Fundamental Concepts Class 6 ICSE — Maths Chapter 12
Fundamental Concepts Class 6 ICSE is where arithmetic begins to use letters. This Maths Chapter 12 explains why symbols like x and y appear beside numbers. You will learn how to name these letters, translate word statements into algebraic form, separate constants from variables, and classify expressions. The chapter builds the vocabulary you need for all later algebra in the Class 6 ICSE syllabus.
Download Fundamental Concepts Class 6 ICSE Mathematics Chapter 12 PDF
Download the Fundamental Concepts Class 6 ICSE Mathematics chapter PDF to revise offline or print for practice. It is the full chapter as printed in the book — 9 pages, 11 sections.
What is inside Chapter 12
9 pages · 11 sections · 4 worked examples
| Section | Title |
|---|---|
| 12.1 | ALGEBRA |
| 12.2 | SIGNS AND SYMBOLS |
| 12.3 | WRITING A GIVEN STATEMENT IN ALGEBRAIC FORM |
| 12.4 | CONSTANTS AND VARIABLES |
| 12.5 | TERM |
| 12.6 | ALGEBRAIC EXPRESSIONS |
| 12.7 | TYPES OF ALGEBRAIC EXPRESSIONS |
| 12.8 | PRODUCTS AND FACTORS |
| 12.9 | COEFFICIENT |
| 12.10 | POWER OF LITERAL QUANTITIES |
| 12.11 | POLYNOMIAL IN ONE VARIABLE AND ITS DEGREE |
12.1 ALGEBRA
Algebra is a generalized form of arithmetic. In arithmetic, numbers like 5, −8 and 0.64 carry definite values. In algebra, letters such as a, b, c, x, y and z appear together with numbers. The letters used in algebra are called variables, literal numbers, or simply literals. They do not have a fixed value.
Why does this matter? A letter can stand for any number in a given situation. The same expression can take many values as the letter changes. That flexibility makes algebra a general tool, while arithmetic handles one fixed case at a time.
Consider the expression 6p - 3. Here 6 is a numeral and p is a literal. When p is 2, the expression equals 9. When p is 5, it equals 27. Because p has no fixed value, 6p - 3 has no single fixed value either.
This simple idea underlies the rest of this ICSE chapter. The full set of chapters sits in the Class 6 ICSE Maths books list, and the syllabus itself is published on the official CISCE website.
12.2 SIGNS AND SYMBOLS
In algebra, the signs +, −, × and ÷ carry the same sense as in arithmetic. Beyond these four, algebra uses a fixed set of relation symbols. Each symbol means the same thing in every branch of mathematics.
| Symbol | Meaning |
|---|---|
| = | is equal to |
| ≠ | is not equal to |
| < | is less than |
| > | is greater than |
| ∴ | therefore |
| ∵ | because or since |
| ~ | difference between |
| ⇒ | implies that |
These symbols appear in every later algebra chapter. The difference symbol ~ is the least familiar. It shows how much two quantities differ. The symbols ∴ and ∵ appear in solutions to show reasoning. Read them as ‘therefore’ and ‘because’ when you meet them.
Worked pattern: read a short chain such as x = 4 \Rightarrow 2x = 8. Say ‘x equals 4, which implies that 2x equals 8’. The ⇒ symbol carries the logical link between the two statements. You will use these symbols constantly in solutions.
12.3 WRITING A GIVEN STATEMENT IN ALGEBRAIC FORM
This section teaches two-way translation. You convert a word statement into algebra. You also convert an algebraic form back into words. Certain operation words signal the operation to use.
- sum, plus, increased by mean addition
- difference, decreased by, subtracted from mean subtraction
- product, multiplied by, times mean multiplication
- divided by means division
- equals, is equal to mean =
- greater than means >; less than means <
Order matters in subtraction and division. ‘x subtracted from 8’ means 8 - x, not x - 8. The phrase tells you which quantity acts first.
Worked pattern, words to algebra. Write ‘6 times x is less than y’ in algebraic form. The phrase ‘6 times x’ becomes 6x. The phrase ‘is less than’ becomes the symbol <. The statement reads 6x \lt y.
Worked pattern, algebra to words. Read p + 3 = 8 as ‘the sum of p and 3 is equal to 8’. The converse direction uses the same operation words in reverse. Practice both directions until the switch feels natural.
12.4 CONSTANTS AND VARIABLES
Algebra uses two kinds of symbols. A constant has a fixed numerical value in every situation. Numbers like 5, 30 and −7 are constants. So are fractions such as \frac{5}{3} and \frac{7}{9}. A variable is a symbol whose value changes with the situation. Letters such as x, y and p are variables.
Here is the key rule of the section: every combination of a constant and a variable is always a variable. The constant keeps its own value. The combination moves whenever the variable moves.
Worked pattern. In 7y, the numeral 7 is a constant and the letter y is a variable. Together, 7y is a variable. If y becomes 3, then 7y becomes 21. If y becomes 9, then 7y becomes 63. Because y changes, 7y changes with it.
The same logic applies to other operations. In 3 + x, the 3 stays fixed while x varies. So the whole expression varies too. Every combination of a constant and a variable remains a variable.
12.5 TERM
A term is a constant, a variable, or a product or quotient of constants and variables. The number 4 is a term because it is a constant. The letter x is a term because it is a variable. The product 4x and the quotient \frac{3}{y} are terms as well.
A term is called a constant term when it contains no literal at all. Each of 3, −20 and \frac{5}{7} is a constant term. Constants and variables combine to form many types of terms.
Like terms have the same literal coefficient. They may differ only in their numeral coefficient. For example, 3pq and -5pq are like terms because both carry the literal pq. Unlike terms do not have the same literal coefficient. For example, 3pq and 3pq^2 are unlike terms because one carries pq and the other carries pq^2.
Worked pattern. Check whether 4ab and -7ab are like terms. Both carry the literal ab. The numerals 4 and −7 differ, but that does not matter. The literal coefficient matches, so the terms are like terms.
12.6 ALGEBRAIC EXPRESSIONS
An algebraic expression is a collection of one or more terms separated by the signs + or −. The plus and minus signs mark the boundaries between terms. To count the terms, count the signs and add one.
Worked pattern. Look at 6p - 2q + 9. The minus and plus signs split the expression into three parts: 6p, 2q and 9. So the expression has three terms.
The term 9 contains no letter. It is the constant term of the expression. A constant term keeps its value no matter what the variables do. In another expression, 5m - 4n + 11, the constant term is 11. Spotting it is easy: find the term with no literal attached.
Expressions can grow long. An expression with four terms, such as 2a + 3b - 4c + 5, still follows the same rule. The plus and minus signs divide it into 2a, 3b, 4c and 5. The constant term here is 5.
12.7 TYPES OF ALGEBRAIC EXPRESSIONS
Expressions are named by their number of terms. The name changes as the count grows from one to three and beyond. This table summarises the five types used in the chapter.
| Type | Number of terms | Example |
|---|---|---|
| Monomial | one term | \frac{2x}{5y} |
| Binomial | two unlike terms | 5x + 2y |
| Trinomial | three unlike terms | ax^2 + bx + c |
| Multinomial | two or more terms | 7 + x - xy + y^2 |
| Polynomial | one or more unlike terms | 3x + 2y |
One rule causes most errors: multiplication and division signs do not separate terms. Only plus and minus signs do. So 3p \times 5z - 7y has two terms, 3p \times 5z and 7y, even though it looks like three pieces.
In the same way, 8 \times 4x \times 7y \div 2z has only one term. All the pieces are joined by multiplication and division. They form a single product.
A polynomial cannot have a variable in a denominator. Forms such as \frac{1}{x} or \frac{3}{x+5} are not polynomials. Every monomial, binomial, trinomial and multinomial is also a polynomial.
12.8 PRODUCTS AND FACTORS
A product is the result of multiplying two or more constants, literals, or both. It is the answer you get from a multiplication. Each constant and each literal multiplied together to form the product is called a factor of that product.
Worked pattern. Take 7ab. It is formed by multiplying 7, a and b together. So 7ab is the product. The numbers and letters 7, a and b are its three factors.
Naming factors matters in later work. When you divide or simplify an expression, you split it into its factors. Knowing which parts make up a product lets you see structure clearly. Every factor is a building block of the product.
Consider 12xy as another case. Its factors include 12, x and y. It also includes groups such as 12x and 3y. Any factor or group of factors is a valid factor of the product. The word ‘product’ always names the result of the multiplication.
12.9 COEFFICIENT
Any factor, or group of factors, of a product is called the coefficient of the remaining factors. This idea works in both directions. If a factor is a numerical quantity, it is a numeral coefficient. If a factor involves letters, it is a literal coefficient.
Worked pattern. In -4xy, the factor −4 is the numeral coefficient of xy. The factor y is a literal coefficient of -4x. The same product offers many coefficients. You just choose which remaining factors to name.
In 3xy, the 3 is the numeral coefficient of xy. The x is a literal coefficient of 3y. The xy is a literal coefficient of 3. Each grouping gives a different coefficient.
When the coefficient is unity, it is usually omitted. The product 1b is written simply as b. The 1 is understood, not shown. This is why the coefficient of x in x alone is 1.
12.10 POWER OF LITERAL QUANTITIES
When a quantity is multiplied by itself any number of times, the product is called a power of that quantity. The index form shows the number of like factors as a small raised numeral.
a \times a = a^2, \quad a \times a \times a = a^3, \quad a \times a \times a \times a \times a = a^5
In a^8, the letter a is the base. The number 8 is the exponent, the index, or the power. The table of products and their index forms builds this idea step by step.
Two rules hold for all values of x. First, x^1 = x, so 5^1 = 5. Second, x^0 = 1, so 5^0 = 1. These rules work for every nonzero number.
Worked pattern, index form. Convert 4 \times m \times m \times p \times p to index form. The factor m appears twice and p appears twice. So the product becomes 4m^2p^2.
Worked pattern, product form. Expand 5q^3 back to product form. It becomes 5 \times q \times q \times q. The exponent 3 tells you how many copies of q to write.
Mnemonic for the zero power. Any nonzero number divided by itself equals 1. When you cancel the factors, the exponents subtract to 0. So x^0 = 1 is just a power divided by itself.
12.11 POLYNOMIAL IN ONE VARIABLE AND ITS DEGREE
A polynomial in one variable is an algebraic expression made of one variable only. The degree of such a polynomial is the greatest of the exponents of its terms.
Worked pattern, one variable. In 6x^5 - 2x^3 + 4, the exponents are 5, 3 and 0. The term 4 has exponent 0 because 4 = 4x^0. The greatest exponent is 5. So the degree of the polynomial is 5.
For polynomials of two or more variables, the method changes. Add the powers of all the variables in each term. Then take the greatest of these sums as the degree.
Worked pattern, two variables. In 3a^2b + 5ab^3, the term 3a^2b has degree 2 + 1 = 3. The term 5ab^3 has degree 1 + 3 = 4. The greatest sum is 4. So the degree of the polynomial is 4.
These degree skills feed directly into the next chapter, Fundamental Operations, in this Class 6 Maths book. Practise the degree method until it feels automatic.
What to practise from Fundamental Concepts Class 6 ICSE
The chapter carries two large exercise sets. One sits inside Section 12.3, the other inside Section 12.11. Each set tests the ideas you have just read.
- Section 12.3, Exercise 1: express each statement in algebraic form — 15 questions.
- Section 12.3, Exercise 2: write each algebraic expression as a statement in words — 10 questions.
- Section 12.11, Exercise 1: separate the constants and the variables — 11 items.
- Section 12.11, Exercise 2: group the like terms together — 4 parts.
- Section 12.11, Exercises 3 and 5: state whether true or false — 18 statements.
- Section 12.11, Exercise 4: state the number of terms — 9 expressions.
- Section 12.11, Exercise 6: state whether each expression is a monomial, binomial or trinomial — 9 expressions.
- Section 12.11, Exercises 7 to 9: write the coefficient and the numeral coefficient — 22 items.
- Section 12.11, Exercise 10: write the degree of each polynomial — 8 polynomials.
- Second exercise set in Section 12.11: statements, like terms, term counts, classification, coefficients and degree — 44 items across 10 exercises.
Fundamental Concepts Class 6 ICSE frequently asked questions
What is the difference between a constant and a variable in fundamental concepts class 6 ICSE?
A constant has a fixed numerical value in all situations, such as 5 or −7. A variable is a symbol whose value changes with the situation, such as x or y. The rule to remember is that any combination of a constant and a variable is always a variable.
How do you write a word statement in algebraic form?
Look for the operation words. ‘Sum’ and ‘increased by’ point to addition, ‘subtracted from’ to subtraction, ‘product’ and ‘times’ to multiplication, and ‘divided by’ to division. Order matters, so read which quantity acts first; ‘x subtracted from 8’ becomes 8 - x.
What are like terms and unlike terms?
Like terms have the same literal coefficient and may differ only in their numeral coefficient, such as 3pq and -5pq. Unlike terms have different literal coefficients, such as 3pq and 3pq^2. Only like terms can be added or subtracted directly.
Why is any number to the power 0 equal to 1?
Any nonzero number divided by itself equals 1. When you divide powers with the same base, the exponents subtract, so x^3 \div x^3 = x^0. Since the division gives 1, x^0 = 1.
How do you find the degree of a polynomial in one variable?
Find the greatest of the exponents of the terms. In 6x^5 - 2x^3 + 4, the exponents are 5, 3 and 0, so the degree is 5. For polynomials with two or more variables, add the powers of all variables in each term and take the greatest sum.
Reference: ICSE Class 6 Mathematics textbook material, chapter 12: Fundamental Concepts — CISCE-aligned.