The shape of the lesson
By the end of the lesson, learners can
- Solving on a balance: In the lab pick an equation and press Next step one move at a time.
- Forming an equation: Choose a letter for the unknown, write each quantity in terms of it, then use a fact you know (angles in a triangle add to 180°, a perimeter, a total) to make an equation.
- Fractional equations (Extended): Clear the fractions first.
- Fractions (Extended): multiply every term by the common denominator first.
- Brackets: expand, watching the sign of the number outside.
- Collect: unknowns on the side where they stay positive, numbers on the other.
- Divide: by the number in front of the unknown.
- Check: substitute into the original equation; answer what was asked.
One picture every idea returns to
Whatever you do to one side of an equation, do to the other. Expand brackets, collect the unknown on one side and the numbers on the other, then divide. Check by substituting your answer back into the original equation.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Keep the scales level.. An equation is a balance: do the same to both sides and it stays true. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Solving on a balance | In the lab pick an equation and press Next step one move at a time. Project tab 1 of the lab (Equations lab, linear equations) and let a student drive it while the class predicts. Model the worked example Two steps on the board, one line at a time. Your turn: Solve \(8 - 3(x - 2) = 2(x + 4)\). | Whole class, then pairs |
| 13 to 21 min | Teach and check: Forming an equation | Choose a letter for the unknown, write each quantity in terms of it, then use a fact you know (angles in a triangle add to 180°, a perimeter, a total) to make an equation. Model the worked example Angles in a triangle on the board, one line at a time. Your turn: The sum of three consecutive whole numbers is 87. Find the numbers. | Pairs |
| 21 to 30 min | Teach and check: Fractional equations (Extended) | Clear the fractions first. Model the worked example Numerical denominators (Extended) on the board, one line at a time. Your turn: (Extended) Solve \(\dfrac{2x + 1}{x} = 4\). | Rally Coach |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 20 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Fractional equations (Extended), never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Rally Coach
Pairs share one worksheet. One partner solves the first question aloud while the other coaches, checking each line and offering a prompt rather than the answer; they swap roles for the next question.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| In \(8 - 3(x - 2)\), the \(-3\) multiplies the \(-2\) as well: \(-3 \times (-2) = +6\). | Writing \(8 - 3x - 6\) is the most common slip on this type. |
| Finding \(x\) is often not the end. | Read the question again: it may ask for an angle, a length or an age built from \(x\). |
| Multiplying only the fractions by 6 and forgetting the 1 on the right gives \(7x + 4 = 1\). | Every term, on both sides, must be multiplied. |
Forming and solving linear equations: worksheet
6 questions, 20 marks. Written in the style of the papers; no past paper question is reproduced.
Solve (a) \(7x - 3 = 25\) (b) \(\dfrac{x}{5} + 6 = 2\).
Solve \(3(2x - 5) = 4x + 9\).
Solve \(8 - 3(x - 2) = 2(x + 4)\).
A rectangle has width \(w\) cm. Its length is 3 cm more than twice its width. The perimeter is 42 cm.
(a) Write an equation in \(w\). (b) Solve it and find the area of the rectangle.
(Extended) Solve \(\dfrac{5}{x + 1} = \dfrac{2}{x - 2}\).
Phone plan A costs $12 per month plus $0.05 per minute of calls. Plan B costs $20 per month plus $0.03 per minute.
(a) For \(m\) minutes of calls, write an equation for when the two plans cost the same.
(b) Solve your equation.
(c) Hana uses 500 minutes a month. Which plan is cheaper, and by how much?
Answer key and marking notes
- (a) \(7x = 28\), \(x = 4\)
- (b) \(\dfrac{x}{5} = -4\), \(x = -20\)
- \(6x - 15 = 4x + 9\)
- \(2x = 24\)
- \(x = 12\)
- \(8 - 3x + 6 = 2x + 8\)
- \(14 - 3x = 2x + 8\), so \(6 = 5x\)
- \(x = 1.2\)
- (a) \(2(w + 2w + 3) = 42\)
- (b) \(6w + 6 = 42\), \(6w = 36\), \(w = 6\)
- Length \(= 2(6) + 3 = 15\) cm
- Area \(= 6 \times 15 = 90\) cm2
- \(5(x - 2) = 2(x + 1)\)
- \(5x - 10 = 2x + 2\), so \(3x = 12\)
- \(x = 4\)
- (a) \(12 + 0.05m = 20 + 0.03m\)
- (b) \(0.02m = 8\), so \(m = 400\) minutes
- (c) A: \(12 + 0.05 \times 500 = \$37\). B: \(20 + 0.03 \times 500 = \$35\)
- Plan B is cheaper by $2.
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Solve \(7x - 4 = 3x + 10\).
- A\(x = 3.5\)
- B\(x = 1.5\)
- C\(x = 1.4\)
- D\(x = 14\)
Solve \(3(2x + 5) = 4(x - 1)\).
- A\(x = -9.5\)
- B\(x = 9.5\)
- C\(x = -4.5\)
- D\(x = -8\)
The angles of a quadrilateral are \(x\)°, \((2x + 10)\)°, \((3x - 20)\)° and \((2x + 50)\)°. Find the size of the largest angle.
- A130°
- B40°
- C100°
- D85°
I think of a number. Multiplying it by 4 and adding 9 gives the same answer as multiplying it by 7 and subtracting 6. What is the number?
- A5
- B1
- C\(-5\)
- D15
Solve \(\dfrac{x - 3}{4} - \dfrac{x + 1}{6} = 1\).
- A\(x = 23\)
- B\(x = 19\)
- C\(x = 12\)
- D\(x = 11.5\)
Solve \(\dfrac{3}{x - 2} = \dfrac{5}{x + 4}\).
- A\(x = 11\)
- B\(x = -13\)
- C\(x = 3\)
- D\(x = -11\)
Quiz key
\(7x - 3x = 10 + 4\), so \(4x = 14\) and \(x = 3.5\). Check: \(7(3.5) - 4 = 20.5\) and \(3(3.5) + 10 = 20.5\). \(x = 1.5\) moves the \(-4\) across without changing its sign (\(4x = 6\)). \(x = 1.4\) adds the \(x\) terms to get \(10x = 14\).
\(6x + 15 = 4x - 4\), so \(2x = -19\) and \(x = -9.5\). \(x = -4.5\) expands \(3(2x + 5)\) as \(6x + 5\). \(x = -8\) expands \(4(x - 1)\) as \(4x - 1\). \(x = 9.5\) loses the minus sign of \(-19\).
\(x + 2x + 10 + 3x - 20 + 2x + 50 = 360\), so \(8x + 40 = 360\), \(8x = 320\), \(x = 40\). The angles are 40°, 90°, 100° and 130°, so the largest is 130°. 40° is \(x\), not the largest angle. 85° comes from using 180°, the angle sum of a triangle.
\(4n + 9 = 7n - 6\). Subtract \(4n\): \(9 = 3n - 6\). Add 6: \(15 = 3n\), so \(n = 5\). Check: \(4(5) + 9 = 29\) and \(7(5) - 6 = 29\). 15 stops before dividing by 3; 1 comes from \(3n = 9 - 6\).
\(\times 12\): \(3(x - 3) - 2(x + 1) = 12\), so \(3x - 9 - 2x - 2 = 12\), \(x - 11 = 12\), \(x = 23\). \(x = 19\) writes \(-2(x + 1)\) as \(-2x + 2\). \(x = 12\) forgets to multiply the right-hand side by 12. \(x = 11.5\) uses 24 on the left but leaves the 1 unchanged.
\(3(x + 4) = 5(x - 2)\), so \(3x + 12 = 5x - 10\), \(22 = 2x\), \(x = 11\). Check: \(\dfrac{3}{9} = \dfrac{1}{3}\) and \(\dfrac{5}{15} = \dfrac{1}{3}\). \(x = -13\) multiplies each numerator by its own denominator. \(x = 3\) drops the brackets: \(3x + 4 = 5x - 2\).
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.