My Curiosity | 4 | Mathematics |1| For the Love of Math’s

For the Love of Math’s

Why didn’t I study Math’s properly? This question always comes to my mind. Mathematics is a really a powerful subject and more importantly it makes sense when you have a equation that time and again do the job for you.

Why mathematics is an important subject to study?

1. It teaches you how to count?
2. It teaches you how to do the comparison?
3. It teaches you, how to quantify something, be it length, size, numbers, distance, power, energy, money, you name it?
4. It gives you equations to do something, like if you want to draw a line, there is an equation, if you want to build a football, there is a equation.
5. It empowers you to identify the probability of something happening, like it helps in doing Forecasting.

My current knowledge in the subject is not that much, that I keep going giving you examples and more examples, but what I want is putting something that help me understand this subject better.

I know, I started such endeavors in past, and they went bust, but there is nothing wrong in initiating one more, I mean who is there to stop me.

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Date: 8th April 2023

Class VIth NCERT Book:

Chapter 1: Knowing your numbers

Why it is important to know numbers? I mean how you will count without it? How you can say, you need just one more candy? How you will say 10% salary raise is not good for me, I want 30-40% raise? How will you say that a 12 inches pizza is better for your group?

You see, it’s always good to have the ability to count, to express yourself in numbers, because we probably don’t know what number ONE is, but for sure we know that it SMALLER than number TWO. So, it is always good to know numbers and there is always exist a number which is greater than the previous number you were aware of. My little one knows the counting till hundred, but he will be mesmerized one day by learning that there is no END to this counting business. There is always one more, one more and one more the number exist which he learned yesterday. Poor guy! 🙂

Apart from counting, numbers also help us expressing the idea around which number is bigger and which is smaller, and based upon knowing this comparison thing, the entire world business runs. If I know that I am purchasing only four candies, but my classmates are more than four, I can ask my mom to give me few extra money so that I can buy more.

So how do you decide, which number is big and which is small?

2345 and 23456 -> Which is a big number? 23456, yes, but why? Because it carries greater positional numbers that 2345, where the highest positional number is 2 which is at the Thousand’s place, but in 23456, 2 is at Ten thousand place.

Ok, what about these two numbers?

2345 and 3456 -> Since they both have same number of digits, 4 each, now the way to compare is looking at the left most digit, since left most digit is sitting at the highest positional value and compare them. So, 2 in 2345, and 3 in 3456, both are at thousand’s position, instead of looking at the positional value, we will look at the digit itself.

Since 3 > 2, we can say that 3456 is greater than 2345. Easy right, it looks easy when we are dealing with the numbers this small, but what if, we need to compare a number like this.:

36472847582967946729461816318360130944464979 and 7527472347242785135123008542698264239640923 ?

You see the problem right and what if the number of these big numbers are also big? I mean what in case you need to do more than ten thousand such comparison dealing with such big numbers and provide the result in next 15 minutes? You and I, we both going to need a computer to do that, and probably some program that can write the program comparing these two numbers.

It is also a good practice to use “comma” (,) notation to express big numbers.

10,00,00,000 It is the way Indians display big numbers,

50,801,592 It is the way Internation system works, where commas are used to mark, thousands, millions, and a billionth place. (So, this given number is Fifty million, eight hundred and one thousand, five hundred and ninety-two.)

20th Nov 2023

Class VI | Ch2:

Whole Numbers

When we start counting anything, our counting never starts from 0 (Zero)! Why? Isn’t 0 a number? No, zero is a very important number, the issue is it simply doesn’t exist in beginning. You can 1 Apple, 2 Pencil, 3 Erasers, but nothing which say 0 Pencil or 0 Apple. Though you can have 0 Apple, once you give all your apples to someone, and left with no or 0 apples. Hence 0 is not part of natural number, but including 0 in your counting, you get a set of Whole Number.

Similarly, there is no largest number, there is no smallest numbest number. You add 1 to any number, you will get next largest, you subtract 1 from any number, you get next smallest number. How cool is this. It’s all so fair in Mathematics till now.

Whole Numbers = 0 + Natural Numbers

Number Line:

Again, a very important concept, addition to 1 to the right increase the count, subtracting by 1, will make your move to left and decrease the count. There is a unit distance between every successive natural number. Let say 3×4, you need to make 4 jumps of 3 unit each to reach to multiplication result.

21st Nov 2023

Class VI | Ch3:

Playing with Numbers

1. Factor of a number is an exact divisor of that number.
2. A factor of a number is equal to or less than the number.
3. A multiple of a number is equal to greater than the number.
4. Prime Numbers: Number with exactly two factors(1, and the number itself)
5. Composite number: Factors more than two.
6. Even numbers: Multiple of 2
7. Odd numbers: Numbers which are not multiple of two.
8. Smallest prime number is 2 and it is also the only prime number which is even.

• Knowing these abstract concepts in Math are important, why because, these abstract concepts help us build bigger things. It is exactly similar to English letters A,B,C…, when they start coming together, they become beautiful words, inspirational stories, lovely poems.
• TESTS for Divisibility:
  • Divisibility by 10: If there is 0 (Zero) at units place, that number is divisible by 10.
  • Divisibility by 5: Number that carry 0/5 at the ones place.
  • Divisibility by 2, having an even number at one’s place.
  • Divisibility by 3, if the sum of the digits is a multiple of 3.
  • Divisibility by 6, if a number is divisible by 2&3 both, then it is divisible by 6.
• HCF (Highest Common Factor) AKA Greatest Common Divisor: Find factors, then highest common factor of numbers is their HCF:
• HCF of 20, 28, 36:
• 20 = 2x2x5 | 28 = 2x2x7 | 36 = 2x2x3x3 => Since 2 is occurring twice in all these 3 numbers hence 2×2 = 4 is the HCF (Highest common factor) that can divide all these numbers perfectly without leaving any remainder.
• Co-prime numbers: Suppose x and y are two positive integers such that they are called co prime numbers if and only if they have 1 as their only common factor and thus HCF (x, y) = 1. In other words, Co-prime numbers are a set of numbers or integers which have only 1 as their common factor i.e. their highest common factor (HCF) will be 1.
• Lowest Common Multiple: The Lowest Common Multiple (LCM) of two or more given numbers is the lowest (or smallest or least) of their common multiples. e.g: For 12 and 18, LCM is 36
• In these prime factorisations, the maximum number of times the prime factor 2 occurs is two; this happens for 12. Similarly, the maximum number of times the factor 3 occurs is two; this happens for 18. The LCM of the two numbers is the product of the prime factors counted the maximum number of times they occur in any of the numbers. Thus, in this case LCM = 2 × 2 × 3 ×3 = 36.

Understanding practical application of GCD/HCF and LCM is very important, it helps in understanding quite a few problems in better way and resolve them quickly. To test yourself whether you have understood it or not, you can try out following problems:

1. Renu purchases two bags of fertilizer of weights 75 kg and 69 kg. Find the maximum value of weight which can measure the weight of the fertilizer exact number of times. (HCF)
2. Three boys step off together from the same spot. Their steps measure 63 cm, 70 cm and 77 cm respectively. What is the minimum distance each should cover so that all can cover the distance in complete steps? (LCM)
3. The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Find the longest tape which can measure the three dimensions of the room exactly. (HCF)
4. Determine the smallest 3-digit number which is exactly divisible by 6, 8 and 12. (LCM)
5. Determine the greatest 3-digit number exactly divisible by 8, 10 and 12. (LCM)
6. The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change simultaneously again? (LCM)
7. Three tankers contain 403 litres, 434 litres and 465 litres of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers exact number of times. (HCF)
   • Their answers: 1. 3 kg| 2. 6930 cm | 3. 75 cm |4. 120 |5.960| 6. 7 minutes 12 seconds past 7 a.m. | 8.31 litres

Ch4: Basic Geometrical Ideas

Introduction:

Geometry has a long and rich history. The term ‘Geometry’ is the English equivalent of the Greek word ‘Geometron’. ‘Geo’ means Earth and ‘metron’ means Measurement. According to historians, the geometrical ideas shaped up in ancient times, probably due to the need in art, architecture and measurement. These include occasions when the boundaries of cultivated lands had to be marked without giving room for complaints. Construction of magnificent palaces, temples, lakes, dams and cities, art and architecture propped up these ideas. Even today geometrical ideas are reflected in all forms of art, measurements, architecture, engineering, cloth designing etc. You observe and use different objects like boxes, tables, books, the tiffin box you carry to your school for lunch, the ball with which you play and so on. All such objects have different shapes. The ruler which you use, the pencil with which you write are straight. The pictures of a bangle, the one-rupee coin or a ball appear round.

1. A point determines a location. It is usually denoted by a capital letter.
2. A line segment corresponds to the shortest distance between two points. The line segment joining points A and B is denoted by AB.
3. A line is obtained when a line segment like AB is extended on both sides indefinitely; it is denoted by AB or sometimes by a single small letter like l.
4. Two distinct lines meeting at a point are called intersecting lines.
5. Two lines in a plane are said to be parallel if they do not meet.
6. A ray is a portion of line starting at a point and going in one direction endlessly.
7. Any drawing (straight or non-straight) done without lifting the pencil may be called a curve. In this sense, a line is also a curve.
8. A simple curve is one that does not cross itself.
9. A curve is said to be closed if its ends are joined; otherwise it is said to be open.
10. A polygon is a simple closed curve made up of line segments. Here, (i) The line segments are the sides of the polygon.
    (ii) Any two sides with a common end point are adjacent sides.
    (iii) The meeting point of a pair of sides is called a vertex.
    (iv) The end points of the same side are adjacent vertices.
    (v) The join of any two non-adjacent vertices is a diagonal.
11. An angle is made up of two rays starting from a common starting/initial point. Two rays OA and OB make ∠AOB (or also called ∠BOA ).
    An angle leads to three divisions of a region: On the angle, the interior of the angle and the exterior of the angle.

Sometimes when we are studying and doing Math, we often forget its overall purpose and application. This gives rise to boredom, interest loss, and monotonous way of studying. You need to keep yourself reminding about power of Math. Mathematics is mother of all science, every great art form, and great product does carry crucial mathematical equation hidden behind. Once you start seeing that, you will never be afraid or bored of Mathematics.

Ch5: Understanding Elementary Shapes

In last chapter we identified few shapes, now start developing understanding towards their measurement.

Measuring Line Segments

Measuring length: The distance between the end points of a line segment is its length.

1. Comparison by observation
2. Comparison by tracing
3. Comparison using Ruler and Divider (Using Divider and Ruler is best way to measure line segments, positioning of eye matters alot in previous two methods)

Angles – ‘Right’ and ‘Straight’

We know that we have four directions, North, East, South, West moving clockwise.

East is at RIGHT ANGLE to North, South is at Right Angle to East, and South is at STRAIGHT ANGLE to North (Angle made on straight line)

Angles — Acute, Obtuse and Reflex

Acute angle: Angle less than Right angle

Obtuse angle: Angle greater than Right angle but less than Straight Angle

Reflex angle: Angle greater than Straight Angle, but less than complete angle.

^^ Image Credit: NCERT MATHS BOOK

Measuring Angles:

Using a tool like D, that is called Protractor. Which is like semi-circle.

One complete revolution divided by 360 equal parts where each part is called Degree. Why 360 degrees, there is no proper answer but a historical convention being followed.

^^ Image credit NCERT

Perpendicular Lines

We read about Parallel lines in our last chapter which never intersect. Perpendicular lines are those that intersect ones, but at Right angle to each other. English letter T represents perpendicular lines intersecting at right angle.

Classification of Triangles

i) Naming based on side measurement: (Use Ruler to measure)

a) All three unequal sides: Scalene Triangle | b) Two equal sides: Isosceles Triangle | c) Three equal sides: Equilateral Triangle

ii) Naming based on Angles:(Use Protractor to measure)

a) If all three angle less than 90 degrees: Acute angled triangle | b) If any one angle 90 degrees: Right angled triangle | c) If any angle greater than 90 degrees: Obtuse angled triangle

Quadrilaterals:

A quadrilateral is a polygon which has four sides.

a) Rectangle: Two sides equal, angles at 90 degrees.

b) Square: All four sides are equal, angles at 90 degrees

c) Parallelogram: Opposite sides are parallel

d) Rhombus: Diagonals are equal

e) Trapezium: One set of opposite sides are equal.

Polygons

3 side polygons: Triangle | 4 side polygon quadrilaterals | 5 side: Pentagon | 6 side: Hexagon | 8 side: Octagon |

CH6: Integers

1,2,3,4…… -> Natural numbers

0,1,2,3,4…. -> Whole numbers

………..-4,-3.-2,-1,0,1,2,3,4….. -> Integers.

1. We have seen that there are times when we need to use numbers with a negative sign. This is when we want to go below zero on the number line. These are called negative numbers. Some examples of their use can be in temperature scale, water level in lake or river, level of oil in tank etc. They are also used to denote debit account or outstanding dues.

2. The collection of numbers…, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, … is called integers.
   So, – 1, – 2, – 3, – 4, … called negative numbers are negative integers and 1, 2, 3, 4,
   … called positive numbers are the positive integers.
3. We have also seen how one more than given number gives a successor and one less
   than given number gives predecessor.
4. We observe that
   (a) When we have the same sign, add and put the same sign.
   (i) When two positive integers are added, we get a positive integer
   [e.g. (+ 3) + ( + 2) = + 5].
   (ii) When two negative integers are added, we get a negative integer
   [e.g. (–2) + ( – 1) = – 3].
   (b) When one positive and one negative integers are added we subtract them as
   whole numbers by considering the numbers without their sign and then put the
   sign of the bigger number with the subtraction obtained. The bigger integer is
   decided by ignoring the signs of the integers [e.g. (+4) + (–3) = + 1 and (–4) +
   ( + 3) = – 1].
   (c) The subtraction of an integer is the same as the addition of its additive inverse.
5. We have shown how addition and subtraction of integers can also be shown on a
   number line.

Ch7: Fractions

A fraction is a number representing part of a whole. The whole may be a single object or a group of objects.

In a proper fraction the denominator shows the number of parts into which the whole is divided, and the numerator shows the number of parts which have been considered. Therefore, in a proper fraction the numerator is always less than the denominator.

The fractions, where the numerator is bigger than the denominator are called improper fractions. Thus, fractions like
3/2, 12/7, 18/5, are all improper fractions.

Mixed Fractions. A mixed fraction has a combination of a whole and apart.

Equivalent Fractions: They represent the same part of a whole.

When you want to compare Fractions, its better to make their denominator equal for comparison.

4/5 and 5/6 Can you compare them?

-> Make denominator equal: 24/30 and 25/30

-> Now you can compare them easily: 25/30 > 24/30

Carrying out operations on fractions is an important part of daily life: e.g: Sharing Medium pizza with six slices among 2 or 3 people.

Ch8: Decimals

A dot represents Decimal, again an important concept in Math. An amount of 3 Rupees and 40 paisa (Though paisa is now a days uncommon to find local exchange) will be represented as 3.40 Rs.

Every physical quantity whether it is length, height, weight, can be expressed in decimals. Since they all carry bigger and small units.

Length: Meter, centimeter, millimeter etc.

Weight: Kg, gram, milli gram etc.

Summary:

1. Every decimal can be written as a fraction.
2. Any two decimal numbers can be compared among themselves. The comparison can
   start with the whole part. If the whole parts are equal then the tenth parts can be
   compared and so on.
3. Decimals are used in many ways in our lives. For example, in representing units of
   money, length and weight.

Ch9: Data Handling

. 1. We have seen that data is a collection of numbers gathered to give some information.

2. To get a particular information from the given data quickly, the data can be arranged
in a tabular form using tally marks

Ch10: Mensuration

Mensuration is a branch of mathematics that deals with the measurement of geometric figures and their parameters such as length, area, and volume.

1. Perimeter is the distance covered along the boundary forming a closed figure
   when you go round the figure once.
2. (a) Perimeter of a rectangle = 2 × (length + breadth)
   (b) Perimeter of a square = 4 × length of its side
   (c) Perimeter of an equilateral triangle = 3 × length of a side
3. Figures in which all sides and angles are equal are called regular closed figures.
4. The amount of surface enclosed by a closed figure is called its area.
5. To calculate the area of a figure using a squared paper, the following conventions
   are adopted :
   (a) Ignore portions of the area that are less than half a square.
   (b) If more than half a square is in a region. Count it as one square.
   (c) If exactly half the square is counted, take its area as 1/2 sq units.
6. (a) Area of a rectangle = length × breadth
   (b) Area of a square = side × side

Ch11: Algebra

The branch of mathematics in which we studied numbers is arithmetic. We have also learnt about figures in two and three dimensions and their properties. The branch of mathematics in which we studied shapes is geometry. Now we begin the study of another branch of mathematics. It is called algebra. The main feature of the new branch which we are going to study is the use of letters. Use of letters will allow us to write rules and formulas in a general way. By using letters, we can talk about any number and not just a particular number. Secondly, letters may stand for unknown quantities. By learning methods of determining unknowns, we develop powerful tools for solving puzzles and many problems from daily life. Thirdly, since letters stand for numbers, operations can be performed on them as on numbers. This leads to the study of algebraic expressions and their properties.

Suppose someone ask you how many match sticks can be used to make English letter L, you will say 2.

How many matchsticks for 2 L your answer will be 4 and there it can go to any number. The powerful part here is once you know the basic relation between single letter L and number of match sticks one needs to make, you can extrapolate and express this relation by an equation.

Number of Matchsticks = 2 X number of L

Or by using x for number of L where x is going to be the variable.

The word ‘variable’ means something that can vary, i.e. change. The value of a variable is not fixed. It can take different values.

1. We looked at patterns of making letters and other shapes using matchsticks. We learnt how to write the general relation between the number of matchsticks required for repeating a given shape. The number of times a given shape is repeated varies; it takes on values 1,2,3,… . It is a variable, denoted by some letter like n.

2. A variable takes on different values, its value is not fixed. The length of a square can have any value. It is a variable. But the number of angles of a triangle has a fixed value. It is not a variable.
3. We may use any letter n, l, m, p, x, y, z, etc. to show a variable.
4. A variable allows us to express relations in any practical situation.
5. Variables are numbers, although their value is not fixed. We can do the operations of addition, subtraction, multiplication and division on them just as in the case of fixed numbers. Using different operations we can form expressions with variables like x – 3, x + 3, 2n, 5m, 3p, 2y + 3, 3l – 5, etc.

Ch12: Ratio and Proportion

In our daily life, many a times we compare two quantities of the same type. I have 4 flowers, you have 6 flowers, so you have 2 flowers (6-4) more than me. We made comparison here by taking the difference between two quantities. Sometimes the difference is so large that it is better tow compare two quantities by doing division rather than taking their difference. For example, comparing lengths of Grasshopper and Ants.

The comparison by division is called RATIO.

I am more than 7 times older than my son. Here we are comparing by division. Father’s age/Son’s age = 7.

Two quantities can be compared only if they are in the same unit.

Proportion

Two friends Ashma and Pankhuri went to market to purchase hair clips. They purchased 20 hair clips for 30. Ashma gave 12 and Pankhuri gave 18. After they came back home, Ashma asked Pankhuri to give 10 hair clips to her. But Pankhuri said, “since I have given more money so I should get more clips. You should get 8 hair clips and I should get 12”. Can you tell who is correct, Ashma or Pankhuri? Why? Ratio of money given by Ashma to the money given by Pankhuri = 12: 18 = 2: 3
According to Ashma’s suggestion, the ratio of the number of hair clips for Ashma to the number of hair clips for Pankhuri = 10: 10 = 1: 1
According to Pankhuri’s suggestion, the ratio of the number of hair clips for Ashma to the number of hair clips for Pankhuri = 8: 12 = 2: 3
Now, notice that according to Ashma’s distribution, ratio of hair clips and the ratio of money given by them is not the same. But according to the Pankhuri’s distribution the two ratios are the same. Hence, we can say that Pankhuri’s distribution is correct.

If two ratios are equal, we say that they are in proportion and use the symbol ‘::’ or “=” to equate the two ratios.

If two ratios are not equal, then we say that they are not in proportion. In a statement of proportion, the four quantities involved when taken in order are known as respective terms. First and fourth terms are known as extreme terms. Second and third terms are known as middle terms.

The method in which first we find the value of one unit and then the value of required number of units is known as Unitary Method.

1. For comparing quantities of the same type, we commonly use the method of taking difference between the quantities.
2. In many situations, a more meaningful comparison between quantities is made by using division, i.e. by seeing how many times one quantity is to the other quantity. This method is known as comparison by ratio.
   For example, Isha’s weight is 25 kg and her father’s weight is 75 kg. We say that Isha’s father’s weight and Isha’s weight are in the ratio 3 : 1.
3. For comparison by ratio, the two quantities must be in the same unit. If they are not, they should be expressed in the same unit before the ratio is taken.
4. The same ratio may occur in different situations.
5. Note that the ratio 3 : 2 is different from 2 : 3. Thus, the order in which quantities are taken to express their ratio is important.

WITH THIS WE END THE DISCUSSION OF CLASS VI MATH OF INDIAN CBSE BOARD EXAMINATION

Yash