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Class 10 Maths · Common Mistakes

Common Mistakes in Real Numbers

Real Numbers looks easy because many questions begin with familiar ideas like HCF, LCM, prime factorisation, and decimals. The marks are usually lost in the reasoning: students know the method, but do not write why a step is allowed or how one fact leads to the next.

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First diagnose the exact Real Numbers gap

Can the student use Euclid's division lemma without mixing up dividend, divisor, quotient, and remainder?
Can they explain why HCF × LCM = product of the two numbers works only for two positive integers?
Can they decide whether a rational number has a terminating decimal by checking the prime factors of the denominator after simplification?
Can they write an irrationality proof in the correct contradiction format instead of only stating that something is irrational?

The mistakes that cost marks in Real Numbers

Using prime factorisation without simplifying the fraction first

For decimal expansion questions, students often inspect the original denominator. The correct check is done after the fraction is reduced to lowest terms.

How to repair it with Eduro

Before deciding terminating or non-terminating, ask Eduro to reduce the fraction, factorise the denominator, and explain why only powers of 2 and 5 matter.

Treating HCF and LCM as separate tricks

Students memorise formulas but do not see HCF as the common part and LCM as the smallest shared multiple built from prime powers.

How to repair it with Eduro

Make the student write prime factorisations side by side, circle the common factors for HCF, and underline the highest powers for LCM.

Skipping the reason in Euclid's algorithm

A solution may show repeated division steps, but the student loses marks when they do not clearly identify the last non-zero remainder as the HCF.

How to repair it with Eduro

Ask Eduro to make the student narrate each division line in words: what is divided, what remains, and why the process continues.

Writing weak irrationality proofs

Many students write 'root 2 is irrational, so this is irrational' without setting up contradiction or showing how the assumption fails.

How to repair it with Eduro

Use a proof skeleton: assume rational, write in lowest terms, square if needed, show both numerator and denominator become divisible by the same prime, then state the contradiction.

Practice prompts that reveal understanding

"Give me two numbers and ask me to find HCF by Euclid's algorithm, then by prime factorisation, and compare both methods."
"Create five decimal expansion questions where I must first reduce the fraction before deciding terminating or non-terminating."
"Give me one wrong Real Numbers solution and ask me to identify the exact line where the mistake starts."
"Ask me to explain why the product of HCF and LCM equals the product of two numbers using prime factors."

Parent note

If a child says Real Numbers is easy, ask them to explain one proof and one decimal expansion rule aloud. If the explanation is only a memorised line, Eduro should slow the chapter down before the student moves to harder algebra.