Did you know? A student can add two whole numbers correctly and still get a fraction question wrong by doing exactly the same thing to its numerator and denominator. The missing idea is that fractions describe parts of a particular size. You have to make the parts comparable before adding them.
For Secondary 1 G2 Mathematics students in Bukit Timah, the fastest useful routine is identify the operation, preserve negative signs, convert mixed numbers when needed, use equivalent fractions for addition or subtraction, and check the result’s sign and size. Below are explanations and worked questions you can practise directly.
These lessons develop the number foundations in the MOE G2 Mathematics syllabus. The local small-group teaching reference is the unchanged eduKateSG Secondary 1 Mathematics tutorial page, describing weekly 90-minute three-pupil tutorials near Sixth Avenue MRT. Actual school chapter order may differ.
Quick rules to remember
- For addition/subtraction: use a common denominator, then combine numerators.
- For multiplication: multiply numerators and denominators; simplify common factors.
- For division: multiply by the reciprocal of the nonzero divisor.
- For mixed numbers: convert to improper fractions when operations become complicated.
- For signs: predict positive or negative before the calculation.
- For final answers: simplify and follow the requested fraction or mixed-number format.
How to tell which fraction method fits
The operation in the question matters. A common denominator is essential to combining fractions as sums or differences, but not to multiplying them. Dividing by a fraction is best understood by asking how many portions of that size fit. And when negatives appear, a number line is a simple independent check.
This distinction helps parents diagnose a mistake more usefully. Instead of saying a child is careless, ask whether the error began with selecting an operation, changing equivalent fractions, using a reciprocal or dropping a sign. Each cause can be repaired with a short example.
Twenty concepts explained with worked reasoning
1. The denominator names the size of a part
Before adding or subtracting fractions, picture them as measurements of the same whole. One third and one fifth are different-sized parts, so adding their numerators as if they were equal pieces changes the value.
Worked example: For 1/3 + 1/5, fifteen is a common denominator. Five fifteenths plus three fifteenths equals 8/15.
Common mistake: A learner writes 2/8 by adding both visible numerators and denominators.
Teaching repair: Use a fraction strip with equal wholes and show why common units are necessary.
2. Equivalent fractions change the notation, not the value
Multiplying numerator and denominator by the same nonzero integer creates another name for the same number. This is the foundation for common denominators, simplifying and fraction comparison.
Worked example: 3/4 = 6/8 = 9/12. All occupy the same point at 0.75 on a number line.
Common mistake: Changing only the denominator produces a different fraction.
Teaching repair: Ask the learner to demonstrate the equivalence by both multiplication and a labelled diagram.
3. The lowest common denominator reduces unnecessary arithmetic
Any common denominator can work, but the lowest common multiple of the denominators often keeps the numbers smaller and corrections easier to inspect.
Worked example: For 5/6 – 1/4, the lowest common denominator is twelve. Ten twelfths minus three twelfths is 7/12.
Common mistake: Students may multiply denominators automatically to get twenty-four, then make avoidable arithmetic errors.
Teaching repair: List multiples briefly, find twelve and use it without claiming larger common denominators are mathematically invalid.
4. Subtracting fractions keeps the operation sign
Fraction subtraction means removing a quantity. With unlike denominators, translate both to common-sized parts before subtracting the numerators.
Worked example: 7/8 – 1/3 = 21/24 – 8/24 = 13/24.
Common mistake: A student may subtract the denominators as well or forget that the order of subtraction matters.
Teaching repair: Estimate whether the result should be between zero and one before calculating.
5. Multiplying fractions is not the same as adding them
A product of fractions can be interpreted as ‘a fraction of another quantity’. Multiply numerators and denominators, then reduce when possible.
Worked example: 2/3 × 3/5 = 6/15 = 2/5. It is two-thirds of three-fifths of a whole.
Common mistake: Some students force a common denominator before multiplication, adding work without need.
Teaching repair: Draw a rectangle partitioned into thirds in one direction and fifths in another to show overlapping parts.
6. Cancellation should simplify factors, not arbitrary digits
Before multiplying fractions, common factors in a numerator and another denominator can be divided out because the multiplication is a product of factors.
Worked example: For 3/8 × 4/9, reduce 3 with 9 and 4 with 8 to get 1/2 × 1/3 = 1/6.
Common mistake: Crossing out digits in sums or differences as though they were factors leads to invalid shortcuts.
Teaching repair: Rewrite numbers as factors first, then show exactly which common factor has been cancelled.
7. Dividing by a fraction asks how many groups fit
Dividing by one-half asks how many half-sized portions fit in the starting amount. This explains why dividing by a number between zero and one can increase the result.
Worked example: 3/4 ÷ 1/2 = 3/4 × 2 = 3/2, because one and a half half-units fit in three quarters.
Common mistake: A learner may assume division always makes a number smaller.
Teaching repair: Use a measuring strip and then justify multiplication by the reciprocal.
8. The reciprocal is an inverse factor
For a nonzero fraction a/b, the reciprocal b/a gives product one. That inverse relationship justifies changing division by a fraction into multiplication by its reciprocal.
Worked example: (2/3) × (3/2) = 1, so 4/5 ÷ 2/3 = 4/5 × 3/2 = 6/5.
Common mistake: Students invert the first fraction instead of the divisor.
Teaching repair: Place a box around the fraction after the division sign before taking its reciprocal.
9. Mixed numbers can be converted to improper fractions
A mixed number such as 1 2/3 means one whole and two thirds. It can be written as 5/3 without changing its value. Improper fractions are usually easier for multiplying or dividing.
Worked example: 1 2/3 × 3/5 = 5/3 × 3/5 = 1.
Common mistake: A student reads 1 2/3 as 1 × 2/3 rather than as 1 + 2/3.
Teaching repair: Say ‘one and two-thirds’ aloud before turning it into five thirds.
10. Addition of mixed numbers can be done in parts
For a positive mixed-number sum, adding wholes and fractions separately is sensible as long as the fractional parts are added using common units and converted when they exceed one.
Worked example: 1 3/4 + 2 1/2 = 3 + 3/4 + 2/4 = 4 1/4.
Common mistake: The fractional total 5/4 may be left incorrectly attached to a whole part without regrouping.
Teaching repair: Convert to improper fractions to cross-check the result.
11. Subtraction of mixed numbers may require regrouping
When the fractional part being subtracted is larger than the fraction present, exchange one whole for equivalent fractional parts. Alternatively convert both mixed numbers to improper fractions.
Worked example: 3 1/4 – 1 3/4 = 13/4 – 7/4 = 6/4 = 1 1/2.
Common mistake: A learner tries to subtract 3/4 from 1/4 and silently changes the sign.
Teaching repair: A fraction strip or improper-fraction method keeps the quantity intact.
12. A negative sign can apply to an entire mixed number
The expression -1 1/2 ordinarily means negative one and a half, or -3/2. In formal working, brackets remove ambiguity when this value is substituted.
Worked example: -1 1/2 + 3/4 = -6/4 + 3/4 = -3/4.
Common mistake: Students treat the minus as applying only to the whole-number part.
Teaching repair: Rewrite as -(1 + 1/2) and then as -3/2 before operating.
13. Signed fraction addition follows the number line
Adding a positive value moves right; adding a negative value moves left. A common denominator makes those movements countable.
Worked example: -2/3 + 1/6 = -4/6 + 1/6 = -3/6 = -1/2.
Common mistake: Dropping the first negative sign when changing denominators turns a sensible negative result into a positive one.
Teaching repair: Draw a short line from -1 to 0 and check that the final point is closer to zero than -2/3.
14. Subtracting a negative fraction means adding its opposite
The minus sign between two quantities is an operation; the negative sign attached to a quantity is part of its value. Both must remain visible until the expression is simplified.
Worked example: 1/4 – (-1/2) = 1/4 + 2/4 = 3/4.
Common mistake: A student treats two separate minus signs as one subtraction and obtains -1/4.
Teaching repair: Use brackets around the negative fraction and translate the operation into words first.
15. Multiplying signed fractions follows sign rules
Multiply the magnitudes as ordinary fractions, then decide the sign. An odd number of negative factors gives a negative product; an even number gives positive.
Worked example: (-2/3) × (-3/4) = 6/12 = 1/2. But (-2/3) × 3/4 = -1/2.
Common mistake: A learner produces a negative answer whenever a minus appears, ignoring the count of negative factors.
Teaching repair: Predict the sign before calculating the fraction.
16. Dividing by a negative fraction changes the sign
The reciprocal retains the sign of the nonzero divisor. The multiplication sign rules then determine the sign of the quotient.
Worked example: (3/5) ÷ (-9/10) = (3/5) × (-10/9) = -2/3.
Common mistake: Forgetting the minus while inverting the divisor gives positive two-thirds.
Teaching repair: Multiply the answer by the original divisor to check that it reconstructs 3/5.
17. Order of operations also applies to fraction expressions
Brackets are handled before operations outside them, and multiplication/division take priority over addition/subtraction at the same level. Fraction bars group entire numerators or denominators.
Worked example: 1/2 + 3/4 × 2/3 = 1/2 + 1/2 = 1. But (1/2 + 3/4) × 2/3 = 5/4 × 2/3 = 5/6.
Common mistake: A student adds the two fractions first despite no brackets.
Teaching repair: Draw a box around the multiplication part that must be evaluated first.
18. Fractions support algebraic coefficients
The rules for rational numbers carry into terms such as 2x/3 and x/6. Combining these terms relies on the same equivalent-fraction reasoning.
Worked example: 2x/3 + x/6 = 4x/6 + x/6 = 5x/6.
Common mistake: A student adds denominators or treats unlike symbolic structures as directly addable.
Teaching repair: First solve the number version 2/3 + 1/6; then attach the common x factor.
19. Estimate whether a fraction answer is plausible
A rough location near zero, one half or one can expose errors before finalising a calculation. This is particularly helpful when denominators are large.
Worked example: For 7/8 – 1/3, the answer should be around 0.9 – 0.3 = 0.6; 13/24 ≈ 0.542 is plausible, whereas 12 is not.
Common mistake: The fraction is calculated mechanically without checking the scale.
Teaching repair: Place the input values approximately on a number line, then compare the answer.
20. Keep the final form appropriate to the problem
A question may request an improper fraction, mixed number, simplest form, decimal or an amount with a unit. A mathematically equal answer may still need to be expressed in the requested form.
Worked example: 9/4 equals 2 1/4. If the task asks for a mixed number, write 2 1/4; if it asks for an improper fraction, keep 9/4.
Common mistake: Stopping at 18/8 despite a request for simplest form creates avoidable mistakes.
Teaching repair: Circle the requested final form before starting and check it at the end.
30 worked fraction exercises
Try each calculation on paper before reading its explanation. If an example is wrong, identify the first invalid transformation and redo a fresh variation without the worked answer in view.
Exercise 1: 1/3 + 1/6
Answer: 1/2. Method: 2/6 + 1/6 = 3/6.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 2: 2/5 + 1/2
Answer: 9/10. Method: 4/10 + 5/10 = 9/10.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 3: 3/4 + 2/3
Answer: 1 5/12. Method: 9/12 + 8/12 = 17/12.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 4: 5/6 – 1/4
Answer: 7/12. Method: 10/12 – 3/12 = 7/12.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 5: 7/8 – 1/3
Answer: 13/24. Method: 21/24 – 8/24 = 13/24.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 6: 1/2 – 3/4
Answer: -1/4. Method: 2/4 – 3/4 = -1/4.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 7: 2/3 × 3/5
Answer: 2/5. Method: 6/15 reduces to 2/5.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 8: 3/8 × 4/9
Answer: 1/6. Method: 12/72 reduces to 1/6.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 9: 5/6 × 9/10
Answer: 3/4. Method: 45/60 reduces to three quarters.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 10: 3/4 ÷ 1/2
Answer: 1 1/2. Method: 3/4 × 2 = 3/2.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 11: 4/5 ÷ 2/3
Answer: 1 1/5. Method: 4/5 × 3/2 = 6/5.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 12: 7/10 ÷ 7/5
Answer: 1/2. Method: 7/10 × 5/7 = 1/2.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 13: 1 2/3 × 3/5
Answer: 1. Method: 5/3 × 3/5 = 1.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 14: 1 3/4 + 2 1/2
Answer: 4 1/4. Method: 7/4 + 10/4 = 17/4.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 15: 3 1/4 – 1 3/4
Answer: 1 1/2. Method: 13/4 – 7/4 = 6/4.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 16: 2 1/2 ÷ 5/4
Answer: 2. Method: 5/2 × 4/5 = 2.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 17: -2/3 + 1/6
Answer: -1/2. Method: -4/6 + 1/6 = -3/6.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 18: -3/4 + 1/2
Answer: -1/4. Method: -3/4 + 2/4 = -1/4.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 19: -1/2 – 2/3
Answer: -1 1/6. Method: -3/6 – 4/6 = -7/6.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 20: 1/4 – (-1/2)
Answer: 3/4. Method: 1/4 + 2/4 = 3/4.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 21: (-2/3) × (-3/4)
Answer: 1/2. Method: Two negative factors give positive 6/12.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 22: (-3/5) × 10/9
Answer: -2/3. Method: -30/45 reduces to -2/3.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 23: (3/5) ÷ (-9/10)
Answer: -2/3. Method: 3/5 × (-10/9) = -2/3.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 24: 1/2 + 3/4 × 2/3
Answer: 1. Method: Multiply to obtain 1/2, then add 1/2.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 25: (1/2 + 3/4) × 2/3
Answer: 5/6. Method: Sum is 5/4; product is 10/12.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 26: 2/3 of 45
Answer: 30. Method: 45 × 2/3 = 30.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 27: 3/4 of 28
Answer: 21. Method: One quarter is seven, so three quarters is twenty-one.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 28: -1 1/2 + 3/4
Answer: -3/4. Method: -6/4 + 3/4 = -3/4.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 29: Distance between -3/4 and 1/4
Answer: 1. Method: One full unit separates the two points.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
Exercise 30: 2x/3 + x/6
Answer: 5x/6. Method: Collect coefficients 4/6 and 1/6.
To check understanding, ask which rule permitted the step, whether the result’s sign and magnitude make sense, and how an inverse operation or equivalent form could verify it. A correct answer with no explainable method deserves a later independent check.
A short four-week improvement plan
Week 1 — equal-sized parts
Use shaded strips, equivalent fractions, common denominators and ten short addition/subtraction tasks. Compare the answer with a simple estimate.
End with one unfamiliar question completed without prompts. That independent transfer is a stronger sign of progress than finishing a long familiar worksheet.
Week 2 — multiplication and division
Begin with fraction-of-a-quantity pictures and dividing into fractional groups. Introduce the reciprocal through its inverse relationship rather than as an unexplained trick.
End with one unfamiliar question completed without prompts. That independent transfer is a stronger sign of progress than finishing a long familiar worksheet.
Week 3 — mixed numbers and negative fractions
Practise converting mixed numbers, attaching negative signs to the entire quantity and using brackets. Draw number lines to check mixed-sign results.
End with one unfamiliar question completed without prompts. That independent transfer is a stronger sign of progress than finishing a long familiar worksheet.
Week 4 — mixed problems without chapter hints
Blend addition, subtraction, multiplication, division and a few word problems. Keep an error ledger and revisit corrected skills after two or three days.
End with one unfamiliar question completed without prompts. That independent transfer is a stronger sign of progress than finishing a long familiar worksheet.
What parents can notice in a small-group tutorial
A three-student Mathematics session can reveal three different problems behind the same wrong fraction answer. One learner does not recognise that the units differ; another loses a minus sign; a third uses a reciprocal on the wrong fraction. A tutor who listens to the working can target the precise cause and check it again later.
The eduKateSG teaching reference describes guided practice and feedback in a three-pupil, 90-minute weekly tutorial near Sixth Avenue MRT. It is a description of the teaching approach, not a promise of exam outcomes or a live statement of class availability.
Frequently asked questions
Why can’t my child add the denominators?
Denominators indicate the size of each part. Different-sized parts must be expressed in a common unit before their counts can be combined.
Do we need a common denominator for multiplication?
No. Multiply numerators and denominators directly, simplifying common factors as appropriate. Common denominators are especially useful for addition, subtraction and comparison.
Why does dividing by one-half make a number bigger?
The question asks how many half-sized groups fit into the amount. More than one half-unit can fit in one whole, so the result can be larger.
What should students do with mixed numbers?
Identify them as wholes plus a fractional part. Improper fractions are often the clearest working form for multiplication, division and mixed signed calculations.
How do negative fractions work?
The same fraction rules apply, with careful attention to signs. Number lines can check additions and subtractions; multiplication sign rules apply to products and quotients.
How do we tell whether an answer is reasonable?
Estimate using fractions near zero, one half or one, check the expected sign and use an inverse operation or decimal conversion where helpful.
Are fractions important for later algebra?
Yes. Fractional coefficients and algebraic fractions build on the same equivalent-fraction and operation principles.
Can a calculator replace fraction models?
It can verify arithmetic but cannot explain why two fractions are equivalent or decide which operation a word problem requires.
What should a tutor diagnose first?
Check whether the learner understands equal parts, equivalent fractions, signed quantities, order of operations and the final form requested.
What does the eduKateSG reference class describe?
The source page describes weekly 90-minute tutorials in groups of three near Sixth Avenue MRT; individual class suitability and current availability should be checked directly.
Official syllabus and related lessons
Check the MOE G2 Mathematics syllabus and the eduKateSG small-group tutorial overview. Continue with these connected explanations:
- Fractions on the number line
- Number skills and estimation
- Fractions, decimals and percentages
- G2 algebra and linear equations
- Prime factors and common multiples
The essential idea is simple: fractions are numbers with meaning. Preserve the unit, choose the appropriate operation and check whether your answer fits the number line.