Table Of ContentALL THAT MATTERS IN
MATH
The ONLY Way To Learn Math And Sharpen Habits of Thinking
Volume I – Part II
KG-VIII Explained Examples
ALL THAT MATTERS IN MATH
Volume I - Part II
KG-VIII Explained Examples
The Only Real Math Book
Copyright© Sandeep Srivastava, 2022
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ISBN: 978-81-953962-1-4
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A to Z Mathematics Series
CONTENT
PART II
Sum of ‘n’ consecutive cubes
L
Difference between the squares of two consecutive
for Long Form triangular numbers
Mathematical expressions that expand
4. Logarithms and Antilogarithms ..............................36
1. Exponents and Numbers ............................................3
Definition of logarithm
Writing and evaluating exponential form of numbers Power rule
Working with zeroth and negative exponents Quotient rule
Comparison of exponential numbers Product rule
Addition of exponents of same base Antilog of numbers
Multiplication of different bases with same exponents Antilog law
Division of same bases with different exponents
5. Ratios ........................................................................40
Division of different bases with same exponents
Idea of ratio
Interpreting power of a power
Simplest form of ratios
Writing daily-life numbers as exponents
Comparison of ratios
Real-life application of exponents
Equivalent ratios – multiples of the given ratio
2. Square of Numbers ...................................................24
6. Rate – A Special Kind of Ratio ..................................50
Square numbers
Idea of rate
Numbers ending with digits 2, 3, 7, or 8 are non-perfect
Rate
squares
Zeros at end of a perfect square 7. Unitary Relationship ................................................54
Square of even number is always an even number
Idea of unitary ratio
Square of odd number is always an odd number
Unitary ratio
Square of numbers ending with 5
8. Percentage ................................................................57
Difference between two consecutive squares
Non-perfect squares between two consecutive square Idea of percentage
numbers How do we convert ratios to percentage?
Sum of first ‘n’ odd natural numbers Difference between “what percentage of x is y?” and “what
Pythagorean triplet percentage is x of y?”
Patterns in square numbers Conversion of decimal into percentage
Quantifying percentage
3. Cube of Numbers .....................................................31
Conversion of percentage into fraction or decimal form
Cubic numbers
9. Proportion ................................................................70
Cube of rational numbers
To make perfect cube–by prime factorisation and Idea of proportion
multiplication Are the numbers in proportion?
To make perfect cube – by prime factorisation and Finding the missing term
division
Continued proportion
M
Mean proportion
Direct proportion for Matter
Inverse proportion More about numbers, exploring factors, multiples
10. Arithmetic Progression .........................................82 16. Factors and Multiples of Numbers ..........................125
Finding missing term in AP Idea of factors
Characterising arithmetic progressions Finding factors of numbers
Using the general term of arithmetic progression Idea of multiples
How to assume terms in AP? Finding multiples of numbers
Sum of n terms of an AP
17. Prime and Composite Numbers ............................129
Applying the concept of arithmetic progression
Defining prime and composite numbers
11. Geometric Progression ..........................................94
Twin prime and co-prime numbers
Finding the term in geometric progression Prime triplets
Finding the geometric progression
18. Fundamental Law/Theorem of Arithmetic ...........132
Finding the sum of geometric progression
Prime factorisation of numbers using division method
Finding the sum to infinity of the geometric progression
Prime factorisation of numbers using factor tree method
Applying the concept of geometric progression
19. Highest Common Factor (or Greatest Common
12. Harmonic Progression ........................................101
Divisor) of Numbers ..............................................134
Finding the term in harmonic progression
Finding HCF using factors
13. Algebraic Expressions .........................................103
Finding HCF using prime factorisation method
Constants, variables/literals and expressions Finding HCF using division method
Creating algebraic expressions Finding HCF using Euclid’s division lemma and algorithm
Finding values of algebraic expressions Application of HCF in real-life
Addition and subtraction of algebraic expressions
20. Least Common Multiple of Numbers ....................140
Multiplication of algebraic expressions
Finding LCM by listing the multiples
Division of algebraic expressions
Finding LCM using prime factorisation method
14. Algebraic Identities .............................................111
Finding LCM using division method
Identity : (a + b) (a – b) = a2 – b2 Application of LCM in real-life
Identity : (a + b)2 = a2 + 2ab + b2
21. Applying Fundamental Theorem of Arithmetic ....144
Identity : (a – b)2 = a2 – 2ab + b2
Relationship between HCF and LCM
Identity : a3 + b3 = (a + b)3 – 3ab (a + b)
Identity : a3 – b3 = (a – b)3 + 3ab (a – b) 22. Square Root of a Number ......................................146
Identity : (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca Finding square root of a number using repeated subtraction
Identity : a3 + b3 + c3 – 3abc = (a + b + c) (a2 + b2 + c2 method
– ab – bc – ca)
Estimating square root of a number
Factorising algebraic expressions
Finding square root of a number using prime factorisation
15. Divisibility Test ....................................................121 method
Finding square root of a number using long division
Utilising divisibility : By 2, 5, 10
method
Utilising divisibility : By 3, 6, 9
Finding square root of fractions
Utilising divisibility : By 4, 11
Finding square root of decimal numbers
23. Cube Root of a Number ........................................153 Zero property of multiplication of positive and negative
numbers
Finding cube root of a number using prime factorisation
method 34. Closure Property of Numbers ............................179
Estimating cube root of a number
Closure property of addition of whole numbers and
Finding cube root of a negative and rational number
integers
24. The Order of Operation ........................................156 Closure property of subtraction of integers
Closure property of multiplication of whole numbers and
Using BODMAS rule to simplify the expressions
integers
N
O
for Negative Numbers
for Organising (data)
The numbers we created
Numericalising geometry, exploring coordinate systems,
25. Visualisation of Negative Numbers .....................161
graphs
Visualising situations involving negative numbers
35. Cartesian System: 1-Dimensional space ...............182
Visualising negative numbers on number lines
Representing situations using integers
26. Operations on Negative Numbers .......................164
Plotting points in 1-dimensional space
Addition of negative numbers
36. Cartesian System: 2-Dimensional space ................183
Subtraction of negative numbers
Multiplication of negative numbers Abscissa, ordinates and quadrants
Division of negative numbers Plotting ordered pairs in quadrants
27. Commutative Property of Numbers ...................169 37. Graph – A Cartesian Plane ...................................185
Commutative law of addition Dependent and independent variables in graph
Commutative law of multiplication Plotting points (or ordered pairs) to make graphs
Finding distance between two points
28. Associative Property of Numbers ......................172
Defining and using section formula
Associative law of addition
Mid-point formula
Associative law of multiplication
Centroid of a triangle - application of section formula
29. Distributive Property of Numbers .......................174 Area of a triangle
Distributive law of multiplication over addition Visualising collinearity of three points
Distributive law of multiplication over subtraction 38. Application of Coordinate System –
30. Additive Identity or Zero Property .....................176 Statistical Graphs ..................................................196
Additive identity law of positive and negative numbers Collecting and organising data
Making pictograph of data
31. Multiplicative Identity Property .........................177
Making bar graph from data
Multiplicative identity law of positive and negative
Making histogram of data
numbers
Making pie charts of data
32. Inverse Property ....................................................177
39. Cumulative Frequency Distribution Tables and
Additive inverse property of positive and negative numbers Curves ....................................................................209
Multiplicative inverse property of positive and negative
Cumulative frequency table (ungrouped data)
numbers
Cumulative frequency table (grouped data)
33. Zero Property of Multiplication ........................179
Introduction to Basics of Other Volumes
40. Comparing Quantities ..........................................214 Finding area between rectangles
Finding area enclosed by two concentric circles
Idea of profit and loss
Finding area of a trapezium
Idea of discount
Finding area of a polygon
Idea of simple interest
Finding volume of a cuboid
Idea of compound interest
Finding volume of a cube
41. Triangle and its Properties ...................................219
Finding volume of a cylinder
Idea of different parts of a triangle Finding volume of a sphere
Classifying triangles on the basis of sides Finding volume of a right circular cone
Classifying triangles on the basis of angles Finding surface area of a cuboid
Idea of a median and altitude of a triangle Finding surface area of a cube
Angle sum property of a triangle Finding surface area of a cylinder
Exterior angle property of a triangle
45. Visualising Solid Shapes .......................................254
Triangle inequality property
Diagnosing solid shapes
Pythagoras theorem
Drawing nets for building 3D shapes
42. Congruency of Triangles ......................................225
Euler’s relation for polyhedrons
Idea of congruency
46. Symmetry ..............................................................257
Idea of SSS congruency
Idea of symmetry
Idea of SAS congruency
Identifying lines of symmetry
Idea of ASA congruency
Idea of reflection symmetry
Idea of AAS congruency
Idea of rotational symmetry
Idea of RHS congruency
47. Probability .............................................................261
43. Understanding Quadrilaterals .............................230
Idea of probability
Idea of quadrilaterals
Probability of throwing of dice
Idea of polygon’s interior angles
Probability of tossing coins
Angle sum property of quadrilaterals
Exterior angles of quadrilaterals/polygons
Properties of parallelograms
PART I
Properties of rectangles
Properties of squares
Properties of rhombuses
A
Properties of kites and trapeziums
for Approximation
44. Mensuration .........................................................240
The world of pre-number quantification
Finding perimeter and area of a triangle
1. Approximate Number System
Finding perimeter and area of a rectangle
Estimating quantities
Finding perimeter and area of a square
Visual estimation
Finding area of a parallelogram and rhombus
Visual isolation and estimation
Finding circumference and area of a circle
Estimating age
Estimating weight Forward or skip counting
Estimating capacity Backward counting
2. Subitising 13. Roman Numerals
Perceptual subitising Formation of Roman numerals
Conceptual subitising
Subitising as key to numeration
C
Number patterns and number conversation
for Calibration
Subitising with five frames
Subitising with ten frames Learning to measure
Subitising with domino tiles 14. Non-Standard Unit of Measurement
Part and whole in five and ten frames
Expressing quantities in non-standard units of
Splat subitising
measurement
3. One to One Correspondence
15. Standard Unit of Measurement
Comparison of quantities using one to one correspondence
Standard unit of measurement for length, weight, volume
Estimating quantities with respect to a standard unit
B
16. Conservation of Measurement
for Bounty
Learning to count Conservation of weight
Conservation of volume
4. Ordinal Numerals
17. Unit of Measurement
Using ordinal numerals to count things
Using ordinal numerals to know the position of things Measuring less than a unit
Measurement using fractional units
5. Cardinal Numerals
What will be counted, what measured?
Using cardinal numerals to show quantity
18. Conversion of Units
6. Why do we count?
Conversion from one unit to another
Difference between ordinal and cardinal numerals
19. Measuring Space
7. Unit is everything in Counting
Exploring area and perimeter by counting number of
Unit of counting
squares
8. Make your own Number System Exploring volume
A number system – all about ‘0’ (starting number) and the ‘1’ 20. Measuring Time
9. How to count? Exploring calendar
Reading time in clock
Counting of like things
Estimating time and time elapsed
Counting of unlike things
Creating timeline
10. What to count?
Differentiate between like and unlike units
D
Converting unlike units to like units
for Decimal Numbers
11. Counting Bigger Quantities
Quantities are expressed as numbers
Counting using tally marks
21. Decimal Number System
12. Types of Counting
0–9 Number system
IX
The idea of ‘10’ Base 2 number system
The idea of ‘0’
30. Ternary Number System
Reaffirming the concept of ‘0’
Base 3 number system
Digits, numerals and numbers
31. Quinary Number System
22. Numbers as Packets
Base 5 number system
Numbers in packets
Place value and face value 32. Octal Number System
23. Concept of Number Names Base 8 number system
Unique names 33. Nonary Number System
Combined (word) names Base 9 number system
Creating number names
34. Dozenal Number System
24. Exploring Numbers
Base 12 number system
Comparison of numbers
35. Hexadecimal Number System
Greatest and smallest number formation
Expanded form of numbers Base 16 number system
Indian and International number system 36. Octodecimal Number System
Odd and even numbers
Base 18 number system
Prime and composite numbers
37. Duotrigesimal Number System
25. Approximation of Numbers
Base 32 number system
Rounding off numbers
26. Implication of Numbers as Packets
F
Carry – a property of numbers
for Fractional Quantities
Borrow – a property of numbers
The real-world numbers
Trailing zeros in multiplication
38. Fractional Quantity
Trailing zeros in division
What is fractional quantity?
27. Graphically Representing Numbers
The unit (1) has to be same for comparison
Number line
39. What is Fraction?
28. Decimal Number Formation
Unit of the fraction
Visualising numbers with a decimal point
Numerator and denominator in fraction
Expanded form of decimal numbers
Identifying equivalent fractions
Representing decimal numbers through blocks
40. Types of Fractions
Like and unlike decimal numbers
Comparison of decimal numbers Like and unlike fractions
Addition of decimal numbers Proper, improper and mixed fractions
Subtraction of decimal numbers
41. Conversion of Mixed and Improper Fractions
Improper fraction to mixed fraction
E
Mixed fraction to improper fraction
for Extended Number System
42. Rational and Irrational Numbers
Playing with numbers
Rational numbers
29. Binary Number System
X
