SECONDARY 1 MATHEMATICS LEARNING GUIDE · GUIDE 25
Directed numbers extend arithmetic beyond zero. Positive and negative values let Mathematics represent opposite directions, gains and losses, elevations, temperatures, credits and debts. Rational numbers extend this system further through fractions, integers and terminating or recurring decimals.
This guide develops number-line order, absolute value, signed addition, subtraction, multiplication and division, rational-number forms, fraction operations, decimal operations, mixed calculations, estimation and common sign errors. It is a learning companion; exact pacing and depth vary across subject levels and schools.
Useful prior guides: Numbers, Number Lines, Approximation and Estimation, Prime Factorisation, HCF, LCM, Squares, Cubes and Roots and Calculator Skills, Exact Values and Input Discipline.
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1. Positive and negative numbers describe direction from zero
On a number line, values increase to the right and decrease to the left.
Therefore −2 > −7 because −2 lies further to the right.
Common trap
Comparing only the digits 2 and 7 ignores direction. Negative order is determined by position, not by unsigned size.
2. Zero is neither positive nor negative
Zero is the reference point separating positive and negative values.
It can represent a neutral balance, sea level reference, no net change or an origin in coordinates.
3. Absolute value measures distance from zero
|−8| = 8 and |8| = 8.
Absolute value ignores direction and records magnitude.
Contrast
−8 and 8 are different numbers but have equal absolute value.
4. Adding a positive number moves right
−5 + 8 = 3.
Start at −5 and move 8 units right.
Number-line reasoning gives a visual meaning to signed addition.
5. Adding a negative number moves left
4 + (−7) = −3.
Adding −7 means a movement of 7 units in the negative direction.
Compression
4 + (−7) is often written 4 − 7.
6. Subtraction can be rewritten as addition of the opposite
a − b = a + (−b).
Worked example
3 − (−5) = 3 + 5 = 8.
Subtracting a negative does not create a special magic rule; it becomes addition of the opposite.
7. Two negative signs can have different jobs
In −(−4), the outer negative means take the opposite of −4, giving 4.
In 7 − (−4), one sign is subtraction and one belongs to the number −4.
Brackets make the roles visible.
8. Multiplication signs follow repeated-direction structure
Positive × positive = positive.
Positive × negative = negative.
Negative × positive = negative.
Negative × negative = positive.
Worked examples
(−6)(4) = −24.
(−6)(−4) = 24.
9. Division uses the same sign pattern as multiplication
−35 ÷ 5 = −7.
−35 ÷ (−5) = 7.
The sign pattern is preserved because division reverses multiplication.
10. Rational numbers can be written as fractions of integers
A rational number can be expressed as p/q where p and q are integers and q ≠ 0.
Examples include 3/4, −7/5, 6, 0.25 and 0.333… recurring.
Integer connection
Every integer is rational because n = n/1.
11. Equivalent fractions represent the same rational number
2/3 = 4/6 = 10/15.
Multiplying numerator and denominator by the same non-zero number preserves the value.
Lowest terms
Divide numerator and denominator by their HCF to simplify fully.
12. Compare rational numbers using a common representation
Compare 5/8 and 3/5.
Common denominator 40 gives 25/40 and 24/40, so 5/8 > 3/5.
Alternatively, decimal forms 0.625 and 0.6 also show the order.
13. Add and subtract fractions only after aligning denominators
2/3 + 5/8.
LCM(3,8)=24.
2/3=16/24 and 5/8=15/24.
Total = 31/24 = 1 7/24.
14. Fraction subtraction needs sign discipline
3/4 − 5/6 = 9/12 − 10/12 = −1/12.
A negative answer is valid when the second quantity is larger.
15. Multiply fractions across numerators and denominators
(−3/5)(10/9).
Cancel first where convenient: 10/5=2 and 3/9=1/3.
Result = −(1×2)/(1×3) = −2/3.
16. Divide by a fraction using its reciprocal
4/7 ÷ 2/3 = 4/7 × 3/2 = 12/14 = 6/7.
Why reciprocal?
Dividing by 2/3 asks how many groups of size 2/3 fit into the original quantity. Multiplying by 3/2 is the inverse operation.
17. Decimal operations should preserve place value
3.75 − 5.2 = −1.45.
Align decimal places before subtracting:
3.75 − 5.20 = −1.45.
Estimate first: about 4−5 = −1, so the sign and scale are sensible.
18. Convert between fractions and decimals when useful
3/8 = 0.375.
0.45 = 45/100 = 9/20.
Choose the representation that makes the relationship easiest to inspect.
19. Order of operations applies to signed numbers too
Evaluate −3 + 2(−5)².
Power first: (−5)² = 25.
Multiply: 2×25=50.
Add: −3+50 = 47.
20. Brackets protect negative bases
(−4)² = 16, but −4² = −16 under standard precedence.
This distinction matters in algebra and calculator entry.
21. Context determines whether negative values are meaningful
A temperature of −6°C is meaningful. A bank balance of −$40 may represent debt. A geometric length of −4 cm is normally not physically valid.
Mathematical solutions must be interpreted in the domain of the problem.
22. Common directed-number errors
| Error | Likely issue | Repair prompt |
|---|---|---|
| −8 > −3 | Digit size used instead of number-line order | Which value is further right? |
| 3−(−5)=−2 | Subtraction of negative misread | Can you rewrite subtraction as addition of the opposite? |
| (−4)²=−16 | Negative base not bracketed conceptually | What entire value is being squared? |
| 2/3+1/4=3/7 | Denominators added directly | Are the parts the same size? |
| −12÷−3=−4 | Division sign rule forgotten | What multiplication reverses this division? |
23. Practice laboratory
- Order −7, 2, −1, 0, 5 from least to greatest.
- Find |−13|.
- Evaluate −6+11.
- Evaluate 7+(−12).
- Evaluate 5−(−9).
- Evaluate −4−7.
- Evaluate (−6)(−5).
- Evaluate −42÷6.
- Simplify −18/24.
- Compare 7/10 and 2/3.
- Evaluate 5/6−3/4.
- Evaluate (−2/3)(9/5).
- Evaluate 7/8÷14/15.
- Convert 0.375 to a fraction in lowest terms.
- Evaluate −2+3(−4)².
- Explain why a negative length may be rejected even when the algebra is correct.
24. Explained answers
1. −7, −1, 0, 2, 5.
2. 13.
3. 5.
4. −5.
5. 14.
6. −11.
7. 30.
8. −7.
9. −3/4.
10. 7/10=21/30 and 2/3=20/30, so 7/10 is greater.
11. 10/12−9/12=1/12.
12. −18/15=−6/5.
13. 7/8×15/14=15/16.
14. 3/8.
15. −2+3×16=46.
16. The mathematical value must still satisfy the physical domain of the quantity.
25. Complete mixed problem
A fictional account begins at −$35, receives $80, then pays $27.50.
Balance = −35+80−27.50 = 45−27.50 = $17.50.
The first transaction crosses zero; the final value is positive.
Estimate: −35+80≈45, then subtract about 28 gives about 17, consistent with the exact result.
26. Teaching signed numbers through movement and structure
Use the number line to establish meaning, then compress to symbolic rules only after direction is secure.
For multiplication and division, connect sign patterns to inverse relationships instead of memorising disconnected slogans.
Changed-case test
If 6−(−4)=10, ask what changes when the second number becomes +4. Comparing the two cases exposes the role of the sign.
27. Questions students often ask
Why is −2 greater than −8?
Because −2 is closer to the right on the number line.
Why does minus a negative become plus?
Because subtraction is addition of the opposite.
Are all decimals rational?
Terminating and recurring decimals are rational. Not every real decimal expansion is rational.
Why do fractions need common denominators for addition?
Because the parts must represent the same unit size before their counts can be combined.
28. Return path and sources
Directed and rational numbers support nearly every later algebraic topic. Revisit Numbers, Number Lines, Approximation and Estimation for wider number sense and Algebraic Expressions and Variables when signed arithmetic enters symbolic work.
Official curriculum reference: MOE Secondary Syllabus Directory. Exact sequencing of directed and rational number operations varies by subject level and school.
Editorial approach: Wintour House V1.0 · CivDJ · eduKate Publishing. Fix the reference point, preserve direction, choose a common representation, execute the operation, estimate the result and return it to the quantity’s domain.
