Class 6 Mathematics · Chapter 1 NotesPatterns in Mathematics

Mathematics is largely the search for patterns and for the explanations as to why those patterns exist. Patterns occur all around us — in nature, in our homes and schools, and in the motion of the sun, moon, and stars. They appear in shopping, cooking, throwing a ball, playing games, understanding weather, and using technology. The search for patterns and their explanations is a fun and creative endeavour, which is why mathematicians think of mathematics both as an art and as a science. Importantly, mathematics aims not just to find patterns but also to explain why they exist. Such explanations can be used in applications far beyond the context in which they were discovered, helping to propel humanity forward. For example, understanding patterns in the motion of stars and planets led to the theory of gravitation, allowing us to launch satellites and send rockets to the Moon and Mars. Similarly, understanding patterns in genomes has helped diagnose and cure diseases.

• Mathematics is the search for patterns and explanations.
• Patterns exist in nature, daily life, and the motion of celestial bodies.
• Mathematics is both an art and a science.
• Explanations of patterns can lead to powerful applications.

Among the most basic patterns 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. Some key number sequences include: all 1's (1, 1, 1, 1, ...), counting numbers (1, 2, 3, 4, ...), odd numbers (1, 3, 5, 7, ...), even numbers (2, 4, 6, 8, ...), triangular numbers (1, 3, 6, 10, 15, ...), squares (1, 4, 9, 16, 25, ...), cubes (1, 8, 27, 64, 125, ...), Virahānka numbers (1, 2, 3, 5, 8, 13, ...), powers of 2 (1, 2, 4, 8, 16, ...), and powers of 3 (1, 3, 9, 27, 81, ...). Each sequence follows a specific rule that determines how the numbers are formed.

• Number theory is the study of patterns in whole numbers.
• Number sequences are basic and fascinating patterns.
• Examplescounting numbers, odd, even, triangular, square, cube, Virahānka, powers of 2 and 3.

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. For example, triangular numbers can be shown as dots arranged in triangles: 1, 3, 6, 10, 15, ... Square numbers can be shown as dots arranged in squares: 1, 4, 9, 16, 25, ... Cube numbers can be shown as dots arranged in cubes: 1, 8, 27, 64, 125, ... The sequence of all 1's can be shown as single dots. Counting numbers, odd numbers, and even numbers can also be represented pictorially. Visualising helps you see why sequences are named as they are and how they grow. For instance, 36 is both a triangular number and a square number — it can be arranged perfectly as a triangle and as a square. This shows that the same number can be represented differently and play different roles depending on the context.

• Pictures and diagrams help understand number sequences.
• Triangular numbers form triangles; square numbers form squares; cube numbers form cubes.
• Some numbers, like 36, belong to more than one sequence.

Sometimes number sequences can be related to each other in surprising ways. For example, when you add up odd numbers starting from 1, you get square numbers: 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, 1 + 3 + 5 + 7 = 16, and so on. This pattern happens forever. A picture can explain why: square numbers are made by counting dots in a square grid, and you can partition the dots into odd numbers of dots: 1, 3, 5, 7, ... This visual explanation shows why adding odd numbers gives square numbers. Another relation: adding counting numbers up and then down also gives square numbers: 1 = 1, 1 + 2 + 1 = 4, 1 + 2 + 3 + 2 + 1 = 9, 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16, and so on. Other relations include: adding the all 1's sequence gives counting numbers; adding pairs of consecutive triangular numbers gives square numbers (1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, ...); adding powers of 2 starting with 1 gives 1, 3, 7, 15, 31, ... and adding 1 to each gives powers of 2 again; multiplying triangular numbers by 6 and adding 1 gives 7, 19, 37, 61, 91, ...; and adding hexagonal numbers gives cube numbers.

• Sum of first n odd numbers = n² (square numbers).
• Adding counting numbers up and down gives square numbers.
• Adding consecutive triangular numbers gives square numbers.
• Adding powers of 2 and then adding 1 gives powers of 2.
• Multiplying triangular numbers by 6 and adding 1 gives hexagonal numbers.

Other important and basic patterns in mathematics are patterns of shapes. These shapes may be in one, two, or three dimensions (1D, 2D, or 3D) — or even more dimensions. The branch of mathematics that studies patterns in shapes is called geometry. Shape sequences are an important type of shape pattern. Some key shape sequences include: regular polygons (triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, decagon), complete graphs (K2, K3, K4, K5, K6), stacked squares, stacked triangles, and the Koch snowflake. Each sequence follows a rule for forming the next shape. For example, in the Koch snowflake, each line segment is replaced by a 'speed bump' shape, and as this is done repeatedly, the changes become tinier and tinier with very small line segments.

• Geometry is the study of patterns in shapes.
• Shape sequences include regular polygons, complete graphs, stacked squares, stacked triangles, and Koch snowflake.
• Shapes can be in 1D, 2D, 3D, or more dimensions.

Often, shape sequences are related to number sequences in surprising ways. Such relationships help in studying and understanding both the shape sequence and the related number sequence. For example, the number of sides in the sequence of regular polygons is given by the counting numbers starting at 3: 3, 4, 5, 6, 7, 8, 9, 10, ... That is why these shapes are called regular triangle, quadrilateral (square), pentagon, hexagon, heptagon, octagon, nonagon, decagon, respectively. The word 'regular' means the shapes have equal-length sides and equal angles. The number of corners in regular polygons is the same as the number of sides, because in any closed figure, the number of sides equals the number of corners (vertices). Other shape sequences also have beautiful relationships with number sequences: complete graphs give triangular numbers (1, 3, 6, 10, 15, ...), stacked squares give square numbers (1, 4, 9, 16, 25, ...), stacked triangles also give square numbers, and the Koch snowflake gives the sequence 3, 12, 48, 192, 768, ... which is 3 times powers of 4.

• Regular polygonsnumber of sides = counting numbers starting at 3.
• Number of sides = number of corners in any closed figure.
• Complete graphs give triangular numbers.
• Stacked squares and stacked triangles give square numbers.
• Koch snowflake gives 3, 12, 48, ... (3 times powers of 4).