The shape of the lesson
By the end of the lesson, learners can
- Table of values, then plot: In the lab, pick a function and fill the table one \(x\) at a time.
- Recognise the shape: Before you plot, you should already know roughly what the graph will look like.
- Exponential growth and decay: Extended.
- Gradient from a tangent: A curve's steepness changes all the time.
- Substitute carefully: put negative \(x\) values in brackets, especially when squaring or cubing.
- Check the pattern: quadratic tables are symmetrical; a value that breaks the pattern is usually a slip.
- Plot and join: one smooth curve through every point; never join reciprocal branches across \(x = 0\).
- Tangent: rule a line touching the curve at the point, then gradient = rise ÷ run using a big triangle.
- Exponential (Extended): \(ab^x\) starts at \(a\) when \(x = 0\) and is multiplied by \(b\) for each step of 1.
One picture every idea returns to
Substitute each \(x\) into the equation (brackets round negatives), plot every point, then join them with ONE smooth curve. The gradient of a curve at a point is the gradient of the tangent drawn there.
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, From table to curve.. Every graph in Paper 4 starts the same way: a table of values, carefully plotted points and a smooth curve. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 11 min | Teach and check: Table of values, then plot | In the lab, pick a function and fill the table one \(x\) at a time. Project tab 1 of the lab (Graphs of functions lab, table of values) and let a student drive it while the class predicts. Model the worked example Completing a table for a quadratic on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Recognise the shape | Before you plot, you should already know roughly what the graph will look like. Your turn: Name the shape of (a) \(y = 4 - x^2\) (b) \(y = -\dfrac{3}{x}\) (c) \(y = 5 - 2x\). | Pairs |
| 17 to 23 min | Teach and check: Exponential growth and decay | Extended. Project tab 4 of the lab (Graphs of functions lab, growth and decay) and let a student drive it while the class predicts. Model the worked example Growth: reading a time from the graph (Extended) on the board, one line at a time. | Pairs |
| 23 to 30 min | Teach and check: Gradient from a tangent | A curve's steepness changes all the time. Model the worked example Gradient of a quadratic at a point on the board, one line at a time. Your turn: A tangent to a curve passes through \((1, 2)\) and \((5, 14)\). What is the gradient of the curve where the tangent touches it? | Jigsaw |
| 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, 17 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 Gradient from a tangent, 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.
Jigsaw
Split the content into expert pieces. Expert groups master one piece each, then regroup so every home team has one expert per piece, and each expert teaches their piece to the team.
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 |
|---|---|
| Typing \(-2^2\) into a calculator gives \(-4\), because it squares 2 and then makes it negative. | Always type \((-2)^2\) with brackets: the answer is \(+4\). One wrong value ruins the shape of the whole curve. |
| Joining the points of a curve with straight ruler lines loses the mark for the curve. | Use one smooth freehand curve that passes through every point, and do not flatten the bottom of a parabola between two equal points. |
| Exponential decay is NOT a straight line: after 3 years the car is worth $6144, not \(12\,000 - 3 \times 2400\) = $4800. | The 20% is taken from a smaller value each year. |
| Use two points on the TANGENT that are far apart, not points on the curve, and read each scale carefully: if the \(y\)-axis goes up in 2s, a rise of 4 squares is 8, not 4. | Return to Gradient from a tangent: ask a pair to state the correct version and show it on the lab before moving on. |
Graphs of functions: tables, plotting and tangents: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
Complete the table of values for \(y = x^2 + x - 4\), and state the equation of the line of symmetry of the graph.
| \(x\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) |
|---|---|---|---|---|---|---|
| \(y\) | \(-2\) | \(-4\) | \(-2\) |
(a) Find \(y = \dfrac{8}{x}\) for \(x = -4, -2, -1, 1, 2, 4\).
(b) Explain why the table has no value for \(x = 0\).
(Extended) Complete the table of values for \(y = x^3 - 4x + 2\).
| \(x\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |
|---|---|---|---|---|---|---|---|
| \(y\) | 2 | 2 | 2 |
A tangent is drawn to a curve at the point \(A\). The tangent passes through \((-1, 6)\) and \((2, -3)\).
Estimate the gradient of the curve at \(A\) and say what the sign tells you.
(Extended) The graph of \(y = a \times b^{x}\), where \(b > 0\), passes through \((0, 3)\) and \((2, 48)\).
Find \(a\) and \(b\), and then find \(y\) when \(x = -1\).
(Extended) A lake is stocked with fish. After \(t\) weeks the number of fish is \(P = 200 \times 1.5^{t}\).
(a) Complete a table of values for \(t = 0, 1, 2, 3, 4\).
(b) Draw the graph and use it to estimate when the number of fish reaches 800.
(c) What does the gradient of a tangent to this graph represent?
Answer key and marking notes
- \(x = -3\): \(9 - 3 - 4 = 2\).
- \(x = -1\): \(1 - 1 - 4 = -4\).
- \(x = 2\): \(4 + 2 - 4 = 2\).
- Equal \(y\) values at \(x = -1\) and \(x = 0\) (and at \(-2\) and \(1\)), so the line of symmetry is halfway: \(x = -0.5\).
- (a) \(-2, -4, -8, 8, 4, 2\).
- (b) \(\dfrac{8}{0}\) is undefined: you cannot divide by zero, so the graph has no point on the \(y\)-axis.
- \(x = -3\): \((-3)^3 - 4(-3) + 2 = -27 + 12 + 2 = -13\).
- \(x = -1\): \(-1 + 4 + 2 = 5\).
- \(x = 1\): \(1 - 4 + 2 = -1\).
- \(x = 3\): \(27 - 12 + 2 = 17\).
- Gradient \(= \dfrac{-3 - 6}{2 - (-1)} = \dfrac{-9}{3} = -3\).
- It is negative, so the curve is decreasing (going downhill) at \(A\).
- At \(x = 0\): \(a \times b^0 = a = 3\).
- At \(x = 2\): \(3b^2 = 48\), so \(b^2 = 16\) and \(b = 4\) (positive).
- \(x = -1\): \(y = 3 \times 4^{-1} = \dfrac{3}{4} = 0.75\).
- (a) \(200, 300, 450, 675, 1012.5\) (about 1010 fish).
- (b) Reading across from \(P = 800\) meets the curve between \(t = 3\) and \(t = 4\), at about \(t = 3.4\) weeks (accept 3.3 to 3.5).
- Check: \(200 \times 1.5^{3.4} \approx 794\), close to 800.
- (c) The rate at which the number of fish is growing, in fish per week, at that moment.
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.
A table of values is being completed for \(y = x^2 + 3x - 4\). What is the value of \(y\) when \(x = -3\)?
- A\(-4\)
- B\(-22\)
- C\(14\)
- D\(0\)
The graph shows a curve drawn for \(-5 \le x \le 5\), \(x \ne 0\). Which could be its equation?
- A\(y = -\dfrac{4}{x}\)
- B\(y = \dfrac{4}{x}\)
- C\(y = -4x\)
- D\(y = x^2 - 4\)
A student's table of values for \(y = 2x^2 - 5\) is shown.
| \(x\) | \(-2\) | \(-1\) | 0 | 1 | 2 |
|---|---|---|---|---|---|
| \(y\) | \(-13\) | \(-3\) | \(-5\) | \(-3\) | 3 |
- AAt \(x = -2\); it should be 3
- BAt \(x = -1\); it should be \(-7\)
- CAt \(x = 0\); it should be 0
- DAt \(x = 2\); it should be \(-13\)
Which of these points lies on the graph of \(y = \dfrac{12}{x}\)?
- A\((-3, -4)\)
- B\((-3, 4)\)
- C\((0, 12)\)
- D\((6, -2)\)
The graph of \(y = 6 - x^2\) is drawn, with the tangent to the curve at the point \((1, 5)\). The tangent crosses the axes at \((0, 7)\) and \((3.5, 0)\). Use the tangent to estimate the gradient of the curve at \(x = 1\).
- A\(-2\)
- B\(2\)
- C\(-0.5\)
- D\(5\)
The curve \(y = a \times b^{x}\), where \(b > 0\), passes through the points \((0, 3)\) and \((2, 12)\). Find the value of \(y\) when \(x = 3\).
- A\(24\)
- B\(48\)
- C\(192\)
- D\(16.5\)
Quiz key
\((-3)^2 + 3(-3) - 4 = 9 - 9 - 4 = -4\). The answer \(-22\) comes from typing \(-3^2\) without brackets, which a calculator reads as \(-9\). The answer 14 treats \(3 \times (-3)\) as \(+9\), and 0 forgets the \(-4\).
Two branches approaching the axes make a reciprocal graph \(y = \dfrac{a}{x}\). When \(x = 1\), \(y = -4\), and when \(x = -4\), \(y = 1\), so the branches are in the second and fourth quadrants and \(a = -4\). The tempting \(y = \dfrac{4}{x}\) has its branches in the first and third quadrants. \(y = -4x\) is a straight line and \(y = x^2 - 4\) is a U-shaped parabola.
\(2 \times (-2)^2 - 5 = 2 \times 4 - 5 = 3\). The student typed \(2 \times -2^2 - 5 = -8 - 5 = -13\). Check the others: \(2(-1)^2 - 5 = -3\), \(2(0)^2 - 5 = -5\), \(2(2)^2 - 5 = 3\), all correct. The values should be symmetrical, 3, \(-3\), \(-5\), \(-3\), 3, which shows at once that \(-13\) is the odd one out. Changing the value at \(x = 2\) to \(-13\) copies the mistake instead of fixing it.
\(\dfrac{12}{-3} = -4\), so \((-3, -4)\) is on the curve. \((-3, 4)\) has the wrong sign: a negative divided by a positive is negative. \((0, 12)\) is impossible because you cannot divide by 0, so the graph never meets the \(y\)-axis. For \(x = 6\), \(y = 2\), not \(-2\).
Gradient of the tangent \(= \dfrac{0 - 7}{3.5 - 0} = \dfrac{-7}{3.5} = -2\). The curve is going downhill at \(x = 1\), so the gradient must be negative; 2 drops the sign. \(-0.5\) divides the change in \(x\) by the change in \(y\), upside down. 5 is the \(y\)-coordinate of the point, not its gradient.
At \(x = 0\): \(a \times b^0 = a = 3\). At \(x = 2\): \(3b^2 = 12\), so \(b^2 = 4\) and \(b = 2\). Then \(y = 3 \times 2^3 = 24\). The answer 192 uses \(b = 4\), forgetting that \(b\) is squared; 48 multiplies by 4 for one step instead of by 2. 16.5 assumes the graph is a straight line rising by 4.5 per unit, but an exponential multiplies by the same number each step.
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.