The shape of the lesson
By the end of the lesson, learners can
- Gradient and area, side by side: Set an initial velocity and an acceleration, then run the trolley.
- Reading the two graphs: Two graphs, four quantities.
- What the shapes mean.
- s-t graph: gradient is velocity.
- v-t graph: gradient is acceleration, area beneath is displacement.
- Vectors: a sign carries the direction; round trips give zero displacement.
One picture every idea returns to
Displacement, velocity and acceleration are vectors; distance and speed are their scalar cousins. On a displacement-time graph the gradient is the velocity. On a velocity-time graph the gradient is the acceleration and the area beneath the line is the displacement. Read the graph correctly and the numbers fall out.
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 story a graph tells.. Motion is the first place where the difference between a scalar and a vector earns its keep. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Gradient and area, side by side | Set an initial velocity and an acceleration, then run the trolley. Project tab 1 of the lab (Linear motion and graphs) and let a student drive it while the class predicts. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Reading the two graphs | Two graphs, four quantities. | Pairs |
| 21 to 30 min | Teach and check: What the shapes mean | What the shapes mean. | 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, 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 What the shapes mean, 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 |
|---|---|
| Match the operation to the right graph. | The area that gives displacement is the area under the velocity-time graph, not under the displacement-time graph, which has no standard meaning. Likewise the gradient of a displacement-time graph is velocity, while the gradient of a velocity-time graph is acceleration. A flat (horizontal) line on a displacement-time graph means the object is stationary, not moving at constant speed. |
Linear motion and graphs: worksheet
6 questions, 15 marks. Written in the style of the papers; no past paper question is reproduced.
A walker travels 6.0 km due north, then 8.0 km due east. State the total distance walked and find the magnitude of the displacement from the start.
A velocity-time graph for a train is a horizontal line at 25 m s⁻¹ lasting 40 s. Use the area to find the distance travelled, and state the acceleration.
A velocity-time graph rises in a straight line from 0 to 18 m s⁻¹ over 6.0 s. Find the acceleration and the distance travelled.
Describe the motion shown by a displacement-time graph that is a horizontal straight line, and then by one that is a straight line sloping downward.
A car accelerates uniformly from 8.0 m s⁻¹ to 20 m s⁻¹ in 4.0 s. Sketch the shape of its velocity-time graph in words, and use the area to find the distance travelled.
A swimmer covers 50 m to the far wall in 25 s, then 50 m back in 25 s, finishing where they started. Find the average speed and the average velocity for the whole swim, and explain why they differ.
Answer key and marking notes
- Distance is the total path: 6.0 + 8.0 = 14 km. ✓
- Displacement is the straight line: √(6.0² + 8.0²) = √100 = 10 km. ✓
- Distance = area = 25 × 40 = 1000 m. ✓
- The line is horizontal, so the gradient and hence the acceleration is zero. ✓
- Acceleration = gradient = 18 / 6.0 = 3.0 m s⁻². ✓
- Distance = area of triangle = ½ × 6.0 × 18 = 54 m. ✓
- Horizontal line: the displacement is not changing, so the object is stationary. ✓
- Straight line sloping downward: constant negative gradient, so constant velocity in the negative direction (moving back toward the origin). ✓
- The graph is a straight line sloping upward from 8.0 to 20 m s⁻¹ over 4.0 s. ✓
- Area = a rectangle (8.0 × 4.0 = 32 m) plus a triangle (½ × 4.0 × 12 = 24 m). ✓
- Total distance = 32 + 24 = 56 m. (Equivalently, average velocity 14 m s⁻¹ × 4.0 s.) ✓
- Average speed = total distance / time = 100 / 50 = 2.0 m s⁻¹. ✓
- Average velocity = displacement / time = 0 / 50 = 0 m s⁻¹. ✓
- They differ because speed uses the path length (a scalar) while velocity uses displacement (a vector), which is zero for a round trip. ✓
17 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 of these quantities is a scalar?
- ADistance
- BDisplacement
- CVelocity
- DAcceleration
The area under an acceleration-time graph represents the:
- Adisplacement
- Bacceleration
- Cchange in velocity
- Ddistance
A horizontal straight line on a velocity-time graph tells you the object has:
- Aconstant acceleration
- Bconstant velocity and zero acceleration
- Cincreasing velocity
- Dzero velocity
The gradient of a velocity-time graph gives the:
- Adisplacement
- Bdistance
- Cspeed
- Dacceleration
A displacement-time graph that is a straight line through the origin sloping upward represents motion at:
- Aconstant velocity
- Bconstant acceleration
- Cincreasing acceleration
- Drest
Average velocity is defined as:
- Adistance divided by time
- Bchange in speed divided by time
- Cdisplacement divided by time
- Dspeed times time
A car covers 120 m in 10 s and then 80 m in 10 s, all in a straight line in the same direction. Its average speed is:
- A20 m s⁻¹
- B10 m s⁻¹
- C4 m s⁻¹
- D2 m s⁻¹
A velocity-time graph rises in a straight line from 0 to 12 m s⁻¹ over 4.0 s. The distance travelled is:
- A48 m
- B3.0 m
- C12 m
- D24 m
A negative gradient on a displacement-time graph means the object is:
- Amoving in the negative direction
- Bspeeding up
- Cstationary
- Daccelerating
The instantaneous velocity at a point on a curved displacement-time graph is found from the:
- Aarea under the curve
- Bvalue of the displacement
- Cgradient of the tangent at that point
- Dtotal time
On a velocity-time graph, an area that lies below the time axis represents:
- Aa negative time
- Bdisplacement in the negative direction
- Czero displacement
- Dan increase in speed
An object moving with constant velocity has an acceleration of:
- A9.8 m s⁻¹
- Ba constant non-zero value
- Cits velocity divided by time
- Dzero
For any journey, the distance travelled is always:
- Agreater than or equal to the magnitude of the displacement
- Bless than the displacement
- Cequal to the displacement
- Dequal to the average velocity
Speed is best described as the:
- Adirection of the velocity
- Brate of change of velocity
- Cmagnitude of the velocity
- Ddisplacement per unit time
A velocity-time line that curves and becomes steadily steeper shows:
- Aconstant acceleration
- Bincreasing acceleration
- Cdecreasing velocity
- Dzero acceleration
A displacement-time graph is a horizontal straight line for 10 s. During this time the object is:
- Amoving at constant speed
- Baccelerating
- Cdecelerating
- Dat rest
The SI unit of acceleration is:
- Am s⁻²
- Bm s⁻¹
- Cm² s⁻¹
- Dm s
Quiz key
Distance is a scalar. Displacement, velocity and acceleration all carry a direction.
Acceleration multiplied by time is a change in velocity, so the area under an a-t graph is the change in velocity.
A horizontal v-t line means the velocity is not changing, so the acceleration (its gradient) is zero.
The gradient of a v-t graph is the rate of change of velocity, which is the acceleration.
A straight sloping s-t line has constant gradient, so the velocity is constant.
Average velocity is displacement divided by the time taken; using distance instead gives average speed.
Average speed = 200 m / 20 s = 10 m s⁻¹.
Distance = area = ½ × 4.0 × 12 = 24 m.
A negative gradient means a negative velocity, so the object is moving back toward the origin.
Instantaneous velocity is the gradient of the tangent to the s-t curve at that instant.
Below the axis the velocity is negative, so that area is displacement in the negative direction.
Constant velocity means the velocity is not changing, so the acceleration is zero.
Distance is the total path length, which can never be less than the straight-line displacement.
Speed is the magnitude of velocity, a scalar with no direction.
A steepening v-t curve has an increasing gradient, so the acceleration is increasing.
A horizontal s-t line means the displacement is not changing, so the object is stationary.
Acceleration is change of velocity (m s⁻¹) per second, giving m s⁻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.