“Seven less than three times a number.”
Is the expression:
7−3x?
Or:
3x−7?
The second is correct.
Three times the number is formed first.
Seven is then removed from that quantity.
Translating words into algebra is not a search for operation keywords. It is the reconstruction of a relationship among quantities.
This is why learners can recognise every word in a sentence and still write the wrong expression.
The challenge is structural:
- What quantities exist?
- Which quantity depends on which?
- What happens first?
- What remains fixed?
- What does the variable represent?
The quick answer: define, relate, write, test
- Define: state exactly what the variable represents, including units where relevant.
- Relate: identify how the quantities are connected before selecting operations.
- Write: translate the relationship into an expression, equation or inequality.
- Test: substitute an easy numerical value and check whether the symbolic form reproduces the verbal meaning.
This four-stage routine is more reliable than memorising that “more means plus” or “of means multiply”.
An expression, equation and inequality perform different jobs
Words:
“Five more than twice a number.”
Expression:
2x+5.
Words:
“Five more than twice a number is 21.”
Equation:
2x+5=21.
Words:
“Five more than twice a number is at least 21.”
Inequality:
2x+5≥21.
The sentence determines what kind of algebraic statement is needed.
Define the variable with meaning
Weak definition:
Let x be apples.
Stronger definition:
Let x be the number of apples in one box.
Or:
Let x kg be the mass of apples purchased.
A variable represents a numerical quantity.
The definition should tell the reader what that quantity measures.
A well-defined variable gives every later symbol a home. A poorly defined variable forces the reader to guess what the algebra is measuring.
Addition phrases: identify the base quantity first
“Eight more than x” means:
x+8.
“The sum of x and 8” also means:
x+8.
“x increased by 8” means:
x+8.
Addition is commutative, so 8+x has the same value.
However, preserving the sentence order can still make the interpretation easier to audit.
Subtraction phrases are directional
“Seven less than x” means:
x−7.
It does not mean 7−x.
“x is seven less than y” means:
x=y−7.
Equivalent statement:
y=x+7.
Subtraction errors often come from following the word order rather than reconstructing which quantity is larger.
Test subtraction with a simple number
“Seven less than a number.”
Let the number be 20.
Seven less than 20 is 13.
Test the candidate expressions:
- x−7 → 20−7=13;
- 7−x → 7−20=−13.
The numerical case exposes the reversal.
Multiplication can be written without a multiplication sign
“Five times x” becomes:
5x.
“The product of 5 and x” becomes:
5x.
“Three groups of x” becomes:
3x.
Algebraic convention places the numerical coefficient before the variable.
Division also has direction
“x divided by 5” means:
x/5.
“5 divided by x” means:
5/x.
These are not equivalent.
The unit can help:
$x shared equally among 5 people gives $x/5 per person.
“Half of” and “one third of” are multipliers
Half of x:
x/2 or (1/2)x.
One third of y:
y/3 or (1/3)y.
Three quarters of n:
(3/4)n.
The phrase describes a multiplicative scale, not subtraction from the whole.
“More than” can be additive or multiplicative depending on the sentence
“Eight more than x” is additive:
x+8.
“Three times as many as x” is multiplicative:
3x.
“Three more than x” and “three times x” are structurally different even though both compare quantities.
Brackets preserve grouped quantities
“Three times the sum of x and 4” means:
3(x+4).
“The sum of three times x and 4” means:
3x+4.
Test x=2:
- 3(x+4)=3(6)=18;
- 3x+4=6+4=10.
The words “the sum” identify a grouped quantity in the first phrase.
Brackets are not optional decoration. They show that an operation acts on an entire quantity rather than on one term alone.
Translate consecutive integers
Let n be an integer.
Three consecutive integers can be written:
n, n+1, n+2.
Three consecutive even integers can be written:
2n, 2n+2, 2n+4.
Three consecutive odd integers can be written:
2n+1, 2n+3, 2n+5.
The variable controls the starting point; the fixed increments preserve the sequence structure.
Translate age relationships
Let s be a son’s current age.
A father is 28 years older:
father’s age = s+28.
In five years:
- son = s+5;
- father = s+33.
The age difference remains 28.
A frequent error is to add five years to only one person or to multiply an age difference that should remain constant.
Translate a fixed-fee cost model
A service charges $6 booking fee plus $2.50 per kilometre.
Let d be distance in kilometres.
Total cost:
C=6+2.5d.
The fixed fee does not multiply by distance.
The distance-dependent rate does.
Check d=0:
C=6.
That matches a fixed booking fee.
Translate a discount model
An item with original price p is discounted by 15%.
Discount amount:
0.15p.
Sale price:
p−0.15p = 0.85p.
Writing p−15 would confuse a percentage with a fixed monetary amount.
Translate an increase
A quantity x increases by 12%.
Increase:
0.12x.
New quantity:
x+0.12x = 1.12x.
“Increased by 12%” and “increased to 12%” do not mean the same thing.
Translate geometry relationships
A rectangle has width w cm and length 5 cm more than its width.
Length:
w+5.
Perimeter:
2w+2(w+5).
Simplified:
4w+10.
Area:
w(w+5).
The same dimensions produce different algebra depending on whether the problem asks for boundary or region.
Translate rate relationships with units
A vehicle travels at 72 km/h for t hours.
Distance:
d=72t km.
If time is given in minutes m:
time in hours=m/60.
Distance:
d=72(m/60)=1.2m km.
Unit conversion becomes part of the algebraic translation.
Translate “at least” and “at most”
“At least 20” means 20 or more:
x≥20.
“At most 20” means 20 or less:
x≤20.
“More than 20” excludes 20:
x>20.
“No more than 20” means:
x≤20.
Boundary words control whether equality is included.
Worked problem: form and solve an equation
Three identical books and a $7 delivery fee cost $52.
Let x be the price of one book in dollars.
Three books:
3x.
Add delivery:
3x+7.
Equation:
3x+7=52.
3x=45.
x=15.
Check in context:
3($15)+$7=$52.
Worked problem: total and difference
Two numbers total 74.
The larger is 18 more than the smaller.
Let x be the smaller number.
Larger=x+18.
Equation:
x+(x+18)=74.
2x+18=74.
2x=56.
x=28.
Larger=46.
Check:
28+46=74 and 46−28=18.
Worked problem: consecutive integers
The sum of three consecutive integers is 72.
Let the first be n.
The integers are:
n, n+1, n+2.
Equation:
n+(n+1)+(n+2)=72.
3n+3=72.
3n=69.
n=23.
The integers are 23, 24 and 25.
Worked problem: mixture of fixed and variable quantities
A gym charges $25 monthly membership plus $4 for each class attended.
Let c be the number of classes.
Total monthly cost:
25+4c.
If the total is $57:
25+4c=57.
4c=32.
c=8.
The fixed fee and per-class rate must remain separate in the expression.
Use tables when the relationship is not yet clear
For the gym model:
- 0 classes → $25;
- 1 class → $29;
- 2 classes → $33;
- 5 classes → $45.
The table reveals:
- a starting value of 25;
- an increase of 4 for each additional class.
That structure becomes 25+4c.
Use a bar model when the comparison is hidden
“A is 12 more than B and their total is 68.”
A bar model can show:
- B=x;
- A=x+12;
- whole=68.
The equation x+(x+12)=68 is then a symbolic compression of the picture.
Test the algebra against the words
For every expression, ask:
- If the variable were zero, what would the expression mean?
- If the variable doubled, which parts should double?
- Does the result have the correct unit?
- Is the expression larger or smaller than the reference quantity as the sentence says?
- Can one easy numerical example verify the order of operations?
This catches many translation errors before solving begins.
Common misconception 1: translate one word at a time
Phrase structure matters.
“Three times the sum” is not the same as “the sum of three times”.
Common misconception 2: “less than” follows written order
Seven less than x is x−7.
Test with a number if unsure.
Common misconception 3: letters are labels for objects
Define the numerical quantity and unit, not merely the noun.
Common misconception 4: every sentence needs one variable
Some models need two variables. Others become simpler when one quantity is expressed in terms of another.
Common misconception 5: a plausible expression is good enough
Substitute a simple value and compare the result with the verbal statement.
Common misconception 6: the equation is complete without units or definitions
Definitions and units explain what the symbols mean in context and whether the final solution is feasible.
A translation diagnostic ladder
- Can the learner define a variable precisely?
- Can the learner distinguish expression, equation and inequality?
- Can the learner translate addition and subtraction phrases?
- Can the learner preserve order in division?
- Can the learner use brackets for grouped quantities?
- Can the learner translate consecutive integers?
- Can the learner model fixed fees plus variable rates?
- Can the learner translate percentage changes with multipliers?
- Can the learner form equations from geometry and age contexts?
- Can the learner test a symbolic form using an easy numerical case?
- Can the learner solve after the surface context changes?
How this fits Secondary Mathematics
Moving from verbal relationships to algebra supports expressions, equations, formulae, inequalities, graphs, coordinate geometry and modelling. The translation demand appears across both G2 and G3 Mathematics, while the complexity and formal expectations differ by subject level and syllabus.
The 2027 SEC syllabuses are published separately for G2 and G3 school candidates. Exact examinable forms should therefore be checked against the relevant current SEAB document rather than treated as one uniform Secondary Mathematics list.
The deeper lesson: algebra preserves relationships while removing the story
A taxi, an age problem, a rectangle and a savings plan can look unrelated.
Algebra removes the surface nouns and keeps the quantity structure.
Successful translation does not begin by asking which symbol matches each word. It begins by asking what quantities exist, how they depend on one another, and which algebraic structure preserves that relationship without distortion.