Secondary 4 Mathematics Classroom | Chapter 6: Numbers and Algebra Revision | SEC G3 K310

SECONDARY 4 MATHEMATICS CLASSROOM · CHAPTER 6 · NUMBERS AND ALGEBRA REVISION · SEC G3 K310

Numbers and Algebra Revision: Rebuild the Whole System, Not Just the Last Topic

In this classroom, you will not revise by reading one chapter after another. You will diagnose the first weak dependency, repair it, reconnect it to later topics, and then test whether the method still works when the question changes form.

Secondary 4 Numbers and Algebra is a connected system. Indices affect algebra. Ratio affects percentage. Percentage affects finance. Rates depend on units. Formulae depend on inverse operations. Graphs depend on substitution and equations. Quadratics can appear as algebra, graphs or real-world models. Sets and matrices return because revision must be cumulative.

Classroom rule: diagnose the first wrong step, not the chapter name at the top of the worksheet.

This classroom follows the current Singapore-Cambridge SEC G3 Mathematics syllabus, K310. Its Number and Algebra strand runs from N1 Numbers and their operations through N9 Matrices, with cumulative expectations in problem solving, reasoning, communication and real-world contexts.

Reference: 2027 SEC G3 syllabuses | SEAB.

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Featured Answer: What Is Numbers and Algebra Revision?

Numbers and Algebra Revision is the process of keeping many mathematical tools simultaneously available and knowing which one a new question requires. It is not a final rereading of definitions. It is repeated retrieval, variation, connection, diagnosis and transfer.

A strong Secondary 4 student should be able to move between:

• number structure and algebraic structure;
• ratio, percentage and multiplicative change;
• rate and gradient;
• expressions, equations, formulae and inequalities;
• functions, tables, graphs and contexts;
• quadratic forms and their graphical features;
• sets, matrices and other representations;
• exact working and sensible estimation.

The Simple Classroom Answer

Revision asks: can you recognise the structure, choose the method, execute it accurately, explain why it works and still do it when the question is disguised?

How to Use This Classroom

1. Attempt the question before reading the explanation.
2. Mark the first line where your work becomes wrong or uncertain.
3. Name the prerequisite behind that error.
4. Repair the prerequisite with a smaller example.
5. Return immediately to the original question.
6. Do one changed version so the method cannot be copied mechanically.
7. Return again later without notes to test retrieval.

1. Build a Dependency Map Before Revision

Teacher: Write these topics across the board: indices, ratio, percentage, rates, algebra, formulae, functions, graphs, equations, inequalities, sets, matrices.

Now connect them with arrows. A reverse-percentage problem may need algebra. A distance-time graph needs rate and gradient. A quadratic graph needs substitution, expansion and equation-solving. A matrix application may use percentage scaling. Mixed problems do not respect chapter boundaries.

The visible topic is not always the true source of the error.

2. Number Control Comes Before Advanced Algebra

Advanced-looking questions often fail because of basic numerical control. Fractions, negative signs, powers, roots, standard form and rounding sit underneath later work.

• Can you preserve signs through brackets?
• Can you simplify fractions before using a calculator?
• Can you estimate the likely size of the result?
• Can you recognise whether a square root answer is sensible?
• Can you keep exact values until rounding is actually required?

Do not label every numerical slip “careless”. Repeated slips usually indicate an unstable routine.

3. Order of Operations Is Structural

The expression 3 + 4 × 5 is not evaluated left to right. Multiplication acts before addition, giving 23.

Brackets change the structure: (3 + 4) × 5 = 35.

Teacher: Make the student annotate which operation owns which part of the expression before calculating.

Your Turn 1

1. 18 − 3 × 4
2. (18 − 3) × 4
3. 24 ÷ 3 × 2
4. 2 + 3² × 4

Answers

6, 60, 16, 38.

4. Approximation Is a Checking Tool, Not Only a Question Type

If a calculator returns 0.0048 when you expected a value around 48, the problem is probably entry, units or powers of ten. Estimation catches this before the answer is submitted.

Round only when appropriate. Keep more accurate intermediate values so final rounding does not accumulate error.

5. Standard Form Is a Scale Language

Standard form writes a non-zero number as a × 10ⁿ where 1 ≤ |a| < 10. It lets very large and very small quantities be compared and calculated cleanly.

For example:

• 4,500,000 = 4.5 × 10⁶;
• 0.00072 = 7.2 × 10⁻⁴.

The sign of the exponent records scale. It does not make the coefficient negative.

6. Indices Compress Repeated Multiplication

Index laws work because they preserve multiplication structure.

• aᵐ × aⁿ = aᵐ⁺ⁿ;
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ, where the division is defined;
• (aᵐ)ⁿ = aᵐⁿ;
• a⁰ = 1 for a ≠ 0;
• a⁻ⁿ = 1/aⁿ;
• a¹⁄ⁿ represents an nth-root relationship where defined.

Index laws belong to multiplication and division structure, not ordinary addition.

7. Misconception Clinic: Adding Powers Across Addition

x² + x³ cannot be simplified to x⁵. The terms are being added, not multiplied.

But x² × x³ = x⁵ because the bases match and multiplication is occurring.

Your Turn 2

1. x⁴ × x³
2. x⁷ ÷ x²
3. (x³)⁴
4. x⁻³
5. 16¹⁄²

Answers

x⁷, x⁵, x¹², 1/x³, 4.

8. Ratio Compares Multiplicatively

If red:blue = 3:5, there are 8 ratio parts in total. A total of 64 objects gives 8 objects per part, so red = 24 and blue = 40.

Do not use the numerical difference 5 − 3 = 2 as the total number of parts. Ratio is a multiplicative partition.

9. Ratio With Fractions Must Be Normalised

A ratio such as 1/2 : 3/4 can be converted by multiplying both terms by a common denominator. Multiply by 4 to obtain 2:3.

The ratio has changed form, not meaning.

10. Scale Is a Ratio Between Representations

A map scale converts representation length into actual length. If 1 cm represents 5 km, a 7.2 cm route represents 36 km.

If two similar figures have length scale factor k, their area scale factor is k² and their volume scale factor is k³. Keep the quantity type visible before applying the factor.

11. Direct Proportion Preserves a Constant Ratio

If y is directly proportional to x, then y = kx for some constant k.

If x doubles, y doubles. If x is multiplied by 5, y is multiplied by 5.

Find k from one known pair, then use the model for new values.

12. Inverse Proportion Preserves a Constant Product

If y is inversely proportional to x, then y = k/x and xy = k.

If x doubles, y halves. The product remains constant.

Teacher: Contrast this with direct proportion using a table so students see the different invariants.

13. Teacher Model 1: Direct Proportion

y is directly proportional to x. When x = 4, y = 18. Find y when x = 10.

y = kx, so 18 = 4k and k = 4.5.

When x = 10:

y = 4.5(10) = 45.

14. Teacher Model 2: Inverse Proportion

y is inversely proportional to x. When x = 6, y = 8. Find y when x = 12.

xy = k, so k = 48.

When x = 12:

y = 48/12 = 4.

15. Percentage Is Always Percentage of a Base

Before every percentage calculation, identify what 100% represents.

Percentage without a base is incomplete information.

A 20% increase means multiply the original by 1.20. A 20% decrease means multiply the current base by 0.80.

16. Successive Percentage Changes Multiply

A 20% increase followed by a 20% decrease gives a total multiplier:

1.20 × 0.80 = 0.96.

The final value is 96% of the original, a 4% decrease overall.

The percentages are equal but the bases are different.

17. Reverse Percentage Means Reconstruct the Original Base

If $84 is 70% of the original amount, then:

0.70x = 84.

x = 84 ÷ 0.70 = 120.

Do not subtract 30% of 84. That uses the final amount as the base instead of reconstructing the original.

Your Turn 3

After a 15% discount, a jacket costs $102. Find the original price.

Answer

85% remains. 0.85x = 102, so x = 120. Original price = $120.

18. Rates Compare Different Units

A rate compares unlike quantities: kilometres per hour, dollars per kilogram, litres per minute or words per second.

The units are part of the mathematical object. If distance is in kilometres and time is in seconds, convert before combining unless the desired rate specifically uses km/s.

19. Average Speed Uses Total Distance and Total Time

Average speed = total distance ÷ total time.

Do not automatically average two speeds arithmetically. If the traveller spends unequal times at the speeds, the simple average is usually wrong.

20. Teacher Model 3: Average Speed

A cyclist travels 18 km in 45 minutes.

45 minutes = 0.75 hours.

Average speed = 18 ÷ 0.75 = 24 km/h.

On a straight segment of a distance-time graph over the same interval, the gradient represents the same rate.

21. Unit Conversion Should Be Visible

Write the conversion line rather than hiding it inside the calculator:

• 3 minutes = 180 seconds;
• 2.5 km = 2500 m;
• 1 hour = 3600 seconds.

Visible unit work often explains the numerical factor and makes checking much easier.

22. Know Which Algebraic Object You Are Holding

Object	Example	Main job
Expression	3x + 7	simplify, expand, factorise or evaluate
Equation	3x + 7 = 19	solve for values satisfying equality
Formula	v = u + at	substitute or change the subject
Identity	(a+b)² = a²+2ab+b²	use as an equality true throughout its domain
Inequality	3x+1 < 10	find a range of values

Do not solve an expression as though it contains an equals sign. Do not factorise if the question asks for evaluation. Object recognition is method selection.

23. Substitution Requires Brackets Around Negative Values

If f(x) = x² − 3x + 2 and x = −2, write:

f(−2) = (−2)² − 3(−2) + 2.

The brackets protect the sign and the power structure.

24. Simplifying Means Combining Like Terms

3x + 5x = 8x because both terms contain the same variable part x.

3x + 5x² cannot be combined because x and x² are different algebraic objects.

25. Expansion Reveals Terms

For:

(x + 3)(x − 2)

expand systematically:

x² − 2x + 3x − 6 = x² + x − 6.

The expanded form exposes coefficients and is often useful in graph and equation work.

26. Factorisation Reveals Multiplicative Structure

x² + x − 6 factorises to (x + 3)(x − 2).

If the expression is set equal to zero, the factorised form immediately reveals roots −3 and 2.

Expansion and factorisation are opposite views of the same expression.

27. Choose the Algebraic Form From the Next Job

• Need roots? Factorised form may help.
• Need coefficients? Expanded form may help.
• Need turning-point structure? Completing-square form may help.
• Need substitution? Use the form that minimises error.

Equivalent forms are tools, not decorations.

28. Algebraic Fractions: Cancel Factors, Not Terms

In:

(x² − 9)/(x − 3),

factorise the numerator:

(x−3)(x+3)/(x−3) = x+3,

for x ≠ 3.

The restriction remains because x = 3 made the original denominator zero.

29. Misconception Clinic: Cancel Across Addition

In (x + 3)/x, you cannot cancel x with the x inside x + 3 because x is not a common factor of the whole numerator.

Cancellation is division by a common factor.

30. Formula Rearrangement Is Controlled Inversion

If y = (3x − 5)/4 and x is required:

4y = 3x − 5

4y + 5 = 3x

x = (4y + 5)/3.

Each line preserves equality. Avoid vague language such as “move it over” when the operation can be stated exactly.

31. Teacher Model 4: Change the Subject With a Product

Given A = πr², make r the subject.

A/π = r².

For a radius, take the non-negative square root:

r = √(A/π).

Context helps choose the physically meaningful root.

32. nth-Term Patterns Turn Sequences Into Rules

A sequence is not only a list. Its nth-term rule lets you generate any required position and test whether a value belongs to the sequence.

For 5, 8, 11, 14, … the common difference is 3. A rule is:

3n + 2.

Check n = 1 gives 5.

33. Functions Connect Inputs to Outputs

A function can be represented as a rule, a table, a graph or a context. Revision should move between all four.

If y = 2x + 3, then:

• the equation gives the rule;
• a table gives sample input-output pairs;
• the graph shows the whole linear relationship;
• a context might interpret 3 as a fixed starting amount and 2 as a rate per unit.

34. Linear Graph Gradient Is a Rate of Change

For y = mx + c, m is the gradient and c is the y-intercept.

Gradient = change in y ÷ change in x. In a context, attach units. On a distance-time graph, the gradient has speed units.

35. Intercepts Have Meaning

A y-intercept is the y-value when x = 0. An x-intercept is an x-value where y = 0.

In context, an intercept can represent an initial fee, starting temperature, break-even point or another boundary condition.

36. Quadratic Graphs Have More Than One Useful Form

A quadratic may appear in expanded form, factorised form or completed-square form. Each exposes different information.

Form	Feature exposed
x² + x − 6	coefficients
(x+3)(x−2)	roots −3 and 2
(x+1/2)² − 25/4	turning-point structure

Move between forms when the question changes.

37. Roots Are Where y = 0

If y = (x+3)(x−2), then y = 0 when x = −3 or x = 2. These are the x-intercepts of the graph.

This links equation-solving to graph-reading.

38. Turning Points Connect Algebra to Optimisation

A quadratic turning point can represent a maximum or minimum in context. The interpretation depends on the variables and the physical domain.

Do not treat the turning point as merely a coordinate to memorise. Ask what it means in the model.

39. Power and Exponential Graphs Have Distinct Shapes

K310 includes selected power functions and exponential functions. Students should recognise their characteristic behaviour and connect shape to the rule.

An exponential y = kaˣ with a positive integer a greater than 1 changes multiplicatively as x increases. This is fundamentally different from the constant additive change of a linear function.

40. Tangent Gradient Estimates Instantaneous Rate of Change

For a curve, the gradient changes from point to point. A tangent at a point gives an estimate of the local gradient there.

Use two well-separated points on the tangent line, not two points on the curve away from the tangent, when estimating the tangent gradient.

41. Equations Are Constraints

An equation states that two expressions represent the same value. Solving finds values that satisfy that constraint.

Every transformation must preserve the solution set, subject to any domain restrictions introduced by fractions or other operations.

42. Linear Equations: Preserve Balance

For 3x + 5 = 20:

3x = 15

x = 5.

Think “subtract 5 from both sides, divide both sides by 3”, not “move 5 over”.

43. Fractional Equations Need Denominator Control

Before clearing denominators, identify values that would make a denominator zero. Then multiply through by an appropriate common denominator carefully.

After solving, check candidate values against the original equation, not only the simplified one.

44. Simultaneous Equations Mean Both Conditions Must Hold

The solution to a pair of simultaneous equations must satisfy both equations.

Algebraically, use elimination or substitution. Graphically, the common solution is the intersection point.

45. Teacher Model 5: Simultaneous Equations

Solve:

x + y = 11

x − y = 3.

Add the equations:

2x = 14 → x = 7.

Then y = 4.

Check both original equations: 7+4=11 and 7−4=3.

46. Quadratic Equations Need Method Selection

• Factorise when the structure is clean.
• Use the quadratic formula systematically when factorisation is inconvenient.
• Complete the square when that form reveals useful structure.
• Use a graph when roots are being interpreted graphically.

The best method is the one that fits the structure and reduces error.

47. Teacher Model 6: Quadratic by Factorisation

Solve x² + x − 6 = 0.

(x+3)(x−2)=0.

Therefore:

x = −3 or x = 2.

The factorised structure exposes the two roots directly.

48. Completing the Square Reveals the Turning Point

x² + 6x + 5 can be written:

(x+3)² − 4.

This immediately exposes a turning point at x = −3 for the corresponding graph, with y-value −4.

49. The Quadratic Formula Is a Systematic Route

For ax² + bx + c = 0, the quadratic formula provides roots when applicable.

Write the values of a, b and c explicitly before substitution, especially when b or c is negative. Brackets prevent sign mistakes.

50. Equations From Context Need a Defined Unknown

Before building an equation, state what x represents and its units where relevant.

This prevents a technically correct equation from being attached to the wrong quantity.

51. Teacher Model 7: Reverse Percentage as an Equation

A device is sold after a 15% discount for $680. Let x be the original price.

After the discount, 85% remains:

0.85x = 680.

x = 800.

Check: 15% of 800 = 120 and 800 − 120 = 680.

52. Inequalities Describe Regions of Values

3x + 1 < 10 gives 3x < 9 and x < 3.

The answer is not one value. It is a set of values.

53. Reverse the Inequality When Multiplying or Dividing by a Negative

If −2x > 6, dividing both sides by −2 reverses the inequality:

x < −3.

This reversal preserves the order relationship.

54. Number-Line Endpoints Communicate Inclusion

An open point excludes the boundary. A closed point includes it.

Do not lose a correct inequality solution by drawing the wrong endpoint type.

55. Simultaneous Inequalities Are an Intersection

If x > 1 and x ≤ 5, both conditions must hold. The solution is:

1 < x ≤ 5.

This is the same logical idea as intersection in Sets: keep only values satisfying both conditions.

56. Practical Graphs: Read the Story in the Axes

When a graph represents distance, cost, temperature, population, resource use or another real quantity, begin with the axes.

• What does x represent?
• What does y represent?
• What are the units?
• What does gradient mean?
• What does the intercept mean?
• What does a horizontal segment mean?
• Is extrapolation beyond the observed range sensible?

A graph is a mathematical story. Every visible feature should correspond to a statement about the situation.

57. Teacher Model 8: Distance-Time Gradient

A straight segment rises from 0 km at 0 h to 18 km at 0.75 h.

Gradient:

18 ÷ 0.75 = 24 km/h.

The graph gradient and the average speed over that straight segment are the same relationship expressed visually and numerically.

58. Sets Must Remain Retrievable

Do not abandon Sets after Chapter 1. Mixed revision should still include:

• membership and subset notation;
• union and intersection;
• complements;
• Venn diagrams;
• counting from the overlap outward.

Return to the Chapter 1 Sets Classroom if this representation is no longer automatic.

59. Matrices Must Remain Retrievable

Mixed revision should still include:

• row-column meaning;
• matrix order;
• scalar multiplication;
• matrix sums;
• matrix products;
• row-by-column multiplication;
• contextual interpretation.

Return to the Chapter 4 Matrices Classroom if compatibility or row-column meaning is unstable.

60. The First-Wrong-Step Diagnostic

A student says, “I cannot do quadratic graphs.” Do not accept the chapter label as the diagnosis. Inspect the first wrong step.

Visible failure	Possible earlier dependency
quadratic graph	substitution, negative signs, scale, factorisation, roots
reverse percentage	base quantity, multiplier, equation setup
speed question	units, ratio, total distance and time
formula rearrangement	fractions, inverse operations, brackets
algebraic fraction	factorisation, common denominators, restrictions
exponential graph	indices, substitution, coordinate plotting
matrix application	row-column labels, compatibility, arithmetic

61. Misconception Clinic: Premature Rounding

Rounding every intermediate value can shift the final answer. Keep exact fractions or fuller calculator values through the working where practical, then round at the requested stage.

62. Misconception Clinic: Percentage Change Uses the Wrong Base

Always write what 100% represents. Successive changes act on successive bases.

63. Misconception Clinic: Average Two Speeds Directly

Average speed comes from total distance divided by total time. The arithmetic mean of two speeds is only valid in special structures.

64. Misconception Clinic: Cancelling Terms Instead of Factors

Factor first. Cancel only a common factor multiplying the whole numerator and denominator.

65. Misconception Clinic: “Move Across and Change Sign”

This shortcut language can fail when fractions, products, roots or brackets appear. State the operation applied to both sides instead.

66. Misconception Clinic: One Quadratic Method for Everything

Factorisation, formula, completing square and graphs have different advantages. Choose from structure rather than habit.

67. Misconception Clinic: Gradient Without Units

Gradient in a context is a rate. Its units come from vertical units divided by horizontal units.

68. Misconception Clinic: Correct Calculator Result, Wrong Model

A calculator can correctly solve the expression you entered even if that expression does not represent the problem. Model first, calculate second.

69. Guided Practice Set A: Indices

1. Simplify x³ × x⁵.
2. Simplify y⁹ ÷ y⁴.
3. Simplify (a²)³.
4. Write p⁻² with positive indices.
5. Evaluate 81¹⁄².

Solutions

x⁸, y⁵, a⁶, 1/p², 9.

70. Guided Practice Set B: Ratio and Percentage

1. Divide $180 in the ratio 2:3.
2. Increase $250 by 12%.
3. Decrease $480 by 15%.
4. After a 20% discount, an item costs $144. Find the original price.

Solutions

$72 and $108. $280. $408. Original price $180.

71. Guided Practice Set C: Rate and Speed

A vehicle travels 150 km in 2 h 30 min.

1. Convert the time to hours.
2. Find the average speed.

Solution

2 h 30 min = 2.5 h. Average speed = 150/2.5 = 60 km/h.

72. Guided Practice Set D: Expansion and Factorisation

1. Expand (x+4)(x−3).
2. Factorise x²+5x+6.
3. Factorise 9x²−25.
4. Expand (2x−3)².

Solutions

x²+x−12. (x+2)(x+3). (3x−5)(3x+5). 4x²−12x+9.

73. Guided Practice Set E: Formulae

1. Given y = 5x−7, find y when x=4.
2. Given y=(2x+3)/5, make x the subject.
3. Given P=2l+2w, make w the subject.

Solutions

13. x=(5y−3)/2. w=(P−2l)/2.

74. Guided Practice Set F: Equations

1. Solve 4x−7=17.
2. Solve x²−5x+6=0.
3. Solve x+y=9 and x−y=1.

Solutions

x=6. x=2 or 3. x=5, y=4.

75. Guided Practice Set G: Inequalities

1. Solve 3x+4<19.
2. Solve −2x≥8.
3. Combine x>−1 and x≤4.

Solutions

x<5. x≤−4. −1<x≤4.

76. Guided Practice Set H: Functions and Graphs

For y = x² − x − 6:

1. factorise the quadratic;
2. state the roots;
3. state the x-intercepts;
4. find y when x=0.

Solutions

(x−3)(x+2). Roots x=3 and x=−2. x-intercepts (3,0) and (−2,0). When x=0, y=−6.

77. Challenge Practice: Successive Percentage Change

A price rises by 25% and then falls by 20%. Find the overall percentage change.

Worked solution

Multiplier = 1.25×0.80=1.00. The final price equals the original price, so overall change is 0%.

78. Challenge Practice: Direct Proportion With a Power

Suppose y is directly proportional to x². When x=3, y=27. Find y when x=5.

Worked solution

y=kx². 27=9k, so k=3. When x=5, y=3(25)=75.

79. Challenge Practice: Reverse an Algebraic Fraction

Simplify (x²−16)/(x−4) and state the restriction.

Solution

(x−4)(x+4)/(x−4)=x+4, with x≠4.

80. Challenge Practice: Mixed Model

A service charges a fixed $12 plus $3.50 per hour.

1. Write a formula C in terms of hours h.
2. Find C when h=8.
3. Find h when C=$47.
4. Interpret the y-intercept of the graph of C against h.

Worked solution

C=12+3.5h. When h=8, C=$40. For C=47: 47=12+3.5h, so h=10. The y-intercept 12 represents the fixed charge before any hourly usage.

81. Examination Method: Write the Mathematical Object First

Before manipulating, label what you have: expression, equation, formula, inequality, graph, proportion or rate.

This one classification step narrows the valid operations.

82. Examination Method: State the Unknown

In a word problem, write:

Let x be the original price in dollars.

The variable now has a meaning and units. The equation can be checked against that meaning.

83. Examination Method: Keep Units Beside Rates

Write km/h, m/s, dollars/kg or the relevant unit beside the final rate. Units help reveal conversion errors.

84. Examination Method: Estimate Before Trusting the Calculator

If 680 is 85% of an original price, the original must be a little larger than 680, not smaller. A result of 80 or 8000 is implausible.

Use scale logic before accepting the numerical output.

85. Examination Method: Separate Model From Calculation

Write the relationship first:

0.85x = 680.

Then solve it. This makes it possible to inspect whether the model is correct even if the arithmetic later fails.

86. Examination Method: Check Solutions in the Original Problem

Substitute equation solutions back where practical. For context problems, check domain restrictions: a negative length, impossible count or excluded denominator value may need rejection.

87. Examination Method: Use Graph Features as Cross-Checks

If a quadratic factorises to roots −2 and 3, the graph should cross or touch the x-axis at those x-values as appropriate. Algebra and graph should tell the same story.

88. Oral Classroom Check

1. Why do index laws not apply across addition?
2. What does 100% represent in a percentage question?
3. How do direct and inverse proportion differ?
4. Why is average speed total distance divided by total time?
5. What is the difference between an expression and an equation?
6. Why can you cancel factors but not arbitrary terms?
7. What does changing the subject of a formula mean?
8. How do roots connect to x-intercepts?
9. Why does an inequality sign reverse after division by a negative?
10. How do you diagnose the first weak dependency in a mixed question?

The student should answer using examples. If a rule can be repeated but not explained, the method is not yet stable.

89. Exit Ticket

1. Simplify x⁴×x⁻².
2. An item is reduced by 20% to $96. Find the original price.
3. Expand (x+5)(x−2).
4. Factorise x²+3x−10.
5. Solve 2x+7=19.
6. Solve x²−x−6=0.
7. Solve −3x<12.
8. For y=2x+5, state the gradient and y-intercept.

Exit-ticket solution

x². Original price $120. x²+3x−10. (x+5)(x−2). x=6. x=3 or −2. x>−4. Gradient 2 and y-intercept 5.

90. Homework: Retrieval, Variation and Transfer

Layer 1 — Retrieval

• Write the main index laws from memory.
• Explain direct and inverse proportion.
• Write percentage multipliers for +12%, −15%, +40% and −7%.
• Define expression, equation, formula and inequality.
• Write the conditions for cancelling an algebraic factor.

Layer 2 — Variation

• Two indices questions.
• One ratio-sharing question.
• One direct and one inverse proportion question.
• One reverse-percentage question.
• One unit-rate question.
• Two expansion/factorisation questions.
• One algebraic-fraction question.
• One formula-rearrangement question.
• One simultaneous equation.
• One quadratic using a method you choose.
• One inequality.
• One graph interpretation.

Layer 3 — Transfer

Create one real-world question that combines at least three of these: percentage, rate, formula, graph, equation or inequality. Solve it, label every unit and identify which prerequisite would fail first if a student made a mistake.

91. The Full Chapter Routine

For mixed Number and Algebra questions, use:

classify → identify dependency → model → execute → interpret → check → vary.

For percentages, use:

identify 100% → choose multiplier → calculate → reverse if needed → check against the base.

For algebra, use:

object type → legal operation → preserve equivalence → simplify → check.

For graphs, use:

axes → units → feature → algebraic link → contextual meaning.

92. Why This Chapter Matters Beyond the Examination

Numbers and algebra teach you to preserve relationships while representations change. A percentage can become a multiplier. A rate can become a gradient. A quadratic can become factors, roots or a graph. A formula can be rearranged to expose a different variable. These are not separate tricks; they are examples of mathematical structure surviving transformation.

This ability matters in science, engineering, economics, computing, finance, statistics and any field where one model must be viewed from several angles without changing its meaning.

93. Connect Back to the Earlier Classrooms

Chapter 6 is deliberately cumulative. Return when necessary:

• Chapter 1 Classroom | Sets
• Chapter 2 Classroom | Probability of Combined Events
• Chapter 3 Classroom | Statistical Data Analysis
• Chapter 4 Classroom | Matrices
• Chapter 5 Classroom | Vectors

94. Ready for Chapter 7?

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

• control signs, fractions, powers and accuracy;
• use standard form and index laws correctly;
• solve ratio and proportion problems;
• identify percentage bases and reverse percentages;
• convert units and calculate rates or average speed;
• distinguish expressions, equations, formulae and inequalities;
• expand, factorise and manipulate algebraic fractions;
• change the subject of a formula;
• move between functions, tables, graphs and contexts;
• solve linear, simultaneous and quadratic equations;
• solve and represent linear inequalities;
• retrieve Sets and Matrices without being told which topic is active; and
• diagnose the first weak dependency in a mixed problem.

If one item is weak, return to the smallest section that owns it, repair it and immediately retry the larger question. If all are stable, continue to Geometry and Measurement Revision, where algebra, ratio, trigonometry, coordinate geometry and vectors reconnect inside diagrams and spatial constraints.

Continue the Secondary 4 Mathematics Classroom

• Secondary 4 Mathematics Chapter-by-Chapter Walkthrough | SEC G3 K310
• Secondary 4 Mathematics Learning Guide | Algebraic Expansion, Factorisation and Equivalent Forms
• Secondary 4 Mathematics Learning Guide | Algebraic Formulae, nth-Term Patterns and Change of Subject
• Secondary 4 Mathematics Learning Guide | Direct and Inverse Proportion, Scale and Rate Models
• Secondary 4 Mathematics Learning Guide | Quadratic Functions
• Previous: Chapter 5 | Vectors
• Next: Chapter 7 | Geometry and Measurement Revision