Fundamental Operations Class 6 ICSE — ICSE Chapter 13
Fundamental Operations Class 6 ICSE applies the four basic arithmetic operations — addition, subtraction, multiplication and division — to algebraic terms and polynomials. This chapter of the Class 6 Mathematics book builds directly on arithmetic you already know. You will learn how to combine like terms, add and subtract polynomials by two layouts, and multiply and divide monomials and polynomials using the laws of exponents.
Download Fundamental Operations Class 6 ICSE Mathematics Chapter 13 PDF
Download the Fundamental Operations Class 6 ICSE Mathematics chapter PDF to revise offline or print for practice. It is the full chapter as printed in the book — 16 pages, 15 sections, 44 questions.
What is inside Chapter 13
16 pages · 15 sections · 27 worked examples · 44 exercise questions
| Section | Title |
|---|---|
| 1 | FUNDAMENTAL OPERATIONS |
| 2 | BASIC CONCEPT |
| 3 | MORE ABOUT ADDITION AND SUBTRACTION |
| 4 | The required addition |
| 5 | Using the column method |
| 6 | (ii) Subtraction in Polynomials |
| 7 | Steps (for the row method) |
| 8 | Alternative method (column method) |
| 9 | Steps (for the column method) |
| 10 | Row method |
| 11 | Column method |
| 12 | MULTIPLICATION |
| 13 | (ii) Multiplication of a polynomial and a monomial |
| 14 | DIVISION |
| 15 | Using exponents (quotient law) |
- EXERCISE 13(A): 8 questions
- EXERCISE 13(B): 12 questions
- EXERCISE 13(D): 4 questions
- EXERCISE 13(E): 5 questions
- REVISION EXERCISE (Chapter 13): 15 questions
1 FUNDAMENTAL OPERATIONS
The chapter opens by placing its topic inside algebra. Its title refers to the four operations — addition (+), subtraction (−), multiplication (×) and division (÷) — now applied to algebraic expressions rather than plain numbers. Everything that follows uses these four operations on terms that contain variables.
2 BASIC CONCEPT
This section names the four fundamental operations and reminds you that you have used them in arithmetic since the lower classes. The new work is applying them to algebraic terms. The first idea to master is the difference between like terms and unlike terms. You first met algebraic terms in the chapter on fundamental concepts.
Like terms have the same literal part, such as 3x and 8x. Adding like terms gives a single like term whose coefficient is the sum of the given coefficients. So 3x + 8x = 11x. The rule works with negative coefficients too: -3xy^2 - 5xy^2 - xy^2 = (-3 - 5 - 1)xy^2 = -9xy^2.
Unlike terms cannot be merged into a single term. For example, 2ab and 4bc stay connected by their sign: 2ab + 4bc. Subtraction works the same way. Since 3 - 7 = -4, you get 3x - 7x = -4x. And unlike terms such as 4bc - 2ab cannot be simplified further.
A common mistake is treating unlike terms as like terms. Check the literal part before adding. 5x^2 and 5x are not like terms and cannot be combined into one term.
Mixed-number coefficients follow the same rule after converting to improper fractions. Consider 2x + 1\frac{1}{4}x:
2x + 1\frac{1}{4}x = \frac{2x}{1} + \frac{5x}{4} = \frac{8x + 5x}{4} = \frac{13x}{4} = 3\frac{1}{4}x
With only negative terms: -2xy - 5xy - 3xy = (-2 - 5 - 3)xy = -10xy. When signs mix, add the positive terms together, add the negative terms together, then combine the two results.
3 MORE ABOUT ADDITION AND SUBTRACTION
Adding whole polynomials introduces two layouts. The row method writes all the polynomials in one row, groups the like terms, then adds them. The column method stacks the polynomials so like terms sit one below the other in a vertical column, then adds column by column. The chapter states that in general the column method is preferred.
| Row method | Column method |
|---|---|
| All polynomials written on one line | Polynomials stacked with like terms aligned |
| Group like terms, then add | Add column by column |
| Suits short expressions | Preferred in general |
Both methods must give the same answer. Add 2a + 3b - c, a - 2b + 4c and 3a + b - c. The worked result is shown under the next two headings.
4 The required addition
This heading shows the row method written out in full. The brackets are removed first, keeping every sign as it is because the operation is addition. The like terms are then gathered together and their coefficients added.
(2a + 3b - c) + (a - 2b + 4c) + (3a + b - c)
= 2a + 3b - c + a - 2b + 4c + 3a + b - c
= (2 + 1 + 3)a + (3 - 2 + 1)b + (-1 + 4 - 1)c = 6a + 2b + 2c
Notice how the a terms, b terms and c terms are collected separately before adding. Each variable is treated as its own group.
5 Using the column method
The same three polynomials can be stacked so each column holds one variable. Missing terms are left as gaps so the columns stay aligned.
\begin{array}{r} 2a + 3b - c \\ a - 2b + 4c \\ 3a + b - c \\ \hline 6a + 2b + 2c \end{array}
Adding each column gives 6a, 2b and 2c. This is the same result as the row method, which confirms both layouts agree.
6 (ii) Subtraction in Polynomials
The chapter now moves from addition to subtraction of polynomials. The same two layouts — the row method and the column method — are used. The new idea is how a bracket with a minus sign in front is removed.
7 Steps (for the row method)
The row method for subtraction follows three steps:
- Enclose the expression to be subtracted in brackets with a minus sign prefixed.
- Remove the bracket by changing the sign of each term inside it.
- Combine the like terms and add.
Step 2 is the one students miss. Whenever a minus sign stands before a bracket, every sign inside the bracket flips. So +(3a) becomes -3a and -(2b) becomes +2b.
The most common sign error is changing only the first term inside the bracket. Change every term, or the answer will be wrong.
Worked example: subtract 2a - 3b + 4c from 5a + b - 2c.
5a + b - 2c - (2a - 3b + 4c)
= 5a + b - 2c - 2a + 3b - 4c = 3a + 4b - 6c
Final answer: 3a + 4b - 6c.
8 Alternative method (column method)
Subtraction can also be set out in columns, exactly as addition was. The expression to be subtracted is written below the first expression with its like terms aligned. The next section gives the steps for this layout.
9 Steps (for the column method)
The column method for subtraction follows three steps:
- Rewrite both expressions in two lines, with the lower line being the expression to be subtracted and like terms one below the other.
- Change the sign of each term in the lower line.
- Add column-wise.
Use the same example as the row method. Write 5a + b - 2c above 2a - 3b + 4c. Change the lower signs to -2a + 3b - 4c, then add: 3a + 4b - 6c. The answer matches the row method.
Where a term is missing in one row, leave a gap so the columns stay aligned. This keeps like terms in the same vertical column.
10 Row method
This section shows a combined problem: add two polynomials, then subtract a third from the sum. The whole expression is written on one line. The bracket before the subtracted part is removed by flipping every sign inside it.
(4x^2 + 3x - 2) + (2x^2 - 5x + 7) - (x^2 + x + 3)
= 4x^2 + 3x - 2 + 2x^2 - 5x + 7 - x^2 - x - 3
= (4 + 2 - 1)x^2 + (3 - 5 - 1)x + (-2 + 7 - 3) = 5x^2 - 3x + 2
Final answer: 5x^2 - 3x + 2.
11 Column method
The same combined problem can be done in columns. Add the first two polynomials column-wise, then subtract the third by changing its signs and adding.
\begin{array}{r} 4x^2 + 3x - 2 \\ 2x^2 - 5x + 7 \\ \hline 6x^2 - 2x + 5 \end{array}
\begin{array}{r} 6x^2 - 2x + 5 \\ -x^2 - x - 3 \\ \hline 5x^2 - 3x + 2 \end{array}
Both layouts give 5x^2 - 3x + 2, the same answer. Working the problem twice is a good check on your signs.
12 MULTIPLICATION
The multiplication of monomials rests on the product law of exponents: a^m \times a^n = a^{m+n}. In multiplication, the powers of like factors are added. The product of monomials is the product of their coefficients times the product of their literals, which is the rule used across the Class 6 ICSE syllabus.
Take three monomials with a negative coefficient:
(-2a^2b) \times 3ab^3c \times 4c^2
= (-2 \times 3 \times 4)(a^2 \times a)(b \times b^3)(c \times c^2) = -24a^3b^4c^3
Final answer: -24a^3b^4c^3.
Any product that contains a zero is zero. So 5x^2 \times 0 = 0 and 4m^2 \times 0 \times 2n^5 = 0.
13 (ii) Multiplication of a polynomial and a monomial
To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial. For example, 3a^2b(2ab - 4a^2b^2) = 6a^3b^2 - 12a^4b^3. Every term inside the bracket is multiplied separately.
To multiply two binomials, multiply each term of the first binomial by each term of the second, then combine like terms. Worked example:
(3x - 2y)(2x + 5y) = 3x(2x + 5y) - 2y(2x + 5y)
= 6x^2 + 15xy - 4xy - 10y^2 = 6x^2 + 11xy - 10y^2
Final answer: 6x^2 + 11xy - 10y^2.
The same product can be set out vertically, with each line holding one partial product and like terms added column-wise.
14 DIVISION
Division of a monomial by a monomial is done by writing each term in expanded form and cancelling factors common to the numerator and the denominator.
\frac{18a^2b^3c}{6abc} = \frac{3 \times 6 \times a \times a \times b \times b \times b \times c}{6 \times a \times b \times c} = 3ab^2
Final answer: 3ab^2.
To divide a polynomial by a monomial, divide each term of the polynomial separately by that monomial:
\frac{15x^4 - 9x^2}{3x^2} = \frac{15x^4}{3x^2} - \frac{9x^2}{3x^2} = 5x^2 - 3
Final answer: 5x^2 - 3.
Every term of the numerator is divided by the same monomial. This mirrors how multiplication distributed the monomial across the polynomial.
15 Using exponents (quotient law)
The quotient law has two cases, depending on which power is larger:
- \frac{a^m}{a^n} = a^{m-n} when m \gt n
- \frac{a^m}{a^n} = \frac{1}{a^{n-m}} when n \gt m
When the larger power is in the numerator, the result stays on top. When the larger power is in the denominator, the result stays below — this is the edge case students miss.
Case 1: \frac{x^8}{x^3} = x^{8-3} = x^5. Case 2: \frac{x^3}{x^7} = \frac{1}{x^{7-3}} = \frac{1}{x^4}.
With coefficients, divide the coefficients separately from the literals:
\frac{24a^5b^2}{6a^2b^7} = \frac{24}{6} \times \frac{a^{5-2}}{b^{7-2}} = \frac{4a^3}{b^5}
Final answer: \frac{4a^3}{b^5}.
What to practise from Fundamental Operations Class 6 ICSE
The practice load for fundamental operations class 6 ICSE sits across four exercises and one revision set. Work through them in order, because each builds on the last. The syllabus is set by the Council for the Indian School Certificate Examinations.
- EXERCISE 13(A) — 8 questions on adding and subtracting like and unlike terms
- EXERCISE 13(B) — 12 questions on sums and differences of polynomials by both layouts
- EXERCISE 13(D) — 4 questions on multiplying monomials and polynomials
- EXERCISE 13(E) — 5 questions on dividing monomials and polynomials
- REVISION EXERCISE (Chapter 13) — 15 questions covering all four operations
For the complete set of Class 6 topics, browse the Class 6 Mathematics book.
Fundamental Operations Class 6 ICSE frequently asked questions
What are the fundamental operations in class 6 ICSE maths?
The four fundamental operations are addition (+), subtraction (−), multiplication (×) and division (÷). This ICSE chapter applies them to algebraic terms and polynomials rather than plain numbers.
How do I subtract one polynomial from another in class 6 ICSE?
Use the row method or the column method. In both, the sign of every term being subtracted is changed before the like terms are combined.
What is the product law of exponents used in fundamental operations class 6 ICSE?
The product law states a^m \times a^n = a^{m+n}. It is used when multiplying monomials, where powers of like factors are added.
When should I use the row method instead of the column method?
The chapter says the column method is generally preferred. The row method suits short expressions, while the column method keeps many terms neatly aligned.
Reference: ICSE Class 6 Mathematics textbook material, chapter 13: Fundamental Operations — CISCE-aligned.