S1 Maths | Linear Equations and Word Problems Lesson Prep Guide (for tutors) | Balancing, brackets and forming equations

2026-11-09

Tutor lesson-prep guides · Secondary Maths (DSE)

An equation is not about moving numbers around; it is about doing the same thing to both sides. Once students hold that sentence, transposing stops being a chant.

Objectives — Students can: (1) solve two-step equations using the both-sides language and state what each move did to each side; (2) handle brackets, denominators and negative coefficients correctly; (3) turn a ticket-style situation into an equation and solve it, scoring at least 9 of 12 in the diagnostic.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Negative arithmetic	What is 5 minus 8?	-3
Substitution	When x is 5, what is 3x plus 5?	20
Fractions	What is half of 8?	4

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-6 min	An equation is a balance	3x plus 5 equals 20: subtract 5 from both sides, then divide both sides by 3	State what each move did
6-16 min	Two-step equations and checking	x is 5; substituting back gives 20, so both sides agree	Solve one and check it
16-28 min	Brackets, denominators, negatives	3(x minus 2) equals 12: divide first; 5 minus 2x equals 11 gives x equals minus 3	Three questions in pairs
28-40 min	Unknowns on both sides	4x minus 7 equals x plus 11; subtract x from both sides, then move the constant, giving 6	Collect the unknown on one side
40-54 min	Forming the equation	Adult tickets 60, child tickets 40, 12 tickets for 560: 4 adults and 8 children	Form and solve independently
54-60 min	Exit check	Three oral questions; any miss points to the stage to reteach	Oral answers

03 Teaching script (read it out)

Scene 1: A balance, not a shuffle

The rule “change the sign when you move it” is only the result; the reason is that both sides change together. From 3x plus 5 equals 20, subtracting 5 from both sides leaves 3x equals 15, and dividing both sides by 3 gives x equals 5. Every move must be spoken as an operation on both sides, otherwise students change a plus into a minus without knowing why.
Teaching move: make students draw a vertical line at the equals sign and annotate “minus 5 both sides”, “divide by 3 both sides”, so a slip is visible at once.

Students do: solve 3x plus 5 equals 20 and substitute the answer back, writing that both sides equal 20.

Scene 2: Brackets first, denominators last

For 3(x minus 2) equals 12 you may divide both sides by 3 to get x minus 2 equals 4, then add 2 to both sides for x equals 6; expanding first also works. Denominators work the other way: x over 2 plus 3 equals 7, so subtract 3 from both sides and multiply both sides by 2 to get x equals 8.
Teaching move: always ask whether the number outside the bracket divides the other side exactly; if it does, divide first and halve the size of every number.

Students do: solve 3(x minus 2) equals 12 twice, once by dividing first and once by expanding.

Scene 3: Negative coefficients: predict the sign first

5 minus 2x equals 11: subtract 5 from both sides to get minus 2x equals 6, then divide both sides by minus 2 for x equals -3. The usual slip is dropping the sign when dividing by a negative number.
Teaching move: before dividing, require one written prediction saying the answer should be negative or positive, then compare after computing.

Students do: solve 5 minus 2x equals 11 and explain why the answer is negative.

Scene 4: Word problems: define the unknown first

Tickets cost 60 and 40; there are 12 tickets worth 560. Let a be the number of adult tickets, so children are 12 minus a, and 60a plus 40(12 minus a) equals 560. This simplifies to 20a plus 480 equals 560, so a equals 4, giving 4 adults and 8 children. Check: 60 times 4 plus 40 times 8 equals 560.
Teaching move: enforce three written lines before any algebra — define the unknown, write the total count, write the total value.

Students do: three consecutive integers sum to 45; let the middle be n, form the equation and find 14, 15 and 16.

04 Question types and worked steps

Type	Example	Steps	Answer
Two-step equation	3x plus 5 equals 20	Add or subtract first, then multiply or divide, annotating each move	x equals 5
With brackets	3(x minus 2) equals 12	Divide first if it divides exactly, otherwise expand	x equals 6
Unknown on both sides	4x minus 7 equals x plus 11	Subtract the smaller x term from both sides to avoid negatives	x equals 6
Word problem	Tickets at 60 and 40, 12 sold for 560	One unknown, the other as the total minus it, then a value equation	4 adults and 8 children

Sanity check (teach this): Sanity checks: substituting the answer back must make both sides equal; every move must be applied to both sides; when clearing denominators every term is multiplied, including terms without a denominator; answers to word problems must fit the situation (ticket counts are positive integers and totals match); 9 out of 12 is 75 percent, not 9 percent.

05 Independent practice (3 questions)

Question	Answer	What to watch
Four two-step equations	Include one negative coefficient and one fraction	Is each move annotated
Two bracket questions	One dividing first, one expanding first	Was the second term multiplied
Two word problems	Tickets and consecutive integers	Is the unknown defined and the unit stated

06 Common mistakes and what to say

Mistake	Student writes	Say this
Forgetting to change the sign	Writing 3x equals 20 plus 5 from 3x plus 5 equals 20	Moving a term is subtracting 5 from both sides, so the right side loses 5 as well; it does not gain it.
Multiplying only the first term	Writing 3(x minus 2) as 3x minus 2	The number outside the bracket multiplies every term inside; draw arrows from it to both terms before expanding.
Dropping the sign on division	Getting a positive answer from minus 2x equals 6	Dividing both sides by a negative number flips the sign of the whole result; say “the unknown is negative” before writing.
No defined unknown in word problems	Writing an equation without saying what the letter means	Without a definition the final number means nothing; three lines: a is adult tickets, children are 12 minus a, the total is 560 dollars.

07 Homework (paper, section and question numbers)

Secondary maths chapter index — See the other chapters in this subject
Lesson prep guides home — Compare related chapters
Redo this chapter — Redo diagnostic questions 1 to 6, annotating each move

08 Exit check (last 5 minutes)

Question	Answer	Decision
x in 3x plus 5 equals 20	5	Answering minus 5 means sign errors on transposing; reteach stage one
x in 4x minus 7 equals x plus 11	6	Answering 4 means x was not subtracted from both sides; reteach stage four
Adults among 12 tickets	4	Answering 8 means the unknown was defined the wrong way round; reteach stage four

09 Tutor FAQ

Must every answer be checked?

In an exam a quick substitution in the margin is enough, but do it: one minute of checking beats losing a whole question.

Can both sides be multiplied?

Yes, by any non-zero number; that is exactly why multiplying both sides by the denominator clears fractions safely.

Must word-problem answers be whole numbers?

Not always, but counts of tickets, people or items must be positive integers; a fraction means the equation or the reading is wrong.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.