The shape of the lesson
By the end of the lesson, learners can
- Conversion graphs: In the lab's conversion tab, choose a pair of units (miles and kilometres, pounds and dollars, Celsius and Fahrenheit).
- Distance-time graphs: In the distance-time tab, build a journey from stages and watch the graph draw itself.
- Speed-time graphs: (Extended) Open the speed-time tab and drag the corners of the graph.
- Curves and tangents: (Extended) When the graph is curved, the rate of change is different at every point.
- Axes: read the quantity, the units and the scale on each axis.
- Gradient: vertical change ÷ horizontal change, with units (km/h, m/s2, $ per £).
- Units: change minutes to hours before working out km/h.
- Area (Extended): split the area under a speed-time graph into triangles, rectangles and trapezia.
- Curves (Extended): draw a tangent and use two far-apart points on it.
One picture every idea returns to
The gradient of a graph is the rate of change of the vertical quantity per unit of the horizontal one. Distance-time: gradient = speed. Speed-time (Extended): gradient = acceleration, and area under the graph = distance travelled.
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, The gradient is a rate.. Real graphs tell stories: how many kilometres in a mile, when a cyclist stopped for a rest, how hard a car braked. 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: Conversion graphs | In the lab's conversion tab, choose a pair of units (miles and kilometres, pounds and dollars, Celsius and Fahrenheit). Project tab 7 of the lab (Graphs of functions lab, conversion) and let a student drive it while the class predicts. Model the worked example Miles and kilometres on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Distance-time graphs | In the distance-time tab, build a journey from stages and watch the graph draw itself. Project tab 5 of the lab (Graphs of functions lab, distance-time) and let a student drive it while the class predicts. Model the worked example Speed of one stage on the board, one line at a time. Your turn: A train travels 45 km in 30 minutes, stops for 15 minutes, then travels a further 30 km in 20 minutes. Find the speed on each moving stage and the average speed for the whole journey. | Pairs |
| 17 to 23 min | Teach and check: Speed-time graphs | (Extended) Open the speed-time tab and drag the corners of the graph. Project tab 6 of the lab (Graphs of functions lab, speed-time) and let a student drive it while the class predicts. Model the worked example A car journey (Extended) on the board, one line at a time. Your turn: (Extended) A cyclist's speed rises uniformly from 4 m/s to 10 m/s in 3 s. Find the acceleration and the distance travelled in that time. | Pairs |
| 23 to 30 min | Teach and check: Curves and tangents | (Extended) When the graph is curved, the rate of change is different at every point. Model the worked example Speed at an instant (Extended) on the board, one line at a time. | Mix-Pair-Share |
| 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, 18 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 Curves and tangents, 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.
Mix-Pair-Share
Students mix around the room until you call pair; they pair with the nearest classmate, discuss the prompt, then mix again. Three prompts of rising difficulty, three different partners.
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 |
|---|---|
| Read the scale on each axis before you read off a value. | On a graph where one small square is 2 units, a point three squares along is 6, not 3. |
| Time read in minutes must be changed to hours for km/h: 40 minutes is \(\dfrac{2}{3}\) h, not 0.4 h. | And the average speed is NOT the mean of the stage speeds: \(\dfrac{18 + 24}{2} = 21\) is wrong because the stages last different times and the stop counts. |
| Check which graph you have. | On a distance-time graph the gradient is speed; on a speed-time graph the gradient is acceleration and a flat line means steady speed. Using the wrong meaning is the commonest lost mark here. |
| Do not use two points on the curve: that gives the AVERAGE speed between them, not the speed at one instant. | Here the average speed from 0 to 4 s is \(16 \div 4 = 4\) m/s, only half the speed at \(t = 4\). |
Conversion, distance-time and speed-time graphs: worksheet
6 questions, 18 marks. Written in the style of the papers; no past paper question is reproduced.
A conversion graph for pounds (£) and dollars ($) is a straight line through \((0, 0)\) and (£80, $100).
(a) Convert £24 to dollars. (b) Convert $65 to pounds. (c) What does the gradient represent?
Describe the motion shown on a distance-time graph by (a) a horizontal line, (b) a straight line steeper than the one before it.
A distance-time graph shows Mia walking 2.4 km in 36 minutes at a steady speed. Find her speed in km/h.
(Extended) A car accelerates uniformly from 5 m/s to 25 m/s in 8 seconds. Find (a) its acceleration, (b) the distance it travels in these 8 seconds.
(Extended) An object moves so that its distance after \(t\) seconds is \(d = t^2\) metres. A tangent to the distance-time curve at \(t = 3\) passes through \((1.5, 0)\) and \((5, 21)\).
(a) Estimate the speed at \(t = 3\). (b) Find the average speed for the first 3 seconds and explain why it is different.
Leila leaves home at 09 00. She drives at 60 km/h for 45 minutes, stops for 30 minutes, then drives a further 30 km in 20 minutes.
(a) How far from home is she after the first stage? (b) At what time does she arrive? (c) Find her speed on the last stage. (d) Find her average speed for the whole journey. (e) Describe how the distance-time graph of the journey would look.
Answer key and marking notes
- Gradient \(= 100 \div 80 = 1.25\), so $ = 1.25 × £.
- (a) \(24 \times 1.25 = 30\), so $30
- (b) \(65 \div 1.25 = 52\), so £52
- (c) The exchange rate: 1.25 dollars for each pound.
- (a) Stationary (not moving).
- (b) Moving at a constant, faster speed.
- 36 min \(= \dfrac{36}{60} = 0.6\) h.
- Speed \(= 2.4 \div 0.6 = 4\) km/h.
- (a) \(\dfrac{25 - 5}{8} = 2.5\) m/s2.
- (b) Trapezium area \(= \dfrac{5 + 25}{2} \times 8 = 120\) m.
- (a) Gradient of tangent \(= \dfrac{21 - 0}{5 - 1.5} = \dfrac{21}{3.5} = 6\) m/s.
- (b) At \(t = 3\), \(d = 9\), so the average speed is \(9 \div 3 = 3\) m/s.
- The object is speeding up, so its speed at \(t = 3\) is greater than its average over the first 3 seconds.
- (a) \(60 \times 0.75 = 45\) km.
- (b) \(45 + 30 + 20 = 95\) min after 09 00, so 10 35.
- (c) \(30 \div \dfrac{20}{60} = 90\) km/h.
- (d) Total 75 km in \(\dfrac{95}{60}\) h: \(75 \div \dfrac{95}{60} = 47.4\) km/h (3 s.f.).
- (e) A straight line from (09 00, 0) to (09 45, 45), horizontal to (10 15, 45), then a steeper line to (10 35, 75).
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 conversion graph between pounds (£) and dollars ($) is a straight line through \((0, 0)\) and (£50, $65). Use it to convert $91 to pounds.
- A£70
- B£118.30
- C£41
- D£65
The distance-time graph shows a walk from home and back. Find the walker's speed, in km/h, on the journey home.
- A6 km/h
- B4 km/h
- C0.1 km/h
- D10 km/h
Use the distance-time graph to find the average speed for the whole walk, in km/h.
- A4 km/h
- B5 km/h
- C4.8 km/h
- D2 km/h
The speed-time graph shows a train journey between two stations. Find the acceleration in the first 30 seconds.
- A0.8 m/s2
- B1.25 m/s2
- C0.6 m/s2
- D360 m/s2
Use the speed-time graph to find the total distance the train travels between the two stations.
- A2280 m
- B3120 m
- C2760 m
- D1440 m
A car's speed-time graph is a curve. A tangent drawn to the curve at \(t = 5\) s passes through the points \((1, 2)\) and \((9, 18)\), with speed in m/s. Estimate the acceleration of the car at \(t = 5\) s.
- A2 m/s2
- B0.5 m/s2
- C2.25 m/s2
- D3.6 m/s2
Quiz key
Gradient \(= \dfrac{65}{50} = 1.3\), so $1.30 for every £1. Dollars to pounds: \(91 \div 1.3 = 70\), so £70. £118.30 multiplies by 1.3, which converts pounds to dollars, the wrong way. Check: £70 \(\times 1.3\) gives $91.
The return section goes from 120 min to 180 min and covers 6 km. Time 60 min \(= 1\) h, so speed \(= 6 \div 1 = 6\) km/h. 4 km/h is the speed on the way OUT (6 km in 1.5 h); 0.1 is 6 km \(\div\) 60 min, which is km per minute, not km/h.
Total distance: 6 km out and 6 km back, 12 km. Total time: 180 min \(= 3\) h, including the 30-minute rest. Average speed \(= 12 \div 3 = 4\) km/h. 5 km/h is the mean of the stage speeds 4 and 6, which ignores how long each stage lasted; 4.8 leaves out the rest (\(12 \div 2.5\)); 2 uses only the one-way distance.
The speed rises from 0 to 24 m/s in 30 s, so acceleration \(= \dfrac{24}{30} = 0.8\) m/s2. 1.25 is \(\dfrac{30}{24}\), the gradient upside down; 0.6 is the deceleration at the end (\(\dfrac{24}{40}\)); 360 is the AREA of the first triangle, which is a distance.
Triangle \(\dfrac{1}{2} \times 30 \times 24 = 360\), rectangle \(60 \times 24 = 1440\), triangle \(\dfrac{1}{2} \times 40 \times 24 = 480\). Total \(360 + 1440 + 480 = 2280\) m. 3120 is \(130 \times 24\), treating the whole shape as a rectangle; 2760 forgets the \(\dfrac{1}{2}\) on the last triangle; 1440 counts only the steady-speed part.
Gradient of the tangent \(= \dfrac{18 - 2}{9 - 1} = \dfrac{16}{8} = 2\) m/s2. 0.5 is \(\dfrac{8}{16}\), change in time over change in speed; 2.25 is \(\dfrac{18}{8}\), forgetting to subtract the 2; 3.6 is \(18 \div 5\), which is not a gradient at all.
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.