Class 6 Maths Chapter 1 Notes Patterns in Mathematics
Class 6 Maths Notes Chapter 1 – Class 6 Patterns in Mathematics Notes
→ 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 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.
What is Mathematics? Class 6 Notes
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 endeavor. 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.
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 satellites and send rockets to the Moon and Mars; similarly, understanding patterns in genomes has helped in diagnosing and curing diseases—among thousands of other such examples.
Patterns in Numbers Class 6 Notes
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.
The table shows some key number sequences that are studied in Mathematics.
Visualizing Number Sequences Class 6 Notes
Many number sequences can be visualized using pictures. Visualizing 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 the Table using the following pictures.
Relations among Number Sequences Class 6 Notes
Sometimes, number sequences can be related to each other in 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
Visualizing 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 numbers of dots: 1, 3, 5, 7,…?
Think about it for a moment before reading further!
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!
Patterns in Shapes Class 6 Notes
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 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. The table shows a few key shape sequences that are studied in Mathematics.
Relation to Number Sequences Class 6 Notes
Often, shape sequences are related to number sequences in surprising ways. Such relationships can help study and understand 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 triangles, quadrilaterals (i.e., square), pentagons, hexagons, heptagons, octagons, nonagons, decagons, 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 the Table also have beautiful relationships with number sequences.
