The shape of the lesson
By the end of the lesson, learners can
- How algebra is written: Open the lab on Substitution, pick a formula and drag the sliders for each letter.
- Substituting safely: Replace every letter with its value, and put each value in brackets, especially negative numbers.
- Formulas in context: A formula is a rule written in algebra.
- Notation: \(3a = 3 \times a\), \(ab = a \times b\), \(3a^{2} = 3 \times a^{2}\).
- Words to algebra: if something happens first, it may need a bracket: \(3(n + 5)\).
- Substitute: write each value in brackets, especially negatives.
- Order: brackets, powers, multiply/divide, add/subtract.
- Check signs: minus a negative is plus; a negative squared is positive.
One picture every idea returns to
A letter stands for a number. To substitute, replace each letter with its value in brackets, then follow the order of operations: brackets, then powers, then multiply and divide, then add and subtract.
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, Letters that stand for numbers.. A letter in algebra is a number you have not met yet. | Whole class, then pairs |
| 5 to 13 min | Teach and check: How algebra is written | Open the lab on Substitution, pick a formula and drag the sliders for each letter. Project tab 1 of the lab (Algebra lab, substitute and simplify) and let a student drive it while the class predicts. Model the worked example Words into algebra on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Substituting safely | Replace every letter with its value, and put each value in brackets, especially negative numbers. Model the worked example Negative values on the board, one line at a time. Your turn: Find the value of \(\dfrac{p^{2} - q}{2p}\) when \(p = -4\) and \(q = 6\). | Pairs |
| 21 to 30 min | Teach and check: Formulas in context | A formula is a rule written in algebra. Model the worked example Writing and using a formula on the board, one line at a time. Your turn: Use \(F = \dfrac{9C}{5} + 32\) to convert \(-15\)°C into degrees Fahrenheit. | Numbered Heads Together |
| 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, 15 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 Formulas in context, 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.
Numbered Heads Together
Teams of four number off. Put up a question, give the team time to make sure every member can answer, then call a number: that student answers for the team, so nobody can leave the thinking to someone else.
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 |
|---|---|
| \(n(n + 1)\) is the product of two consecutive numbers, but not necessarily even ones. | Writing \(2n\) guarantees the number is even. |
| \(2x^{2}\) means \(2 \times x^{2}\), not \((2x)^{2}\). | With \(x = -3\): \(2x^{2} = 18\) but \((2x)^{2} = 36\). And a calculator reads \(-3^{2}\) as \(-9\): type \((-3)^{2}\) with the brackets. |
| Money answers need two decimal places: $127.5 should be written $127.50. | Return to Formulas in context: ask a pair to state the correct version and show it on the lab before moving on. |
Algebraic notation and substitution: worksheet
6 questions, 15 marks. Written in the style of the papers; no past paper question is reproduced.
Write an expression for (a) 7 less than \(y\) (b) \(x\) divided by 4, then add 3.
Without a calculator, find the value of each expression when \(a = 5\) and \(b = -3\).
(a) \(a - b\) (b) \(ab^{2}\) (c) \(\dfrac{a + b}{a - b}\)
The time, \(T\) seconds, for one swing of a pendulum of length \(l\) metres is \(T = 2\pi\sqrt{\dfrac{l}{g}}\).
Find \(T\) when \(l = 0.8\) and \(g = 9.8\).
Sam says that \(2x^{2}\) and \((2x)^{2}\) are always equal. Use \(x = 3\) to show that Sam is wrong.
Use the formula \(F = \dfrac{9C}{5} + 32\) to find \(F\) when \(C = -15\).
A taxi charges $3.50 plus $1.20 for each kilometre travelled.
(a) Write a formula for the cost, \(C\) dollars, of a journey of \(d\) kilometres.
(b) Find the cost of a 12 km journey.
(c) A journey costs $11.30. How long was it?
Answer key and marking notes
- (a) \(y - 7\)
- (b) \(\dfrac{x}{4} + 3\)
- (a) \(5 - (-3) = 8\)
- (b) \(b^{2} = 9\), so \(5 \times 9 = 45\)
- (c) \(\dfrac{5 + (-3)}{8} = \dfrac{2}{8} = \dfrac{1}{4}\)
- \(\dfrac{0.8}{9.8} = 0.08163...\), and \(\sqrt{0.08163...} = 0.2857...\)
- \(T = 2\pi \times 0.2857... = 1.795...\), so \(T = 1.80\) s (3 s.f.)
- \(2x^{2} = 2 \times 9 = 18\)
- \((2x)^{2} = 6^{2} = 36\). \(18 \ne 36\), so Sam is wrong.
- \(\dfrac{9 \times (-15)}{5} = -27\)
- \(F = -27 + 32 = 5\)
- (a) \(C = 3.5 + 1.2d\)
- (b) \(C = 3.5 + 1.2 \times 12 = 3.5 + 14.4 = 17.9\), so $17.90
- (c) \(3.5 + 1.2d = 11.3\), so \(1.2d = 7.8\)
- \(d = 7.8 \div 1.2 = 6.5\) km
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.
Which expression means 'subtract 4 from a number \(n\), then multiply the result by 5'?
- A\(5(n - 4)\)
- B\(5n - 4\)
- C\(4 - 5n\)
- D\(5(4 - n)\)
Find the value of \(5a - 2b\) when \(a = -2\) and \(b = -3\).
- A\(-4\)
- B\(-16\)
- C\(4\)
- D\(16\)
Find the value of \(4x^{2}\) when \(x = -3\).
- A\(36\)
- B\(144\)
- C\(-36\)
- D\(-24\)
Use the formula \(v = u + at\) to find \(v\) when \(u = 12\), \(a = -3\) and \(t = 5\).
- A\(-3\)
- B\(27\)
- C\(45\)
- D\(3\)
Ravi is \(y\) years old. His father's age is 4 less than three times Ravi's age. Which expression gives the sum of their ages in 5 years' time?
- A\(4y + 6\)
- B\(4y + 1\)
- C\(4y - 4\)
- D\(3y + 6\)
Find the value of \(\dfrac{a^{2} - 3b}{a - b}\) when \(a = -4\) and \(b = 2\).
- A\(-\dfrac{5}{3}\)
- B\(\dfrac{11}{3}\)
- C\(-5\)
- D\(\dfrac{5}{3}\)
Quiz key
Subtracting 4 from \(n\) gives \(n - 4\). Multiplying the whole of that by 5 needs a bracket: \(5(n - 4)\). \(5n - 4\) multiplies first and subtracts second, which is the wrong order. \(5(4 - n)\) subtracts \(n\) from 4 instead of 4 from \(n\).
\(5(-2) - 2(-3) = -10 - (-6) = -10 + 6 = -4\). The answer \(-16\) comes from treating \(-2 \times -3\) as \(-6\): minus times minus is plus, so \(-2b\) is \(+6\).
\(x^{2} = (-3)^{2} = 9\), so \(4x^{2} = 4 \times 9 = 36\). 144 is \((4x)^{2} = (-12)^{2}\): the 4 is not inside the square. \(-36\) comes from typing \(-3^{2}\) without brackets, which a calculator reads as \(-9\). \(-24\) doubles instead of squares.
\(at = (-3)(5) = -15\), so \(v = 12 + (-15) = -3\). 45 comes from \((u + a)t = 9 \times 5\), adding before multiplying. 27 ignores the negative sign of \(a\).
Father now: \(3y - 4\). In 5 years: Ravi \(y + 5\), father \(3y - 4 + 5 = 3y + 1\). Sum: \(y + 5 + 3y + 1 = 4y + 6\). \(4y + 1\) adds the 5 years only once, but both people age. \(4y - 4\) is the sum of their ages now.
Top: \((-4)^{2} - 3(2) = 16 - 6 = 10\). Bottom: \(-4 - 2 = -6\). So the value is \(\dfrac{10}{-6} = -\dfrac{5}{3}\). \(\dfrac{11}{3}\) comes from \(-4^{2} = -16\) without brackets, giving \(\dfrac{-22}{-6}\). \(-5\) uses \(a + b = -2\) on the bottom.
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.