Secondary 1 Mathematics Classroom | Chapter 2: Directed Numbers, Rational and Real Numbers, Number Lines, Approximation and Estimation | G2/G3

SECONDARY 1 MATHEMATICS CLASSROOM · CHAPTER 2 · DIRECTED NUMBERS, RATIONAL AND REAL NUMBERS · G2/G3

Directed Numbers, Rational and Real Numbers, Number Lines, Approximation and Estimation: Locate the Number Before You Operate on It

In this classroom, you will not begin by memorising “two negatives make a positive”. You will begin by locating numbers on a line, deciding what operation is actually happening, and preserving the direction of every change.

Secondary Mathematics expands the number system students used in Primary School. Values can lie below zero. Fractions and decimals can be positive or negative. Inequalities describe whole regions of the number line rather than one answer. Approximation replaces an exact value with a nearby one for a stated purpose. Estimation gives you an independent sense of the size an answer should have before you trust the calculator.

Classroom rule: position first, operation second, sign third, estimate before trust.

The current Secondary One G2 and G3 Mathematics syllabuses include work with negative and rational numbers, number-line ordering, arithmetic operations, approximation and estimation. The exact sequence and depth can differ by subject level and school, so this classroom teaches the shared core first and marks deeper number-classification ideas as extension where appropriate.

Official reference: MOE G2 and G3 Mathematics Syllabuses.

Navigate: number line · addition and subtraction · multiplication and division · rational numbers · real numbers · fractions and decimals · inequalities · approximation · estimation · misconception clinic · guided practice · examination transfer · exit ticket.

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Featured Answer: What Is a Directed Number?

A directed number carries position relative to zero. Positive numbers lie to the right of zero on a horizontal number line. Negative numbers lie to the left. The sign is therefore not a decorative mark attached to the number. It is part of the number’s position and meaning.

For example, +6 and −6 are equally far from zero but lie in opposite directions. In a context, they might represent 6 metres above and below a reference level, a gain and loss of 6 dollars, or temperatures 6 degrees above and below zero.

The Simple Classroom Answer

The number line is the operating map: right means greater, left means smaller, and every arithmetic move must preserve the direction described by the operation.

• Position tells you whether a number is positive, zero or negative.
• Ordering compares positions from left to right.
• Addition combines a signed change with the starting value.
• Subtraction means adding the opposite change.
• Multiplication and division combine magnitudes with a direction/sign rule.
• Rational numbers can be expressed as one integer divided by another non-zero integer.
• Approximation replaces an exact value by a nearby stated value.
• Estimation predicts the rough scale of an answer before exact calculation.

How to Use This Classroom

1. Draw or imagine a number line whenever a sign rule feels uncertain.
2. Attempt every Your Turn question before opening the worked answer.
3. Put brackets around substituted negative values.
4. Separate the operation sign from the sign of the number.
5. If an answer is wrong, identify the first wrong move, not only the final arithmetic.
6. Estimate before using the calculator on multi-step numerical work.
7. Return later without notes to test retrieval.

1. Start the Lesson With Zero as a Reference Point

Teacher: Draw a horizontal number line from −8 to 8 and mark zero clearly. Ask: “Which number is larger, −2 or −6?”

The answer is −2 because it lies further to the right. The common Primary-School instinct “6 is bigger than 2” is no longer enough once signs matter.

On a number line, greater means further right.

2. Positive Numbers Lie to the Right of Zero

The positive numbers 1, 2, 3, … lie to the right of zero. A positive sign may be written explicitly as +5 or omitted and written simply as 5.

In many contexts, positive numbers represent increase, gain, height above a reference level or movement in a chosen positive direction.

3. Negative Numbers Lie to the Left of Zero

The negative numbers −1, −2, −3, … lie to the left of zero. The further left you go, the smaller the number becomes.

Therefore:

−9 < −4 < 0 < 3 < 11.

4. Zero Is Neither Positive Nor Negative

Zero is the reference separating positive and negative values. It is neither positive nor negative.

Do not classify zero as positive merely because it is not negative.

Your Turn 1

Arrange these numbers in ascending order:

5, −2, 0, −7, 3, −1.

Check your answer

−7, −2, −1, 0, 3, 5.

5. Distance From Zero Is Different From Signed Position

The numbers −5 and +5 lie five units from zero in opposite directions. Their distance from zero is the same, but the numbers themselves are different.

This distinction prepares students for later ideas such as magnitude, coordinate distance and absolute value, even if the formal notation is introduced later in the school sequence.

6. Translate Context Into Signed Position

Context	Possible signed model
6°C above zero	+6
4°C below zero	−4
$25 credit	+25
$25 debt relative to zero balance	−25
12 m above reference level	+12
3 m below reference level	−3

The reference point must be defined. A signed number is meaningful only relative to a chosen zero or direction.

7. Addition Means Apply a Signed Change

Start at 3 and add 4. Move four units right:

3 + 4 = 7.

Start at 3 and add −4. A negative change moves four units left:

3 + (−4) = −1.

The number being added already contains its direction.

8. Same-Sign Addition Reinforces One Direction

For 5 + 7, both contributions are positive, so the result is 12.

For −5 + (−7), both contributions are negative. Move five units left, then seven more units left:

−5 + (−7) = −12.

A useful compressed rule is: add the magnitudes and keep the common sign. But the number line explains why.

9. Opposite-Sign Addition Is a Competition of Magnitudes

For −9 + 4, the negative contribution has magnitude 9 and the positive contribution magnitude 4. They oppose each other. Five units of negative movement remain:

−9 + 4 = −5.

For 12 + (−7), five positive units remain:

12 + (−7) = 5.

Opposite signs: compare magnitudes, subtract the smaller magnitude from the larger, keep the sign of the larger magnitude.

10. Teacher Model 1: Temperature Change

The temperature is −3°C. It rises by 8°C.

Model the rise as +8:

−3 + 8 = 5.

The final temperature is 5°C.

Check on the number line: start at −3 and move eight places right.

11. Subtraction Means Add the Opposite

Subtraction can be rewritten as addition of the opposite:

a − b = a + (−b).

For example:

7 − 10 = 7 + (−10) = −3.

The rewrite is not a trick. Subtracting 10 means applying a change of −10.

12. Subtracting a Negative Reverses the Direction of the Subtraction

Consider:

5 − (−3).

Rewrite subtraction as addition of the opposite:

5 + 3 = 8.

The two negative symbols have different jobs: the first is the subtraction operation; the second belongs to the number −3.

13. Teacher Model 2: Separate Operation Sign From Number Sign

Evaluate:

−4 − (−9).

First number: −4.

Operation: subtract.

Second number: −9.

Rewrite:

−4 + 9 = 5.

Your Turn 2

1. −7 + 12
2. 8 + (−13)
3. −6 + (−9)
4. 5 − 11
5. 5 − (−11)
6. −8 − (−3)

Answers

5, −5, −15, −6, 16, −5.

14. Use a Two-Step Translation for Word Problems

1. Choose the starting value and reference direction.
2. Translate each change into a signed number before calculating.

A bank balance is −$18 relative to zero. A deposit of $25 is made:

−18 + 25 = 7.

The new balance is $7 above zero.

15. Multiplication Combines Magnitude With Direction

For positive factors, multiplication behaves as before:

4 × 3 = 12.

When signs appear, calculate the magnitude and determine the sign separately.

Signs	Product sign
positive × positive	positive
positive × negative	negative
negative × positive	negative
negative × negative	positive

16. One Negative Factor Gives a Negative Product

(−4) × 6 = −24.

7 × (−3) = −21.

There is one direction reversal in the sign structure.

17. Two Negative Factors Give a Positive Product

(−4) × (−6) = 24.

One way to see why is to preserve distributive structure:

Since 0 = (−4) × 0 = (−4)[6 + (−6)], distributing gives:

0 = (−4)(6) + (−4)(−6) = −24 + (−4)(−6).

Therefore (−4)(−6) must be +24 to make the total zero.

This explanation shows that the sign rule preserves the ordinary laws of arithmetic.

18. Count Negative Factors in a Longer Product

If an even number of negative factors is multiplied, the final sign is positive. If an odd number is multiplied, the final sign is negative.

For example:

(−2)(−3)(−5) = −30

because there are three negative factors.

19. Division Uses the Same Sign Pattern

• 24 ÷ 6 = 4;
• −24 ÷ 6 = −4;
• 24 ÷ (−6) = −4;
• −24 ÷ (−6) = 4.

Same signs give a positive quotient. Different signs give a negative quotient.

20. Zero in Multiplication and Division

Any number multiplied by zero is zero.

Zero divided by any non-zero number is zero.

Division by zero is not defined. There is no real number x such that 0 × x = 5, so 5 ÷ 0 cannot be assigned an ordinary real-number value.

Your Turn 3

1. (−7)(5)
2. (−8)(−4)
3. (−2)(3)(−5)
4. 54 ÷ (−6)
5. −72 ÷ (−8)
6. 0 ÷ 9

Answers

−35, 32, 30, −9, 9, 0.

21. Order of Operations Still Governs Directed Numbers

Signs do not suspend the order of operations.

Evaluate:

−3 + 4 × (−2).

Multiply first:

4 × (−2) = −8.

Then:

−3 + (−8) = −11.

22. Brackets Protect Negative Values

If x = −3, then x² means:

(−3)² = 9.

When substituting a negative value into an expression, put the value in brackets first. This habit becomes essential in algebra and graphs.

23. Rational Numbers Include More Than Fractions Written With a Fraction Bar

A rational number can be expressed as p/q where p and q are integers and q ≠ 0.

Examples include:

• 5 = 5/1;
• −3 = −3/1;
• 2/7;
• −11/4;
• 0.25 = 1/4;
• 0.333… = 1/3.

Integers are therefore rational numbers because every integer can be divided by 1.

24. Terminating Decimals Are Rational

A terminating decimal can be converted to a fraction with a power of 10 in the denominator.

0.375 = 375/1000 = 3/8.

Therefore 0.375 is rational.

25. Recurring Decimals Are Rational

A recurring decimal repeats a fixed pattern and can be represented exactly as a fraction.

For example:

0.333… = 1/3.

The decimal representation continues, but the value is exact.

26. Fractions Can Be Negative in Several Equivalent Forms

The following represent the same rational number:

−3/5 = 3/(−5) = −(3/5).

It is usually clearest to place the negative sign in front of the fraction.

27. Ordering Rational Numbers May Require a Common Representation

To compare −2/3 and −0.7, convert one form:

−2/3 ≈ −0.666….

On the number line, −0.7 lies further left, so:

−0.7 < −2/3.

Among negative numbers, the value with the greater magnitude can be smaller.

Your Turn 4

Arrange in ascending order:

−3/4, 0.2, −0.6, 1/5, −1.

Answer

−1, −3/4, −0.6, 0.2 and 1/5 are last two equal because 1/5 = 0.2. A complete ordered statement may show −1 < −3/4 < −0.6 < 0.2 = 1/5.

28. The Real Number Line Contains Rational and Irrational Numbers

All ordinary numbers represented on the continuous number line are real numbers. Rational numbers are one part of the real numbers. Irrational numbers form the other part.

This classification is useful for understanding the number system. The depth at which irrational numbers are formally developed can vary by subject level and school sequence, so treat the next sections as a conceptual bridge where necessary.

29. Irrational Numbers Cannot Be Written as a Ratio of Integers

Numbers such as √2 and π are irrational. Their decimal expansions do not terminate and do not repeat in a fixed recurring pattern.

They still have exact meaning. Writing √2 is exact; writing 1.414 is an approximation.

30. Not Every Square Root Is Irrational

√49 = 7, which is rational. √81 = 9, also rational. A square root is irrational only when the value cannot simplify to a rational number.

This links directly back to Chapter 1: recognising perfect squares helps classify roots.

31. Exact Form and Decimal Approximation Are Different Objects

√2 and 1.414 are not exactly equal. The second is a rounded approximation of the first.

Use an equality sign only when values are equal. Use an approximation sign where the notation is available and appropriate, or write “≈”.

32. The Number Hierarchy Helps You Avoid False Categories

• Whole numbers include 0, 1, 2, 3, ….
• Integers include negative whole numbers, zero and positive whole numbers.
• Rational numbers include all integers and fractions of integers with non-zero denominator.
• Real numbers include rational and irrational numbers.

A number can belong to several nested categories. The number 5 is whole, integer, rational and real.

33. Four Operations With Fractions Still Need Sign Control

Negative signs do not change the denominator logic.

For addition and subtraction, use a common denominator.

For multiplication, multiply numerators and denominators after simplifying common factors where possible.

For division, multiply by the reciprocal of the divisor, then apply sign rules.

34. Teacher Model 3: Add Fractions With Opposite Signs

Evaluate:

−3/4 + 5/6.

LCM of 4 and 6 is 12:

−3/4 = −9/12

5/6 = 10/12.

Therefore:

−9/12 + 10/12 = 1/12.

35. Teacher Model 4: Subtract a Negative Fraction

Evaluate:

2/3 − (−5/8).

Rewrite subtraction of a negative as addition:

2/3 + 5/8.

Common denominator 24:

16/24 + 15/24 = 31/24 = 1 7/24.

36. Teacher Model 5: Multiply Signed Fractions

Evaluate:

(−4/9)(3/10).

One negative factor means the product is negative. Simplify before multiplying:

−(4×3)/(9×10) = −12/90 = −2/15.

37. Teacher Model 6: Divide Signed Fractions

Evaluate:

−5/6 ÷ 10/9.

Multiply by the reciprocal:

−5/6 × 9/10.

Simplify:

−3/4.

38. Convert Decimals and Fractions Deliberately

Use the form that makes the next operation easiest.

• 0.75 = 3/4;
• −1.25 = −5/4;
• 3/8 = 0.375;
• −7/20 = −0.35.

Do not switch to decimals automatically if an exact fraction keeps the calculation simpler.

39. Decimal Multiplication Needs Place-Value Control

For −1.2 × 0.4, first determine the sign: negative. Then calculate the magnitude 1.2 × 0.4 = 0.48.

−1.2 × 0.4 = −0.48.

An estimate 1 × 0.4 ≈ 0.4 confirms that 4.8 would be implausibly large.

40. Decimal Division Can Be Checked by Inverse Multiplication

If −3.6 ÷ 0.6 = −6, check:

−6 × 0.6 = −3.6.

Use the inverse operation to verify the quotient.

41. Mixed Numbers Should Usually Be Converted Before Multiplication or Division

For −1 1/2 × 2/3, convert:

−1 1/2 = −3/2.

Then:

−3/2 × 2/3 = −1.

Do not multiply the whole-number and fraction parts separately.

Your Turn 5

1. −2/5 + 3/10
2. 7/8 − (−1/4)
3. (−3/7)(14/15)
4. −5/6 ÷ (−10/9)
5. −1.5 + 0.8
6. −2.4 × (−0.5)

Answers

−1/10, 9/8 or 1 1/8, −2/5, 3/4, −0.7, 1.2.

42. Comparison Symbols Describe Position

• a < b means a lies to the left of b.
• a > b means a lies to the right of b.
• a ≤ b includes equality.
• a ≥ b includes equality.

The symbol should be read as a relationship, not as a decorative arrow.

43. Negative Comparisons Are Easier on a Number Line

Compare −8 and −3. Since −8 lies further left:

−8 < −3.

Do not compare only the unsigned digits 8 and 3.

44. A Number Line Can Represent a Range of Values

The statement x > 2 represents every real value to the right of 2. The statement x ≤ 5 represents every real value up to and including 5.

When number-line endpoint notation is introduced, an included boundary is shown differently from an excluded boundary. Follow the notation used by your school and explain the inclusion in words.

45. Write Compound Position Statements Carefully

If x is greater than −2 and not greater than 4:

−2 < x ≤ 4.

This statement describes one continuous region of the number line.

46. Teacher Model 7: Translate Context Into an Inequality

A lift can carry at most 600 kg. Let m be the total mass loaded.

“At most” includes 600:

m ≤ 600.

The inequality symbol is determined by the language and inclusion of the boundary.

47. Common Boundary Language

Phrase	Typical inequality
more than 5	x > 5
at least 5	x ≥ 5
less than 5	x < 5
at most 5	x ≤ 5
between 2 and 7, not including endpoints	2 < x < 7
from 2 to 7 inclusive	2 ≤ x ≤ 7

Later algebra chapters will solve inequalities. Here the priority is reading and representing number-line relationships correctly.

48. Approximation Replaces an Exact Value With a Nearby Stated Value

Rounding is not arbitrary shortening. A rounded value must state the required place value or number of significant figures.

For 7.386:

• to 1 decimal place: 7.4;
• to 2 decimal places: 7.39;
• to 3 significant figures: 7.39.

The last two happen to match numerically here, but decimal places and significant figures are different instructions.

49. Decimal Places Count From the Decimal Point

For 23.746:

• 1 decimal place looks at the tenths digit and then the hundredths digit for rounding: 23.7;
• 2 decimal places gives 23.75.

Decimal-place counting begins immediately after the decimal point.

50. Significant Figures Begin at the First Non-Zero Digit

For 0.004826:

The first significant digit is 4, not the leading zeros.

• 1 significant figure: 0.005;
• 2 significant figures: 0.0048;
• 3 significant figures: 0.00483.

Leading zeros locate the decimal point; they do not count as significant figures.

51. Zeros Between Significant Digits Do Count

In 3.04, all three digits 3, 0 and 4 are significant because the zero lies between non-zero significant digits.

In 0.0304, the leading zeros do not count, but the zero between 3 and 4 does. The number has three significant figures.

52. Teacher Model 8: Round a Large Number

Round 58,746 to 3 significant figures.

The first three significant digits are 5, 8 and 7. The next digit is 4, so 7 stays unchanged:

58,700.

The zeros preserve place value in the rounded number.

53. Teacher Model 9: Round a Small Decimal

Round 0.006784 to 2 significant figures.

The first significant digit is 6 and the second is 7. The next digit is 8, so round the 7 upward:

0.0068.

54. Negative Numbers Round by Magnitude and Keep Their Sign

To round −7.386 to 1 decimal place, round the magnitude 7.386 to 7.4 and keep the negative sign:

−7.4.

Do not think “round up means more positive”. Rounding acts on place value; the sign remains attached to the number.

55. Do Not Round Intermediate Values Too Early

In a multi-step calculation, early rounding can accumulate error. Keep exact fractions or fuller calculator values through the working where practical and round at the required final stage.

If a question explicitly instructs you to use rounded data, follow that instruction. Otherwise preserve accuracy.

Your Turn 6

1. Round 18.764 to 1 decimal place.
2. Round 18.764 to 2 decimal places.
3. Round 18.764 to 3 significant figures.
4. Round 0.003746 to 2 significant figures.
5. Round 725,481 to 2 significant figures.
6. Round −9.857 to 2 decimal places.

Answers

18.8, 18.76, 18.8, 0.0037, 730,000, −9.86.

56. Estimation Predicts the Scale Before Exact Calculation

Estimate 49.8 × 19.7.

Round to convenient values:

50 × 20 = 1000.

The exact answer should be near 1000. If a calculator returns 98.106 or 98,106, the estimate tells you to inspect the entry.

57. One Significant Figure Often Makes a Good Quick Estimate

For many estimation questions, rounding each input to one significant figure creates simple arithmetic while preserving scale.

Estimate:

198 × 3.96 ÷ 0.502.

Use 200 × 4 ÷ 0.5 = 1600.

The exact result should be around 1600.

58. Estimation Is Not Random Rounding

Choose approximations that make the calculation simpler while preserving the rough size of each quantity.

For 397 ÷ 19.8, using 400 ÷ 20 = 20 is sensible. Using 100 ÷ 10 = 10 changes the scale too much to be a useful estimate.

59. Estimate Fractions Before Trusting a Decimal

If a result is 17/19, you know it should be slightly less than 1 before converting to a decimal.

If the calculator shows 8.947, the estimate disproves the entry immediately.

60. Estimate Signed Results Too

For −49.8 × 19.7, one negative factor means the answer is negative and the magnitude is near 1000.

So expect about −1000. Estimation predicts both sign and scale.

61. Teacher Model 10: Estimate a Multi-Step Expression

Estimate:

(79.4 × 0.203) ÷ 4.97.

Round:

80 × 0.2 ÷ 5 = 16 ÷ 5 = 3.2.

Therefore the exact answer should be near 3.2.

62. Estimation Can Identify the Correct Multiple-Choice Option

If the exact calculation is long but the options are 0.32, 3.2, 32 and 320, the estimate above often identifies the correct scale before exact work.

Do not replace required exact working with estimation unless the question asks for an estimate. Use estimation as a control layer.

63. Estimation and Approximation Are Related but Different

Idea	Main job
approximation	replace a value by a nearby value to a stated accuracy
estimation	obtain a rough result to predict scale or check reasonableness

A rounded value may be the final requested answer. An estimate is often deliberately rougher and used for judgement.

64. Build a Calculator Preflight

1. Predict the sign.
2. Estimate the scale.
3. Enter brackets around negative values.
4. Check the operation symbols.
5. Read the display with units or context.
6. Reject an output that violates sign or scale.

This routine prepares students for later calculator-heavy work without surrendering mathematical control to the device.

65. Misconception Clinic: −8 Is Greater Than −3 Because 8 Is Greater Than 3

Compare positions, not unsigned digits. −8 lies to the left of −3, so −8 < −3.

66. Misconception Clinic: Zero Is Positive

Zero is neither positive nor negative. It is the reference separating the two directions.

67. Misconception Clinic: A Minus Sign Always Means Subtraction

In 5 − (−3), the first minus is the subtraction operation. The second belongs to the number −3. Separate operation sign from number sign.

68. Misconception Clinic: Two Negatives Always Make a Positive

The phrase is dangerously incomplete. In multiplication or division, two negative signs give a positive result. In addition, −4 + (−5) = −9. Always identify the operation first.

69. Misconception Clinic: Subtracting a Negative Means Ignore Both Signs

Do not cross out signs mechanically. Rewrite subtraction as addition of the opposite: a − (−b) = a + b.

70. Misconception Clinic: Negative Squared Is Always Negative

(−3)² = 9 because the entire negative number is multiplied by itself. Brackets preserve what is being squared.

By contrast, −3² is conventionally read as −(3²) = −9 because the power acts before the leading negative sign. This is why substitution brackets matter.

71. Misconception Clinic: Every Decimal Is Irrational

Terminating decimals and recurring decimals are rational. Irrational decimals are non-terminating and non-recurring.

72. Misconception Clinic: Every Square Root Is Irrational

√64 = 8 is rational. The classification depends on the value, not the presence of a root symbol alone.

73. Misconception Clinic: Rounding to 2 Decimal Places Means 2 Significant Figures

Decimal places count digits after the decimal point. Significant figures begin at the first non-zero significant digit. They answer different accuracy instructions.

74. Misconception Clinic: Leading Zeros Are Significant

In 0.0048, the leading zeros locate the decimal place and are not significant figures. The number has two significant figures: 4 and 8.

75. Misconception Clinic: Rounding a Negative Number “Up” Must Make It More Positive

Round the magnitude according to place value and keep the sign. −7.386 to 1 decimal place is −7.4.

76. Misconception Clinic: Estimate Means Calculate Exactly With Rounded Inputs

An estimate deliberately sacrifices precision to gain speed and scale awareness. Choose convenient nearby numbers; do not pretend the estimate is exact.

77. Misconception Clinic: If the Calculator Gives a Decimal, It Must Be Correct

A calculator can return a perfectly computed result for the wrong expression. Compare the sign and scale with your estimate and context.

78. Guided Practice Set A: Order and Number-Line Position

1. Arrange −4, 3, −9, 0 and 6 in ascending order.
2. Insert <, > or = between −7 and −2.
3. Insert <, > or = between −3/4 and −0.8.
4. Which is further from zero, −11 or 7?
5. Which is greater, −0.25 or −1/5?

Solutions

−9, −4, 0, 3, 6. −7<−2. −3/4=−0.75, so −3/4>−0.8. −11 is further from zero. −1/5=−0.2, so −1/5>−0.25.

79. Guided Practice Set B: Addition and Subtraction

1. −8 + 13
2. 11 + (−17)
3. −6 + (−12)
4. 7 − 15
5. 7 − (−15)
6. −14 − (−9)
7. −3 − 8 + 5

Solutions

5, −6, −18, −8, 22, −5, −6.

80. Guided Practice Set C: Multiplication, Division and Order

1. (−8)(6)
2. (−7)(−9)
3. (−2)(−3)(−4)
4. −63 ÷ 7
5. −72 ÷ (−8)
6. −5 + 3(−4)
7. (−3)² − 5

Solutions

−48, 63, −24, −9, 9, −17, 4.

81. Guided Practice Set D: Fractions and Decimals

1. −2/3 + 5/6
2. 3/4 − (−5/8)
3. (−5/12)(18/25)
4. −7/9 ÷ 14/27
5. −1.7 + 2.4
6. −3.2 × 0.5

Solutions

1/6, 11/8 or 1 3/8, −3/10, −3/2, 0.7, −1.6.

82. Guided Practice Set E: Number Classification

State whether each number is definitely rational. Where relevant, also state whether it is an integer.

1. −7
2. 3/11
3. 0.125
4. 0.666… recurring
5. √49
6. √2

Solutions

−7 is integer and rational. 3/11 is rational. 0.125=1/8, rational. 0.666… recurring is rational. √49=7, integer and rational. √2 is irrational; treat formal irrational-number classification according to your school sequence.

83. Guided Practice Set F: Inequality Language

1. Write “x is greater than 4” as an inequality.
2. Write “y is at most 12”.
3. Write “m is at least −3”.
4. Write “t is between 2 and 9 inclusive”.
5. Write “p is greater than −5 but less than or equal to 1”.

Solutions

x>4. y≤12. m≥−3. 2≤t≤9. −5<p≤1.

84. Guided Practice Set G: Approximation

1. Round 36.784 to 1 decimal place.
2. Round 36.784 to 2 decimal places.
3. Round 36.784 to 2 significant figures.
4. Round 0.006583 to 2 significant figures.
5. Round 583,491 to 3 significant figures.
6. Round −12.746 to 1 decimal place.

Solutions

36.8, 36.78, 37, 0.0066, 583,000, −12.7.

85. Guided Practice Set H: Estimation

1. Estimate 39.7 × 20.4.
2. Estimate 598 ÷ 19.9.
3. Estimate 203 × 4.08 ÷ 0.498.
4. Predict the sign and rough size of −51.2 × 9.8.

Possible estimates

40×20≈800. 600÷20≈30. 200×4÷0.5≈1600. −50×10≈−500.

86. Challenge Practice: A Temperature Journey

A temperature begins at −6°C. It rises by 11°C, falls by 8°C and then rises by 4°C.

1. Write the calculation using signed numbers.
2. Find the final temperature.
3. Find the total upward change and total downward change separately.

Worked solution

−6+11−8+4=1. Final temperature 1°C. Total upward change 15°C; total downward change 8°C.

87. Challenge Practice: Exact and Approximate Values

A square has side length √2 cm.

1. State the exact side length.
2. Give the side length to 3 decimal places.
3. Find the exact area.
4. Explain why using 1.414 for the side before calculating area can introduce rounding error.

Worked solution

Exact side √2 cm. To 3 d.p., about 1.414 cm. Exact area = (√2)² = 2 cm². Using 1.414 first produces 1.999396 cm², which is close but no longer exact.

88. Challenge Practice: Ordering Mixed Representations

Arrange in ascending order:

−0.72, −7/10, 0, 2/5, 0.39.

Worked solution

−7/10=−0.7 and 2/5=0.4. Therefore −0.72<−0.7<0<0.39<0.4.

89. Challenge Practice: Estimate Before Exact Calculation

Without using a calculator first, estimate:

(61.2 × 19.6) ÷ 3.98.

Worked solution

Use 60×20÷4=300. The exact answer should therefore be near 300. This gives a strong calculator-entry check.

90. Examination Method: Put Brackets Around Negative Substitutions

If x = −4 and an expression contains x², write (−4)² before evaluating. This habit prevents sign ambiguity and carries directly into algebra and graph substitution.

91. Examination Method: Rewrite Subtraction Before You Rush

When two negative signs appear together, rewrite:

−5 − (−8) = −5 + 8.

This makes the operation visible and reduces sign slips.

92. Examination Method: Convert to the Most Useful Representation

• Use fractions when exact arithmetic is cleaner.
• Use decimals when the context or calculator work makes them natural.
• Use a number line when ordering negatives is uncertain.
• Use an inequality when a whole range of values is described.

93. Examination Method: Distinguish Exact From Approximate

Write √2 when the exact value is useful. Write a rounded decimal only when the question asks for an approximation or a decimal is needed for practical interpretation.

94. Examination Method: State the Requested Accuracy

Before rounding, underline the instruction: decimal places, significant figures or another stated accuracy. Do not round to your habitual number of decimal places.

95. Examination Method: Estimate Before Calculator Work

Predict sign and scale before pressing equals. After calculation, compare the output against the estimate. A large discrepancy means inspect the entry, units or brackets.

96. Examination Method: Keep Units Attached in Context

Negative 4 may mean −4°C, −$4 relative to zero balance, or 4 metres below a reference level. Keep units and reference meaning visible so the sign is interpreted correctly.

97. Examination Method: Use an Independent Check

• Addition/subtraction: verify on a number line or by inverse operation.
• Multiplication/division: check sign separately from magnitude.
• Fraction result: compare its rough size with nearby benchmark values.
• Rounded value: check the next digit used for rounding.
• Calculator result: compare with an estimate.

98. Oral Classroom Check

1. Why is −8 less than −3?
2. Why is zero neither positive nor negative?
3. What is the difference between the subtraction sign and a negative-number sign?
4. Why does subtracting a negative become addition?
5. When do two negative signs give a positive result?
6. What makes a number rational?
7. Why is a terminating decimal rational?
8. Why is √49 rational but √2 irrational?
9. What is the difference between decimal places and significant figures?
10. Why should you estimate before trusting a calculator?

The student should answer with a numerical example. If the explanation collapses into a slogan such as “minus minus plus”, return to the relevant operation.

99. Exit Ticket

Complete without notes.

1. Arrange −5, 2, −1, 0 and −8 in ascending order.
2. Evaluate −7 + 12 − 9.
3. Evaluate −4 − (−11).
4. Evaluate (−3)(−5)(2).
5. Evaluate −3/4 + 5/8.
6. Write “x is at least −2” as an inequality.
7. Round 0.007864 to 2 significant figures.
8. Estimate 49.6 × 20.3.
9. State whether 0.375 is rational and justify.

Exit-ticket solution

−8, −5, −1, 0, 2. −7+12−9=−4. −4−(−11)=7. (−3)(−5)(2)=30. −3/4+5/8=−6/8+5/8=−1/8. x≥−2. 0.0079. Estimate 50×20≈1000. 0.375=3/8, so it is rational.

100. Homework: Retrieval, Variation and Transfer

Layer 1 — Retrieval

• Draw a number line from −10 to 10 and mark six values.
• Explain addition and subtraction of directed numbers without using only a sign slogan.
• Write the multiplication/division sign table.
• Define rational number.
• Write the difference between decimal places and significant figures.
• Write a six-step calculator preflight.

Layer 2 — Variation

• Five ordering questions mixing negatives, fractions and decimals.
• Five addition/subtraction questions.
• Five multiplication/division questions.
• Three signed-fraction questions.
• Three inequality-language questions.
• Four rounding questions.
• Three estimation questions.

Layer 3 — Transfer

Create a real-world journey using temperature, altitude or account balance. Include at least four signed changes, one inequality condition and one final value rounded to a stated accuracy. Solve it and add an estimate or number-line check.

101. The Seven-Day Return Cycle

1. Day 0: complete models and guided practice.
2. Day 1: order ten mixed signed values and solve four arithmetic questions.
3. Day 3: solve one signed-fraction question, one inequality translation and one estimate without notes.
4. Day 7: repeat the exit ticket with changed values and explain each sign decision aloud.

102. A 60-Minute Teaching Lesson

1. 5 minutes: number-line retrieval.
2. 10 minutes: addition and subtraction through movement.
3. 10 minutes: multiplication and division sign structure.
4. 10 minutes: rational numbers, fractions and decimals.
5. 10 minutes: inequalities and number-line language.
6. 10 minutes: rounding and estimation.
7. 5 minutes: exit ticket and return date.

103. A 90-Minute Teaching Lesson

1. 10 minutes: signed-number diagnostic.
2. 15 minutes: number-line and addition/subtraction modelling.
3. 15 minutes: multiplication, division and order of operations.
4. 15 minutes: signed fractions and decimals.
5. 10 minutes: rational/real-number classification.
6. 10 minutes: inequalities.
7. 10 minutes: approximation and estimation.
8. 5 minutes: exit ticket.

104. The Full Chapter Routine

For number-line ordering:

common representation → locate → left/right compare → write inequality.

For directed addition/subtraction:

starting value → operation → signed change → rewrite subtraction if needed → move/check.

For multiplication/division:

sign count → magnitude calculation → attach sign → inverse check.

For approximation:

identify requested accuracy → locate final retained digit → inspect next digit → round → preserve place value.

For estimation:

predict sign → round to convenient scale → calculate mentally → compare exact output.

105. Why This Chapter Matters Beyond Chapter 2

Directed-number control sits underneath almost every later Secondary Mathematics topic. Algebra requires negative substitution and sign preservation. Coordinates use negative positions. Graphs cross positive and negative regions. Inequalities describe number-line intervals. Statistics may contain negative changes. Trigonometry and geometry depend on calculator discipline and approximation. Scientific and financial models rely on knowing whether a value is exact, rounded or estimated.

The deeper habit is this: before calculating, decide what the number represents, where it lies, how the operation changes it and what scale the answer should have.

106. Connect Back to Chapter 1

Chapter 1 decomposed positive whole numbers into prime factors. Chapter 2 expands the number system and teaches signed position, rational representation and approximation. The two chapters reconnect whenever fractions need common denominators, roots are classified or numerical structure is estimated before exact calculation.

Return to Secondary 1 Mathematics Classroom | Chapter 1: Number Structure →

107. Ready for Chapter 3?

You are ready to move on when you can do all of the following without prompts:

• order positive and negative values on a number line;
• compare negative numbers correctly;
• add and subtract directed numbers by interpreting the operation;
• multiply and divide signed numbers accurately;
• use brackets for negative substitution;
• work with signed fractions and decimals;
• recognise rational numbers in fraction, integer and decimal forms;
• distinguish exact values from approximations;
• read and write simple inequality relationships;
• round to stated decimal places and significant figures;
• estimate the sign and scale of a calculation;
• use estimation to reject implausible calculator output.

If one item is weak, return to the smallest section that owns it and complete a changed example. If all are stable, continue to Ratio and Proportion, where multiplicative comparison becomes the next major Secondary 1 structure.

Continue the Secondary 1 Mathematics Learning Route

• Secondary 1 Mathematics Learning Guide | Directed Numbers, Rational Numbers and Four Operations
• Secondary 1 Mathematics Learning Guide | Numbers, Number Lines, Approximation and Estimation
• Secondary 1 Mathematics Learning Guide | Calculator Skills, Exact Values and Input Discipline
• Secondary 1 Mathematics Learning Guide | Indices, Powers and Standard Form
• Secondary 1 Mathematics Learning Guide | Ratio and Proportion
• Secondary Mathematics Sengkang | S1–S4 Capability Map