of the division of a cm is 1mm. How many millimetres make one centimetre? Since 1cm = 10 mm, how will we write 2 cm? 3mm? What do we mean by 7.7 cm? Place the zero mark of the ruler at A. Read the mark against B. This gives the length of AB . Suppose the length is 5.8 cm, we may write, Length AB = 5.8 cm or more simply as AB = 5.8 cm. There is room for errors even in this procedure. The thickness of the ruler may cause difficulties in reading off the marks on it. Ruler Divider 1 mm is 0.1 cm. 2 mm is 0.2 cm and so on . 2.3 cm will mean 2 cm and 3 mm. 2020-21 MATHEMATICS 88 Think, discuss and write 1. What other errors and difficulties might we face? 2. What kind of errors can occur if viewing the mark on the ruler is not proper? How can one avoid it? Positioning error To get correct measure, the eye should be correctly positioned, just vertically above the mark. Otherwise errors can happen due to angular viewing. Can we avoid this problem? Is there a better way? Let us use the divider to measure length. Open the divider. Place the end point of one of its arms at A and the end point of the second arm at B. T aking care that opening of the divider is not disturbed, lift the divider and place it on the ruler. Ensure that one end point is at the zero mark of the ruler. Now read the mark against the other end point. EXERCISE 5.1 1. What is the disadvantage in comparing line segments by mere observation? 2. Why is it better to use a divider than a ruler, while measuring the length of a line segment? 3. Draw any line segment, say AB . Take any point C lying in between A and B. Measure the lengths of AB, BC and AC. Is AB = AC + CB? [Note : If A,B,C are any three points on a line such that AC + CB = AB, then we can be sure that C lies between A and B.] 1. T ake any post card. Use the above technique to measure its two adjacent sides. 2. Select any three objects having a flat top. Measure all sides of the top using a divider and a ruler. 2020-21 Fractions and Fractions and Fractions and Fractions and Fractions and Integers Integers Integers Integers Integers Fractions and Fractions and Fractions and Fractions and Fractions and Integers Integers Integers Integers Integers UNDERSTANDING ELEMEN TA RY SHAPES 89 4. If A,B,C are three points on a line such that AB = 5 cm, BC = 3 cm and AC = 8 cm, which one of them lies between the other two? 5. Verify, whether D is the mid point of AG . 6. If B is the mid point of AC and C is the mid point of BD , where A,B,C,D lie on a straight line, say why AB = CD? 7. Draw five triangles and measure their sides. Check in each case, if the sum of the lengths of any two sides is always less than the third side. 5.3 Angles – ‘Right’ and ‘Straight’ You have heard of directions in Geography. We know that China is to the north of India, Sri Lanka is to the south. We also know that Sun rises in the east and sets in the west. There are four main directions. They are North (N), South (S), East (E) and West (W). Do you know which direction is opposite to north? Which direction is opposite to west? Just recollect what you know already. We now use this knowledge to learn a few properties about angles. Stand facing north. Turn clockwise to east. We say, you have turned through a right angle. Follow this by a ‘right-angle-turn’, clockwise. You now face south. If you turn by a right angle in the anti-clockwise direction, which direction will you face? It is east again! (Why?) Study the following positions : Do This You stand facing north By a ‘right-angle-turn’ clockwise, you now face east By another ‘right-angle-turn’ you finally face south. 2020-21 Page 5 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. 5.2 Measuring Line Segments We have drawn and seen so many line segments. A triangle is made of three, a quadrilateral of four line segments. A line segment is a fixed portion of a line. This makes it possible to measure a line segment. This measure of each line segment is a unique number called its “length”. We use this idea to compare line segments. T o compare any two line segments, we find a relation between their lengths. This can be done in several ways. (i) Comparison by observation: By just looking at them can you tell which one is longer? You can see that AB is longer. But you cannot always be sure about your usual judgment. For example, look at the adjoining segments : 5.1 Introduction Chapter 5 Chapter 5 Chapter 5 Chapter 5 Chapter 5 Understanding Understanding Understanding Understanding Understanding Elementary Elementary Elementary Elementary Elementary Shapes Shapes Shapes Shapes Shapes 2020-21 Fractions and Fractions and Fractions and Fractions and Fractions and Integers Integers Integers Integers Integers Fractions and Fractions and Fractions and Fractions and Fractions and Integers Integers Integers Integers Integers UNDERSTANDING ELEMEN TA RY SHAPES 87 The difference in lengths between these two may not be obvious. This makes other ways of comparing necessary. In this adjacent figure, AB and PQ have the same lengths. This is not quite obvious. So, we need better methods of comparing line segments. (ii) Comparison by Tracing To compare AB and CD , we use a tracing paper, trace CD and place the traced segment on AB . Can you decide now which one among AB and CD is longer? The method depends upon the accuracy in tracing the line segment. Moreover, if you want to compare with another length, you have to trace another line segment. This is difficult and you cannot trace the lengths everytime you want to compare them. (iii) Comparison using Ruler and a Divider Have you seen or can you recognise all the instruments in your instrument box? Among other things, you have a ruler and a divider. Note how the ruler is marked along one of its edges. It is divided into 15 parts. Each of these 15 parts is of length 1cm. Each centimetre is divided into 10subparts. Each subpart of the division of a cm is 1mm. How many millimetres make one centimetre? Since 1cm = 10 mm, how will we write 2 cm? 3mm? What do we mean by 7.7 cm? Place the zero mark of the ruler at A. Read the mark against B. This gives the length of AB . Suppose the length is 5.8 cm, we may write, Length AB = 5.8 cm or more simply as AB = 5.8 cm. There is room for errors even in this procedure. The thickness of the ruler may cause difficulties in reading off the marks on it. Ruler Divider 1 mm is 0.1 cm. 2 mm is 0.2 cm and so on . 2.3 cm will mean 2 cm and 3 mm. 2020-21 MATHEMATICS 88 Think, discuss and write 1. What other errors and difficulties might we face? 2. What kind of errors can occur if viewing the mark on the ruler is not proper? How can one avoid it? Positioning error To get correct measure, the eye should be correctly positioned, just vertically above the mark. Otherwise errors can happen due to angular viewing. Can we avoid this problem? Is there a better way? Let us use the divider to measure length. Open the divider. Place the end point of one of its arms at A and the end point of the second arm at B. T aking care that opening of the divider is not disturbed, lift the divider and place it on the ruler. Ensure that one end point is at the zero mark of the ruler. Now read the mark against the other end point. EXERCISE 5.1 1. What is the disadvantage in comparing line segments by mere observation? 2. Why is it better to use a divider than a ruler, while measuring the length of a line segment? 3. Draw any line segment, say AB . Take any point C lying in between A and B. Measure the lengths of AB, BC and AC. Is AB = AC + CB? [Note : If A,B,C are any three points on a line such that AC + CB = AB, then we can be sure that C lies between A and B.] 1. T ake any post card. Use the above technique to measure its two adjacent sides. 2. Select any three objects having a flat top. Measure all sides of the top using a divider and a ruler. 2020-21 Fractions and Fractions and Fractions and Fractions and Fractions and Integers Integers Integers Integers Integers Fractions and Fractions and Fractions and Fractions and Fractions and Integers Integers Integers Integers Integers UNDERSTANDING ELEMEN TA RY SHAPES 89 4. If A,B,C are three points on a line such that AB = 5 cm, BC = 3 cm and AC = 8 cm, which one of them lies between the other two? 5. Verify, whether D is the mid point of AG . 6. If B is the mid point of AC and C is the mid point of BD , where A,B,C,D lie on a straight line, say why AB = CD? 7. Draw five triangles and measure their sides. Check in each case, if the sum of the lengths of any two sides is always less than the third side. 5.3 Angles – ‘Right’ and ‘Straight’ You have heard of directions in Geography. We know that China is to the north of India, Sri Lanka is to the south. We also know that Sun rises in the east and sets in the west. There are four main directions. They are North (N), South (S), East (E) and West (W). Do you know which direction is opposite to north? Which direction is opposite to west? Just recollect what you know already. We now use this knowledge to learn a few properties about angles. Stand facing north. Turn clockwise to east. We say, you have turned through a right angle. Follow this by a ‘right-angle-turn’, clockwise. You now face south. If you turn by a right angle in the anti-clockwise direction, which direction will you face? It is east again! (Why?) Study the following positions : Do This You stand facing north By a ‘right-angle-turn’ clockwise, you now face east By another ‘right-angle-turn’ you finally face south. 2020-21 MATHEMATICS 90 From facing north to facing south, you have turned by two right angles. Is not this the same as a single turn by two right angles? The turn from north to east is by a right angle. The turn from north to south is by two right angles; it is called a straight angle. (NS is a straight line!) Stand facing south. Turn by a straight angle. Which direction do you face now? You face north! To turn from north to south, you took a straight angle turn, again to turn from south to north, you took another straight angle turn in the same direction. Thus, turning by two straight angles you reach your original position. Think, discuss and write By how many right angles should you turn in the same direction to reach your original position? Turning by two straight angles (or four right angles) in the same direction makes a full turn. This one complete turn is called one revolution. The angle for one revolution is a complete angle. We can see such revolutions on clock-faces. When the hand of a clock moves from one position to another, it turns through an angle. Suppose the hand of a clock starts at 12 and goes round until it reaches at 12 again. Has it not made one revolution? So, how many right angles has it moved? Consider these examples : From 12 to 6 From 6 to 9 From 1 to 10 1 2 of a revolution. 1 4 of a revolution 3 4 of a revolution or 2 right angles. or 1 right angle. or 3 right angles. 2020-21Read More

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