Welcome dear students! Today we are going to learn about Patterns in Mathematics from Class 6 Maths.
Let us begin with section 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.
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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. Now let us think about the questions in your Figure it Out section. First, can you think of other examples where mathematics helps us in our everyday lives? Think about how we use numbers when we buy vegetables at the market, measure ingredients while baking a cake, or check the time before leaving for school. Mathematics helps us count, measure, compare, and plan our daily routines. Second, how has mathematics helped propel humanity forward? Mathematics has been essential in carrying out scientific experiments, running our economy and democracy, building bridges, houses or other complex structures, and making televisions, mobile phones, computers, bicycles, trains, cars, planes, calendars, and clocks. Without mathematical calculations, none of these modern achievements would be possible.
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Let us move to section 1.2, Patterns in Numbers. Among the most basic patterns that occur in mathematics are 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. Let us look at the examples of number sequences from your textbook. The first sequence is 1, 1, 1, 1, 1, 1, 1, ... representing All 1's. The second is 1, 2, 3, 4, 5, 6, 7, ... representing Counting numbers. The third is 1, 3, 5, 7, 9, 11, 13, ... representing Odd numbers. The fourth is 2, 4, 6, 8, 10, 12, 14, ... representing Even numbers. The fifth is 1, 3, 6, 10, 15, 21, 28, ... representing Triangular numbers. The sixth is 1, 4, 9, 16, 25, 36, 49, ... representing Squares. The seventh is 1, 8, 27, 64, 125, 216, ... representing Cubes. The eighth is 1, 2, 3, 5, 8, 13, 21, ... representing Virahanka numbers. The ninth is 1, 2, 4, 8, 16, 32, 64, ... representing Powers of 2. The tenth is 1, 3, 9, 27, 81, 243, 729, ... representing Powers of 3.
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Now let us answer your Figure it Out questions for this section. First, can you recognise the pattern in each of the sequences? Yes. The All 1's sequence repeats the same number. The counting numbers increase by 1 each time. The odd numbers increase by 2 each time, starting from 1. The even numbers increase by 2 each time, starting from 2. The triangular numbers are formed by adding consecutive counting numbers to the previous term. The squares are formed by multiplying a number by itself. The cubes are formed by multiplying a number by itself three times. The Virahanka numbers are formed by adding the two previous numbers together. The powers of 2 are formed by multiplying the previous number by 2. The powers of 3 are formed by multiplying the previous number by 3. Second, let us rewrite each sequence with the next three numbers and state the rule. For All 1's, the next three are 1, 1, 1. Rule: repeat 1. For counting numbers, the next three are 8, 9, 10. Rule: add 1. For odd numbers, the next three are 15, 17, 19. Rule: add 2. For even numbers, the next three are 16, 18, 20. Rule: add 2. For triangular numbers, the next three are 36, 45, 55. Rule: add the next counting number. For squares, the next three are 64, 81, 100. Rule: multiply the position number by itself. For cubes, the next three are 343, 512, 729. Rule: multiply the position number by itself three times. For Virahanka numbers, the next three are 34, 55, 89. Rule: add the last two terms. For powers of 2, the next three are 128, 256, 512. Rule: multiply by 2. For powers of 3, the next three are 2187, 6561, 19683. Rule: multiply by 3.
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Now we come to section 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 describe the pictorial representations of the first seven sequences. For the All 1's sequence, each term is shown as a single dot. For the counting numbers, the first term is 1 dot, the second is 2 dots in a row, the third is 3 dots in a row, the fourth is 4 dots, and the fifth is 5 dots. For the odd numbers, the first term is 1 dot, the second is 3 dots arranged in an L shape, the third is 5 dots in an L shape, the fourth is 7 dots, and the fifth is 9 dots. For the even numbers, the first term is 2 dots side by side, the second is 4 dots in a 2 by 2 arrangement, the third is 6 dots in two rows of 3, the fourth is 8 dots in two rows of 4, and the fifth is 10 dots in two rows of 5. For the triangular numbers, the first term is 1 dot, the second is 3 dots forming a triangle, the third is 6 dots forming a larger triangle, the fourth is 10 dots, and the fifth is 15 dots. For the squares, the first term is 1 dot, the second is a 2 by 2 grid of 4 dots, the third is a 3 by 3 grid of 9 dots, the fourth is a 4 by 4 grid of 16 dots, and the fifth is a 5 by 5 grid of 25 dots. For the cubes, the first five terms are represented by three dimensional cube drawings containing 1, 8, 27, 64, and 125 unit cubes respectively.
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Let us work through your Figure it Out questions for this section. First, copy the pictures and draw the next one. For All 1's, draw 1 dot. For counting numbers, draw 6 dots in a row. For odd numbers, draw 11 dots in an L shape. For even numbers, draw 12 dots in two rows of 6. For triangular numbers, draw 21 dots forming a triangle. For squares, draw a 6 by 6 grid of 36 dots. For cubes, draw a 6 by 6 by 6 cube containing 216 unit cubes. Second, why are they called triangular, square, and cube numbers? They are called triangular numbers because the dots can be arranged perfectly into equilateral triangles. They are called square numbers because the dots can be arranged perfectly into square grids. They are called cubes because the units can be stacked perfectly into three dimensional cubes. Third, 36 is both a triangular and a square number. You can draw 36 dots arranged in a triangle with 8 dots on each side, and you can also draw 36 dots arranged in a 6 by 6 square grid. This shows that the same number can be represented differently depending on the context. Fourth, the sequence 1, 7, 19, 37 is called hexagonal numbers. The next number is 61. You can draw these by arranging dots into hexagon shapes. Fifth, to visualise powers of 2, you can draw a single dot for 1, two dots connected by a line for 2, four dots forming a square for 4, eight dots forming a cube for 8, and sixteen dots representing a four dimensional projection for 16.
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Now let us look at the Math Talk section about powers of 2. Here is one possible way of thinking about powers of 2. Imagine a single dot labeled 1. Then imagine two dots connected by a line labeled 2. Next, imagine four dots forming a square labeled 4. Then imagine eight dots forming a cube labeled 8. Finally, imagine sixteen dots forming a four dimensional hypercube projection labeled 16. This shows how each power of 2 can be seen as doubling the previous geometric dimension.
Let us move to section 1.4, Relations among Number Sequences. Sometimes, number sequences can be related to each other in surprising ways. Let us look at the first worked 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. 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. We can partition the dots in a square grid into odd numbers of dots by drawing L shaped borders around each previous square. The first dot is 1. Adding 3 dots around it makes a 2 by 2 square. Adding 5 dots around that makes a 3 by 3 square. This picture makes it evident that adding odd numbers gives square numbers. Because such a picture can be made for a square of any size, this explains why the pattern continues forever.
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Now let us answer the questions. By drawing a similar picture, what is the sum of the first ten odd numbers? The sum of the first ten odd numbers equals 10², which is 100. By imagining a similar picture, what is the sum of the first one hundred odd numbers? The sum equals 100², which is 10000. Let us look at another example. Adding up and down. 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 gives yet another way of getting the square numbers by adding the counting numbers up and then down.
Let us solve the Figure it Out questions for this section. First, can you find a pictorial explanation for why adding counting numbers up and down gives square numbers? Yes. Imagine a square grid. If you count the dots diagonally from one corner to the opposite corner, you get 1, then 2, then 3, and so on, up to the middle, and then back down. This diagonal counting matches the up and down addition pattern. Second, what is the value of 1 + 2 + 3 + ... + 99 + 100 + 99 + ... + 3 + 2 + 1? Following the pattern, this equals 100², which is 10000. Third, which sequence do you get when you add the All 1's sequence up? You get the counting numbers. Which sequence do you get when you add the All 1's sequence up and down? You get the sequence of even numbers starting from 2, or you can see it as doubling the counting numbers minus 1 depending on how you arrange it, but visually it forms a symmetric pattern that relates to squares. Fourth, which sequence do you get when you add the counting numbers up? You get the triangular numbers. A pictorial explanation is stacking rows of dots: one dot, then two below it, then three, forming a triangle. Fifth, what happens when you add pairs of consecutive triangular numbers? 1 + 3 = 4. 3 + 6 = 9. 6 + 10 = 16. 10 + 15 = 25. You get the square numbers. You can explain this with a picture by taking two triangles of different sizes and fitting them together to form a square. Sixth, what happens when you add up powers of 2 starting with 1? 1 = 1. 1 + 2 = 3. 1 + 2 + 4 = 7. 1 + 2 + 4 + 8 = 15. Now add 1 to each of these numbers. You get 2, 4, 8, 16. These are the next powers of 2. This happens because each sum of powers of 2 is exactly one less than the next power of 2. Seventh, what happens when you multiply the triangular numbers by 6 and add 1? 1 × 6 + 1 = 7. 3 × 6 + 1 = 19. 6 × 6 + 1 = 37. 10 × 6 + 1 = 61. You get the hexagonal numbers. You can explain this by arranging six triangular dots around a central dot to form a hexagon. Eighth, what happens when you add up hexagonal numbers? 1 = 1. 1 + 7 = 8. 1 + 7 + 19 = 27. 1 + 7 + 19 + 37 = 64. You get the cube numbers. You can explain this using a picture of a cube by seeing how each hexagonal layer builds the next cubic shell. Ninth, find your own patterns. You might notice that the difference between consecutive square numbers is always an odd number, or that the sum of the digits of powers of 3 often adds up to 9. Explaining these with pictures or algebra deepens your understanding.
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Now we come to section 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, 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. Let us look at the examples. The first sequence is Regular Polygons, which goes from triangle to quadrilateral to pentagon to hexagon to heptagon to octagon to nonagon to decagon. The second sequence is Complete Graphs, labeled K₂, K₃, K₄, K₅, K₆. The third is Stacked Squares. The fourth is Stacked Triangles. The fifth is Koch Snowflake.
Let us answer the Figure it Out questions. First, can you recognise the pattern? Regular polygons add one side each time. Complete graphs add one vertex and connect it to all existing vertices. Stacked squares add a new row and column of squares. Stacked triangles add a new row of triangles. Koch snowflake replaces each straight edge with a smaller triangular bump. Second, redraw each and draw the next shape. For regular polygons, the next is an eleven sided polygon called a hendecagon. For complete graphs, the next is K₇, connecting seven points. For stacked squares, the next adds a sixth row and column. For stacked triangles, the next adds a sixth row. For Koch snowflake, the next iteration adds smaller bumps to every existing edge. The rule for each is to increase the number of sides or vertices by one, or to recursively add geometric subdivisions.
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Now we reach section 1.6, Relation to Number Sequences. Often, shape sequences are related to number sequences in surprising ways. Let us look at the example. The number of sides in the shape sequence of Regular Polygons is given by the counting numbers starting at 3, which is 3, 4, 5, 6, 7, 8, 9, 10. That is why these shapes are called, respectively, regular triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon. The word regular refers to the fact that these shapes have equal length sides and also equal angles. We will discuss angles in more depth in the next chapter.
Let us solve the Figure it Out questions for this section. First, count the number of sides in each shape in the sequence of Regular Polygons. You get 3, 4, 5, 6, 7, 8, 9, 10. What about the number of corners? You get the exact same sequence. This happens because every polygon has exactly one corner for every side. Second, count the number of lines in each shape in the sequence of Complete Graphs. K₂ has 1 line. K₃ has 3 lines. K₄ has 6 lines. K₅ has 10 lines. K₆ has 15 lines. You get the triangular numbers. This happens because each new point connects to all previous points, so the number of lines equals the sum of counting numbers up to that point. Third, how many little squares are there in each shape of the sequence of Stacked Squares? 1, 4, 9, 16, 25. You get the square numbers. This happens because each shape forms an n by n grid. Fourth, how many little triangles are there in each shape of the sequence of Stacked Triangles? 1, 4, 9, 16, 25. You get the square numbers. This happens because the number of upward pointing triangles plus downward pointing triangles in each row forms a perfect square grid pattern. Fifth, to get from one shape to the next in the Koch Snowflake sequence, each line segment is replaced by a speed bump. The number of total line segments is 3, 12, 48, 192. The corresponding number sequence is 3 times the powers of 4. This happens because each segment splits into 4 new segments at every step.
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Let us review the summary of this chapter. 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, Virahanka 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. Visualising 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.
Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]