Secondary 1 G3 Mathematics Tutor Bukit Timah | Rational and Irrational Numbers, Fractions and Number Lines

Did you know? −3/4 is smaller than −2/3, even though three quarters is a bigger fraction than two thirds. Negative numbers turn the usual visual comparison around: −0.75 lies farther left on the number line than approximately −0.667. The challenge is not learning a new trick. It is understanding what the minus sign does to the order of numbers.

If you are looking for a Secondary 1 G3 Mathematics tutor in Bukit Timah, rational numbers provide an excellent early diagnostic. Fractions, integers, decimals and negative signs appear in later algebra and geometry, so a fragile understanding here can make other chapters seem unexpectedly difficult.

In this guide, I’ll show how the number families fit together, why some decimals end while others repeat, how to compare negative fractions and how to check mixed calculations without relying entirely on a calculator.

The quick answer: what is a rational number?

A rational number can be written as p/q, where p and q are integers and q is not zero. Examples include 4, −7, 1/3, −5/8, 0.25 and 0.333… with the 3 repeating forever.

An irrational number cannot be expressed as a ratio of two integers. Familiar examples are √2 and π. Their decimal representations neither terminate nor repeat in a fixed recurring block.

Real numbers include both rational and irrational numbers. They can be represented as points on the real number line. Some are familiar fractions; others, such as √2, cannot be written exactly as ordinary integer fractions.

For Secondary 1, a useful goal is to identify the number’s family and select an appropriate form for the calculation. A fraction is often better for exact working, while a decimal may be helpful for estimation or placing values approximately on a number line.

Whole numbers, integers and rational numbers

Start with whole numbers: 0, 1, 2, 3 and so on. Integers extend this list to include negatives such as −1 and −2. Every integer is rational because, for example, −7 can be expressed as −7/1.

Fractions such as 3/5 and −11/4 are also rational. They do not need to describe parts of an object; they can represent exact signed quantities on a number line.

A decimal such as 0.625 is rational because 0.625 = 625/1000 = 5/8. Decimal appearance alone does not tell you whether a number is rational. You need to recognise whether the expansion terminates or repeats.

For example, 0.121212… is rational when the block 12 repeats without end: it equals 12/99 = 4/33. The dots only establish a recurring decimal if the repeated pattern is stated or clear.

Worked example 1: ordering negative fractions

Arrange −3/4, −2/3 and −1/2 from smallest to largest.

Use a common denominator of 12. Rewrite the values as −9/12, −8/12 and −6/12. The smallest is −9/12 because it lies farthest left on the number line. Therefore the correct ascending order is −3/4, −2/3, −1/2.

Another check is to use decimals: −0.75, about −0.667, and −0.5. The ordering agrees.

A common mistake is to ignore the sign and arrange fractions by their positive magnitudes. Ask the pupil to locate −1 on the number line and place all three fractions between −1 and 0. The left-to-right arrangement becomes easier to explain.

Worked example 2: adding fractions with different denominators

Calculate 3/4 − 5/6.

The denominators 4 and 6 need a common multiple. The lowest common multiple is 12. Convert 3/4 to 9/12 and 5/6 to 10/12. The difference is 9/12 − 10/12 = −1/12.

Check the sign before accepting the answer. Three quarters is 0.75 while five sixths is about 0.833. Subtracting the larger number from the smaller should produce a negative result.

A wrong method is to subtract numerator from numerator and denominator from denominator, getting (3 − 5)/(4 − 6) = 1. Denominators tell us the size of the parts; the parts must be made comparable before addition or subtraction.

Worked example 3: multiplying signed fractions

Calculate (−2/3) × (9/4).

Multiply numerators and denominators: −18/12. Simplify by dividing by 6 to obtain −3/2, or −1.5.

The sign is negative because a negative factor multiplied by a positive factor gives a negative product. This is the same signed-number rule used with integers.

You could simplify first: 9 ÷ 3 = 3 and 2 ÷ 4 = 1/2. Then the remaining product is −3/2. Both routes are valid because cancelling occurs between factors in a multiplication.

Do not cancel digits or remove a term across addition. Cancelling factors in multiplication is not permission to cross out any repeated symbol you notice.

Worked example 4: division by a fraction

Calculate (−5/6) ÷ (2/3).

Dividing by a nonzero number is equivalent to multiplying by its reciprocal. Therefore (−5/6) × (3/2) = −15/12 = −5/4.

Check with an approximate estimate: −5/6 is about −0.833. Dividing by 2/3, a number less than one but greater than zero, makes the magnitude larger. A result of −1.25 is plausible.

The rule only applies when the divisor is nonzero. Division by zero is undefined, whether written as 5 ÷ 0 or 5/0.

A tutor should explain the reciprocal as a consequence of division, not merely tell students to “flip the second fraction” without conditions.

Mixed calculations and order of operations

Consider 1/2 + (−3/4) × 2.

Perform multiplication before addition: (−3/4) × 2 = −3/2. Then 1/2 + (−3/2) = −2/2 = −1.

A rushed pupil may add 1/2 to −3/4 first and multiply everything by 2. That answers a different expression, namely [1/2 + (−3/4)] × 2.

Brackets, multiplication and addition control what the expression means. An explicit line of working helps avoid an invisible change in the question.

Terminating and recurring decimals

A terminating decimal has a finite number of decimal places. Examples are 0.5, 0.25 and 0.125.

A recurring decimal repeats one digit or a block of digits. One third is 0.333… and two elevenths is 0.181818…, with the indicated digits repeating forever.

Why do terminating decimals work as fractions? Every decimal with finitely many places can be written over a power of ten: 0.125 = 125/1000 = 1/8.

A repeating decimal also represents an exact rational value even though writing its decimal expansion never ends. For example, let x = 0.272727… . Then 100x = 27.272727… . Subtract x to obtain 99x = 27, so x = 27/99 = 3/11.

This last algebra method is an optional demonstration of why repetition implies rationality. A Secondary 1 learner does not need it to compare simple recurring decimals correctly.

Why √2 is different

The square root of 49 is exactly 7, so √49 is rational. The square root of 2 cannot be expressed as a ratio of two integers and is irrational.

A calculator may show √2 as approximately 1.41421356. That display is a rounded approximation, not the complete exact value. Writing 1.4142 does not turn √2 into a rational number; it replaces the exact irrational value with a rational approximation.

Likewise π is irrational, while 22/7 is rational. The fraction 22/7 can approximate π, but it does not equal π exactly.

When a question asks for an exact result, leave √2 or π in exact notation when appropriate. When it requests a decimal to a specified accuracy, provide an approximation with the required units or significant figures.

Where irrational numbers appear on a number line

Since 1² = 1 and 2² = 4, the positive number √2 lies between 1 and 2. Since 1.4² = 1.96 and 1.5² = 2.25, √2 lies between 1.4 and 1.5.

The estimate becomes more precise as you test narrower intervals. You do not need to pretend that a rounded decimal is the exact square root.

A real number line holds rational and irrational numbers together. Understanding this prevents the mistaken belief that every number between two integers must be a simple fraction.

A useful word problem: sharing a negative balance

Imagine a fictional account balance of −$18 shared equally as a liability among three people. Each person’s share of the balance is −18 ÷ 3 = −$6.

The negative sign describes a debt or negative balance, not an instruction to treat the absolute value as positive income. A calculation that gives +$6 has changed the meaning of the situation.

Signed fractions can model temperatures below zero, changes in elevation and other quantities with reference points. The arithmetic should preserve the interpretation of the sign.

Mini diagnostic: ten questions

• A. Is −9 a rational number? Explain.
• B. Is 0.125 rational? Write it as a simplified fraction.
• C. Which is smaller: −2/5 or −3/4?
• D. Calculate 2/3 + 1/6.
• E. Calculate 3/4 − 5/6.
• F. Calculate (−2/3) × (9/4).
• G. Calculate (−5/6) ÷ (2/3).
• H. Is √49 irrational or rational?
• I. Give an irrational number between 1 and 2.
• J. Calculate 1/2 + (−3/4) × 2.

Answers and checks

A. Yes. −9 = −9/1, a ratio of integers with nonzero denominator.

B. Yes. 0.125 = 125/1000 = 1/8.

C. −3/4 = −0.75 is smaller than −2/5 = −0.4.

D. 2/3 + 1/6 = 4/6 + 1/6 = 5/6.

E. 3/4 − 5/6 = 9/12 − 10/12 = −1/12.

F. −18/12 = −3/2.

G. −5/6 × 3/2 = −15/12 = −5/4.

H. Rational. √49 = 7.

I. √2 is between 1 and 2 and is irrational.

J. Perform multiplication first: −3/2 + 1/2 = −1.

An incorrect answer tells us where to look. If C is wrong, revisit the number line. If E is wrong, review common denominators. If J is wrong, inspect calculation order before discussing arithmetic speed.

A fifteen-minute repair routine

Start with a number-line drawing showing −1, 0 and 1. Ask the child to place −3/4, −1/2 and 1/4. Then give one fraction addition and one fraction multiplication question, requesting verbal explanations.

Next show a terminating decimal, a recurring decimal and √2. Ask the learner to classify each without guessing based on the number of visible digits. End with a changed problem containing negative fractions and brackets.

Return to a similar problem in a later lesson without giving the model. Independent correct reasoning is a stronger sign of progress than copying a fully worked solution.

What Bukit Timah parents should ask a Mathematics tutor

A useful tutor can distinguish a weak fraction concept from a sign error, a denominator mistake and an order-of-operations mistake. The same wrong final answer can arise from different causes.

The original eduKateSG Secondary 1 Mathematics small-group tutorial guide describes focused three-student tutorials near Sixth Avenue MRT. The Bukit Timah Secondary 1 G3 Mathematics tuition overview gives the corresponding location-specific programme context. Parents should confirm suitability, arrangements and availability directly.

The MOE G2 and G3 Mathematics syllabus is a useful curriculum reference for number concepts. Individual schools may sequence and assess content differently.

Frequently asked questions

Is every integer a rational number?

Yes. Any integer n can be written as n/1.

Is 0.333… an irrational number because it never ends?

No. When the 3 repeats indefinitely, it represents the exact rational value 1/3. Irrational decimal expansions are nonterminating and nonrecurring.

Is −3/4 greater than −2/3 because 3/4 is greater than 2/3?

No. The negative signs reverse their positions on the number line. −3/4 is farther left and therefore smaller.

Can I always replace √2 with 1.41?

Only if an approximation to that accuracy is appropriate. √2 is exact; 1.41 is an approximate value.

What should a struggling child practise first?

Begin with signed numbers, fraction equivalence and common denominators before attempting lengthy mixed calculations. The negative-number sign guide can help identify that foundation.

Continue learning

• Expressions with fractional coefficients
• Linear equations containing fractions
• Significant figures and estimation
• Algebraic expressions and brackets

A number is not difficult merely because it is written as a fraction, decimal or square root. The skill is to know what the form means and whether it represents an exact value or an approximation.