In this guide
Ask a class which is bigger, 1/3 or 1/5, and several children will say 1/5, "because 5 is bigger". Ask which is bigger, 0.3 or 0.25, and many will say 0.25. Neither answer is careless. Both come from rules that worked perfectly for whole numbers. Fractions and decimals are where children first discover that numbers do not always behave the way they expect, which is exactly why CTET uses them so often in pedagogy questions.
Paper I content is based on the NCERT Class 1–5 books, which introduce halves, quarters and parts of a whole, so fractions can appear there at that level, and even more often in pedagogy questions. Paper II adds operations on fractions and decimals. This guide covers both and puts the misconceptions at the centre.
What a fraction means
A fraction has more than one meaning, and children need to meet all of them.
| Meaning | Example | Model |
|---|---|---|
| Part of a whole | 3/4 of a roti: the roti cut into 4 equal parts, 3 taken | Circles, paper folding |
| Part of a collection | 1/3 of 12 marbles = 4 marbles | Counters in groups |
| A number on the number line | 3/4 lies between 0 and 1, three steps of 1/4 from 0 | Number line |
| Division | 3/4 = 3 ÷ 4: three rotis shared equally among four children | Sharing real objects |
| Ratio | 3 boys to 4 girls | Comparing two sets |
The word equal is the heart of the idea. A roti torn into two unequal pieces has not been cut into halves.
Types of fractions
- Proper: numerator smaller than denominator (3/5).
- Improper: numerator equal to or greater than denominator (7/4).
- Mixed number: a whole and a proper fraction (1 3/4 = 7/4).
- Unit fraction: numerator 1 (1/6).
- Like fractions: same denominator (2/7, 5/7). Unlike: different denominators.
- Equivalent fractions: name the same amount (1/2 = 2/4 = 3/6 = 4/8).
Equivalent fractions and simplest form
Multiply or divide the numerator and denominator by the same non-zero number and the value does not change.
Worked example. Write 18/24 in its simplest form. HCF of 18 and 24 is 6. 18 ÷ 6 = 3 and 24 ÷ 6 = 4, so 3/4.
On fraction strips, a child can see that 3/4 and 6/8 cover exactly the same length. That picture is more convincing than the rule.
Comparing fractions
- Same denominator: compare numerators. 5/9 > 2/9.
- Same numerator: the one with the smaller denominator is larger, because the parts are bigger. 3/5 > 3/8.
- Different both: convert to a common denominator, or cross-multiply.
Worked example. Which is larger, 5/7 or 7/10? Cross-multiply: 5 × 10 = 50 and 7 × 7 = 49. Since 50 > 49, 5/7 is larger. Check with decimals: 5/7 ≈ 0.714 and 7/10 = 0.7.
Operations on fractions
Addition and subtraction need a common denominator.
- 2/3 + 3/4 = 8/12 + 9/12 = 17/12 = 1 5/12.
- 5/6 − 1/4 = 10/12 − 3/12 = 7/12.
- 2 1/2 + 1 2/3 = 5/2 + 5/3 = 15/6 + 10/6 = 25/6 = 4 1/6.
Multiplication: multiply numerators and denominators. 3/4 × 2/9 = 6/36 = 1/6. "Of" means multiply: 3/5 of 45 = 3 × 45 ÷ 5 = 27.
Division: multiply by the reciprocal. 3/4 ÷ 3/8 = 3/4 × 8/3 = 2. Read it as "how many 3/8s fit in 3/4?" and the answer makes sense.
Word problem. A ribbon is 3 1/2 m long. How many pieces of 1/4 m can be cut from it? 7/2 ÷ 1/4 = 7/2 × 4 = 14 pieces.
Word problem. One child eats 1/3 of a cake and another eats 1/4. What fraction is left? 1/3 + 1/4 = 4/12 + 3/12 = 7/12 eaten, so 5/12 is left.
Decimals
Decimals extend place value to the right of the ones place.
| Tens | Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|
| 1 | 2 | . | 3 | 4 | 5 |
12.345 = 10 + 2 + 3/10 + 4/100 + 5/1000.
Conversions to know: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, 1/8 = 0.125, 3/8 = 0.375, 7/8 = 0.875, 3/20 = 0.15.
Comparing: make the number of decimal places equal. 0.3 = 0.30, and 0.30 > 0.25.
Operations:
- Addition and subtraction: line up the decimal points. 4.5 + 0.35 = 4.50 + 0.35 = 4.85.
- Multiplication: multiply as whole numbers, then count the total decimal places. 0.4 × 0.3: 4 × 3 = 12, two decimal places, so 0.12. Also 1.25 × 0.4 = 0.5.
- Division: make the divisor a whole number by shifting both points. 2.4 ÷ 0.6 = 24 ÷ 6 = 4; 0.5 ÷ 0.05 = 50 ÷ 5 = 10.
Decimals connect naturally to money (₹12.50) and measurement (1.5 km = 1,500 m).
Children's misconceptions
| Misconception | Example | Where it comes from |
|---|---|---|
| Larger denominator means larger fraction | 1/8 > 1/4 | Whole-number thinking: 8 > 4 |
| Add tops and bottoms | 1/2 + 1/3 = 2/5 | Treating numerator and denominator as separate numbers |
| Longer decimal is larger | 0.25 > 0.3 | Reading digits after the point as a whole number |
| Parts need not be equal | Calling any two pieces "halves" | Everyday language ("the bigger half") |
| Multiplying always makes bigger | Expects 1/2 × 8 to be more than 8 | Generalised from whole numbers |
| Dividing always makes smaller | Expects 6 ÷ 1/2 to be 3 | Generalised from whole numbers |
| Decimal places add up wrongly | 0.3 × 0.2 = 0.6 | Not counting decimal places, or not seeing that tenths × tenths gives hundredths |
How to teach fractions and decimals
- Concrete to pictorial to symbolic. Fold paper, cut rotis or fruit, then draw, then write 3/4.
- Fraction strips and a fraction wall show equivalence and comparison at a glance: the 1/8 strip is visibly shorter than the 1/4 strip.
- Set models with counters for "fraction of a collection".
- The number line shows that a fraction is a number, and helps with ordering and with improper fractions.
- 10 × 10 grids for decimals: shade 3 columns (0.3) and 25 squares (0.25) and compare.
- Money and measuring tapes give decimals meaning.
- Language: say "three-fourths means three of four equal parts" and ask children to explain in their own words.
- Estimation before calculation: "Is 1/2 + 1/3 more or less than 1?" A child who estimates will reject 2/5 at once, because 2/5 is smaller than 1/2 alone.
See maths teaching aids and errors and remedial teaching for more.
Worked pedagogy questions
Question 1
A child writes 1/2 + 1/3 = 2/5. The best way to help is to:
(a) make the child memorise the rule for adding fractions
(b) ask the child to show 1/2 and 1/3 with fraction strips, put them together and compare with 2/5
(c) mark it wrong and give ten more sums
(d) tell the child to use a calculator
Answer: (b). The strips show that the sum is more than 1/2, while 2/5 is less than 1/2. The contradiction makes the child rethink the rule, and the teacher can then introduce common denominators (3/6 + 2/6 = 5/6).
Question 2
Asked to arrange 0.5, 0.25 and 0.125, a child writes 0.5 < 0.25 < 0.125. The child's difficulty is with:
(a) addition of decimals
(b) decimal place value
(c) multiplication
(d) reading
Answer: (b). The child treats the digits after the point as whole numbers (5 < 25 < 125). Shading on a 10 × 10 grid, or writing all three with three decimal places (0.500, 0.250, 0.125), corrects it.
Question 3
A teacher asks, "Is this square divided into halves?" and shows a square split into one large and one small piece. The purpose is to develop the idea that:
(a) fractions are always less than 1
(b) the parts of a whole must be equal
(c) squares are easy to divide
(d) halves are the only fractions
Answer: (b).
Practice set
- Which is larger: 2/3 or 3/5?
- What is 3/5 of 45?
- Arrange in ascending order: 0.3, 0.25, 0.5, 0.05.
- Simplify 28/42.
- Write 7/8 as a decimal.
- 0.6 × 0.2 = ?
- 2 1/4 ÷ 3/4 = ?
- Arrange in descending order: 1/2, 3/8, 5/6.
- A child says 1/6 > 1/3 because 6 > 3. What does the child misunderstand, and which aid would help?
Answers
- 2/3 = 10/15 and 3/5 = 9/15, so 2/3.
- 27.
- 0.05, 0.25, 0.3, 0.5.
- HCF is 14, so 2/3.
- 0.875.
- 6 × 2 = 12, two decimal places: 0.12.
- 9/4 × 4/3 = 3.
- 5/6 ≈ 0.83, 1/2 = 0.5, 3/8 = 0.375, so 5/6, 1/2, 3/8.
- That a larger denominator means smaller parts. Fraction strips or folding the same sheet into 3 and into 6 parts shows it.
What to do next
- Make a paper fraction wall with halves, thirds, quarters, sixths and eighths.
- Write the misconceptions table from memory with one teaching fix for each.
- Learn the eight fraction-decimal conversions above by heart.
- Revise numbers and operations, then continue with geometry and shapes.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Central Board of Secondary Education website .
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