Primary 3 fractions mark a major change in mathematical thinking. Whole numbers count complete units. Fractions describe equal parts of a whole, and different-looking fractions can represent the same amount. At the same time, money introduces decimal notation in a familiar real-world setting, asking students to preserve place value across dollars and cents.
This is Guide 2 in the Primary 3 Mathematics Learning Hub. It develops two related capabilities: seeing quantity beyond whole numbers, and keeping notation connected to meaning.
A fraction is not two whole numbers separated by a line. It is one quantity defined by a numerator, a denominator and a whole.
What Primary 3 Students Need to Learn
Primary 3 students learn equivalent fractions, simplest form, comparison and ordering of unlike fractions where the denominators stay within the syllabus range, and addition or subtraction of two related fractions within one whole. In money, students add and subtract amounts written in decimal notation.
For the official curriculum reference, see the MOE Primary Mathematics Syllabus.
The First Question in Fractions | What Is the Whole?
Every fraction depends on a whole. If a pizza is divided into 8 equal slices and 3 slices are eaten, the eaten amount is 3/8 of that pizza. But if the whole changes, the same fraction name can describe a different absolute amount. Half of a small cake is not necessarily the same amount of cake as half of a large cake.
Students should therefore train themselves to identify the whole before reasoning about the fraction. This habit becomes increasingly important in later ratio, percentage and algebraic reasoning.
Numerator and Denominator Have Different Jobs
| Part | Meaning | In 3/8 |
|---|---|---|
| Denominator | How many equal parts the whole is divided into | 8 equal parts |
| Numerator | How many of those equal parts are being counted | 3 parts |
The phrase equal parts matters. A shape split into pieces of different sizes does not create a valid fraction model merely because several pieces are visible.
Equivalent Fractions | Different Names for the Same Quantity
Fractions are equivalent when they represent the same amount of the same whole. For example, 1/2 = 2/4 = 3/6. The number of pieces changes, but the total portion does not.
One way to understand this is to imagine subdividing every piece equally. If one half is split into two equal smaller pieces, there are now two quarters. The amount has not changed; only the description has.
Equivalent fractions change the partition, not the quantity.
Generate Equivalent Fractions by Scaling Both Parts
If the numerator and denominator are multiplied by the same non-zero whole number, the fraction keeps the same value. For example:
- 2/3 × 2/2 = 4/6
- 2/3 × 3/3 = 6/9
- 3/5 × 2/2 = 6/10
The expression “multiply by 2/2” is useful conceptually because 2/2 equals one. Multiplying a quantity by one does not change its value, even though its written form changes.
Simplest Form | Remove a Common Scale Factor
A fraction is in simplest form when its numerator and denominator no longer share a common factor greater than 1. For example, 6/8 can be simplified by dividing both numerator and denominator by 2, giving 3/4.
Students should understand that simplifying does not make the fraction smaller. It makes the notation simpler while preserving the same quantity.
Worked Example: Simplify 8/12.
- 8 and 12 are both divisible by 2, giving 4/6.
- 4 and 6 are both divisible by 2 again, giving 2/3.
- 2 and 3 share no common factor greater than 1.
Therefore 8/12 = 2/3.
Comparing Fractions With the Same Denominator
If the denominator is the same, the pieces are the same size. Compare how many pieces are taken. For example, 5/8 > 3/8 because five eighths is more than three eighths.
Comparing Fractions With the Same Numerator
If the numerator is the same, the number of pieces is the same, but the piece size changes. The larger the denominator, the smaller each equal piece.
Therefore 3/5 > 3/8. Three fifths contains three larger pieces; three eighths contains three smaller pieces.
Comparing Unlike Fractions | Build a Common View
When both numerator and denominator differ, students need a common reference. Equivalent fractions are one reliable route.
Worked Example: Compare 2/3 and 3/4.
Express both with denominator 12: 2/3 = 8/12 and 3/4 = 9/12. Since 9/12 > 8/12, 3/4 > 2/3.
The goal is not to memorise a comparison trick. The goal is to make the pieces comparable.
Use Benchmarks When They Make the Comparison Easier
Benchmarks such as 0, 1/2 and 1 can provide quick reasoning. For example, 5/8 is greater than 1/2 because half of 8 is 4 and 5 eighths is one eighth more. Meanwhile 3/8 is less than 1/2.
This kind of benchmark reasoning develops fraction sense and reduces dependence on long calculations.
Ordering Several Fractions
Ordering fractions requires consistent comparison. Students can convert related fractions to a common denominator or use clear benchmarks when appropriate.
Example: Order 1/2, 3/4 and 2/8 from smallest to largest.
2/8 = 1/4. Therefore 1/4 < 1/2 < 3/4. So the order is 2/8, 1/2, 3/4.
Adding Related Fractions
Fractions can be added directly when they are expressed in equal-sized parts. If the denominators already match, add the numerators and keep the denominator.
For example, 2/7 + 3/7 = 5/7.
When denominators are related, convert one fraction to an equivalent form first.
Worked Example: 1/2 + 1/4.
One half equals two quarters, so 1/2 + 1/4 = 2/4 + 1/4 = 3/4.
Why We Do Not Add Denominators
A common error is 1/2 + 1/4 = 2/6. This fails because the denominator describes the size of the equal parts. Adding one half and one quarter does not create sixths automatically. The parts must first be described using the same-sized units.
Before adding or subtracting fractions, make sure the pieces mean the same thing.
Subtracting Related Fractions
Worked Example: 5/6 − 1/3.
Convert 1/3 to sixths: 1/3 = 2/6. Then 5/6 − 2/6 = 3/6 = 1/2.
The subtraction happens after both quantities are written in the same fractional unit.
Fraction Models Should Reveal the Relationship
Area models, fraction strips and number lines can all support understanding. The best representation depends on the learning job.
- Area model: useful for seeing equal parts of a whole.
- Fraction strip: useful for comparing equivalence and size.
- Number line: useful for seeing fractions as numbers with positions and order.
- Bar model: useful when a word problem combines fractional parts and known quantities.
Students should eventually move between representations rather than depend permanently on one picture type.
Worked Fraction Word Problem
Question: Mei used 1/4 of a ribbon in the morning and 1/2 of the same ribbon in the afternoon. What fraction of the ribbon did she use altogether?
The whole is the same ribbon. Convert 1/2 to quarters: 1/2 = 2/4. Then 1/4 + 2/4 = 3/4.
The phrase “same ribbon” protects the whole. Without the same whole, the fractions could not be added so simply.
Money | Decimal Notation With Meaning
Money gives Primary 3 students a practical encounter with decimal notation. In Singapore currency, $4.35 means 4 dollars and 35 cents. The decimal point separates whole dollars from the fractional part represented by cents.
Students often already understand that $1 = 100 cents from daily life. The mathematical task is to preserve that relationship when calculating.
Align Dollars With Dollars and Cents With Cents
When adding or subtracting money in decimal notation, align the decimal points. This keeps place values matched.
Worked Example: $12.75 + $6.80 = $19.55.
The zero in $6.80 is useful. It shows that there are 80 cents. Writing $6.8 represents the same amount mathematically, but at Primary 3 the two-decimal-place money notation can help students keep cents visible and aligned.
Subtraction Across One Dollar
Worked Example: $10.00 − $3.68.
One dollar can be regrouped as 100 cents. The result is $6.32. Students who understand the dollars–cents relationship can see why regrouping works instead of memorising a decimal subtraction routine.
Money Word Problems | Identify the Financial Relationship
- Total cost: add prices.
- Change: amount paid − cost.
- Difference in price: larger amount − smaller amount.
- Repeated equal cost: use multiplication when several identical items cost the same.
- Shared cost: use division if a total is split equally.
Keywords can help, but the relationship decides the operation. “How much more” may indicate a difference; “altogether” may indicate a total; yet students should still read the full situation.
Worked Money Problem
Question: A book costs $8.75 and a pen costs $2.60. Jia pays with $20. How much change does she receive?
First find the total cost: $8.75 + $2.60 = $11.35. Then find the change: $20.00 − $11.35 = $8.65.
The two operations represent different stages of the story. The total cost must be known before the change can be calculated.
Common Fraction Misconceptions
- “A bigger denominator means a bigger fraction.” For the same numerator and whole, more equal parts means smaller pieces.
- “Equivalent fractions are different amounts.” Their notation changes while their value remains the same.
- “Simplifying makes the fraction smaller.” Simplifying preserves value.
- “Add the top and add the bottom.” Fractions require common-sized parts before addition or subtraction.
- “The picture looks about right.” Fraction parts must be equal.
- “The whole does not matter.” Every fraction is defined relative to a whole.
Common Money Misconceptions
- Aligning digits instead of decimal points.
- Treating 35 cents as 0.35 dollars without understanding the 100-cent relationship.
- Dropping the currency unit in the final answer.
- Finding change before finding the total cost.
- Confusing $4.05 with $4.50.
- Reading $6.80 as “six dollars and eight cents”.
Diagnostic Check | Fractions
- Can the student identify the whole?
- Can the learner explain numerator and denominator roles?
- Can the student generate an equivalent fraction?
- Can the learner simplify a fraction and explain why the value is unchanged?
- Can the student compare fractions using a common denominator or benchmark?
- Can the learner add and subtract related fractions?
- Can the student reject the incorrect rule of adding denominators?
Diagnostic Check | Money
- Can the student state how many cents are in one dollar?
- Can the learner read $7.04 and $7.40 correctly?
- Can the student align decimal notation accurately?
- Can the learner add and subtract money with regrouping?
- Can the student find total cost and then change in a two-step problem?
- Can the learner estimate whether a money answer is reasonable?
How to Practise Fractions Well
Use concrete and visual representations early, but do not stop there. Ask students to connect the picture to symbols and then explain the relationship in words. Mix questions so they must decide whether the job is equivalence, simplification, comparison, addition or subtraction.
A useful variation exercise is to keep the fractions similar while changing the question: “Which is larger?”, “Write an equivalent fraction”, “Simplify”, “Find the missing numerator”, or “Find the total”. This trains discrimination instead of routine copying.
How to Practise Money Well
Use realistic price lists, receipts and change problems. Ask the student to estimate before calculating exactly. For example, if two items cost about $9 and $3, a total near $12 is reasonable. An answer of $1.20 or $120 should trigger a recheck.
Exam Craft | Protect the Whole, Unit and Decimal Point
For fractions, identify the whole and check whether the parts being compared or combined are equal-sized. For money, line up decimal points and write the dollar sign in the final answer. In multi-step problems, label intermediate values so that a total cost does not accidentally become the amount of change.
Fractions protect the whole. Money protects place value. Good working protects both.
Checkpoint | Is the Fraction-and-Money System Stable?
- Equivalent fractions are understood as equal quantities.
- Simplest form is produced without changing value.
- Unlike fractions can be compared with a clear reason.
- Related fractions can be added and subtracted correctly.
- Money amounts are read accurately in dollars and cents.
- Decimal points are aligned in calculations.
- Total cost, difference and change are distinguished.
- Answers are checked against sensible benchmarks.
How This Connects to the Rest of Primary 3 Mathematics
Fraction equivalence prepares students for later fraction operations, ratio and percentage. Money strengthens decimal place-value awareness. Both topics depend on the whole-number engine from Guide 1: Whole Numbers and Operations and both appear inside multi-step problems developed in Guide 4: Word Problems, Models and Bar Graphs.
Continue with Guide 3: Measurement, Time, Area, Perimeter and Geometry.
Final Thought
Fractions teach students that mathematical quantity can keep the same value while changing its representation. Money teaches them that notation must remain anchored to place value and unit meaning. These are not isolated Primary 3 topics. They are early lessons in mathematical equivalence, precision and representation.
Do not teach the notation as though it were the idea. Teach the quantity first, then make the notation carry it accurately.
Return to the Primary 3 Mathematics Learning Hub.
