Algebra for CTET Paper II: expressions, equations, ratio and misconceptions

Ask a Class 6 student what 5x means when x = 2, and some will say 52. Ask them to simplify 2a + 3b and some will write 5ab. Ask them to fill the box in 3 + 5 = ☐ + 2 and many will write 8. None of these students is weak at arithmetic. Each has a specific, wrong idea about what letters, terms or the equals sign mean. Paper II asks the algebra itself, and the pedagogy questions ask whether you can spot these ideas and fix them.

The Paper II syllabus lists introduction to algebra and ratio and proportion, at the NCERT Class 6–8 level. That covers expressions, simple and linear equations in one variable, and the standard identities. This guide takes each in turn, then the misconceptions and teaching.

Letters, expressions and equations

• A variable is a letter that stands for a number, which can take different values. In "n squares need 3n + 1 matchsticks", n can be 1, 2, 3 or any number.
• A constant has a fixed value: 5, −3, ½.
• An expression combines numbers and variables with operations: 3x + 2, 4ab − b².
• Terms are the parts joined by + or −. In 7x²y − 3xy + 5 there are three terms, so it is a trinomial. The coefficient of x²y is 7 and the constant term is 5.
• Like terms have the same variables with the same powers: 3ab and −5ab are like; 3ab and 3a²b are not.
• An equation states that two expressions are equal, and it is true only for certain values: 3x + 2 = 11 is true only when x = 3.
• An identity is true for every value: (a + b)² = a² + 2ab + b².

Value of an expression. Find 2x² − 3x + 1 when x = 3: 2 × 9 − 9 + 1 = 10. When x = −1: 2 × 1 + 3 + 1 = 6.

Simplifying. Combine only like terms.

• 3a + 2b − a + 4b = 2a + 6b.
• 5p + 3q − 2p − q = 3p + 2q.
• 4(x + 3) − 2x = 4x + 12 − 2x = 2x + 12.

Solving equations

The rule is simple: do the same thing to both sides, so the balance stays level. "Transposing" (moving a term across and changing its sign) is a shortcut for this, and should come after the idea is understood.

Worked examples.

1. 3x + 5 = 20. Subtract 5: 3x = 15. Divide by 3: x = 5.
2. 2(x − 3) = 10. Divide by 2: x − 3 = 5, so x = 8.
3. x/3 + 4 = 10. Subtract 4: x/3 = 6. Multiply by 3: x = 18.
4. 5x − 7 = 3x + 9. Subtract 3x: 2x − 7 = 9. Add 7: 2x = 16, so x = 8. Check: 5 × 8 − 7 = 33 and 3 × 8 + 9 = 33.

Always check by putting the answer back into the equation. It turns algebra from a ritual into something the student can verify.

Word problems

Consecutive numbers. The sum of three consecutive numbers is 51. Let them be n, n + 1 and n + 2. Then 3n + 3 = 51, so 3n = 48 and n = 16. The numbers are 16, 17 and 18.

Ages. A parent is three times as old as their child. In 10 years the parent will be twice as old as the child. Let the child be x, so the parent is 3x. In 10 years: 3x + 10 = 2(x + 10), so 3x + 10 = 2x + 20 and x = 10. The child is 10 and the parent 30. Check: in 10 years they will be 20 and 40.

Doubling. A number doubled and increased by 6 gives 20. So 2n + 6 = 20, 2n = 14, and n = 7.

Identities

Ratio and proportion

• A ratio compares two quantities of the same kind: 12 boys to 18 girls is 12 : 18 = 2 : 3.
• Four numbers are in proportion if a : b = c : d, that is, if a × d = b × c. 2 : 3 and 8 : 12 are in proportion because 2 × 12 = 3 × 8 = 24.
• The unitary method finds the value of one unit first. If 6 pens cost ₹90, one costs ₹15, so 10 cost ₹150.

Dividing in a ratio. Divide ₹720 between two people in the ratio 5 : 3. Total parts = 8, so one part = ₹90. Shares: ₹450 and ₹270.

Children's misconceptions in algebra

Teaching algebra so it makes sense

• Start from patterns. Matchstick rows (3n + 1), table patterns and "think of a number" tricks let children discover why a letter is useful. See patterns.
• Generalise arithmetic. Children already know 4 + 7 = 7 + 4 and 9 + 6 = 6 + 9. Writing a + b = b + a says the same thing for all numbers at once.
• Use a balance. A pan balance, drawn or real, with bags of unknown weight and unit weights, makes "do the same to both sides" concrete.
• Algebra tiles or strips of paper for x and small squares for 1 show why 2x + 3 cannot become 5x.
• Substitute and check. Ask children to test an expression with several values: if 2(x + 3) = 2x + 3, try x = 1. They get 8 on one side and 5 on the other.
• Talk in words first. "Three more than twice a number" before 2n + 3.

For how children's errors should be diagnosed and treated, see errors and remedial teaching.

Worked pedagogy question

A student simplifies 4x + 3 as 7x. The most useful response from the teacher is to:

(a) tell the student to learn the rule for like terms
(b) ask the student to substitute x = 2 in both 4x + 3 and 7x and compare
(c) give twenty similar questions
(d) move on, since this is a careless slip

Answer: (b). 4 × 2 + 3 = 11, but 7 × 2 = 14. The student sees the contradiction and has a reason to rethink. This is a misconception, not a careless slip.

Practice set

1. Solve 4x − 7 = 13.
2. Simplify 5p + 3q − 2p − q.
3. Solve 3(x + 2) = 2x + 11.
4. Find the value of a² − 2ab + b² when a = 7 and b = 4.
5. Use an identity to find 49 × 51.
6. Divide ₹1,200 in the ratio 7 : 5.
7. If 8 notebooks cost ₹240, what do 5 notebooks cost?
8. A student says 3 + 4 = ☐ + 2 has the answer 7. What does this show, and how would you help?

Answers

1. 4x = 20, so x = 5.
2. 3p + 2q.
3. 3x + 6 = 2x + 11, so x = 5. Check: 3 × 7 = 21 and 10 + 11 = 21.
4. (a − b)² = 3² = 9. Check directly: 49 − 56 + 16 = 9.
5. (50 − 1)(50 + 1) = 2,500 − 1 = 2,499.
6. 12 parts, one part ₹100: ₹700 and ₹500.
7. One costs ₹30, so ₹150.
8. The student reads = as "write the answer" rather than "is equal to". Use a balance: 3 + 4 on one side, ☐ + 2 on the other. The box must be 5.

What to do next

• Learn the four identities and use each for one quick mental calculation.
• Solve ten equations, checking every answer by substitution.
• Write the misconceptions table from memory with one teaching fix for each.
• Continue with the nature of mathematics.