Two steps
2x + 3 = 11
Take 3 from both sides: 2x = 8
Divide both sides by 2: x = 4
Solving linear equations What can you do to an equation without changing its answer?
Foundation and Higher · AQA · 45 minutes
Students solve a balance puzzle with bags of counters, then learn the one rule behind every step: do the same to both sides. Brackets, unknowns on both sides, fractions and equations from a context follow, with working checked line by line. Slides to present, a vote that shows who's stuck, practice and homework.
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The sloping line is y = 2x + 3 and the flat one is y = c. Where they cross, 2x + 3 = c. Move c to 11: what x solves 2x + 3 = 11? Is there a c with no answer?
They cross at (2, 7)
The idea in one minute
An equation says two sides are equal, like a balance. Whatever you do to one side, do to the other, and it still balances: the answer doesn't change. Solving is choosing what to do: take away the smaller x-term from both sides, undo the adding or taking away, then divide. Brackets are expanded (or both sides divided), and fractions are cleared by multiplying every term.
The lesson goes picture first. Students solve a puzzle of bags and counters on a balance before any algebra, then see the same steps written as equations, as bars of equal length and as two lines that cross. Practice is working written line by line, each line checked as it is written, so a slip shows on the line where it happens. The core is on both tiers' papers; the stretch, with two fractions, is for Higher.
By the end, students can
- solve one- and two-step equations by doing the same to both sides
- solve equations with brackets and with the unknown on both sides
- clear a fraction by multiplying every term
- form an equation from a context, solve it and check the answer
Key words
- equation
- solve
- unknown
- both sides
- expand
- inverse operation
Before this lesson
Students should collect like terms, expand a single bracket such as 3(x + 2) and work with negative numbers. Solving is taught here, from the balance.
The lesson, step by step
60 minutes in total. Each step says what changes for this length.
5 min 1 · Hook · students think
Bags on a balance
Students find how many counters are in a bag before anyone explains.
Say“Every bag holds the same number of counters, and the two pans balance. How many counters are in a bag? Don't show anyone yet.”
Ask- What could you take off both pans and keep them balanced?
- How do you know your answer is right?
Watch forGuessing without checking: ask them to count both pans with their number.
For the teacher
Give two minutes of silent work, then ask how they found it. Many guess and check; some take two bags off both pans, then four counters.
Don't write any algebra yet. The question is the point: what can you take off a balance without tipping it?
Gather how they found it with Ask the class. Taking two bags off both pans is the next step's first line.
Shorter: Solve the puzzle together; skip the silent round (−2 min).
12 min 2 · Explain · you explain
Keep the balance
The balance as equations and as bars; two steps on a graph; brackets, fractions, both sides, a context.
Say“One rule: whatever you do to one side, do to the other. Say it each time before you write the line.”
Ask- 2x + 3 = 11: what do you undo first, the + 3 or the × 2? Why?
- 4x + 1 = x + 13: which x-term would you take away from both sides?
Watch forMoving a term across without changing what it does: to undo − 3, add 3 to both sides.
Unknowns on both sides
4x + 1 = x + 13
Take x from both sides: 3x + 1 = 13
Take 1 from both sides: 3x = 12, so x = 4
From a context
A taxi costs £3 plus £2 a mile. A trip costs £19.
2x + 3 = 19, so 2x = 16 and x = 8 miles
For the teacher
First slide: the puzzle as algebra. A bag is x: 3x + 4 = 2x + 9. Take 2x from both sides, then 4: x = 5. Each line says what was done to both sides.
Second slide: the same equation as two bars of equal length. Cover two x's on each bar, and what is left still matches: x + 4 = 9.
Third slide: two steps. Undo in reverse order: the + 3 first, then the × 2. On the graph, y = 2x + 3 meets y = 11 where x = 4; move c and the answer moves.
Fourth slide: brackets (expand, or divide both sides first) and fractions (multiply both sides, every term, by the denominator). Fifth: unknowns on both sides, where two lines cross. Sixth: form the equation from words, solve it, check it in the words.
Vote before moving c (Ask the class): where will the lines cross when c is 15?
Shorter: Skip the bars slide; one worked example (−4 min).
8 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
Say“Watch me do one. Then you do its twin, which is almost the same. Say what you do to both sides.”
Ask- What will you do to both sides first? Why that?
Watch forMultiplying only the first term in a bracket: every term inside is multiplied.
Two steps
x/3 + 2 = 6
Take 2 from both sides: x/3 = 4
Times both sides by 3: x = 12
Then theirs
x/5 + 4 = 7
Brackets
2(x + 3) = 14
Expand: 2x + 6 = 14
2x = 8, so x = 4
Then theirs
2(x − 3) = 10
Both sides
5x + 2 = 3x + 12
Take 3x from both sides: 2x + 2 = 12
2x = 10, so x = 5
Then theirs
6x + 5 = 2x + 29
For the teacher
Work the first example aloud, saying what you do to both sides before writing each line. Then students solve its twin while you watch who says it first.
Do the same for each pair. Each twin changes only the numbers. For the last, ask which x-term is smaller before anyone writes: take that one from both sides.
Open the answers slide once most have finished their twins.
Shorter: Do the first two pairs only (−3 min).
25 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then working checked line by line, in levels, and a stretch.
For this length: The stretch, then: what do you multiply by to clear two fractions at once?
Say“Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- Put your answer back into 5x − 3 and into 2x + 9. Are they equal?
Watch forMoving a term across the equals sign without saying what was done to both sides.
Write one equation a line, and say to yourself what you did to both sides. Each line is checked as you write it. End with x = … (write x = 3, not 3 = x). The stretch is optional.
The question that shows who's stuck
- Solve 5x − 3 = 2x + 9.
In your Classroom, as they answer (an example) - A6
- B12
- C5
- D4
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- Solve 3x + 7 = 22, one step a line.They solve 3x + 7 = 22 line by line, each line checked as they write it.
- Solve x/4 − 3 = 2, one step a line.They solve x/4 − 3 = 2 line by line, each line checked as they write it.
- Solve 5(x − 2) = 3x + 4, one step a line.They solve 5(x − 2) = 3x + 4 line by line, each line checked as they write it.
- Solve 7 − 2x = 3x − 8, one step a line.They solve 7 − 2x = 3x − 8 line by line, each line checked as they write it.
- Each line was written as a first step for 3x + 4 = 2x + 9. Which keep the balance, and which break it?They sort 6 items into Keeps the balance and Breaks it.
- Solve 4x + 3 = 5. Then drag the marker to the solution on the number line.They drag a marker on a number line from -2 to 3.
- Solve (x + 5)/3 = x − 1, one step a line.They solve (x + 5)/3 = x − 1 line by line, each line checked as they write it.
- A rectangle is x cm wide and (x + 4) cm long. Its perimeter is 30 cm. Form an equation and solve it to find x. Give your answer as a decimal.
- I think of a number. Three times it, take away 4, is the same as the number add 10. My number is ___.
- Stretch: solve (2x − 1)/3 − (x + 2)/4 = 1, one step a line. Your answer may be a fraction.They solve (2x − 1)/3 − (x + 2)/4 = 1 line by line, each line checked as they write it.
Support · For question 3: expand the bracket first, multiplying both terms inside by 5. Then take 3x from both sides, then add 10 to both sides, then divide.
For the teacher
Everyone chooses before anyone explains. Whatever they chose, ask them to put it back into both sides of the equation: only one option makes them equal. Then ask what was done to both sides in each line.
In the working questions each line is checked as it is written, so a slip shows on its own line. Questions 1–2 build fluency; 3–4 change the look (a bracket, a negative x-term); 5–6 ask for reasons (sort first steps, a solution on a number line); 7–9 are exam-style. The stretch has two fractions.
Students write their working line by line in the quiz; questions 8 and 9 keep only the answer, so work those on paper.
If the class splits, have everyone put each option back into both sides before they practise.
Shorter: After the vote, questions 1–5 only (−6 min).
Longer: The stretch, then: what do you multiply by to clear two fractions at once? (+10 min).
10 min 5 · Exit · students show
Solve one, say why
One equation solved line by line, and one sentence, show who keeps the balance.
For this length: Check your answer on a sketch: where do y = 4(x − 1) and y = 2x + 6 cross?
Say“Solve it line by line. Then say in words what you did to both sides in your first line.”
Watch forExpanding 4(x − 1) as 4x − 1.
One equation a line, ending with x = …. One sentence for the second question.
- Solve 4(x − 1) = 2x + 6, one step a line.They solve 4(x − 1) = 2x + 6 line by line, each line checked as they write it.
- In words: what did you do to both sides in your first line, and why does that keep the answer the same?
For the teacher
Solve 4(x − 1) = 2x + 6, one step a line.
Then answer the second exit question in one sentence.
The sentences show who thinks of both sides and who moves terms by rule: start the next lesson with the bags if few do. Set the homework.
Longer: Check your answer on a sketch: where do y = 4(x − 1) and y = 2x + 6 cross? (+5 min).
One to one: ask the student to say what they will do to both sides before writing each line, and to check every answer by putting it back into the first line.
Homework: keep the balance
Write one step a line, and beside each line what you did to both sides. Check every answer by putting it back into the first line. For question 6, form the equation first.
- Solve 4x − 5 = 23
- Solve 3(x + 4) = 27
- Solve 6x − 7 = 2x + 9
- Solve 10 − x = 2x + 1
- Solve (3x − 1)/4 = 5
- The angles of a triangle are x°, (2x + 10)° and (3x − 40)°. Form an equation, solve it, and find the size of each angle.
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching solving linear equations: common questions
Should I teach the balance method or 'change side, change sign'?
Start with the balance: do the same to both sides. It explains every step, works for brackets, fractions and unknowns on both sides, and carries on into quadratics and rearranging formulae. 'Change side, change sign' is a shortcut of it; students who use it without the balance often move a term without changing what it does, or can't handle a fraction or a minus in front of x.
Which side should the x-terms go on?
Take away the smaller x-term from both sides. In 5x + 2 = 3x + 12, take 3x from both sides, which leaves 2x on the left with a positive number in front. With 7 − 2x = 3x − 8, add 2x to both sides: the x-terms end on the right, and that is fine, as 15 = 5x means x = 3.
How does the working line by line check students' answers?
Each line a student writes is compared with the line before: if it has the same solution, it is right; if not, it is wrong on that line, without the answer being shown. So a slip in the second line is found in the second line, not at the end. The last line has to be the answer, x = …, and a question can ask for at least two or three lines.
Is this linear equations lesson for Foundation or Higher GCSE?
Both. Two-step equations, brackets, unknowns on both sides, a fraction and forming an equation from a context are on both tiers' papers, often inside a bigger question (angles, perimeter, ages). Higher students can move quickly through the core and spend longer on questions 7–9 and the stretch, with two fractions.
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Curriculum reference: AQA GCSE Mathematics 8300: A17 (solve linear equations in one unknown, including brackets and the unknown on both sides) and A21 (form an equation from a situation, solve it and interpret the solution).
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