Patterns in Mathematics (Grade 6 PDF)

1 PATTERNS IN MATHEMATICS 1.1 What is Mathematics? Mathematics is, in large part, the search for patterns, and for the explanations as to why those pat...

1 PATTERNS IN MATHEMATICS 1.1 What is Mathematics? Mathematics is, in large part, the search for patterns, and for the explanations as to why those patterns exist. Such patterns indeed exist all around us — in nature, in our homes and schools, and in the motion of the sun, moon, and stars. They occur in everything that we do and see, from shopping and cooking, to throwing a ball and playing games, to understanding weather patterns and using technology. The search for patterns and their explanations can be a fun and creative endeavour. It is for this reason that mathematicians think of mathematics both as an art and as a science. This year, we hope that you will get a chance to see the creativity and artistry involved in discovering and understanding mathematical patterns. It is important to keep in mind that mathematics aims to not just find out what patterns exist, but also the explanations for why they exist. Such explanations can often then be used in applications well beyond the context in which they were discovered, which can then help to propel humanity forward. Chapter 1_Patterns in Mathematics.indd 1 10-08-2024 11:55:05  Ganita Prakash | Grade 6 For example, the understanding of patterns in the motion of stars, planets, and their satellites led humankind to develop the theory of gravitation, allowing us to launch our own satellites and send rockets to the Moon and to Mars; similarly, understanding patterns in genomes has helped in diagnosing and curing diseases—among thousands of other such examples. Figure it Out 1. Can you think of other examples where mathematics helps us in our everyday lives? Math Talk 2. How has mathematics helped propel humanity forward? (You might think of examples involving: carrying out scientific experiments; running our economy and democracy; building bridges, houses or other complex structures; making TVs, mobile phones, computers, bicycles, trains, cars, planes, calendars, clocks, etc.) 1.2 Patterns in Numbers Among the most basic patterns that occur in mathematics are patterns of numbers, particularly patterns of whole numbers: 0, 1, 2, 3, 4,... The branch of Mathematics that studies patterns in whole numbers is called number theory. Number sequences are the most basic and among the most fascinating types of patterns that mathematicians study. Table 1 shows some key number sequences that are studied in Mathematics. 2 Chapter 1_Patterns in Mathematics.indd 2 10-08-2024 11:55:05  Patterns in Mathematics Table 1 Examples of number sequences 1, 1, 1, 1, 1, 1, 1,... (All 1’s) 1, 2, 3, 4, 5, 6, 7,... (Counting numbers) 1, 3, 5, 7, 9, 11, 13,... (Odd numbers) 2, 4, 6, 8, 10, 12, 14,... (Even numbers) 1, 3, 6, 10, 15, 21, 28,... (Triangular numbers) 1, 4, 9, 16, 25, 36, 49,... (Squares) 1, 8, 27, 64, 125, 216,... (Cubes) 1, 2, 3, 5, 8, 13, 21,... (Virahānka numbers) 1, 2, 4, 8, 16, 32, 64,... (Powers of 2) 1, 3, 9, 27, 81, 243, 729,... (Powers of 3) Figure it Out 1. Can you recognize the pattern in each of the sequences in Table 1? Math 2. Rewrite each sequence of Table 1 in your notebook, along Talk with the next three numbers in each sequence! After each sequence, write in your own words what is the rule for forming the numbers in the sequence. 1.3 Visualising Number Sequences Many number sequences can be visualised using pictures. Visualising mathematical objects through pictures or diagrams can be a very fruitful way to understand mathematical patterns and concepts. Let us represent the first seven sequences in Table 1 using the following pictures. 3 Chapter 1_Patterns in Mathematics.indd 3 10-08-2024 11:55:05  Ganita Prakash | Grade 6 Table 2 Pictorial representation of some number sequences All 1’s 1 1 1 1 1 Counting 1 2 3 4 5 numbers Odd numbers 1 3 5 7 9 Even numbers 2 4 6 8 10 Triangular numbers 1 3 6 10 15 Squares 1 4 9 16 25 Cubes 1 8 27 64 125 4 Chapter 1_Patterns in Mathematics.indd 4 10-08-2024 11:55:06  Patterns in Mathematics Figure it Out 1. Copy the pictorial representations of the number sequences Math in Table 2 in your notebook, and draw the next picture for Talk each sequence! 2. Why are 1, 3, 6, 10, 15, … called triangular numbers? Why are 1, 4, 9, 16, 25, … called square numbers or squares? Why are 1, 8, 27, 64, 125, … called cubes? 3. You will have noticed that 36 is both a triangular number and a square number! That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make pictures in your notebook illustrating this! This shows that the same number can be represented differently, and play different roles, depending on the context. Try representing some other numbers pictorially in different ways! 4. What would you call the following sequence of numbers? 1 7 19 37 That’s right, they are called hexagonal numbers! Draw these in your notebook. What is the next number in the sequence? 5. Can you think of pictorial ways to visualise the sequence of Powers of 2? Powers of 3? 5 Chapter 1_Patterns in Mathematics.indd 5 10-08-2024 11:55:06  Ganita Prakash | Grade 6 Here is one possible way of thinking about Powers of 2: 1 2 4 8 16 1.4 Relations among Number Sequences ometimes, number sequences can be related to each other in S surprising ways. Example: What happens when we start adding up odd numbers? 1 = 1 1 + 3 = 4 1+3+5=9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25 1 + 3 + 5 + 7 + 9 + 11 = 36... This is a really beautiful pattern! Why does this happen? Do you think it will happen forever? The answer is that the pattern does happen forever. But why? As mentioned earlier, the reason why the pattern happens is just as important and exciting as the pattern itself. A picture can explain it Visualising with a picture can help explain the phenomenon. Recall that square numbers are made by counting the number of dots in a square grid. How can we partition the dots in a square grid into odd Math numbers of dots: 1, 3, 5, 7,... ? Talk Think about it for a moment before reading further! 6 Chapter 1_Patterns in Mathematics.indd 6 10-08-2024 11:55:06  Patterns in Mathematics Here is how it can be done: This picture now makes it evident that 1 + 3 + 5 + 7 + 9 + 11 = 36. Because such a picture can be made for a square of any size, this explains why adding up odd numbers gives square numbers. By drawing a similar picture, can you say what is the sum of the first 10 odd numbers? Now by imagining a similar picture, or by drawing it partially, as needed, can you say what is the sum of the first 100 odd numbers? Another example of such a relation between sequences: Adding up and down Let us look at the following pattern: 1 = 1 1+2+1=4 1+2+3+2+1=9 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25 1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1 = 36... This seems to be giving yet another way of getting the square numbers— by adding the counting numbers up and then down! 7 Chapter 1_Patterns in Mathematics.indd 7 10-08-2024 11:55:06  Ganita Prakash | Grade 6 Can you find a similar pictorial explanation? Figure it Out 1. Can you find a similar pictorial explanation for why adding Try counting numbers up and down, i.e., 1, 1 + 2 + 1, 1 + 2 + 3 + This 2 + 1, …, gives square numbers? 2. By imagining a large version of your picture, or drawing it partially, as needed, can you see what will be tha value of 1 + 2 + 3 +... + 99 + 100 + 99 +... + 3 + 2 + 1? 3. Which sequence do you get when you start to add the All 1’s sequence up? What sequence do you get when you add the All 1’s sequence up and down? 4. Which sequence do you get when you start to add the Counting numbers up? Can you give a smaller pictorial explanation? 5. What happens when you add up pairs of consecutive triangular numbers? That is, take 1 + 3, 3 + 6, 6 + 10, 10 + 15, … ? Which sequence do you get? Why? Can you explain it with a picture? 6. What happens when you start to add up powers of 2 starting with 1, i.e., take 1, 1 + 2, 1 + 2 + 4, 1 + 2 + 4 + 8, … ? Now add 1 to each of these numbers — what numbers do you get? Why does this happen? 8 Chapter 1_Patterns in Mathematics.indd 8 10-08-2024 11:55:06  Patterns in Mathematics 7. What happens when you multiply the triangular numbers by 6 and add 1? Which sequence do you get? Can you explain it with a picture? 8. What happens when you start to add up hexagonal numbers, i.e., take 1, 1 + 7, 1 + 7 + 19, 1 + 7 + 19 + 37, … ? Which sequence do you get? Can you explain it using a picture of a cube? 9. Find your own patterns or relations in and among the sequences in Table 1. Can you explain why they happen with a picture or otherwise? 1.5 Patterns in Shapes Other important and basic patterns that occur in Mathematics are patterns of shapes. These shapes may be in one, two, or three dimensions (1D, 2D, or 3D) — or in even more dimensions. The branch of Mathematics that studies patterns in shapes is called geometry. Shape sequences are one important type of shape pattern that mathematicians study. Table 3 shows a few key shape sequences that are studied in Mathematics. 9 Chapter 1_Patterns in Mathematics.indd 9 10-08-2024 11:55:06  Ganita Prakash | Grade 6 Table 3 Examples of shape sequences Triangle Quadrilateral Pentagon Hexagon Regular Polygons Heptagon Octagon Nonagon Decagon Complete Graphs K2 K3 K4 K5 K6 Stacked Squares Stacked Triangles Koch Snowflake 10 Chapter 1_Patterns in Mathematics.indd 10 10-08-2024 11:55:06  Patterns in Mathematics Figure it Out  1. Can you recognise the pattern in each of the sequences in Table 3? Math 2. Try and redraw each sequence in Table 3 in your notebook. Talk Can you draw the next shape in each sequence? Why or why not? After each sequence, describe in your own words what is the rule or pattern for forming the shapes in the sequence. 1.6 Relation to Number Sequences Often, shape sequences are related to number sequences in surprising ways. Such relationships can be helpful in studying and understanding both the shape sequence and the related number sequence. Example: The number of sides in the shape sequence of Regular Polygons is given by the counting numbers starting at 3, i.e., 3, 4, 5, 6, 7, 8, 9, 10,.... That is why these shapes are called, respectively, regular triangle, quadrilateral (i.e., square), pentagon, hexagon, heptagon, octagon, nonagon, decagon, etc., respectively. The word ‘regular’ refers to the fact that these shapes have equal-length sides and also equal ‘angles’ (i.e., the sides look the same and the corners also look the same). We will discuss angles in more depth in the next chapter. The other shape sequences in Table 3 also have beautiful relationships with number sequences. Figure it Out 1. Count the number of sides in each shape in the sequence Try of Regular Polygons. Which number sequence do you get? This What about the number of corners in each shape in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens? 2. Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why? 11 Chapter 1_Patterns in Mathematics.indd 11 10-08-2024 11:55:07  Ganita Prakash | Grade 6 3. How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain why? 4. How many little triangles are there in each shape of the sequence of Stacked Triangles? Which number sequence does this give? Can you explain why? (Hint: In each shape in Try This the sequence, how many triangles are there in each row?) 5. To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment­ ‘—’ by a ‘speed bump’. As one does this more and more times, the changes become tinier and tinier with very very small line segments. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence? (The answer is 3, 12, 48,..., i.e. 3 times Powers of 4; this sequence is not shown in Table 1) Summary  Mathematics may be viewed as the search for patterns and for the explanations as to why those patterns exist.  Among the most basic patterns that occur in mathematics are number sequences.  Some important examples of number sequences include the counting numbers, odd numbers, even numbers, square numbers, triangular numbers, cube numbers, Virahānka numbers, and powers of 2.  Sometimes number sequences can be related to each other in beautiful and remarkable ways. For example, adding up the sequence of odd numbers starting with 1 gives square numbers.  Visualizing number sequences using pictures can help to understand sequences and the relationships between them.  Shape sequences are another basic type of pattern in mathematics. Some important examples of shape sequences include regular polygons, complete graphs, stacked triangles and squares, and Koch snowflake iterations. Shape sequences also exhibit many interesting relationships with number sequences. 12 Chapter 1_Patterns in Mathematics.indd 12 10-08-2024 11:55:08  2 Lines and Angles In this chapter, we will explore some of the most basic ideas of geometry including points, lines, rays, line segments and angles. These ideas form the building blocks of ‘plane geometry’, and will help us in understanding more advanced topics in geometry such as the construction and analysis of different shapes. 2.1 Point Mark a dot on the paper with a sharp tip of a pencil. The sharper the tip, the thinner will be the dot. This tiny dot will give you an idea of a point. A point determines a precise location, but it has no length, breadth or height. Some models for a point are given below. The tip of a The sharpened The pointed compass end of a pencil end of a needle If you mark three points on a piece of paper, Z P you may be required to distinguish these three points. For this purpose, each of the three points T may be denoted by a single capital letter such as Chapter 2_Lines and Angles.indd 13 08-08-2024 17:31:20  Ganita Prakash | Grade 6 Z, P and T. These points are read as ‘Point Z’, ‘Point P’ and ‘Point T’. Of course, the dots represent precise locations and must be imagined to be invisibly thin. 2.2 Line Segment A Fold a piece of paper and unfold it. Do you see a crease? This gives the idea of a line segment. It has two end points, A and B. Mark any two points A and B on a sheet of paper. Try to connect A to B by various B routes (Fig. 2.1). What is the shortest route from A to B? B This shortest path from point A to Point B (including A and B) as shown here is called the line segment from A to B. It is denoted by A either AB or BA. The points A and B are called Fig. 2.1 the end points of the line segment AB. 2.3 Line m Imagine that the line segment from A to B (i.e., B AB) is extended beyond A in one direction and beyond B in the other direction without any end (see Fig 2.2). This is a model for a line. Do A you think you can draw a complete picture of Fig. 2.2 a line? No. Why? A line through two points A and B is written as AB. It extends forever in both directions. Sometimes a line is denoted by a letter like l or m. Observe that any two points determine a unique line that passes through both of them. 14 Chapter 2_Lines and Angles.indd 14 08-08-2024 17:31:20  Lines and Angles 2.4 Ray A ray is a portion of a line that starts at one point (called the starting point or initial point of the ray) and goes on endlessly in a direction. The following are some models for a ray: Beam of light from a Ray of light from a torch Sun rays lighthouse Look at the diagram (Fig. 2.3) of a ray. Two points are marked on it. One is the starting point A and the other P is a point P on the path of the ray. We then denote the ray by AP. A Fig. 2.3 Figure it Out 1. Rihan marked a point Sheetal marked two points on a piece of paper. on a piece of paper. How How many lines can he many different lines can draw that pass through she draw that pass through the point? both of the points? Can you help Rihan and Sheetal find their answers? 15 Chapter 2_Lines and Angles.indd 15 08-08-2024 17:31:20  Ganita Prakash | Grade 6 2. Name the line segments in Fig. 2.4. Which of the five marked points are on exactly one of the line segments? Which are on two of the line segments? Q M R P L Fig. 2.4 3. Name the rays shown in Fig. 2.5. Is T the starting point of each of these rays? A T N B Fig. 2.5 4. Draw a rough figure and write labels appropriately to illustrate each of the following: a. OP and OQ meet at O. b. XY and PQ intersect at point M. c. Line l contains points E and F but not point D. d. Point P lies on AB.  5. In Fig. 2.6, name: a. Five points B b. A line O C c. Four rays E d. Five line segments D Fig. 2.6 16 Chapter 2_Lines and Angles.indd 16 08-08-2024 17:31:20  Lines and Angles 6. Here is a ray OA (Fig. 2.7). It starts at O and A passes through the point A. It also passes B through the point B. a. Can you also name it as OB ? Why? O b. Can we write OA as AO ? Why or why not? Fig. 2.7 2.5 Angle D m An angle is formed by two rays having a ar common starting point. Here is an angle B formed by rays BD and BE where B is vertex the common starting point (Fig. 2.8). arm The point B is called the vertex of the E angle, and the rays BD and BE are called Fig. 2.8 the arms of the angle. How can we name this angle? We can simply use the vertex and say that it is the Angle B. To be clearer, we use a point on each of the arms together with the vertex to name the angle. In this case, we name the angle as Angle DBE or Angle EBD. The word angle can be replaced by the symbol ‘∠’, i.e., ∠DBE or ∠EBD. Note that in specifying the angle, the vertex is always written as the middle letter. To indicate an angle, we use a small curve at the vertex (refer to Fig. 2.9). Vidya has just opened her book. Let us observe her opening the cover of the book in different scenarios. Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 17 Chapter 2_Lines and Angles.indd 17 08-08-2024 17:31:21  Ganita Prakash | Grade 6 Do you see angles being made in each of these cases? Can you mark their arms and vertex? Which angle is greater—the angle in Case 1 or the angle in Case 2? Just as we talk about the size of a line based on its length, we also talk about the size of an angle based on its amount of rotation. So, the angle in Case 2 is greater as in this case she needs to rotate the cover more. Similarly, the angle in Case 3 is even larger than that of Case 2, because there is even more rotation, and Cases 4, 5, and 6 are successively larger angles with more and more rotation. The size of an angle is the amount of rotation or turn that is needed about the vertex to move the first ray to the second ray. Final position of ray Amount of turn is the size of the angle Vertex Initial position of ray Fig. 2.9 Let’s look at some other examples where angles arise in real life by rotation or turn: In a compass or divider, we turn the arms to form an angle. The vertex is the point where the two arms are joined. Identify the arms and vertex of the angle. A pair of scissors has two blades. When we open them (or ‘turn them’) to cut something, the blades form an angle. Identify the arms and the vertex of the angle. 18 Chapter 2_Lines and Angles.indd 18 08-08-2024 17:31:22  Lines and Angles Look at the pictures of spectacles, wallet and other common objects. Identify the angles in them by marking out their arms and vertices. Do you see how these angles are formed by turning one arm with respect to the other? Teacher’s Note Teacher needs to organise various activities with the students to recognise the size of an angle as a measure of rotation. Figure it Out 1. Can you find the angles in the given pictures? Draw the rays forming any one of the angles and name the vertex of the angle. B C A D 19 Chapter 2_Lines and Angles.indd 19 08-08-2024 17:31:23  Ganita Prakash | Grade 6 2. Draw and label an angle with arms ST and SR. 3. Explain why ∠APC cannot be labelled as ∠P. Math A Talk P B C 4. Name the angles marked in the given figure. P Q T R 5. Mark any three points on your paper that are not on one line. Label them A, B, C. Draw all possible lines going through pairs of these points. How many lines do you get? Name them. How many angles can you name using A, B, C? Write them down, and mark each of them with a curve as in Fig. 2.9. 20 Chapter 2_Lines and Angles.indd 20 08-08-2024 17:31:23  Lines and Angles 6. Now mark any four points on your paper so that no three of them are on one line. Label them A, B, C, D. Draw all possible lines going through pairs of these points. How many lines do you get? Name them. How many angles can you name using A, B, C, D? Write them all down, and mark each of them with a curve as in Fig. 2.9. 2.6 Comparing Angles Look at these animals opening their mouths. Do you see any angles here? If yes, mark the arms and vertex of each one. Some mouths are open wider than others; the more the turning of the jaws, the larger the angle! Can you arrange the angles in this picture from smallest to largest? Is it always easy to compare two angles? Math Talk Here are some angles. Label each of the angles. How will you compare them? Draw a few more angles; label them and compare. 21 Chapter 2_Lines and Angles.indd 21 08-08-2024 17:31:23  Ganita Prakash | Grade 6 Comparing angles by superimposition Any two angles can be compared by placing them one over the other, i.e., by superimposition. While superimposing, the vertices of the angles must overlap. After superimposition, it becomes clear which angle is smaller and which is larger. P P A A B C Q R Q (B) RC The picture shows the two angles superimposed. It is now clear that ∠PQR is larger than ∠ABC. Equal angles. Now consider ∠AOB and ∠XOY in the figure. Which is greater? X X A A O B O Y O B Y The corners of both of these angles match and the arms overlap with each other, i.e., OA ↔ OX and OB ↔ OY. So, the angles are equal in size. The reason these angles are considered to be equal in size is because when we visualise each of these angles as being formed out of rotation, we can see that there is an equal amount of rotation needed to move OB to OA and OY to OX. From the point of view of superimposition, when two angles are superimposed, and the common vertex and the two rays of both angles lie on top of each other, then the sizes of the angles are equal. 22 Chapter 2_Lines and Angles.indd 22 08-08-2024 17:31:24  Lines and Angles Where else do we use superimposition to compare? Math Talk Figure it Out 1. Fold a rectangular sheet of paper, then draw a line along the fold created. Name and compare the angles formed between the fold and the sides of the paper. Make different angles by folding a rectangular sheet of paper and compare the angles. Which is the largest and smallest angle you made? 2. In each case, determine which angle is greater and why. a. ∠AOB or ∠XOY X b. ∠AOB or ∠XOB A Y c. ∠XOB or ∠XOC Discuss with your friends on how O C you decided which one is greater. B Math 3. Which angle is greater: ∠XOY or ∠AOB? Give reasons. Talk X A Y O B Comparing angles without superimposition Two cranes are arguing about who can open their mouth wider, i.e., who is making a bigger angle. Let us first draw their angles. How do we know which one is bigger? As seen Fig. 2.10 23 Chapter 2_Lines and Angles.indd 23 08-08-2024 17:31:24  Ganita Prakash | Grade 6 before, one could trace these angles, superimpose them and then check. But can we do it without superimposition? Suppose we have a transparent circle which can be moved and placed on figures. Can we use this for comparison? Let us place the circular paper on the angle made by the first crane. The circle is placed in such a way that its centre is on the vertex of the angle. Let us mark the points A and B on the edge circle at the points where the arms of the angle pass through the circle. B B O O A A Can we use this to find out if this angle is greater than, or equal to or smaller than the angle made by the second crane? Let us place it on the angle made by the second crane so that the vertex coincides with the centre of the circle and one of the arms passes through OA. B Y O A X Can you now tell which angle is bigger? 24 Chapter 2_Lines and Angles.indd 24 08-08-2024 17:31:24  Lines and Angles Which crane was making the bigger angle? If you can make a circular piece of transparent paper, try this method to compare the angles in Fig. 2.10 with each other. Teacher’s Note A teacher needs to check the understanding of the students around the notion of an angle. Sometimes students might think that increasing the length of the arms of the angle increases the angle. For this, various situations should be posed to the students to check their understanding on the same. 2.7 Making Rotating Arms Let us make ‘rotating arms’ using two paper straws and a paper clip by following these steps: 1. Take two paper straws and a paper clip.  2. Insert the straws into the arms of the paper clip. 3. Your rotating arm is ready! Make several ‘rotating arms’ with different angles between the arms. Arrange the angles you have made from smallest to largest by comparing and using superimposition. Passing through a slit: Collect a number of rotating arms with different angles; do not rotate any of the rotating arms during this activity. 25 Chapter 2_Lines and Angles.indd 25 08-08-2024 17:31:25  Ganita Prakash | Grade 6 Take a cardboard and make an angle-shaped slit as shown below by tracing and cutting out the shape of one of the rotating arms. Now, shuffle and mix up all the rotating arms. Can you identify which of the rotating arms will pass through the slit? The correct one can be found by placing each of the rotating arms over the slit. Let us do this for some of the rotating arms: Slit angle is greater than Slit angle is less than the Slit angle is equal to the the arms’ angle. The arms arms’ angle. The arms arms’ angle. The arms will will not go through the will not go through the go through the slit. slit. slit. Only the pair of rotating arms where the angle is equal to that of the slit passes through the slit. Note that the possibility of passing through the slit depends only on the angle between the rotating arms and not on their lengths (as long as they are shorter than the length of the slit). 26 Chapter 2_Lines and Angles.indd 26 08-08-2024 17:31:26  Lines and Angles Challenge: Reduce this angle. Angle The angle is still reduced. the same! 2.8 Special Types of Angles Let us go back to Vidya’s notebook and observe her opening the cover of the book in different scenarios. She makes a full turn of the cover when she has to write while holding the book in her hand. She makes a half turn of the cover when she has to open it on her table. In this case, observe the arms of the angle formed. They lie in a straight line. Such an angle is called a straight angle. A O B Let us consider a straight angle ∠AOB. Observe that any ray OC divides it into two angles, ∠AOC and ∠COB. 27 Chapter 2_Lines and Angles.indd 27 08-08-2024 17:31:28  Ganita Prakash | Grade 6 Is it possible to draw OC such that the two angles are Math equal to each other in size? Talk Let’s Explore We can try to solve this problem using a piece of paper. Recall that when a fold is made, it creates a crease which is straight. Take a rectangular piece of paper and on one of its sides, mark the straight angle AOB. By folding, try to get a line (crease) passing through O that divides ∠AOB into two equal angles. How can it be done? Fold the paper such that OB overlaps with OA. Observe the crease and the two angles formed. 28 Chapter 2_Lines and Angles.indd 28 08-08-2024 17:31:28  Lines and Angles Justify why the two angles are equal. Is there a way to superimpose and check? Can this superimposition be done by folding? Each of these equal angles formed are called right angles. So, a straight angle contains two right angles. Because they're Why shouldn't you always right. argue with a 90 ̊ angle? If a straight angle is formed by half of a full turn, how much of a full turn will form a right angle?  Observe that a right angle resembles the shape of an ‘L’. An angle is a right angle only if it is exactly half of a straight angle. Two lines that meet at right angles are called perpendicular lines. Figure it Out 1. How many right angles do the windows of your classroom contain? Do you see other right angles in your classroom? 29 Chapter 2_Lines and Angles.indd 29 08-08-2024 17:31:29  Ganita Prakash | Grade 6 2. Join A to other grid points in the figure by a straight line to get a straight angle. What are all the different ways of doing it? A B A B 3. Now join A to other grid points in the figure by a straight line to get a right angle. What are all the different ways of doing it? A B A B Hint: Extend the line further as shown in the figure below. To get a right angle at A, we need to draw a line through it that divides the straight angle CAB into two equal parts. C A B 30 Chapter 2_Lines and Angles.indd 30 08-08-2024 17:31:29  Lines and Angles 4. Get a slanting crease on the paper. Now, try to get another crease that is perpendicular to the slanting crease. a. H  ow many right angles do you have now? Justify why the angles are exact right angles. b. D  escribe how you folded the paper so that any other person who doesn’t know the process can simply follow your description to get the right angle. Classifying Angles Angles are classified in three groups as shown below. Right angles are shown in the second group. What could be the common feature of the other two groups? In the first group, all angles are less than a right angle or in other words, less than a quarter turn. Such angles are called acute angles. In the third group, all angles are greater than a right angle but less than a straight angle. The turning is more than a quarter turn and less than a half turn. Such angles are called obtuse angles. Figure it Out 1. Identify acute, right, obtuse and straight angles in the previous figures. 2. Make a few acute angles and a few obtuse angles. Draw them in different orientations. 31 Chapter 2_Lines and Angles.indd 31 08-08-2024 17:31:29  Ganita Prakash | Grade 6 3. Do you know what the words acute and obtuse mean? Acute means sharp and obtuse means blunt. Why do you think these words have been chosen? 4. Find out the number of acute angles in each of the figures below. What will be the next figure and how many acute angles will it have? Do you notice any pattern in the numbers? 2.9 Measuring Angles We have seen how to compare two angles. But can we actually quantify how big an angle is using a number without having to compare it to another angle? We saw how various angles can be compared using a circle. Perhaps a circle could be used to assign measures for angles? Fig. 2.12 To assign precise measures to angles, mathematicians came up with an idea. They divided the angle in the centre of the circle into 32 Chapter 2_Lines and Angles.indd 32 08-08-2024 17:31:29  Lines and Angles 360 equal angles or parts. The angle measure of each of these unit parts is 1 degree, which is written as 1°. This unit part is used to assign measure to any angle: the measure of an angle is the number of 1° unit parts it contains inside it. For example, see this figure: 30 units It contains 30 units of 1° angle and so we say that its angle measure is 30°. Measures of different angles: What is the measure of a full turn in degrees? As we have taken it to contain 360 degrees, its measure is 360°. What is the measure of a straight angle in degrees? A straight angle is half of a full turn. As a full-turn is 360°, a half turn is 180°. What is the measure of a right angle in degrees? Two right angles together form a straight angle. As a straight angle measures 180°, a right angle measures 90°. 180 units A O B A O B A A 90 units O B O B A pinch of history A full turn has been divided into 360°. Why 360? The reason why we use 360° today is not fully known. The division of a circle into 360 33 Chapter 2_Lines and Angles.indd 33 08-08-2024 17:31:29  Ganita Prakash | Grade 6 parts goes back to ancient times. The Rigveda, one of the very oldest texts of humanity going back thousands of years, speaks of a wheel with 360 spokes (Verse 1.164.48). Many ancient calendars, also going back over 3000 years—such as calendars of India, Persia, Babylonia and Egypt—were based on having 360 days in a year. In addition, Babylonian mathematicians frequently used divisions of 60 and 360 due to their use of sexagesimal numbers and counting by 60s. Perhaps the most important and practical answer for why mathematicians over the years have liked and continued to use 360 degrees is that 360 is the smallest number that can be evenly divided by all numbers up to 10, aside from 7. Thus, one can break up the circle into 1, 2, 3, 4, 5, 6, 8, 9 or 10 equal parts, and still have a whole number of degrees in each part! Note that 360 is also evenly divisible by 12, the number of months in a year, and by 24, the number of hours in a day. These facts all make the number 360 very useful. The circle has been divided into 1, 2, 3, 4, 5, 6, 8, 9 10 and 12 parts below. What are the degree measures of the resulting angles? Write the degree measures down near the indicated angles. Degree measures of different angles How can we measure other angles in degrees? It is for this purpose that we have a tool called a protractor that is either a circle divided into 360 equal parts as shown in Fig. 2.12 (on page 32), or a half circle divided into 180 equal parts. 34 Chapter 2_Lines and Angles.indd 34 08-08-2024 17:31:29  Lines and Angles Unlabelled protractor Here is a protractor. Do you see the straight angle at the center divided into 180 units of 1 degree? Only part of the lines dividing the straight angle are visible, though! Starting from the marking on the rightmost point of the base, there is a long mark for every 10°. From every such long mark, there is a medium sized mark after 5°. Figure it out 1. Write the measures of the following angles: K a. ∠ KAL Notice that the vertex of this angle coincides with the centre of L the protractor. So the number of units of 1 degree angle between KA A and AL gives the measure of ∠KAL. By counting, we get ∠KAL = 30°. Making use of the medium sized and large sized marks, is it possible to count the number of units in 5s or 10s? W b. ∠WAL c. ∠TAK T 35 Chapter 2_Lines and Angles.indd 35 08-08-2024 17:31:30  Ganita Prakash | Grade 6 Labelled protractor This is a protractor that you find in your geometry box. It would appear similar to the protractor above except that there are numbers written on it. Will these make it easier to read the angles? 80 90 100 70 110 100 90 80 12 60 70 0 110 13 0 60 50 12 0 30 50 1 14 40 0 0 40 14 15 30 0 0 30 15 160 20 160 20 170 10 170 10 180 180 0 0 There are two sets of numbers on the protractor: one increasing from right to left and the other increasing from left to right. Why does it include two sets of numbers? Name the different angles in the figure and write their measures. R S 80 90 100 70 110 100 90 80 12 60 70 0 110 50 12 0 60 13 0 Q 0 50 13 14 40 0 0 40 14 T 15 30 0 0 30 15 160 20 160 20 170 10 170 10 P U 180 180 0 0 O 36 Chapter 2_Lines and Angles.indd 36 08-08-2024 17:31:30  Lines and Angles Did you include angles such as ∠TOQ? Which set of markings did you use - inner or outer? What is the measure of ∠TOS? Can you use the numbers marked to find the angle without counting the number of markings? Here, OT and OS pass through the numbers 20 and 55 on the outer scale. How many units of 1 degree are contained between these two arms? Can subtraction be used here? How can we measure angles directly without having to subtract? Place the protractor so the center is on the vertex of the angle. Align the protractor so that one the arms passes through the 0º mark as in the picture below. A 80 90 100 70 110 100 90 80 12 60 70 0 110 13 0 60 50 12 0 30 50 1 14 40 0 0 40 14 15 30 0 0 30 15 160 20 160 20 170 10 170 10 180 180 0 0 O B What is the degree measure of ∠AOB? Make your own Protractor! You may have wondered how the different equally spaced markings are made on a protractor. We will now see how we can make some of them! 1. Draw a circle of a convenient radius on a sheet of paper. Cut out the circle (Fig. 2.13). A circle or one full turn is 360°. 2. Fold the circle to get two equal halves and cut it through the crease to get a semicircle. Write ‘0°’ in the bottom right corner of the semi-circle. 37 Chapter 2_Lines and Angles.indd 37 08-08-2024 17:31:30  Ganita Prakash | Grade 6 Fig. 2.13 The measure of half a circle is 12 of a full turn. (Fig. 2.14) So, the measure of 180 units half a turn = 12 of ____ = 180°. Thus, write 180° A O B Fig. 2.14 in the left bottom corner of the semicircle. 3. Fold the semi-circular sheet in half as shown in Fig. 2.15 to form a quarter circle. The measure of a quarter circle is 14 of a full turn. The measure of a 1 1 A 4 turn = 4 of 360° = ________. Or, the measure of 90 units a 14 turn = 12 of a half O turn = 12 of 180° = B ______. Fig. 2.15 Thus, mark 90° at the top of the semicircle. 38 Chapter 2_Lines and Angles.indd 38 08-08-2024 17:31:31  Lines and Angles 4. Fold the sheet again as shown in Figs. 2.16 and 2.17: 90O 135O 45O 180O 0O Fig. 2.16 Fig. 2.17 When folded, this is 18 of the circle, or 18 of a turn, or 18 of 360°, or 14 of 180° or 12 of 90° = ________________________. The new creases formed give us measures of 45° and 180°−  45° = 135° as shown. Write 45° and 135° at the correct places on the new creases along the edge of the semicircle. 5. Continuing with another half fold as shown in Fig. 2.18, we get an angle of measure ________________________. Fig. 2.18 6. Unfold and mark the creases as OB, OC,..., etc., as shown in Fig. 2.19 and Fig. 2.20. E F D 90O 112 5O G C 135O.5 67. O 45O H B 157.5 O O 5 22. 180O 0O I A O Fig. 2.20 Fig. 2.19 39 Chapter 2_Lines and Angles.indd 39 08-08-2024 17:31:31  Ganita Prakash | Grade 6 Think ! I n Fig. 2.20, we have ∠AOB = ∠BOC = ∠COD = ∠DOE = ∠EOF = ∠FOG = ∠GOH = ∠HOI=_____. Why? Angle Bisector At each step, we folded in halves. This process of getting half of a given angle is called bisecting the angle. The line that bisects a given angle is called the angle bisector of the angle. Identify the angle bisectors in your handmade protractor. Try to make different angles using the concept of angle bisector through paper folding. Figure it Out 1. Find the degree measures of the following angles using your protractor. H I I J H J J I H G K 2. Find the degree measures of different angles in your classroom using your protractor. Teacher’s Note It is important that students make their own protractor and use it to measure different angles before using the standard protractor so that they know the concept behind the marking of the standard protractor. 40 Chapter 2_Lines and Angles.indd 40 08-08-2024 17:31:31  Lines and Angles 3. Find the degree measures for the angles given below. Check if your paper protractor can be used here! H I H c J J I 4. How can you find the degree measure of the angle given below using a protractor? 5. Measure and write the degree measures for each of the following angles: a. b. 41 Chapter 2_Lines and Angles.indd 41 08-08-2024 17:31:31  Ganita Prakash | Grade 6 c. d. e. f. 6. Find the degree measures of ∠BXE, ∠CXE, ∠AXB and ∠BXC. C B 90 70 80 A 100 110 100 90 80 12 60 70 0 110 13 20 60 50 1 0 0 50 13 14 40 0 0 40 14