Algebra | ICSE Class 6 Maths Notes

This note covers number patterns, variables, constants and unknowns, algebraic expressions, terms, factors and coefficients, like and unlike terms, polynomials and degree, formulae, substitution, and simple linear equations in one variable.

How does algebra use letters to describe numbers?

Algebra uses letters to represent numbers and express mathematical relationships. Start with a familiar relationship: Shabnam is 3 years older than Aftab. To find her age, add 3 to his age. The calculation changes with Aftab’s age, but the relationship remains the same.

Let a represent Aftab’s age in years and s represent Shabnam’s age in years. Then s = a + 3. The sign = means “is equal to”, and + means addition. Read this as “Shabnam’s age equals Aftab’s age plus three”.

What are literal numbers, variables and constants?

A literal number, also called a letter-number, is a letter used to represent a number. A variable is a letter representing a number that can take different values. Here, a and s are variables because the ages change.

A constant has a fixed value. In the age relationship, the number 3 is constant: Shabnam remains 3 years older. The ages and the difference between the ages play different roles, even though all of them are quantities measured in years.

Worked example 1. Shabnam is 3 years older than Aftab. Find Shabnam’s age when Aftab is 23 years old.

Answer: Aftab’s age a = 23. Replace a by 23 in s = a + 3. This gives s = 23 + 3 = 26. Shabnam is 26 years old.

Two columns pair Aftab’s ages 4, 10, 23, ?, and a with the expressions 4 + 3, 10 + 3, 23 + 3, ? + 3, and a + 3 for Shabnam’s age. The question mark represents an unspecified age.

See Fig. 4.1 in your NCERT textbook

The letter does not stand for the person’s name in the calculation. It stands for the numerical age. Defining what each letter represents makes the relationship understandable and tells us which number to use when an age is supplied.

How do number patterns lead to general rules?

A pattern follows a recognisable rule. To generalise is to express that rule so that it covers different cases. Letters help us move from calculating one particular case to describing how every case in the pattern is formed.

Consider separate letter L shapes made with matchsticks. Each L needs 2 sticks. Let n be the number of L shapes. The total number of sticks is 2 × n, where × means multiplication. We usually write this more compactly as 2n.

The drawing shows groups of one, two and three L shapes. Each L consists of one upright matchstick and one horizontal matchstick. The shapes repeat beside one another.

See Fig. 4.2 in your NCERT textbook

Worked example 2. Each separate L shape requires 2 matchsticks. How many sticks are needed for 5 L shapes?

Answer: Use 2n with n = 5. The number of matchsticks is 2 × 5 = 10. Multiplication counts the same requirement of 2 sticks for each of the 5 shapes.

Result: Positive even numbers have the form 2n

A natural number is a counting number: 1, 2, 3, and so on. An even number is divisible by 2 without a remainder. A positive even number can be written as 2n, where n is a natural number. Here n represents a number, not a matchstick shape.

Result: The first n odd numbers have sum n²

An odd number is not divisible by 2 without a remainder. The first positive odd numbers are 1, 3, 5 and 7. Their sums follow the pattern 1 + 3 = 2², 1 + 3 + 5 = 3², and 1 + 3 + 5 + 7 = 4².

The notation n², read “n squared”, means n × n. In the rule “sum of the first n odd numbers = n²”, n is the number of odd numbers being added. It does not represent the last odd number in the sum.

How can words be translated into algebraic expressions?

An algebraic expression combines numbers and letters using operations such as addition, subtraction and multiplication. An operation is a mathematical action performed on numbers. For example, a + 3 is the expression for Shabnam’s age when a represents Aftab’s age in years.

The sign − indicates subtraction or a negative value. To write “3 less than Shabnam’s age”, use s − 3, where s represents her age in years. The expression starts with the age from which 3 is taken away.

How do operation words affect the order?

Let d represent a number in the following table. “More than” tells us to add. “Less than” tells us to subtract from the quantity named afterwards. “Times” tells us to multiply. Read the complete phrase before choosing the expression.

Worked example 3. A pipe is 20 metres long. Another pipe of length k metres is joined to it. Write the expression for the combined length, where k represents the second pipe’s length.

Answer: Add the two lengths. The combined length is (20 + k) metres. The brackets group the expression, so the unit “metres” applies to the whole length.

Writing an expression does not require a numerical value for every letter. The purpose is to describe the relationship. The expression 20 + k remains useful for different possible lengths of the second pipe, with the original 20 metres kept fixed.

What are terms, factors and coefficients?

A term is a part of an expression separated from other parts by addition or subtraction. When identifying terms, keep the sign belonging to each term. A factor is a number or expression multiplied by another to form a product; a product is the result of multiplication. The variable part of a term consists of its letter factors.

A coefficient is the factor multiplying a specified variable or variable part. The numerical coefficient is its number factor. In 4x, where x represents a number, 4 and x are factors, and 4 is the numerical coefficient of x.

How do we separate an expression into its parts?

Consider 4x + 5y + 3. Here x is the number of red boxes, each containing 4 pens, and y is the number of blue boxes, each containing 5 pencils. With 3 extra pens, the expression gives the total number of pens and pencils.

A constant term is a term containing no variable. In this expression, changing the numbers of boxes changes 4x and 5y. It does not change the extra 3 pens. Terms describe the separate contributions that are added to obtain the total.

Worked example 4. In 4x + 5y + 3, x and y are variables. Identify the terms, their numerical coefficients, and the constant term.

Answer: The terms are 4x, 5y and 3. The coefficient of x is 4; the coefficient of y is 5. The constant term is 3. The product 4x is one term, although it has the factors 4 and x.

This distinction matters when counting terms. Multiplication within a term does not split it into separate terms. First locate the addition or subtraction between parts; then examine the factors within each part.

How are like terms different from unlike terms?

Like terms have the same variable part, including the same powers of the variables. A power shows how many times a variable is used as a factor in repeated multiplication. Terms whose variable parts differ are called unlike terms.

For example, let c represent the price of a pencil and d the price of an eraser. The terms 5c, 3c and 10c are like terms: each is a number multiplied by c. The terms 18c and 11d are unlike terms.

Why can like terms be collected?

To simplify an expression means to rewrite it in a simpler form without changing its value. Adding 5 lots of c, 3 lots of c and 10 lots of c gives 18 lots of c. The variable part remains c.

Worked example 5. A shop sells 5, 3 and 10 pencils on three days. Each pencil costs c rupees. Write and simplify the total amount earned from pencils.

Answer: The daily amounts are 5c, 3c and 10c rupees. Their sum is 5c + 3c + 10c = (5 + 3 + 10)c = 18c rupees. Add the number factors because the price c is the same throughout.

If the eraser sales contribute 11d rupees, the combined expression is 18c + 11d. These unlike terms remain separate. The prices c and d represent different quantities, so their coefficients cannot simply be added to make a single term with one of those letters.

A rectangle is a four-sided shape with four right angles and equal opposite sides. A right angle is an angle of 90 degrees, where degrees measure the size of an angle. Let l be the rectangle’s length and b its breadth. Its boundary lengths add as l + b + l + b. Collecting the two l terms and the two b terms gives 2l + 2b. Both expressions represent the same total length.

Note: Look at the complete variable part when identifying like terms. Matching the numerical coefficients is not the test. The letters and their powers determine which terms can be collected.

What do polynomial and degree mean?

A polynomial in one variable is a finite sum of terms involving number coefficients and whole-number powers of that variable. A whole number is one of 0, 1, 2, 3, and so on. Constants can also be terms of a polynomial.

The degree of a non-zero polynomial in one variable is the highest power of that variable with a non-zero coefficient after like terms have been collected. “Non-zero” means not equal to zero. The degree describes a power, not the number of terms.

How are powers read?

Let x, y and z represent numbers in the examples below. The notation x² means x × x, and y³ means y × y × y. A variable written without a visible power, such as z in 3z, has power 1.

A linear polynomial has degree 1, a quadratic polynomial has degree 2, and a cubic polynomial has degree 3. A non-zero constant polynomial, such as 8, has degree 0. These names describe the highest power present.

Worked example 6. Identify the degree, coefficients and constant term of 5y³ + y² + 2y − 1, where y is the variable.

Answer: The highest power of y is 3, so the degree is 3. The coefficients of y³, y² and y are 5, 1 and 2 respectively. The constant term is −1. The unwritten number factor in y² is 1.

The coefficient and the power answer different questions. In 5y³, the coefficient 5 tells us how many lots of y³ are present. The power 3 tells us that y is used as a factor three times: y × y × y. It is this power that determines the degree here.

How do formulae express rules for shapes?

A formula expresses a mathematical relationship using symbols. A square has four equal sides and four right angles. Its perimeter is the total distance around its boundary. Adding the four equal side lengths gives a rule that works for squares of different sizes.

Result: A square’s perimeter is four times its side

Let P represent the perimeter of a square and x its side length, measured in the same length unit. Then P = 4x. The number 4 is fixed by the number of sides; the side length x can vary.

Worked example 7. A square has side length 7 centimetres, written 7 cm. Find its perimeter.

Answer: Use P = 4x, where P is perimeter and x is side length. With x = 7 cm, P = 4 × 7 cm = 28 cm. The result is a length because it measures the boundary.

How are a rectangle’s perimeter and area expressed?

Let l represent its length and b its breadth. These are the lengths of two adjacent sides, meaning sides that meet at a corner.

If p represents the rectangle’s perimeter, then p = l + b + l + b = 2l + 2b. The letters must represent lengths in the same unit before the lengths are added. This formula counts both pairs of opposite sides.

Area measures the surface enclosed by a boundary. Let A represent the rectangle’s area. Then A = lb, where lb means l × b. If both lengths are in centimetres, the area is measured in square centimetres, written cm².

The perimeter formula adds boundary lengths, while the area formula multiplies adjacent side lengths. Defining the symbols helps distinguish these two measurements. The same letters can be used in different problems, but their meanings must be stated in each problem.

How is an algebraic expression evaluated by substitution?

To evaluate an expression is to find its numerical value. Substitution means replacing a variable by a given number. Once the replacement is made, use the ordinary rules of arithmetic to complete the calculation.

What steps make substitution clear?

1. Write the expression and identify what each letter represents.
2. Record the given value of each variable before calculating.
3. Replace each occurrence of that variable by its given value.
4. Carry out the operations, respecting brackets and multiplication before addition or subtraction.

Worked example 8. Evaluate 7k when the variable k has value 4.

Answer: The expression 7k means 7 × k. Substitute k = 4 to obtain 7 × 4 = 28. Writing the multiplication sign during substitution makes the calculation clear.

Worked example 9. Evaluate 5m + 3 when the variable m has value 2.

Answer: Replace m by 2, giving 5 × 2 + 3. Multiply first: 5 × 2 = 10. Then add 3, giving 13. The coefficient 5 multiplies m, while the constant 3 is added afterwards.

How does substitution work with two variables?

For the expression 2l + 2b, let l and b represent a rectangle’s length and breadth in the same unit. If l = 3 and b = 4, substitution gives 2 × 3 + 2 × 4 = 14. Both variable values are needed.

The longer expression l + b + l + b gives 3 + 4 + 3 + 4 = 14 for the same values. This illustrates that simplifying the expression has preserved its value. Each occurrence of l receives the same value, as does each occurrence of b.

Note: Replacing a letter does not mean joining digits together. For k = 4, the expression 7k becomes 7 × 4. A number written next to a letter indicates multiplication.

If a variable has no supplied value, an expression containing it may remain as an expression. Writing a rule, simplifying a rule and evaluating a rule are related tasks, but they do not ask for the same kind of answer.

How is an equation different from an expression?

An equation states that two expressions are equal. It contains an equality sign. An expression gives a quantity or a calculation; an equation makes a statement about equality. For example, 5m + 3 is an expression, where m represents a number.

An unknown is a value we need to find from the information in a problem. A letter can represent an unknown in an equation. The letter has a specified role: we seek a value that makes the equality true.

What are the two sides of an equation?

The left-hand side, abbreviated LHS, is the expression to the left of the equality sign. The right-hand side, abbreviated RHS, is the expression to its right. Both sides must have equal values when a correct answer is substituted.

For example, in 2e = 6, let e represent an unknown number. The LHS is 2e and the RHS is 6. This equation says that twice the unknown number equals 6. The number factor 2 multiplies e.

A solution is a value of the unknown that makes the equation true. Solving an equation means finding such a value. A linear equation in one variable can be written with a degree-one polynomial in one variable equal to a constant.

The equation 2e = 6 is linear: e has power 1. There is one variable, e. By contrast, an expression such as a + 3, with a representing a number, does not by itself tell us what value a must have.

Note: In substitution, the variable’s value is given and the expression’s value is found. In solving an equation, an equality is given and the unknown value is found. Checking a solution uses substitution again.

How can simple equations be solved and checked?

Inverse operations undo one another. Addition and subtraction form an inverse pair; multiplication and division form another. The sign ÷ means division. For a simple equation, use the inverse operation to find the unknown while keeping the equality true.

How is the equality preserved?

Adding the same number to both sides preserves equality, as does subtracting the same number from both sides. Multiplying both sides by the same number also preserves equality. Division by the same non-zero number preserves equality; division by zero is not allowed.

Worked example 10. Solve 2e = 6, where e is an unknown number, and check the answer.

Answer: Divide both sides by 2. Then e = 6 ÷ 2 = 3. Check by replacing e with 3 in the original equation: 2 × 3 = 6. The two sides are equal, so e = 3 is the solution.

Worked example 11. Solve 20 = y − 3, where y is the unknown, and check the result.

Answer: Add 3 to both sides: 20 + 3 = y − 3 + 3. Thus y = 23. Checking in the original equation gives 23 − 3 = 20, which matches its left-hand side.

What should a complete solution show?

1. State which letter is the unknown and write the given equation.
2. Choose an inverse operation that removes the operation attached to the unknown.
3. Apply the operation to both sides and calculate the unknown’s value.
4. Substitute that value into the original equation and compare its two sides.

Another method is trial and error: try values and check whether the sides become equal. It can be inefficient. A trial value is not a confirmed solution until it satisfies the original equation. The final check connects the answer directly to the given information.

Notice that an unknown can occur on either side of the equality sign. In 20 = y − 3, the unknown expression is on the right. The meaning of equality still requires the value on that side to match 20.

Glossary

• Algebra — A branch of mathematics using letters to express numbers, patterns and mathematical relationships.
• Literal number — A letter used to represent a number in a mathematical expression or relationship.
• Variable — A letter representing a number that can take different values in a mathematical relationship.
• Constant — A number whose value stays fixed in the expression or relationship being considered.
• Unknown — A value to be found from the information given in a mathematical problem.
• Algebraic expression — A combination of numbers and letters connected by mathematical operations such as addition and multiplication.
• Term — A part of an expression separated by addition or subtraction, with its associated sign.
• Factor — A number or expression multiplied by another to form a product.
• Numerical coefficient — The number factor multiplying the variable part of an algebraic term.
• Like terms — Terms with the same variable part, including the same powers of all variables involved.
• Polynomial in one variable — A finite sum of terms formed from number coefficients and whole-number powers of one variable.
• Degree — The highest variable power with a non-zero coefficient in a simplified, non-zero polynomial in one variable.
• Substitution — Replacing each occurrence of a variable by its given value in an expression.
• Equation — A mathematical statement that two expressions have equal values, written using an equality sign.
• Solution — A value of the unknown that makes the two sides of an equation equal.

Common errors and misconceptions

• Misconception: A letter in algebra names an object rather than a number. Correct: In the age relationship, a represents Aftab’s numerical age in years. Define the quantity represented before calculating with the letter.
• Misconception: Writing a number next to a letter joins their digits. Correct: Adjacent number and letter symbols indicate multiplication. When k = 4, the expression 7k means 7 × 4 and has value 28.
• Misconception: “4 less than d” means 4 − d. Correct: With d representing a number, subtract 4 from that number. The expression is d − 4. The order matters in subtraction.
• Misconception: Every factor is a separate term. Correct: In 4x + 5y + 3, where x and y are variables, 4x is one term. Its factors 4 and x are multiplied within that term.
• Misconception: Unlike terms can be collected by adding their coefficients. Correct: With c and d representing different prices, 18c and 11d have different variable parts. Their sum remains 18c + 11d.
• Misconception: Degree means the number of terms. Correct: For a non-zero polynomial in one variable, degree is the highest variable power remaining. The polynomial x² + 5x + 1 has degree 2, although it has three terms.
• Misconception: An expression is an equation even without an equality statement. Correct: An equation states that two expressions are equal. Finding an expression’s value and solving an equation are different tasks.
• Misconception: A plausible value is enough to solve an equation. Correct: Check it in the original equation. For 2e = 6, substituting e = 3 gives 2 × 3 = 6 and confirms the equality.

Exam-style questions with model answers

Q1. Shabnam is 3 years older than Aftab. Let a and s represent their respective ages in years, with a for Aftab and s for Shabnam. Write the relationship and find s when a = 23. [2 marks]

1. The relationship is s = a + 3 because Shabnam’s age is obtained by adding 3 to Aftab’s age.
2. Substitute a = 23: s = 23 + 3 = 26. Shabnam is 26 years old.

Q2. Each separate L shape uses 2 matchsticks. Let n be the number of L shapes. Write a general expression, calculate the number of sticks for 5 L shapes, and explain the roles of n and 2. [3 marks]

1. The expression for the total number of matchsticks is 2n, meaning 2 multiplied by n. It counts two sticks for each separate shape.
2. For 5 L shapes, substitute n = 5. The number of sticks is 2 × 5 = 10.
3. The variable n records how many L shapes are made. The constant 2 records the fixed number of sticks needed for each L.

Q3. In the expression 4x + 5y + 3, x and y are variables. List the terms, identify both variable coefficients, and identify the constant term. [4 marks]

1. The expression has three terms: 4x, 5y and 3. Addition joins these separate parts of the expression.
2. The coefficient of x is 4 because 4x means 4 multiplied by x. Multiplication keeps these factors within one term.
3. The coefficient of y is 5 because the variable y is multiplied by the number factor 5.
4. The constant term is 3. It contains no variable and remains fixed when the values assigned to x and y change.

Q4. Evaluate 5m + 3 when m = 2. Explain the multiplication notation, substitute, and calculate in the correct order. [3 marks]

1. The term 5m means 5 × m. The coefficient 5 multiplies the variable; the constant 3 is added to that product.
2. Replace m by its given value 2. The expression becomes 5 × 2 + 3, with the same operations as in the original expression.
3. Perform multiplication before addition. The product is 10, so the final value is 10 + 3 = 13.

Q5. For the polynomial 5y³ + y² + 2y − 1, where y is the variable, give the degree, the coefficients of y³, y² and y, and the constant term. Explain each answer. [5 marks]

1. The degree is 3. This is the highest power of y present with a non-zero coefficient, so the polynomial is cubic.
2. The coefficient of y³ is 5. The term 5y³ means five lots of y³, where y³ means y × y × y.
3. The coefficient of y² is 1. Although no number is written before y², the term means one lot of y².
4. The coefficient of y is 2. In the term 2y, the number factor 2 multiplies the variable y, whose power is 1.
5. The constant term is −1. It includes the subtraction sign and does not depend on the value assigned to the variable y.

Q6. Solve 2e = 6, where e is an unknown number. Explain the inverse operation, apply it to both sides, state the solution, check it, and explain why it solves the equation. [5 marks]

1. The equation says that twice e equals 6. To undo multiplication by 2, choose division by 2 as the inverse operation.
2. Divide both sides by the same non-zero number 2. This preserves the equality and leaves the unknown e on the left.
3. Calculate 6 ÷ 2 = 3, so e = 3. This gives the value of the unknown, rather than the value of twice the unknown.
4. Check the original equation by substituting e = 3. Its left-hand side becomes 2 × 3 = 6, while its right-hand side is 6.
5. The two sides have equal values after substitution. Therefore e = 3 satisfies the original equation and is its solution.

Q7. A shop sells 5 pencils, then 3 pencils, then 10 pencils over three days. Every pencil costs c rupees. Write each day’s amount and simplify the total, explaining why the terms combine. [4 marks]

1. The first day’s amount is 5c rupees, because 5 pencils are sold at the same price c rupees each.
2. The second day’s amount is 3c rupees and the third day’s amount is 10c rupees, using the same price.
3. The total is 5c + 3c + 10c = (5 + 3 + 10)c = 18c rupees.
4. The terms combine because they are like terms with the same variable part c. Adding their coefficients counts the total number of pencils sold.

Key takeaways

• Algebra uses letters to represent numbers, allowing one expression to describe a relationship across different numerical cases.
• Define each variable before using it, and distinguish a changing quantity from a constant that stays fixed.
• Terms are the parts joined by addition or subtraction; factors are multiplied within a term.
• Like terms have the same variable part and powers, so their numerical coefficients can be combined.
• A polynomial’s degree is its highest remaining variable power, rather than the number of terms it contains.
• Substitution replaces variables by given values, after which the expression is evaluated using the rules of arithmetic.
• A formula expresses a relationship; an equation states that two expressions have equal values.
• Solve a simple equation using inverse operations on both sides, then check the answer by substitution.

Test yourself

What does 2n mean when n is the number of separate L shapes and each L needs 2 matchsticks?

It means 2 multiplied by n, giving the total number of matchsticks needed.

In the rule “sum of the first n positive odd numbers = n²”, what does n represent?

It represents how many odd numbers are added, starting with 1.

Let d represent a number. How do you write “4 less than the number”?

Write d − 4 because 4 is subtracted from the number represented by d.

When x represents a number, why is 4x one term rather than two?

The factors 4 and x are multiplied together within one term.

Let x be a variable. What is the degree of x² + 5x + 1?

The degree is 2 because the highest power of x is 2.

Evaluate 7k when the variable k has value 4.

Replace k by 4: 7k becomes 7 × 4, which equals 28.

What is the difference between evaluating an expression and solving an equation?

Evaluation finds an expression’s value from given variable values. Solving finds an unknown value that makes an equation true.

For the equation 20 = y − 3, does y = 23 satisfy the equality?

Yes. Substituting 23 for y gives 23 − 3 = 20, matching the left-hand side.