My Curiosity|4|Mathematics|2|For the Love of Math’s

For the Love of Math’s Part 2, continued from: My Curiosity Part 4: Mathematics |1| For the Love of Math’s – Logic Searcher

These blogs are school level based, once we reach to college level, we will update the same.

Class VII NCERT INDIA Math Book

Ch1: Integers
After learning about what integers are in class VI, we will discuss their properties.

1.1 Properties of Addition and Subtraction of Integers

1.1.1 Closure under Addition

What is the Closure property in Mathematics?
The closure property in Mathematics states that when a set of numbers is closed under an arithmetic operation, operating on any two numbers in the set always results in a number
belonging to the same group of numbers.

Now that you understand what a Closure property is, we have learned that the sum of two whole numbers is, again, a whole number.
2 + 3 =5
-4+5=1

1.1.2 Closure under Subtraction

Integers follow Closure under subtraction as well.
3-4=-1
6-4=2

1.1.3 Commutative Property

Commutative property is a mathematical rule that says that the order of terms doesn’t matter when adding or multiplying two numbers.

3+5=+3=8; the whole numbers can be added in any order.

- Addition is commutative for integers.
- Subtraction is not commutative for integers.

1.1.4 Associative Property

When more than two numbers are added or multiplied, the result remains the same, irrespective of how they are grouped.

- Addition is associative for integers.

1.1.5 Additive Identity

We get the same whole number when we add zero to any whole number. Zero is an additive identity for whole numbers.

1.2 Multiplication of Integers
1.2..1 Multiplication of a Positive and a Negative Integer

Multiplication of whole numbers is repeated addition.
e.g.: 5 + 5 + 5 = 3 x 5 = 15

Multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (-) before the product.

1.2.2 Multiplication of two Negative Integers

The product of two negative integers is a positive integer. We multiply the two negative integers as whole numbers and put the positive sign before the product.

1.3 Properties of Multiplication of Integers

1.3.1 Closure under Multiplication
The product of two integers is again an integer. So, integers are closed under multiplication.

axb is an integer for all integers a and b.

1.3.2 Commutative of Multiplication

Multiplication is commutative for integers.

- Multiplication by zero
- Multiplicative identity

0 is the additive identity, whereas 1 is the integer multiplicative identity. We get the additive inverse of an integer a when a multiply (-1) to a, i.e., ax(-1) = (-1)xa = -a.

1.3.5 Associativity for Multiplication

The product of three integers does not depend upon the grouping of integers, and this is called the associative property for multiplication of integers.

1.3.6 Distributive Property

Multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together.

a x (b + c) = a x b + a x c

1.4 Division of Integers

4×3 = 12 => 12 / 4 = 3 and 12/3 = 4
For each multiplication statement of whole numbers, there are two division statements.

- When we divide a negative integer by a positive integer, we divide them as whole numbers and then put a minus sign (-) before the quotient.
- When we divide them as whole numbers and then put a minus sign (-) before the quotient.
- When we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+)

Summary:

1. We now study the properties satisfied by addition and subtraction.
   (a) Integers are closed for addition and subtraction. A + b and a – b are again integers, where a and b are any integers.
   (b) Addition is commutative for integers, i.e., a + b = b + a for all integers a and b.
   (c) Addition is associative for integers, i.e., (a + b) + c = a + (b + c) for all integers a, b and c.
   (d) Integer 0 is the identity under addition. That is, a + 0 = 0 + a = a for every integer a.
2. We studied how integers could be multiplied and found that the product of a positive and a negative integer is a negative integer. In contrast, the product of two negative integers is a positive integer. For example, – 2 × 7 = – 14 and – 3 × – 8 = 24.
3. The product of an even number of negative integers is positive, whereas the product of an odd number of negative integers is negative.
4. Integers show some properties under multiplication.
   (a) Integers are closed under multiplication. A × b is an integer for any two integers a and b.
   (b) Multiplication is commutative for integers. A × b = b × a for any integers a and b.
   (c) The integer 1 is the identity under multiplication, i.e., 1 × a = a × 1 = a for any integer a.
   (d) Multiplication is associative for integers, i.e., (a × b) × c = a × (b × c) for any three integers a, b and c.
5. Under addition and multiplication, integers show distributive property. A × (b + c) = a × b + a × c for any three integers a, b and c.
6. The properties of commutativity, associativity under addition and multiplication, and the distributive property help us to make our calculations easier.
7. We also learned how to divide integers. We found that,
   (a) When a positive integer is divided by a negative integer, the quotient obtained is negative and vice-versa.
   (b) Division of a negative integer by another negative integer gives positive as quotient.
8. For any integer a, we have (a) a ÷ 0 is not defined, (b) a ÷ 1 = a

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Ch2: Fractions and Decimals

2.1 Multiplication of Fractions

To multiply a whole number with a proper or an improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator same.
– To multiply a mixed fraction to a whole number, first convert the mixed fraction to an improper fraction and then multiply.

When two proper fractions are multiplied, the product is less than each of the fractions. Or, we say the value of the product of two proper fractions is smaller than each of the two fractions.

The product of two improper fractions is greater than each of the two fractions.

Proper Fraction: A fraction less than one, with the numerator less than the denominator.
Improper Fraction: A fraction that is more than one, with the numerator greater than the denominator.

2.2 Division of Fractions

2.2.1 Division of Whole Number by a Fraction

1 / 1/2 = 1 x 2 = 2

- The non-zero numbers whose product with each other is 1, are called the reciproals of each other. 
- 9/5 x 5/9 = 1 

2.2.2 Division of a Fraction by a Whole Number
– 3/4 / 3 = 0.25

2.2.3 Division of a Fraction by Another Fraction

2.3 Multiplication of Decimal Numbers

SUMMARY

1. We have learned how to multiply fractions. Two fractions are multiplied by multiplying their numerators and denominators separately and writing the product as a product of numerators/product of denominators.
2. A fraction acts as an operator ‘of.’ For example, 1 2 of 2 is 1 2 × 2 = 1. 3.
   (a) The product of two proper fractions is less than each of the fractions that are multiplied.
   (b) The product of a proper and an improper fraction is less than the improper fraction and greater than the proper fraction.
   (c) The product of two improper fractions is greater than the two fractions.
3. A fraction’s reciprocal is obtained by inverting it upside down.
4. We have seen how to divide two fractions.
   (a) While dividing a whole number by a fraction, we multiply the whole number with the reciprocal of that fraction.
   (b) While dividing a fraction by a whole number, we multiply the fraction by the reciprocal of the whole number.
   (c) While dividing one fraction by another fraction, we multiply the first fraction by the reciprocal of the other.
5. We also learned how to multiply two decimal numbers. While multiplying two decimal numbers, first multiply them as whole numbers. Count the number of digits to the right of the decimal point in both decimal numbers. Add the number of digits counted. Put the decimal point in the product by counting the digits from its rightmost place. The count should be the sum obtained earlier. For example, 0.5 × 0.7 = 0.35 Rationalised 2023-24 FRACTIONS AND DECIMALS 43
6. To multiply a decimal number by 10, 100, or 1000, we move the decimal point in the number to the right by as many places as there are zeros over 1. Thus 0.53 × 10 = 5.3, 0.53 × 100 = 53, 0.53 × 1000 = 530
7. We have seen how to divide decimal numbers.
   (a) To divide a decimal number by a whole number, we first divide them as whole numbers. Then, place the decimal point in the quotient as in the decimal number. For example, 8.4 ÷ 4 = 2.1. Note that here, we consider only those divisions in which the remainder is zero.
   (b) To divide a decimal number by 10, 100, or 1000, shift the digits in the decimal number to the left by as many places as there are zeros over 1 to get the quotient. So, 23.9 ÷ 10 = 2.39,23.9 ÷ 100 = 0 .239, 23.9 ÷ 1000 = 0.0239
   (c) While dividing two decimal numbers, first shift the decimal point to the right by an equal number of places in both to convert the divisor to a whole number. Then divide. Thus, 2.4 ÷ 0.2 = 24 ÷ 2 = 12.

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Ch3: Data Handling

3.1 Representative Values

An average is a number that represents or shows the central tendency of a group of observations or data. Since the average lies between the highest and the lowest value of the given data so,
We say average is a measure of the central tendency of the group of data. Different forms of data need different forms of representative or central value to describe it. One of the representative values is the
“Arithmetic mean”.

3.2 Arithmetic Mean

The average or Arithmetic Mean (A.M.) or mean is defined as follows:

Mean = Sum of all observations/number of observations

3.2.1 Range

The difference between the highest and the lowest observation gives us an idea of the spread of the observations. This can be found by subtracting the lowest observation from the highest observation. We call the result the
Range of the observation.

3.3 MODE

The mode of a set of observations is the observation that occurs most often.

3.3.1 Mode of Large Data

e.g. 1,1,2,4,3,2,1,2,2,4
Ans: 1,1,1,2,2,2,2,3,4,4
Mode of this data is 2 because it occurs more frequently than other observations.

Terms:
Central Tendency: Central Tendency is a statistical measure that describe the center or middle point of a data sets a summary of the central or typical value around which the data points tend to cluster.

Mean:
– The mean, also known as the average, is calculated by adding up all the values in a dataset and then dividing the sum by number of values.
– It is sensitive to extreme values, also known as outliers, because it takes into account the magnitude of considers dataset
– The mean is commonly used when dealing with continuous data.
Mode:
– mode is the value that occurs most frequently in a dataset.
– It is not affected by extreme values because it only considers the frequency of values, not their magnitude.
– A dataset may have one mode (unimode), more than one mode (multimodal), or no mode if all values occur with the same frequency.
– Unlike the mean, the mode is not necessarily unique; a dataset can have multiple modes.
– Mode is often used for categorical or discrete data.

For the following statements, decide whether the mean or the mode is better.
i) You must decide upon the number of chapattis needed for 25 people called for a feast.- Mean would be useful
ii) A shopkeeper selling has decided to replenish her stock. – Mode
iii) We need to find the door height needed in our house. – Find the Range based on the maximum value and decide the height.
iv) When going on a picnic, if only one fruit can be bought for everyone, which is the fruit that we would get. – Mode

3.4 MEDIAN

The median refers to the value that lies in the middle of the data (when arranged in an increasing or decreasing order), with half of the observations above it and the other half below it.
Median is a valuable statistical measure because it offers robustness in the presence of outliers, is applicable to ordinal and nominal data, and provides a meaningful central value in skewed or non-normally distributed datasets.

Ch4: Simple Equations

4.1 A Mind-Reading Game!

You can tell your friend to think about a number let’s say x, then add 2 in it and then multiply by 4, ask your friend what is the result, if he reply that the result is 12, then you can tell that the original number you thought of was, 1! How? You were able to read what was in your friend mind?

4.2 Setting up of an Equation

In the above example let’s say your friend think about an imaginary number, x,
If you add 2 in it: x + 2
If you multiply by 4: 4(x+2)

4(x+2) = 12
Divide both side by 4: x + 2 = 3
Subtract both side by 2: x = 1

Hence the value of x is 1.

4.3 Review of what we know

An equation is a condition on a variable.
The word variable means something that can vary, i.e. change. A variable takes on different numerical values; its value is not fixed. Variables are denoted usually by expressions.

4.4 What Equation Is?

An equation is a condition on a variable. The condition is that two expressions should have equal value. Note that at least one of the two expressions must contain the variable.

4.4.1 Solving an Equation

An equation is like a weighing balance with equal weights on both its pans, in which case the arm of the balance is exactly horizontal. If we add the same weights to both the pans, the arm remains horizontal.
Similarly, if we remove the same weights from both the pans, the arm remains horizontal. On the other hand if we add different weights to the pans or remove different weights from them, the balance is tilted;
That is , the arm of the balance does not remain horizontal.
Thus if we fail to do the same mathematical operation with same number on both sides of an equality, the equality may not hold.

4.5 More Equations

While solving equations one commonly used operation is adding or subtracting the same number on both sides of the equation. Transposing a number (i.e., changing the side of the number) is the same as adding or subtracting the number from both sides. In doing so, the sign of the number has to be changed. What applies to numbers also applies to expressions.

4.6 Applications of Simple Equations to Practical Situations

Summary:

1. An equation is a condition on a variable such that two expressions in the variable should have equal value.
2. The value of the variable for which the equation is satisfied is called the solution of the equation.
3. An equation remains the same if the LHS and the RHS are interchanged.
4. In case of the balanced equation, if we
   (i) add the same number to both the sides, or
   (ii) subtract the same number from both the sides, or
   (iii) multiply both sides by the same number, or
   (iv) divide both sides by the same number, the balance remains undisturbed, i.e., the value of the LHS remains equal to the value of the RHS
5. The above property gives a systematic method of solving an equation. We carry out a series of identical mathematical operations on the two sides of the equation in such a way that on one of the sides we get just the variable. The last step is the solution of the equation. Rationalised 2023-24 SIMPLE EQUATIONS 73
6. Transposing means moving to the other side. Transposition of a number has the same effect as adding same number to (or subtracting the same number from) both sides of the equation. When you transpose a number from one side of the equation to the other side, you change its sign. For example, transposing +3 from the LHS to the RHS in equation x + 3 = 8 gives x = 8 – 3 (= 5). We can carry out the transposition of an expression in the same way as the transposition of a number.
7. We also learnt how, using the technique of doing the same mathematical operation (for example adding the same number) on both sides, we could build an equation starting from its solution. Further, we also learnt that we could relate a given equation to some appropriate practical situation and build a practical word problem/puzzle from the equation.

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Ch5 Lines and Angles

5.1 Introduction

Line segment has two end points.

Angle is formed when lines or line segment meet.

5.2 Related Angles

5.2.1 Complementary Angles
When the sum of the measure of two angles is 90(degree), the angles are called complementary angles.

5.2.2 Supplementary Angles

Sum of the measures of the angles in each of the above pairs comes out to be 180(degree). Such pairs of angles are called supplementary angles.

5.3 Pairs of Lines

5.3.1 Intersecting Lines

Two lines intersect if they have a point in common.

5.3.2 Transversal

A line that intersects two or more lines at distinct points is called a transversal.

5.3.4 Transversal of Parallel Lines

They are lines on a plane that do not meet anywhere.

5.4 Checking for Parallel Lines

When a transversal cuts two lines, such that pairs of corresponding angles are equal, the lines must be parallel. When a transversal cuts two lines, such that pairs of alternate interior angles are equal, the lines have to be parallel.

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Ch6 The Triangles and its Properties

Triangles are classified based on i) sides ii) angles

i) Based on Sides: Scalene, Isosceles and Equilateral triangles.
ii)  Based on Angles: Acute-angled, Obtuse-angled and Right-angled triangles.

6.2 Medians of a Triangle

A median connects a vertex of a triangle to the mid-point of the opposite side.

6.3 Altitudes of a Triangle

An altitude has one end point at a vertex of the triangle and the other on the line containing the opposite side. Through each vertex, an altitude can be drawn.

6.4 Exterior Angle of a Triangle and its Property

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

6.5 Angle SUM Property of a Triangle

The total measure of the three angles of a triangle is 180 degrees.

Food for thought

1. Can you have a triangle with two right angles?: No
2. Can you have a triangle with two obtuse angles?: No
3. Can you have a triangle with two acute angles?: Yes
4. Can you have a triangle with all the three angles greater than 60º?: No
5. Can you have a triangle with all the three angles equal to 60º?: Yes
6. Can you have a triangle with all the three angles less than 60º?: No

6.6 Two Special Triangles: Equilateral and Isosceles

A triangle in which all the three sides are of equal lengths is called an equilateral triangle.

In an Equilateral triangle:

a) All sides have same length.

b) Each angle has measure 60 degrees.

A triangle in which two sides are of equal lengths is called an isosceles triangles.

a) Two sides have same length.

b) base angles opposite to the equal sides are equal.

6.7 SUM of the Lengths of Two Sides of a Triangle

The sum of the lengths of any two sides of a triangle is greater than the third side.

6.8 Right-angled Triangles and Pythagoras property

In a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the legs.

SUMMARY:

1. The six elements of a triangle are its three angles and the three sides.
2. The line segment joining a vertex of a triangle to the mid-point of its opposite side is called a median of the triangle. A triangle has 3 medians.
3. The perpendicular line segment from a vertex of a triangle to its opposite side is called an altitude of the triangle. A triangle has 3 altitudes.
4. An exterior angle of a triangle is formed, when a side of a triangle is produced. At each vertex, you have two ways of forming an exterior angle.
5. A property of exterior angles:
   The measure of any exterior angle of a triangle is equal to the sum of the measures of its interior opposite angles.
6. The angle sum property of a triangle:
   The total measure of the three angles of a triangle is 180°.
7. A triangle is said to be equilateral, if each one of its sides has the same length. In an equilateral triangle, each angle has measure 60°.
8. A triangle is said to be isosceles, if atleast any two of its sides are of same length. The non-equal side of an isosceles triangle is called its base; the base angles of an isosceles triangle have equal measure.
9. Property of the lengths of sides of a triangle:
   The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
   The difference between the lengths of any two sides is smaller than the length of the third side. This property is useful to know if it is possible to draw a triangle when the lengths of the three sides are known.
10. In a right-angled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are called its legs.
11. Pythagoras property:
    In a right-angled triangle, the square on the hypotenuse = the sum of the squares on its legs. If a triangle is not right-angled, this property does not hold good. This property is useful to decide whether a given triangle is right-angled or not.

Ch7: Comparing Quantities

7.1 Percentage – Another way of Comparing Quantities

Two students from different schools, comparing their performance based on total marks that they have scored in their examination. One student scored 450, and other scored just 400. So can we say one who scored 450 has done better in his exam?

No, we can’t, till the time we don’t calculate their respective percentages, we cannot tell who has done better, the one who got 450, got them out of 900 marks, making his percentage mere 50%, whereas, the other student who got 400 got them out of 500, that is 80%, so the other student with 400 marks has done far better than the 450 one.

Percentages are numerators of fractions with denominator 100 and have been used in comparing results.

7.1.1 Meaning of Percentage

Percent is derived from Latin word ‘per centum’. meaning ‘per hundred’.

1% means 1/100 = 0.01 percent.

Percentage when total is not hundred

When total is not hundred, in such cases we need to convert the fraction to an equivalent with denominator 100.

7.1.2 Converting Fractional Numbers to Percentage

1/3 x 100/100 = 1/3 x 100% = 100%/3 = 33×1/3%

7.1.3 Converting Decimals to Percentage

a) 0.75 => 0.75 x 100% = 75%

7.1.4 Converting Percentages to Fractions or Decimals

1% => 1/100 => 0.01

7.1.5 Fun with Estimation

Percentages help us to estimate the parts of an area.

7.2 Use of Percentages

7.2.1 Interpreting Percentages

If we say that Ravi is saving 5% of his income every month, that means that for every 100rs that Ravi earns, he saves 5Rs.

7.2.2 Converting Percentages to “How many”

Example: A survey of 40 children showed that 25% liked playing football. How many children liked playing football.

25/100 x 40 = 10

7.2.3 Ratios to Percents

Sometimes, parts given in ratio, needs to be converted to percentages.

Example: To make idlis, you must take 2 parts rice and 1 part urad dal. What percentage of such a mixture would be rice and what percentage would be urad dal?

Rice: Urad dal = 2: 1

Percentage of rice= 2/3 x 100% = 66×2/3 %

7.2.4 Increase or Decrease as Per Cent

Example: A school team won 6 games this year against 4 games won last year. What is the per cent increase?

Percentage increase = amount of change / original amount or base x 100

= increase in the number of wins/original number of wins x 100 = 2/4 x 100 = 50%

7.3 Prices Related to an Item or Buying and Selling

The buying price of any item is known as its cost price. It is written in short as CP. The price at which you sell is known as the selling price or in short SP.

If CP< SP => Profit | CP = SP => No profit no loss | CP > SP => Loss

7.3.1 Profit or Loss as a Percentage

The profit or loss can be converted to a percentage. It is always calculated on the CP.

7.4 Charge given on borrowed money or simple interest

The money one borrows is known as sum borrowed or principal. This money would be used by the borrower for some time before it is returned. For keeping this money for some time the borrower has to pay some extra money to the bank. This is known as Interest.

On Rs. P borrowed, the interest paid for T years with interest rate R would be PxRxT/100

Ch8: Rational Numbers

Study of numbers start with counting objects around you.

8.2 Need for Rational Numbers

Most of the real-life situations that deal with numbers which are not whole numbers, they are in fractions. Rational numbers are better in expressing such situations.

8.3 What are rational numbers?

Ratio of two integers p and q (q /= 0), i.e., p:q can be written in the form p/q. This is the form in which rational numbers are expressed.

Rational numbers include integers and fractions.

Equivalent rational numbers

2/3 = 4/6 = 10/15 are equivalent rational numbers.

By multiplying the numerator and denominator of a rational number by the same nonzero integer, we obtain another rational number equivalent to the given rational number. This is exactly like obtaining equivalent fractions.

8.4 Positive and Negative Rational Numbers

8.5 Rational numbers on a number line

8.6 Rational Numbers In Standard Form

3/5,-5/8,2/7,-7/11 The denominator of these rational numbers are positive integers and 1 is the only common factor between the numerators and denominators. Further the negative sign occurs only in the numerator. Such rational numbers are said to be in standard form.

A rational number is said to be in the standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than 1.

If a rational number is not in the standard form, then it can be reduced to the standard form.

Thus, to reduce the rational number to its standard form, we divide its numerator and denominator by their HCF ignoring the negative sign, if any.

8.7 Comparison of Rational Numbers

• The case of pairs of negative rational number is similar. To compare two negative rational numbers, we compare them ignoring their negative signs and then reverse the order.
• Comparison of a negative and a positive rational number is obvious. A negative rational number is to the left of zero whereas a positive rational number is to the right of zero on a number line. So, a negative rational number will always be less than a positive rational number.

8.8 Rational Numbers between Two Rational Numbers

We can find unlimited number of rational numbers between any two rational numbers.

8.9 Operations on Rational Numbers

8.9.1 Addition

While adding rational numbers with same denominator, we add the numerators keeping the denominators same.

In case the denominators are different, we first find the LCM of the two denominators. Then we find the equivalent rational numbers of the given rational numbers with this LCM as the denominator. Then, add the two rational numbers.

Additive Inverse

In case of integers, we call -2 as the additive inverse of 2 and 2 as the additive inverse of -2.

For rational numbers also, we call -4/7 as the additive inverse of 4/7 and 4/7 as the additive inverse of -4/7.

8.9.2 Subtraction

We say while subtracting two rational numbers, we add the additive inverse of the rational number that is being subtracted, to the other rational number.

8.9.3 Multiplication

While multiplying a rational number by a positive integer, we multiply the numerator by that integer, keeping the denominator unchanged.

8.9.4 Division

To divide one rational number by the other non-zero rational number we multiply the rational number by the reciprocal of the other.

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Ch9: Perimeter and Area

9.1 Area of a Parallelogram

A Parallelogram is a shape with parallel opposite sides.

Area of Parallelogram = Length of sides x Perpendicular distance between opposite sides. (Just like a rectangle)

9.2 Area of a Triangle

Area of Triangle = 1/2 x base x height (Or say half of Parallelogram Area of same sides measurements.)

All the congruent triangles are equal in area but the triangles equal in area need not be congruent.

9.3 Circles

9.3.1 Circumference of a circle

Circumference of a Circle = 2x pi x r (r = radius of circle / Pi = 3.14)

9.3.2 Area of Circle

Area of circle = 2 x pi x r x r

Summary:

1. Area of a parallelogram = base × height
2. Area of a triangle = 1/2 x (area of the parallelogram generated from it) = 1/2 x base × height
3. The distance around a circular region is known as its circumference. Circumference of a circle = πd, where d is the diameter of a circle and π = 22/7 or 3.14 (approximately).
4. Area of a circle = π x r x r, where r is the radius of the circle

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Ch10: Algebraic Expressions

10.1 Introduction

In class VI we come across simple algebraic expressions. We observed that these expressions help in formulating interesting puzzles and quizzes.

10.2 How are expressions formed?

A variable can take various values. It is some letter like x,y,z…etc. we use for the component in expressions which is yet to be find out. On the other hand, a constant has a fixed value.

We combine variables and constants to make algebraic expressions.

10.3 Terms of an expression

Expression 4x + 5 contain two terms 4x and 5. Terms are added to form expression.

We can represent the terms and factors of the terms of an expression conveniently and elegantly by a tree diagram. In tree diagram, dotted lines are for factors and continuous lines for terms.

Coefficients

The numerical coefficient or simply the coefficient of the term.

10.4 Like and Unlike terms

When terms have the same algebraic factors, they are like terms. When terms have different algebraic factors, they are unlike terms.

e.g: 2xy – 3x + 5xy – 4, here 2xy and 5xy are like terms and -3x,4,2xy are unlike terms, and so does 5xy,-3x are unlike terms.

10.5 Monomials, Binomials, Trinomials, and Polynomials

An expression with one term is called monomial e.g: 7xy. An expression which contains two unlike terms is called a binomial. e.g: x + y. An expression which contain three terms is called a trinomial e.g: x + y + 4. In general an expression with more than one term is called polynomial.

10.6 Finding the value of an expression

We need to find the value of expression when we want to find out area of square or rectangle etc. Sometimes we need to find the value to check whether a particular value of variable satisfies a given equation or not.

Summary:

1. Algebraic expressions are formed from variables and constants. We use the operations of addition, subtraction, multiplication and division on the variables and constants to form expressions. For example, the expression 4xy + 7 is formed from the variables x and y and constants 4 and 7. The constant 4 and the variables x and y are multiplied to give the product 4xy and the constant 7 is added to this product to give the expression.
2. Expressions are made up of terms. Terms are added to make an expression. For example, the addition of the terms 4xy and 7 gives the expression 4xy + 7.
3. A term is a product of factors. The term 4xy in the expression 4xy + 7 is a product of factors x, y and 4. Factors containing variables are said to be algebraic factors.
4. The coefficient is the numerical factor in the term. Sometimes anyone factor in a term is called the coefficient of the remaining part of the term.
5. Any expression with one or more terms is called a polynomial. Specifically, a one term expression is called a monomial; a two-term expression is called a binomial; and a three-term expression is called a trinomial.
6. Terms which have the same algebraic factors are like terms. Terms which have different algebraic factors are unlike terms. Thus, terms 4xy and – 3xy are like terms; but terms 4xy and – 3x are not like terms.
7. In situations such as solving an equation and using a formula, we have to find the value of an expression. The value of the expression depends on the value of the variable from which the expression is formed. Thus, the value of 7x – 3 for x = 5 is 32, since 7(5) – 3 = 35 – 3 = 32

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Ch11: Exponents and Powers

11.1 Introduction

Mass of Earth is: 5,970,000,000,000,000,000,000,000 kg

Mass of Uranus: 86,800,000,000,000,000,000,000,000 kg

If someone asks you which one is bigger, it will take time to come up with answer due to big number involved here. The very large numbers are difficult to read, understand and compare. To make these numbers easy to read, understand and compare, we use exponents.

11.2 Exponents

10,000 = 10x10x10x10 = 10^4

Here 10 is base and 4 is the exponent.

11.3 Laws of Exponents

11.3.1 Multiplying powers with the same base

i) 2^3 x 2^4 = 2^7 (Powers add up)

11.3.2 Dividing powers with the same base

i) 2^4 / 2^2 = 2^2 (Powers subtracts in division)

11.3.3 Taking power of a power

i) 2^3^2 = 2^6 (Powers get multiplied)

11.3.4 Multiplying powers with the same exponents

i) 2^3 x 3^3 = (2×3)^3 = 6^3 (Base gets multiplied)

11.3.5 Dividing powers with the same exponents

i) 2^4 / 3^4 = (2/3)^4

Numbers with exponent zero

2^0 = 1 | 3^0 = 1 | 4^0 = 1

Ch12: Symmetry

12.1 Introduction

Symmetry is a geometrical concept. Everyone loves symmetry, you want to have designs symmetrical on your walls, you want there has to be symmetry between right and left half of your car. It refers to a balanced and proportionate similarity found in two halves of an object. It means one half of the object is the mirror image of the other half. This can be observed in both regular and irregular shapes, such as squares, rectangles, circles, and triangles.

The line of symmetry is an imaginary line or axis along which you can fold a figure to obtain the symmetrical halves. It essentially divides an object into two mirror-image halves. The line of symmetry can be vertical, horizontal, or diagonal. In mathematics, symmetry defines that one shape is exactly like the other share when it is moved, rotated, or flipped.

Symmetry is also used in everyday life to refer to a sense of harmonious and beautiful proportion and balance. Symmetric objects are found all around us, in nature, architecture, and art.

A figure has a line symmetry, if there is a line about which the figure may be folded so that the two parts of the figure will coincide.

12.2 Lines of Symmetry for Regular Polygon

A polygon is a closed figure made of several line segments. The polygon made up of the least number of line segments is the triangle. A polygon is regular if its sides and angles are of equal measure. Thus equilateral triangle is a regular polygon of three sides.

One interesting thing I noticed, as the number of sides increase the angle between two sides in a regular polygon also keeps on increasing.

Triangle – 60 degrees

Rectangle/Square – 90 degrees

Pentagon – 108 degrees

Octagon – 120 degrees

The regular polygons are symmetrical figures and hence their lines of symmetry are quite interesting, each regular polygon has as many lines of symmetry as it has sides. Triangles has 3, square/rectangle have four and so on. The concept of line symmetry is closely related to mirror reflection. A shape has line symmetry when one half of it is the mirror image of the other half.

12. 3 Rotational Symmetry

When an object rotates, its shape and size do not change. The rotation turns an object about a fixed point. This fixed point is the Centre of rotation. The angle of turning during rotation is called the angle of rotation.

Square has a rotational symmetry of order 4: What does it mean is Square has rotational symmetry, after every 90 degrees turns it looks same, and after four such 90 degrees rotations it will come to its original position.

12.4 Line Symmetry And Rotational Symmetry

Some shapes have only line symmetry, some have only rotational symmetry, and some have both line symmetry and rotational symmetry. Square has both. You can pass 4 lines across which Square has symmetrical shape (2 across diagonals, and 2 from mid-points of adjacent sides. It also has rotational symmetry across its plane, you rotate it 90 degrees and you will find resultant shape symmetrical to the one before, we cannot say the same for rectangle. I guess you understand now why?

Circle is the most perfect symmetrical figure, because it can be rotated around its Centre through any angle and at the same time it has unlimited number of lines of symmetry. Observe any circle pattern. Every line through the Centre (that is every diameter) forms a line of (reflectional) symmetry and it has rotational symmetry around the Centre for every angle.

Ch13: Visualizing Solid Shapes

13.1 Introduction: Plane Figures And Solid Shapes

In this chapter, we will classify figures you have seen in terms of what is known as dimension. In our day-to-day life, we see several objects like books, balls, ice-cream cones etc., around us which have different shapes. One thing common about most of these objects is that they all have some length, breadth, and height or depth. Hence due to this, they occupy space and have three dimensions. Hence, they are called three dimensional shapes.

3-D shapes: Cuboid, Sphere, Cylinder, Cube, Cone, Pyramid.

2- D Shapes: Rectangle, Circle, Square, Quadrilateral, Triangle.

13.2 Faces, Edges and Vertices

We can notice that the two-dimensional figures can be identified as the faces of the three-dimensional shapes. For example, the 3-D shaped pyramids have 2-d shaped 4 triangles.

13.3 Nets for building 3-d shapes

Take a cardboard box. Cut the edges to lay the box flat. You have now a net for that box.

Small exercise for you:

13.4 Drawing Solids On a Flat Surface

Your drawing surface is paper, which is flat. When you draw a solid shape, the images are somewhat distorted to make them appear three-dimensional. It is a visual illusion.

13.4.1 Oblique Sketches

Here is a picture of a cube. It gives a clear idea of how the cube looks like, when seen from the front. You do not see certain faces. In the drawn picture, the lengths are not equal, as they should be in a cube. Still, you are able to recognise it as a cube. Such a sketch of a solid is called an oblique sketch.

13.4.2 Isometric Sketches

Isometric dot sheet divides the paper into small equilateral triangles made up of dots or lines. To draw sketches in which measurements also agree with those of the solid, we can use isometric dot sheets.

13.4.3 Visualising Solid Objects

13.5 Viewing Different Sections Of A Solid

Summary:

1. The circle, the square, the rectangle, the quadrilateral and the triangle are examples of plane figures; the cube, the cuboid, the sphere, the cylinder, the cone and the pyramid are examples of solid shapes.
2. Plane figures are of two-dimensions (2-D) and the solid shapes are of three-dimensions (3-D).
3. The corners of a solid shape are called its vertices; the line segments of its skeleton are its edges; and its flat surfaces are its faces.
4. A net is a skeleton-outline of a solid that can be folded to make it. The same solid can have several types of nets.
5. Solid shapes can be drawn on a flat surface (like paper) realistically. We call this 2-D representation of a 3-D solid.
6. Two types of sketches of a solid are possible: (a) An oblique sketchdoes not have proportional lengths. Still it conveys all important aspects of the appearance of the solid.
   (b) An isometric sketch is drawn on an isometric dot paper, a sample of which is given at the end of this book. In an isometric sketch of the solid the measurements kept proportional.
7. Visualising solid shapes is a very useful skill. You should be able to see ‘hidden’ parts of the solid shape.
8. Different sections of a solid can be viewed in many ways:
   (a) One way is to view by cutting or slicing the shape, which would result in the cross-section of the solid.
   (b) Another way is by observing a 2-D shadow of a 3-D shape.
   (c) A third way is to look at the shape from different angles; the front-view, the side-view and the top-view can provide a lot of information about the shape observed