Secondary 1 Algebraic Expressions, Formulae and nth-Term Patterns

SECONDARY 1 MATHEMATICS CLASSROOM · CHAPTER 6 · ALGEBRAIC LANGUAGE, EXPRESSIONS, FORMULAE AND NTH-TERM PATTERNS · G2/G3

Algebraic Language, Expressions, Formulae and nth-Term Patterns: Preserve the Meaning While the Symbols Change

In this classroom, you will not begin by “moving letters around”. You will begin by deciding what each symbol means, what operation connects it to the others, and whether a new expression still represents the same relationship.

Algebra is a language for compressing patterns and relationships. A variable can stand for a changing quantity. A coefficient tells how many copies of a variable are present. An expression can be rewritten into an equivalent form. A formula connects several quantities. An nth-term rule replaces repeated counting with a direct rule for position n.

Classroom rule: define the symbols → read the structure → transform only by legal operations → verify equivalence → return the algebra to its meaning.

The current Secondary One G2 and G3 Mathematics syllabuses include algebraic representation, manipulation, substitution, formulae and generalisation, with exact sequencing and depth varying by subject level and school. This classroom teaches the common algebraic foundation and uses rearrangement or reverse-position work only as clearly marked bridges where the learner’s course has reached them.

Official reference: MOE G2 and G3 Mathematics Syllabuses.

Navigate: algebraic language · words to expressions · substitution · simplification · brackets · common factors · formulae · nth-term patterns · misconception clinic · guided practice · examination transfer · exit ticket.

Featured Answer: What Is Algebra?

Algebra is a system for representing quantities and relationships with symbols. It allows one rule to describe many numerical cases at once.

For example:

C = 3n + 5

can describe a quantity C made from three copies of n plus a fixed 5. At n = 1, C = 8. At n = 10, C = 35. The same algebraic rule controls every allowed input.

The Simple Classroom Answer

Algebra replaces repeated numerical cases with a structure that can be read, transformed, checked and reused.

Variable: a symbol representing a number or quantity.
Constant: a fixed numerical value.
Coefficient: the numerical factor multiplying a variable term.
Term: one signed part of an expression at the outer addition/subtraction level.
Expression: a mathematical phrase with no assertion of equality.
Equation: a statement that two expressions are equal.
Formula: a compact relationship among quantities.
Equivalent expressions: different forms producing the same value for every allowed input.
nth-term rule: a direct expression for the term at position n.

How to Use This Classroom

Read each expression aloud before manipulating it.
Circle coefficients, underline variable parts and box constants when structure is unclear.
Attempt every Your Turn question before opening the solution.
Put substituted negative values in brackets.
Use substitution or reverse expansion to test suspicious simplifications.
For patterns, separate position from value.
Return after a delay and reproduce the method without notes.

1. Start the Lesson by Reading Symbols as a Sentence

Teacher: Write 3x + 5. Ask the student to read it without saying “three x plus five”.

A stronger reading is:

three copies of x, then add five.

This makes the multiplication and addition visible.

2. A Variable Represents a Quantity, Not Merely a Letter

In C = 4n + 7, n might represent the number of items and C a total cost in an invented model.

The letter acquires meaning from its definition. Once defined, it should be used consistently.

3. A Constant Does Not Depend on the Variable

In 5x + 9, the 9 is a constant term. Changing x changes 5x but does not change the 9.

Constants often represent fixed starting amounts, fixed lengths, offsets or other quantities not varying with the chosen input.

4. A Coefficient Multiplies the Variable Part

In −6x, the coefficient of x is −6.

in x, the coefficient is 1;
in −x, the coefficient is −1;
in 4x², the coefficient is 4.

The sign belongs with the coefficient.

5. Terms Are Separated by Outer Addition or Subtraction

In:

5x − 3y + 7,

the terms are 5x, −3y and 7.

Keep the sign attached to the term when identifying it.

6. Factors Multiply Inside a Term

In 6xy, the factors include 6, x and y.

This is different from the three separate terms 6 + x + y.

7. Algebra Usually Omits the Multiplication Sign

3×x is written 3x;
x×y is written xy;
2×a×b is written 2ab.

The omitted sign still means multiplication.

8. Numerical Coefficients Are Written Before Variables

Write 7x rather than x7. Standard notation places the numerical coefficient before the variable part.

Consistent notation makes expressions easier to compare and simplify.

9. Powers Describe Repeated Multiplication

x² means x×x. x³ means x×x×x.

3x² means 3×x×x.

It is not the same as (3x)² = 9x².

10. An Expression Is Not an Equation

3x + 5 is an expression. It has no equals sign and makes no claim that x has one particular value.

3x + 5 = 17 is an equation. It states that two expressions are equal for the value or values of x satisfying the statement.

Chapter 7 will develop equations and problem formation in depth.

11. A Formula Describes a Relationship Among Quantities

A = lw is a formula connecting the area A of a rectangle with length l and width w.

A formula can generate many numerical cases without being a question to solve by itself.

12. An Identity Is an Equality True for Every Allowed Value

The statement:

2(x+3) = 2x+6

is true for every real x. It expresses equivalence created by the distributive law.

At this level, the important idea is not the identity symbol itself but the difference between “true for every x” and “true only for particular x-values”.

Your Turn 1

For 4x² − 7x + 9:

list the terms;
state the coefficient of x²;
state the coefficient of x;
state the constant term.

Answer

Terms: 4x², −7x, 9. Coefficient of x² is 4. Coefficient of x is −7. Constant is 9.

13. Translation Into Algebra Begins With the Quantity Being Described

“Five more than x” means start with x and add 5:

x + 5.

Do not translate words one at a time without preserving their relationship.

14. “Five Less Than x” Means x − 5

The reference is x and five is removed:

x − 5.

15. “x Less Than 5” Means 5 − x

The order of subtraction follows the language:

5 − x.

This is different from x−5.

16. “Twice x” Means 2x

Twice x means two copies of x:

2x.

Three times x is 3x. Half of x is x/2.

17. “Twice x, Then Add 3” Means 2x + 3

The multiplication applies only to x.

18. “Twice the Sum of x and 3” Means 2(x + 3)

The bracket shows that the entire sum is doubled.

Compare:

2x+3;
2(x+3)=2x+6.

The wording changes the structure.

19. “The Product of a and b” Means ab

The word product signals multiplication.

“The quotient of a and b” means a/b, with b non-zero.

20. Translate a Perimeter Before Substituting Numbers

A rectangle has length x+3 and width x.

Perimeter:

P = 2(x+3)+2x.

Then simplify:

P = 4x+6.

21. Teacher Model 1: Build an Expression From a Story

An invented service has a fixed $5 charge plus $3 for each unit n.

Variable part: 3n.

Fixed part: 5.

C = 3n+5.

The expression records both the variable and fixed components.

Your Turn 2

Write seven more than n.
Write four less than x.
Write x less than four.
Write three times the sum of p and 2.
Write half of y, then add 6.

Answers

n+7. x−4. 4−x. 3(p+2). y/2+6.

22. Substitution Replaces a Variable With Its Complete Value

If y = 4x−7 and x=5:

y = 4(5)−7 = 13.

The value 5 replaces x everywhere x appears.

23. Negative Substitution Requires Brackets

Evaluate x²+3x when x=−4:

(−4)² + 3(−4) = 16−12 = 4.

The brackets show that the complete signed value −4 replaces x.

24. −4² and (−4)² Are Different

By the usual order of operations:

−4² = −(4²) = −16;
(−4)² = 16.

Substitution brackets prevent this ambiguity.

25. Fractions Can Be Substituted Exactly

Evaluate y=3x+1 when x=2/3:

y=3(2/3)+1=3.

Keeping the exact fraction may be cleaner than converting to a decimal.

26. Several Variables Must Each Be Replaced Correctly

Evaluate P=2a+3b for a=4 and b=7:

P=2(4)+3(7)=29.

Match each value with the correct variable before calculating.

27. Simplifying Before Substitution Can Reduce Work

Evaluate 5x+3x−2x+7 at x=4.

First simplify:

6x+7.

Then substitute:

6(4)+7=31.

28. Direct and Simplified Routes Should Agree

The original expression at x=4 is 20+12−8+7=31.

Agreement provides an independent equivalence check.

Your Turn 3

Evaluate 3x+8 for x=5.
Evaluate 2x²−3x for x=−2.
Evaluate 4a−2b for a=−3 and b=5.
Evaluate 5p+1 for p=3/5.

Answers

23. 14. −22. 4.

29. Simplification Changes Form Without Changing Value

A simplification step is a claim that the new expression is equivalent to the old one for every allowed value of the variables.

This makes verification part of algebra, not an optional extra.

30. Like Terms Have the Same Variable Structure

3x and −8x are like terms;
4x² and 7x² are like terms;
5x and 5x² are not like terms;
2ab and 9ba are like terms because ab=ba.

31. Collect Like Terms by Combining Coefficients

7x+3x−5x = (7+3−5)x = 5x.

The variable part stays x because we are counting copies of the same algebraic object.

32. Constants Combine With Constants

3x+7−2x+5 = x+12.

Combine variable terms with matching variable terms and constants with constants.

33. Unlike Terms Cannot Be Merged

4x+3y cannot become 7xy or 7x.

The variable structures differ.

34. Addition of Like Terms Is Not Multiplication of Powers

5x²+2x² = 7x²,

not 7x⁴. The terms are being added, so coefficients add while the shared variable part remains x².

35. Teacher Model 2: Collect Several Families of Terms

Simplify:

5x + 3y − 2x + 4y − 7.

x-terms: 5x−2x=3x;
y-terms: 3y+4y=7y;
constant: −7.

Answer: 3x+7y−7.

Your Turn 4

7x+4x−3x.
9a−3a+2a.
4x+3y−x+5y.
6p²−2p²+p.
3m+8−m+5.

Answers

8x. 8a. 3x+8y. 4p²+p. 2m+13.

36. Brackets Group an Expression as One Object

3(x+4) means three copies of the whole bracket.

Therefore:

3(x+4)=3x+12.

37. Distribute the Multiplier to Every Term

For a(b+c):

a(b+c)=ab+ac.

Skipping a term changes the value.

38. A Negative Multiplier Changes the Products Accordingly

−2(x−5)=−2x+10.

The second product is positive because −2×−5=+10.

39. Expand Before Collecting When That Makes the Structure Clear

3(x+2)+4x = 3x+6+4x = 7x+6.

Separate the two operations: expand, then collect.

40. Teacher Model 3: Expand a Negative Bracket

Simplify:

5x−2(x−3).

Expand:

5x−2x+6.

Collect:

3x+6.

41. Equivalent Expressions Can Look Very Different

2(x+5), 2x+10 and 10+2x are all equivalent.

They produce the same value for every x.

42. Substitution Can Detect a False Expansion

Someone claims:

3(x+2)=3x+2.

Test x=1.

original = 3(3)=9;
claimed form = 3+2=5.

The expressions are not equivalent.

Your Turn 5

Expand 4(x+3).
Expand −3(x−5).
Simplify 2(x+4)+3x.
Simplify 5(2x−1)−3x.
Simplify 4−2(3−x).

Answers

4x+12. −3x+15. 5x+8. 7x−5. 4−6+2x=2x−2.

43. Factorisation Reverses Expansion

Since:

3(2x+3)=6x+9,

we can reverse the process:

6x+9=3(2x+3).

44. Extract a Common Numerical Factor

8x+12 = 4(2x+3).

The factor 4 divides every term.

45. Use the Greatest Common Factor for Full Common-Factor Factorisation

12x+18 = 6(2x+3).

Writing 3(4x+6) is correct but has not extracted the greatest numerical common factor.

46. Variable Factors Can Also Be Common

8x²+12x = 4x(2x+3).

Both terms contain the common factor 4x.

47. A Negative Common Factor Can Create a Cleaner Bracket

−6x+12 = −6(x−2).

Expand to verify.

48. Every Factorisation Should Be Checked by Expansion

If you write 15a−20 = 5(3a−4), expand:

15a−20.

The original expression is recovered exactly.

49. Teacher Model 4: Factorise With a Variable Common Factor

Factorise 18x²−12x.

Greatest numerical factor = 6.

Both terms contain x.

18x²−12x = 6x(3x−2).

Check by expansion.

Your Turn 6

Factorise 8x+12.
Factorise 15a−20.
Factorise 12x²+18x.
Factorise −10y+15 using a negative common factor.

Answers

4(2x+3). 5(3a−4). 6x(2x+3). −5(2y−3).

50. A Formula Gives Each Variable a Defined Role

In A=lw:

A represents area;
l represents length;
w represents width.

The letters are not interchangeable once the quantities have been defined.

51. A Formula Can Produce Many Cases

For C=3n+5:

n=0 gives C=5;
n=4 gives C=17;
n=10 gives C=35.

The formula is a reusable input-output rule.

52. Formulae Carry Units as Well as Numbers

For d=vt, if v is measured in km/h and t in hours, d is in kilometres.

Using minutes directly with km/h would make the units incompatible unless the time is converted.

53. Teacher Model 5: Substitute Into a Geometric Formula

Triangle area:

A = 1/2 bh.

For b=14 cm and h=9 cm:

A = 1/2×14×9 = 63 cm².

The unit is square centimetres because area is being calculated.

54. Formula Meaning Comes Before Substitution

The triangle formula requires h to be perpendicular to the chosen base b. Substitution cannot repair a wrong interpretation of height.

Match the quantities to the formula first.

55. Formulae Connect Back to Rates

Chapter 5 used:

d=vt.

The rate relationship becomes algebraic notation. The units still provide a check.

56. Bridge: Rearrangement Changes the Subject but Preserves the Formula

Where your course has introduced simple change of subject, consider:

y=x+6.

Subtract 6 from both sides:

x=y−6.

The relationship is unchanged. Only the isolated variable changes.

57. Bridge: Undo Operations in Reverse Structural Order

For y=3x+5, to isolate x:

subtract 5;
divide the entire result by 3.

x=(y−5)/3.

Chapter 7 will develop the equality logic behind this operation in greater depth.

58. A Formula Rearrangement Can Be Checked Forward

If C=4n+7 and n=(C−7)/4, choose n=5.

Then C=27.

Use the rearranged form:

(27−7)/4=5.

The numerical check supports the rearrangement.

Your Turn 7

Evaluate P=2a+5b for a=−2, b=3.
Evaluate A=1/2 bh for b=12, h=7.
Evaluate d=vt for v=18 km/h, t=2/3 h.
Bridge: make x the subject of y=x+9.
Bridge: make x the subject of y=4x−3.

Answers

11. 42. 12 km. x=y−9. x=(y+3)/4.

59. A Sequence Has Values Attached to Positions

For the sequence:

5, 8, 11, 14, …

5 is the first term, 8 the second, 11 the third and 14 the fourth.

Position and value are different quantities.

60. Constant First Difference Signals an Arithmetic Sequence

For 5,8,11,14,… the difference is always +3.

For 30,24,18,12,… the difference is always −6.

Check several gaps before declaring the difference constant.

61. Recursive Rules Tell You the Next Step

“Start at 5 and add 3 each time” is a recursive description.

It is useful for generating nearby terms but inefficient for finding the 100th term.

62. A Direct Rule Gives the Term at Position n

The nth term of 5,8,11,14,… is:

3n+2.

At n=1 it gives 5; at n=4 it gives 14.

63. Use First Term Plus n−1 Jumps

For an arithmetic sequence with first term a and common difference d:

nth term = a+(n−1)d.

The first term requires zero jumps; position n requires n−1 jumps.

64. Teacher Model 6: Build the nth Term From First Term and Difference

Sequence:

5,8,11,14,…

a=5, d=3.

5+3(n−1)=5+3n−3=3n+2.

65. Another Route Starts With the Common Difference Times n

For 5,8,11,14,… start with 3n:

n	3n	actual term
1	3	5
2	6	8
3	9	11

The actual term is always 2 greater, giving 3n+2.

66. Decreasing Arithmetic Sequences Use Negative Differences

For 41,35,29,23,…:

a=41 and d=−6.

nth term = 41−6(n−1)=47−6n.

67. Decimal Differences Are Still Valid

For 1.5,2.0,2.5,3.0,…:

d=0.5.

nth term = 0.5n+1.

68. Use the Direct Rule for Distant Terms

If T=7n−3, the 50th term is:

7(50)−3=347.

There is no need to list the first 49 terms.

69. Verify an nth-Term Rule Against Several Positions

A rule that matches the first term alone may still be wrong.

For 5,8,11,… the rule n+4 gives 5 at n=1 but gives 6 at n=2, so it fails.

70. Shape Patterns Can Be Converted Into Algebra

Suppose Figures 1,2,3,4 use 4,7,10,13 tiles.

The pattern grows by 3 each time.

T=3n+1.

A structural explanation might say: three new tiles are associated with each stage plus one fixed tile.

71. Different Decompositions Can Give Equivalent Pattern Rules

A shape might be counted as 2n+n+4 or as 3n+4.

These are equivalent because 2n+n=3n.

72. Not Every Sequence Is Linear

1,4,9,16,25,… has differences 3,5,7,9,…, which are not constant.

The sequence is n², not of the linear form an+b.

Do not force the arithmetic-sequence method onto a non-linear pattern.

73. The Domain of a Figure Number Is Usually Positive Integers

If n represents Figure n, ordinary positions are n=1,2,3,….

The expression 3n+2 can be evaluated algebraically at n=2.5, but Figure 2.5 may not exist in the original pattern.

74. Bridge: Reverse-Position Questions Become Equations

Does 145 occur in the sequence T=6n+1?

Set the rule equal to the target:

6n+1=145.

This is now an equation. Solving gives n=24, a valid positive whole-number position.

Chapter 7 will take over from here and develop equation solving systematically.

Your Turn 8

Find the nth term of 4,7,10,13,…
Find the nth term of 20,15,10,5,…
Find the 40th term of 6n−1.
Find the nth term of 2.5,3.0,3.5,4.0,…
A tile pattern uses 5,9,13,17 tiles. Find a rule for Figure n.

Answers

3n+1. 25−5n. 239. 0.5n+2. 4n+1.

75. Misconception Clinic: x Means One Particular Unknown Number Everywhere

A variable may represent a changing input, a measured quantity or an unknown depending on context. Read its role from the statement rather than assuming every letter is an equation unknown.

76. Misconception Clinic: 3x Means 3+x

3x means three multiplied by x. Algebra often omits the multiplication sign.

77. Misconception Clinic: The Sign Is Separate From the Term

In 5x−3y+7, the second term is −3y. Carry its sign with it when collecting or rearranging.

78. Misconception Clinic: 4x+3y=7xy

Unlike terms cannot be merged. Their variable structures differ.

79. Misconception Clinic: 5x²+2x²=7x⁴

Add coefficients and keep the common variable part: 7x².

80. Misconception Clinic: 2(x+5)=2x+5

The multiplier applies to every term in the bracket, giving 2x+10.

81. Misconception Clinic: −3(x−2)=−3x−6

The second product is (−3)(−2)=+6, so the expansion is −3x+6.

82. Misconception Clinic: A Factorisation Is Correct Because It Looks Shorter

Expand it. A correct factorisation must recover the original expression exactly.

83. Misconception Clinic: Negative Substitution Does Not Need Brackets

Brackets preserve the complete signed value, especially under powers and multiplication.

84. Misconception Clinic: A Formula Is Just a Formula Triangle or Recipe

A formula is a relationship among quantities. Define variables and units before substituting.

85. Misconception Clinic: The Common Difference Is the nth Term

The common difference controls growth, but an adjustment is usually needed to match the first term.

86. Misconception Clinic: If a Rule Matches One Term, It Is Correct

Test several positions. One matching input is weak evidence.

87. Misconception Clinic: Every Pattern Is Linear

Check first differences. If they are not constant, a linear nth-term rule may not apply.

88. Guided Practice Set A: Algebraic Vocabulary

For −5x²+3x−8:

list the terms;
state the coefficient of x²;
state the coefficient of x;
state the constant.

Solution

Terms −5x², 3x, −8. Coefficients −5 and 3. Constant −8.

89. Guided Practice Set B: Translation

four more than n;
seven less than x;
x less than seven;
three times the sum of a and 4;
half of p minus 5.

Solutions

n+4. x−7. 7−x. 3(a+4). p/2−5.

90. Guided Practice Set C: Like Terms

7x+4x.
9a−3a+2a.
5x+3y−2x+4y.
6p²−2p²+p.
4m+7−m+5.

Solutions

11x. 8a. 3x+7y. 4p²+p. 3m+12.

91. Guided Practice Set D: Expansion

4(x+3).
−3(x−5).
2(x+4)+3x.
5(2x−1)−3x.
3(2a+5)−4a.

Solutions

4x+12. −3x+15. 5x+8. 7x−5. 2a+15.

92. Guided Practice Set E: Common-Factor Factorisation

8x+12.
15a−20.
12x²+18x.
14pq+21p.

Solutions

4(2x+3). 5(3a−4). 6x(2x+3). 7p(2q+3).

93. Guided Practice Set F: Substitution

3x+5 for x=4.
2x²−3x for x=−2.
3a+2b for a=−4, b=5.
x²+y for x=−3, y=7.

Solutions

17. 14. −2. 16.

94. Guided Practice Set G: Formulae

A=lw for l=8 cm, w=5 cm.
P=2l+2w for l=7 m, w=4 m.
d=vt for v=12 km/h, t=0.75 h.
C=3n+5 for n=12.

Solutions

40 cm². 22 m. 9 km. 41.

95. Guided Practice Set H: nth-Term Rules

Find nth term: 4,7,10,13,…
Find nth term: 20,15,10,5,…
Find nth term: 2.5,3.0,3.5,4.0,…
Find the 25th term of 4n+1.

Solutions

3n+1. 25−5n. 0.5n+2. 101.

96. Guided Practice Set I: Equivalence Checks

Is 3(x+4) equivalent to 3x+12?
Is 2(x−5) equivalent to 2x−5?
Is 6x+9 equivalent to 3(2x+3)?

Solutions

Yes. No; correct expansion is 2x−10. Yes.

97. Challenge Practice: Equivalent Forms

Simplify 3(2x+4)−2(x−5).

Worked solution

6x+12−2x+10 = 4x+22.

98. Challenge Practice: Pattern From Structure

A shape pattern has 7,12,17,22,… tiles in Figures 1,2,3,4. Find the nth-term rule and the number of tiles in Figure 50.

Worked solution

Difference 5. Rule = 5n+2. Figure 50 uses 252 tiles.

99. Challenge Practice: Formula and Units

A rectangle has length l=2.4 m and width w=75 cm. Find its area in square metres using A=lw.

Worked solution

Convert 75 cm=0.75 m. Then A=2.4×0.75=1.8 m².

100. Challenge Practice: Reverse Pattern as an Equation Bridge

A sequence has nth term 4n+3. Which position has term 99?

Worked solution

Set 4n+3=99. Then 4n=96 and n=24. This is a bridge into Chapter 7 equation solving.

101. Examination Method: Read the Expression Before Simplifying It

Identify terms, coefficients, powers and brackets before making a transformation.

102. Examination Method: Keep Signs Attached to Terms

Write −3y as one signed term. Many algebra errors begin when the minus sign is separated from the quantity it belongs to.

103. Examination Method: Use Brackets for Negative Substitution

If x=−4, write (−4) wherever x appears before evaluating powers or products.

104. Examination Method: Separate Expansion From Collection

Write one clean expansion line, then one collection line. Combining both mentally makes sign errors harder to diagnose.

105. Examination Method: Check Factorisation by Re-Expanding

A factorised form is trustworthy when expansion returns exactly to the original expression.

106. Examination Method: Keep Formula Variables and Units Defined

Do not substitute a sloping side into a formula requiring perpendicular height. Symbols are attached to quantities, not merely spaces in a formula.

107. Examination Method: Verify an nth-Term Rule at n=1 and a Later Position

This tests both the starting adjustment and the repeated change.

108. Examination Method: Do Not Force a Linear Rule Onto a Non-Linear Pattern

Check whether first differences are constant before using an+b.

109. Examination Method: Use a Different Representation to Check

simplification: substitute a convenient value;
factorisation: expand;
formula: substitute the result forward;
sequence: test several positions;
word translation: read the algebra back as a sentence.

110. Oral Classroom Check

What is a variable?
What is the difference between a coefficient and an exponent?
What makes two terms like terms?
What is the difference between an expression and an equation?
Why must a bracket multiplier act on every term?
How do you check a factorisation?
Why do negative substitutions need brackets?
What is a formula?
What is the difference between a recursive sequence rule and an nth-term rule?
How do you test whether an arithmetic-sequence rule is correct?

The student should answer with a small example. A memorised slogan without a valid example is not yet secure.

111. Exit Ticket

State the coefficient of x in 5x−7.
Write “twice the sum of n and 3” algebraically.
Simplify 7x+3x−4x.
Expand −3(x−4).
Factorise 12x+18.
Evaluate 2x²−3x for x=−2.
Evaluate A=1/2 bh for b=10 cm and h=7 cm.
Find the nth term of 6,10,14,18,….
Find the 20th term of that sequence.

Exit-ticket solution

Coefficient 5. 2(n+3). 6x. −3x+12. 6(2x+3). 14. 35 cm². nth term 4n+2. 20th term 82.

112. Homework: Retrieval, Variation and Transfer

Layer 1 — Retrieval

Define variable, coefficient, constant, term, expression and formula.
Write five word phrases as algebraic expressions.
State the rule for collecting like terms.
State the distributive law in words.
Explain factorisation as the reverse of expansion.
Write the arithmetic-sequence nth-term structure.

Layer 2 — Variation

five like-term simplifications;
five bracket expansions;
four common-factor factorisations;
four substitutions, including negative and fractional values;
three formula evaluations with units;
four nth-term questions;
one shape-pattern generalisation.

Layer 3 — Transfer

Create one real-world formula with a fixed term and a variable term, one geometric formula substitution and one arithmetic sequence. Define every variable, calculate three cases and explain how you would verify each algebraic result.

113. The Seven-Day Return Cycle

Day 0: complete teacher models and guided practice.
Day 1: translate five phrases, simplify three expressions and substitute two values.
Day 3: expand, factorise and derive two nth-term rules without notes.
Day 7: repeat the exit ticket with changed expressions and explain each equivalence aloud.

114. A 60-Minute Teaching Lesson

5 minutes: algebra vocabulary retrieval.
10 minutes: words to expressions.
10 minutes: substitution and negative-value control.
10 minutes: like terms and equivalent expressions.
10 minutes: brackets and common factors.
10 minutes: formulae and nth-term rules.
5 minutes: exit ticket.

115. A 90-Minute Teaching Lesson

10 minutes: algebra-language diagnostic.
15 minutes: translation and term structure.
15 minutes: substitution.
20 minutes: simplification, expansion and factorisation.
10 minutes: formulae and units.
10 minutes: arithmetic sequences and nth terms.
5 minutes: oral explanation.
5 minutes: exit ticket and return date.

116. The Full Algebraic-Language Routine

define variable → identify terms and operations → translate meaning → choose representation → calculate or transform → read back.

117. The Full Simplification Routine

identify signed terms → expand brackets if needed → group like terms → combine coefficients → verify by substitution.

118. The Full Factorisation Routine

inspect every term → find greatest common factor → divide each term → write bracket → expand to verify.

119. The Full Formula Routine

define variables and units → match known values → bracket substitutions → follow operations → attach output unit → forward-check.

120. The Full nth-Term Routine

label positions → find repeated change → build direct rule → simplify → test n=1 and another position → use the rule.

121. Why This Chapter Matters Beyond Chapter 6

Algebra is the compression language of later Secondary Mathematics. Equations use expressions connected by equality. Graphs use formulas as input-output rules. Geometry uses symbolic measurements and formulae. Rates become d=vt. Percentages become multipliers. Sequences become functions of position. Factorisation becomes a central tool in later quadratic and algebraic work.

The deeper habit is this: a symbol does not replace meaning. It carries meaning more compactly. Every transformation must preserve the relationship the symbols represent.

122. Connect Back to Chapters 2–5

Chapter 2 supplies signed arithmetic and exact substitution. Chapter 3 supplies multiplicative structure. Chapter 4 supplies multipliers and reference quantities. Chapter 5 supplies rate formulas and unit checks. Chapter 6 compresses these ideas into symbolic relationships.

123. Ready for Chapter 7?

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

identify variables, constants, coefficients, terms and factors;
distinguish expressions, equations and formulae;
translate common verbal relationships into algebra;
substitute positive, negative, fractional and decimal values correctly;
collect like terms without combining unlike structures;
expand brackets completely;
factorise by extracting common factors;
verify equivalent expressions by substitution or reverse expansion;
evaluate formulae while keeping variables and units attached;
recognise simple rearrangement as preservation of equality where taught;
identify an arithmetic sequence from constant first differences;
derive and verify a linear nth-term rule;
use an nth-term rule for distant positions;
recognise when a pattern is not linear.

If one item is weak, return to the smallest section that owns it and complete a changed example. If all are stable, continue to Linear Equations and Problem Formation, where equality becomes the central constraint and algebraic expressions are used to model unknown quantities.

Small Group Tutorials

Secondary 1 Mathematics Classroom | Chapter 6: Algebraic Language, Expressions, Formulae and nth-Term Patterns | G2/G3

Featured Answer: What Is Algebra?

The Simple Classroom Answer

How to Use This Classroom

1. Start the Lesson by Reading Symbols as a Sentence

2. A Variable Represents a Quantity, Not Merely a Letter

3. A Constant Does Not Depend on the Variable

4. A Coefficient Multiplies the Variable Part

5. Terms Are Separated by Outer Addition or Subtraction

6. Factors Multiply Inside a Term

7. Algebra Usually Omits the Multiplication Sign

8. Numerical Coefficients Are Written Before Variables

9. Powers Describe Repeated Multiplication

10. An Expression Is Not an Equation

11. A Formula Describes a Relationship Among Quantities

12. An Identity Is an Equality True for Every Allowed Value

Your Turn 1

13. Translation Into Algebra Begins With the Quantity Being Described

14. “Five Less Than x” Means x − 5

15. “x Less Than 5” Means 5 − x

16. “Twice x” Means 2x

17. “Twice x, Then Add 3” Means 2x + 3

18. “Twice the Sum of x and 3” Means 2(x + 3)

19. “The Product of a and b” Means ab

20. Translate a Perimeter Before Substituting Numbers

21. Teacher Model 1: Build an Expression From a Story

Your Turn 2

22. Substitution Replaces a Variable With Its Complete Value

23. Negative Substitution Requires Brackets

24. −4² and (−4)² Are Different

25. Fractions Can Be Substituted Exactly

26. Several Variables Must Each Be Replaced Correctly

27. Simplifying Before Substitution Can Reduce Work

28. Direct and Simplified Routes Should Agree

Your Turn 3

29. Simplification Changes Form Without Changing Value

30. Like Terms Have the Same Variable Structure

31. Collect Like Terms by Combining Coefficients

32. Constants Combine With Constants

33. Unlike Terms Cannot Be Merged

34. Addition of Like Terms Is Not Multiplication of Powers

35. Teacher Model 2: Collect Several Families of Terms

Your Turn 4

36. Brackets Group an Expression as One Object

37. Distribute the Multiplier to Every Term

38. A Negative Multiplier Changes the Products Accordingly

39. Expand Before Collecting When That Makes the Structure Clear

40. Teacher Model 3: Expand a Negative Bracket

41. Equivalent Expressions Can Look Very Different

42. Substitution Can Detect a False Expansion

Your Turn 5

43. Factorisation Reverses Expansion

44. Extract a Common Numerical Factor

45. Use the Greatest Common Factor for Full Common-Factor Factorisation

46. Variable Factors Can Also Be Common

47. A Negative Common Factor Can Create a Cleaner Bracket

48. Every Factorisation Should Be Checked by Expansion

49. Teacher Model 4: Factorise With a Variable Common Factor

Your Turn 6

50. A Formula Gives Each Variable a Defined Role

51. A Formula Can Produce Many Cases

52. Formulae Carry Units as Well as Numbers

53. Teacher Model 5: Substitute Into a Geometric Formula

54. Formula Meaning Comes Before Substitution

55. Formulae Connect Back to Rates

56. Bridge: Rearrangement Changes the Subject but Preserves the Formula

57. Bridge: Undo Operations in Reverse Structural Order

58. A Formula Rearrangement Can Be Checked Forward

Your Turn 7

59. A Sequence Has Values Attached to Positions

60. Constant First Difference Signals an Arithmetic Sequence

61. Recursive Rules Tell You the Next Step

62. A Direct Rule Gives the Term at Position n

63. Use First Term Plus n−1 Jumps

64. Teacher Model 6: Build the nth Term From First Term and Difference

65. Another Route Starts With the Common Difference Times n

66. Decreasing Arithmetic Sequences Use Negative Differences

67. Decimal Differences Are Still Valid

68. Use the Direct Rule for Distant Terms