Mathematics · Class 10 · Numbers and Algebraic Structures · Term 1

Decimal Expansions of Rational Numbers

Students will explore the conditions for terminating and non-terminating repeating decimal expansions of rational numbers.

• CBSE Learning Outcomes
• NCERT: Real Numbers - Class 10

About This Topic

Decimal expansions of rational numbers are either terminating or non-terminating repeating. Students examine fractions in lowest terms to find that terminating decimals occur when the denominator has prime factors of only 2 and 5. For example, 1/8 = 0.125 terminates, while 1/3 = 0.333... repeats. They practise predicting the type by factorising denominators, avoiding lengthy division, and link this to the structure of rational numbers.

This topic anchors the Real Numbers unit in CBSE Class 10 Mathematics. It builds skill in prime factorisation and number theory, essential for distinguishing rationals from irrationals, whose decimals neither terminate nor repeat. Mastery here supports solving problems in algebra and geometry, fostering logical reasoning.

Active learning excels for this abstract rule. Sorting fraction cards into categories, predicting in pairs then verifying with calculators, or charting patterns from long division makes the criterion concrete. Group discussions resolve errors, reinforcing the denominator rule through shared discovery.

Key Questions

1. Explain the relationship between the prime factors of a denominator and the nature of a decimal expansion.
2. Predict whether a given rational number will have a terminating or non-terminating decimal without division.
3. Differentiate between the decimal representations of rational and irrational numbers.

Learning Objectives

• Analyze the prime factors of the denominator of a rational number to predict the nature of its decimal expansion.
• Calculate the decimal expansion of a rational number and classify it as terminating or non-terminating repeating.
• Differentiate between the decimal representations of rational numbers and irrational numbers.
• Explain the condition under which a rational number's decimal expansion terminates.
• Identify rational numbers that will result in non-terminating repeating decimal expansions based on their denominator's prime factors.

Before You Start

Prime Factorization

Why: Students need to be proficient in finding the prime factors of a number to analyze the denominator of a rational number.

Fractions and Simplification

Why: Understanding how to express fractions in their lowest terms is essential before applying the rule about denominator factors.

Basic Division

Why: While the topic aims to avoid long division, a foundational understanding of division is necessary for grasping the concept of decimal expansions.

Key Vocabulary

Rational Number	A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Its decimal expansion is either terminating or non-terminating repeating.
Terminating Decimal	A decimal expansion that ends after a finite number of digits. For a rational number p/q in its lowest terms, this occurs if the prime factors of q are only 2 and/or 5.
Non-terminating Repeating Decimal	A decimal expansion that continues infinitely with a pattern of digits repeating. For a rational number p/q in its lowest terms, this occurs if the prime factors of q include primes other than 2 and 5.
Prime Factorization	The process of finding the prime numbers that multiply together to make the original number. This is crucial for examining the denominator's factors.

Watch Out for These Misconceptions

Common Misconception

All short decimals terminate; long ones repeat.

What to Teach Instead

Length does not determine the type; only denominator factors matter after lowest terms. Hands-on sorting cards helps students test examples like 1/2 (short, terminating) versus 1/6 (longer but terminating), shifting focus to the rule.

Common Misconception

Numerator factors decide terminating or repeating.

What to Teach Instead

Denominator primes alone govern this, post-simplification. Prediction races in pairs reveal this when students simplify first, compare predictions, and verify, correcting the error through repeated practice.

Common Misconception

Repeating decimals of rationals always repeat from the start.

What to Teach Instead

They may have non-repeating digits before repeating, like 1/6 = 0.1666.... Group division relays expose these patterns, helping students identify blocks accurately via peer comparison.

Active Learning Ideas

Card Sort: Terminating vs Repeating Fractions

Prepare 20 fraction cards in lowest terms. Small groups sort them into terminating and repeating piles based on denominator factors. Groups then select five from each to verify using long division or calculators, noting patterns.

35 min·Small Groups

Prediction Challenge: Factor and Classify

List 15 fractions on board. Pairs predict terminating or repeating by factorising denominators, record predictions. Switch pairs to verify one prediction each via division, discuss matches or surprises.

30 min·Pairs

Pattern Hunt: Long Division Relay

In small groups, assign fractions for relay division. First student starts long division, passes to next after three decimal places to continue and identify repeating blocks. Group compiles a class chart of findings.

40 min·Small Groups

Decimal Match-Up: Whole Class Game

Display fractions and decimal expansions shuffled. Whole class calls out matches, justifying with denominator rule. Use projector for visibility, award points for correct predictions before revealing.

25 min·Whole Class

Real-World Connections

• Financial analysts use decimal expansions to represent interest rates and currency conversions precisely. For instance, calculating compound interest on a loan involves precise decimal calculations, and understanding terminating versus repeating decimals ensures accuracy in financial reporting.
• Engineers designing digital circuits often work with binary representations of numbers, which are essentially decimal expansions. The ability to predict the nature of these expansions helps in optimizing data storage and processing efficiency for devices like smartphones and computers.

Assessment Ideas

Quick Check

Present students with a list of rational numbers (e.g., 3/16, 7/12, 5/20, 11/30). Ask them to write down the prime factors of each denominator and then predict whether the decimal expansion will terminate or repeat, without performing division.

Exit Ticket

Give each student a card with a rational number like 13/40. Ask them to: 1. State the prime factors of the denominator. 2. Determine if the decimal expansion terminates or repeats. 3. Write the decimal expansion to verify their prediction.

Discussion Prompt

Pose the question: 'Can a rational number with a denominator containing prime factors of 3 and 7 have a terminating decimal expansion?' Facilitate a class discussion where students justify their answers by referring to the prime factorization rule.

Frequently Asked Questions

How do you determine if a rational number has a terminating decimal expansion?

Simplify the fraction to lowest terms, then check the denominator's prime factors. It terminates if factors are only 2 and/or 5, like 3/40 (denominator 2^3 * 5). Practise by factorising quickly; this avoids full division and builds confidence in predictions.

What causes non-terminating repeating decimals in rational numbers?

If the denominator in lowest terms has prime factors other than 2 or 5, such as 3 or 7, the decimal repeats. For 1/7 = 0.142857142857..., the block repeats. Students learn the length of repeat ties to the denominator's factors through pattern observation.

How can active learning help students understand decimal expansions?

Activities like card sorts and prediction challenges engage students in classifying fractions by denominator rules before verifying. Pair work and relays promote discussion of errors, making abstract criteria tangible. Class charts of real examples reinforce patterns, improving retention over rote memorisation.

How to predict decimal type without long division?

Factorise the denominator after simplifying the fraction. Terminating if only 2s and 5s; else repeating. Examples: 17/80 (2^4 * 5, terminates), 1/11 (11, repeats). Quick factor trees in groups speed this skill, linking to NCERT exercises effectively.