Mathematics Tutor Singapore | Secondary 1 negative numbers are where many students first discover that an answer can be smaller than zero and that subtracting a negative number can increase a total. A learner may confidently calculate 7−5 but hesitate at −7−(−5). I do not want them to memorise “two minuses make a plus” without knowing what operation that phrase describes.
Integers are positions and directed changes on a number line. They support temperature differences, gains and losses, coordinate geometry, algebraic expressions and inequalities. When a Sec 1 Mathematics student sees how direction, magnitude and the operation interact, the sign rules become consistent rather than a collection of exceptions.
This guide offers original worked examples of integer addition, subtraction, multiplication, division, order of operations, absolute value and real-world interpretation, with diagnostics for G1/G2/G3 learners and parent advice. Check the current school syllabus for the precise level and assessment expectations.
Explore the Secondary 1 integers learning route
- Why the number line extends below zero
- Which negative number is greater?
- Adding integers: direction and total change
- Subtraction means adding the opposite
- Worked example: −7 − (−12)
- Why negative times negative is positive
- Division and signed numbers
- The difference between −3² and (−3)²
- Distance and absolute value
- Temperature, lifts and money as models
- Signed numbers underlie Secondary algebra
- Common errors and targeted repairs
- Fourteen worked Sec 1 integer questions
- An achievable revision routine
- Full SBB and Singapore SEC Mathematics
- Frequently asked questions
- Official syllabus and Mathematics study routes
Why the number line extends below zero
A number line places zero at a reference point. Positive numbers lie to the right and negative numbers to the left when the line is drawn in the conventional direction. The number −4 represents a position four units left of zero, not an error or an impossible quantity.
Negative numbers are useful in stories involving departures from a reference: four degrees below zero, a balance of negative four dollars, or four floors below ground when a building uses that convention. Different stories use different reference meanings, but the mathematics orders signed quantities consistently.
I ask students to mark −5, −2, 0, 3 and 6 on the same line. Then I ask which number is greatest and which pair is furthest apart. The visual answer is clear: six is greatest, while the distance from −5 to 6 is eleven units.
Which negative number is greater?
On the number line, −2 is greater than −5 because it lies to the right. The magnitude of −5 is larger, but the number itself is smaller. A learner may mistakenly compare the digits five and two and conclude the opposite. That is a place-value habit applied in a new context.
| Comparison | Correct relationship | Explanation |
|---|---|---|
| −2 and −5 | −2 > −5 | −2 is nearer zero and farther right |
| −10 and −1 | −10 < −1 | −10 is farther left |
| 0 and −3 | 0 > −3 | Zero lies to the right |
| −4 and +4 | −4 < 4 | Signs give opposite sides of zero |
A quick test is to order −8, −3, 2 and −1. From least to greatest the values are −8, −3, −1, 2. The student should be able to justify their order through location rather than memorising “reverse everything when negative”.
Adding integers: direction and total change
On the number line, adding a positive value moves right, while adding a negative value moves left. Start at −4 and add 7. Moving seven units right crosses zero and ends at +3. Thus −4+7=3.
Now calculate −4+(−7). Start at −4 and move another seven units left. The result is −11. The signs tell us which direction each movement takes; the absolute numbers tell us how far.
| Expression | Number-line story | Answer |
|---|---|---|
| −4+7 | Start −4, move 7 right | 3 |
| −4+(−7) | Start −4, move 7 left | −11 |
| 6+(−9) | Start 6, move 9 left | −3 |
| −8+8 | Opposite movements cancel | 0 |
Students often ask whether the operation of addition always makes a number bigger. It does not: adding a negative number decreases the value. The name of the operation does not decide the direction without knowing the sign of what is added.
Subtraction means adding the opposite
Subtracting a positive quantity moves left. But subtracting a negative quantity reverses a leftward change, which can move us right. Algebraically, a−b=a+(−b). When b itself is negative, −b becomes positive.
For example, 3−(−4) equals 3+4=7. The expression −7−(−5) equals −7+5=−2. In both questions, removing a negative term has the effect of adding its opposite, but the final number can be positive or negative depending on the starting value.
I avoid saying only “two minuses make plus” because students sometimes apply it to unrelated signs. In 3+(−4), the adjacent symbols are plus and negative, and the value is −1, not +7. We must first identify the operation and what is being added or subtracted.
| Expression | Rewrite as addition | Answer |
|---|---|---|
| 3−(−4) | 3+4 | 7 |
| −7−(−5) | −7+5 | −2 |
| 3+(−4) | 3−4 | −1 |
| −2−6 | −2+(−6) | −8 |
| −2−(−6) | −2+6 | 4 |
Worked example: −7 − (−12)
Start at −7. The operation asks us to subtract negative twelve. Rewrite the subtraction as adding the opposite: −7−(−12)=−7+12. From −7 move twelve positions right, passing zero after seven moves and ending at five. The answer is 5.
A reliable check is to reverse the relation: 5+(−12)=−7. If the proposed answer were −19, adding negative twelve would give −31, not −7. The inverse check catches a wrong sign without any need to remember a rule by rote.
Some learners find a lift-floor story helpful: imagine a balance of negative seven units and cancelling a debt of twelve units. But financial metaphors have limits and can introduce language confusion. The number line is the most general model for signed arithmetic.
Why negative times negative is positive
The multiplication sign rule can be understood through a pattern. Consider multiplying negative three by a sequence of integers: (−3)×3=−9, (−3)×2=−6, (−3)×1=−3, (−3)×0=0. Each time the second factor decreases by one, the product increases by three. Continuing the pattern gives (−3)×(−1)=+3.
| Expression | Product | Pattern |
|---|---|---|
| (−3)×3 | −9 | Start |
| (−3)×2 | −6 | Increase by 3 |
| (−3)×1 | −3 | Increase by 3 |
| (−3)×0 | 0 | Increase by 3 |
| (−3)×(−1) | 3 | Increase by 3 |
| (−3)×(−2) | 6 | Increase by 3 |
There is a structural argument too. Since zero equals (−1)+1, multiplying by (−1) gives 0=(−1)[(−1)+1]=(−1)(−1)+(−1). Therefore (−1)(−1)=1. The familiar negative-times-negative-positive rule follows from ordinary multiplication and addition laws.
A Sec 1 student need not reproduce a formal proof every time, but an explanation helps them distinguish multiplication signs from subtracting a negative term.
Division and signed numbers
Division reverses multiplication. If (−4)×6=−24, then −24÷6=−4. If (−3)×(−8)=24, then 24÷(−3)=−8. A positive divided by a negative is negative; a negative divided by a negative is positive.
| Division | Related multiplication | Answer |
|---|---|---|
| −24÷6 | −4×6=−24 | −4 |
| 24÷(−3) | −8×(−3)=24 | −8 |
| −24÷(−3) | 8×(−3)=−24 | 8 |
| −36÷9 | −4×9=−36 | −4 |
The divisor must not be zero. Dividing a nonzero number by zero is undefined, because no finite number multiplied by zero could recover the original nonzero amount. A result such as 8÷0 should not be given the number zero or infinity as though it were an ordinary quotient.
The difference between −3² and (−3)²
This is a useful extension once signed-number arithmetic is stable. The expression −3² means −(3²) under conventional operation precedence and equals −9. The expression (−3)² squares the entire negative number and equals +9.
| Expression | Interpretation | Value |
|---|---|---|
| −3² | −(3×3) | −9 |
| (−3)² | (−3)×(−3) | 9 |
| −2³ | −(2×2×2) | −8 |
| (−2)³ | (−2)×(−2)×(−2) | −8 |
A student who says “anything squared is positive” without identifying the base may obtain the wrong result for −3². The base of the power is three unless brackets include the negative sign. I teach children to write the multiplication represented by the exponent before calculating.
Distance and absolute value
The absolute value |x| is the distance of x from zero on a number line, so |−5|=5 and |5|=5. Distance is nonnegative even when the starting or ending position is negative. This is a conceptual bridge to coordinate distance later.
The distance between −4 and 3 is |3−(−4)|=|7|=7 units. The distance between −8 and −2 is |−2−(−8)|=|6|=6 units. A student who says the gap from −4 to 3 is one because they subtract four from three has not accounted for crossing zero.
Absolute value can also help check an answer: if one point is −10 and another is +2, the distance must exceed ten because the points lie on opposite sides of zero. A reported eight would not match the number line.
Temperature, lifts and money as models
Suppose the temperature is −2°C and rises by 7°C. It becomes +5°C. If it then falls by 9°C, the new temperature is −4°C. Those operations can be shown as rightward and leftward movements, but we should distinguish a temperature reading from a temperature change.
A building might number basement floors −1, −2 and −3, though actual floor labels differ. Travelling from B3 represented as −3 to Level 4 represented as +4 crosses seven floor-number steps: −3 to zero is three, zero to four is four. The specific physical count of floors in a real building depends on its numbering convention, so treat this as a mathematical number-line illustration.
A bank balance of −$25 may represent owing money rather than having cash. Depositing $40 changes the balance to +$15. The same signed arithmetic is used as with temperature, but the meaning of zero depends on the context.
Signed numbers underlie Secondary algebra
Later, expressions such as 4−(−2x) or −3(2x−5) require the same sign discipline. Expanding −3(2x−5) gives −6x+15 because the negative multiplier acts on each term. A student who learns signs through meaningful operations can check these algebraic steps without guessing.
Negative coefficients also matter for graphs and inequalities. The gradient −2 describes a line falling by two vertical units for every unit moved right. Multiplying both sides of an inequality by a negative reverses its direction, a later rule that becomes easier to understand when the learner can order numbers on a number line.
Common errors and targeted repairs
| First wrong move | Likely concept gap | Next test |
|---|---|---|
| Thinks −8 > −3 because 8 > 3 | Ordering of signed numbers | Place both on a line |
| Treats 3+(−4) as 3+4 | Sign of addend ignored | Move left from 3 |
| Treats −7−(−5) as −12 | Opposite of subtracted number missed | Rewrite as −7+5 |
| Claims negative×negative is negative | Multiplication pattern missing | Continue product pattern through zero |
| Says −3² equals +9 | Exponent base not identified | Expand with and without brackets |
| Calls |−6|=−6 | Distance vs signed position confused | Measure distance from zero |
| Divides by zero and writes zero | Division meaning unclear | Check with inverse multiplication |
A student may understand signed multiplication while making an unrelated bracket mistake. Another may be accurate with procedures but unable to explain negative number order. I recommend practice targeted to the first recurring mistake rather than repeating the entire topic indiscriminately.
Fourteen worked Sec 1 integer questions
1. Order integers
Arrange −6, 2, −1, 0 from least to greatest: −6, −1, 0, 2.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
2. Add across zero
−5+9=4 because nine rightward steps from −5 end at four.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
3. Add two negatives
−7+(−4)=−11 because both changes move left.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
4. Subtract positive
−3−8=−11.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
5. Subtract negative
−7−(−12)=−7+12=5.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
6. Subtract negative again
4−(−9)=13.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
7. Multiply different signs
(−6)×4=−24.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
8. Multiply two negatives
(−6)×(−4)=24.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
9. Divide two negatives
−30÷(−5)=6.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
10. Evaluate order of operations
3+(−2)×4=3−8=−5; multiplication comes first.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
11. Compare powers
−3²=−9 while (−3)²=9.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
12. Absolute distance
Distance from −4 to +3 is |3−(−4)|=7.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
13. Temperature
−2°C rising 7°C gives +5°C; falling 9°C after that gives −4°C.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
14. Expand signed bracket
−2(3x−4)=−6x+8.
Check the result with a number line, an opposite operation or a simple estimated sign. A short explanation is more reliable than the phrase “two minuses” without identifying what was subtracted or multiplied.
A manageable Secondary 1 revision routine
| Session | Focus | Evidence |
|---|---|---|
| 1 | Locate negative and positive values | Orders signed integers |
| 2 | Compare distances to zero | Explains −2 > −5 |
| 3 | Add signed numbers | Predicts number-line direction |
| 4 | Subtract a negative | Rewrites as adding opposite |
| 5 | Multiply and divide signed integers | Checks with inverse operations |
| 6 | Handle brackets and powers | Identifies the base |
| 7 | Use temperature and money stories | Names what zero represents |
| 8 | Solve signed algebra terms | Distributes negative coefficients |
| 9 | Mixed work without headings | Chooses valid operation |
| 10 | Delayed review of earlier errors | Explains independently |
Some children need only a focused explanation and a short sequence of examples. Others may have gaps in primary arithmetic or English word-problem interpretation. Quality tuition should recognise those differences rather than add the same number of worksheets to every student’s week.
Full SBB and Singapore SEC Mathematics
Singapore’s Full Subject-Based Banding allows Secondary students to study at appropriate G1, G2 or G3 subject levels. The 2027 SEC G3 Mathematics K310 syllabus includes negative numbers, integers, rational and real numbers, their operations and ordering on the number line. The precise sequencing of topics within Secondary One depends on the school, and a G1 or G2 learner should use the relevant subject-level specification.
Read SEAB’s 2027 G3 Mathematics syllabus index and the main SEC syllabus page for the applicable cohort. We should not imply that a single tuition plan fits all Mathematics subject levels or exam years.
Frequently asked questions
Why is −2 greater than −5?
On the number line, −2 lies to the right of −5 and is closer to zero.
Why does subtracting a negative increase the answer?
Subtraction can be rewritten as adding the opposite. For example 3−(−4)=3+4.
Why does adding a negative make a number smaller?
A negative addend represents a movement left along the conventional number line.
Why is a negative times a negative positive?
It is consistent with the distributive law and the multiplication pattern that continues through zero.
Is −3² the same as (−3)²?
No. The first is −9 under standard precedence; the second is +9 because the whole negative base is squared.
Can distance between negative and positive positions be negative?
No. Distance is nonnegative and can be expressed using absolute value.
Should every Secondary 1 pupil have Maths tuition?
Not necessarily. Seek targeted extra help when a persistent gap is evident and other teaching support is insufficient.
Will mastering integers guarantee A-Math or JC success?
No. Signed-number confidence is an important foundation, but many other concepts and education choices influence future pathways.
Official syllabus and connected Mathematics lessons
Start with the Mathematics Learning Hub for the Primary-to-JC route. The next algebra step is explained in Secondary Algebra and Fraction Equations. For coordinate distance and gradients, visit Secondary 3 Coordinate Geometry.
The unchanged Secondary 1 Mathematics Tutor Clementi small-group teaching reference remains a separate programme example. Families can arrange an eduKate Singapore consultation and bring a correct signed-addition question alongside one mistake involving subtraction or brackets. The first wrong operation is often the most useful place to begin.