The shape of the lesson
By the end of the lesson, learners can
- Build the diagonal: Set the horizontal and vertical forces below to change the diagonal.
- Two ways to find it.
- Magnitude: R = √(x² + y²) for two perpendicular vectors.
- Direction: θ = tan⁻¹(y ÷ x), and state which reference it is measured from.
- Always give direction: a resultant is a vector, so size alone is incomplete.
One picture every idea returns to
Two perpendicular vectors combine into a single resultant. Its magnitude is found from Pythagoras, R = √(x² + y²), and its direction from trigonometry, θ = tan⁻¹(y ÷ x). It can also be found by an accurate scale drawing.
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, Two pulls become one.. A boat pushed east by its engine and north by the current does not travel along either arrow. | Whole class, then pairs |
| 5 to 17 min | Teach and check: Build the diagonal | Set the horizontal and vertical forces below to change the diagonal. Project tab 3 of the lab (Resultant vectors) and let a student drive it while the class predicts. | Whole class, then pairs |
| 17 to 30 min | Teach and check: Two ways to find it | Two ways to find it. | 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, 16 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 Two ways to find it, 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 |
|---|---|
| A 3 N force and a 4 N force at right angles do not give 7 N. | Perpendicular vectors add by Pythagoras, not by simple addition, so the resultant is √(3² + 4²) = 5 N. Adding the magnitudes only works when the vectors point the same way; subtract them only when they point in exactly opposite directions. |
Resultant of perpendicular vectors: worksheet
6 questions, 16 marks. Written in the style of the papers; no past paper question is reproduced.
Two forces act on a point at right angles: 3.0 N pointing east and 4.0 N pointing north. Calculate the magnitude of the resultant force.
A horizontal force of 12 N and a vertical force of 5.0 N act on an object at right angles. Find the magnitude and direction of the resultant.
A student adds a 3 N force and a 4 N force that act at right angles and writes "resultant = 7 N." Explain why this is wrong and state the correct magnitude.
Describe how to find the resultant of two perpendicular forces using an accurate scale drawing.
A boat's engine drives it east at 8.0 m/s. A current carries it north at 6.0 m/s. Find the magnitude and direction of the boat's resultant velocity.
A question asks for the resultant of two perpendicular forces. A student correctly calculates 10 N but writes only "10 N" as the final answer. State what is missing and why it costs a mark.
Answer key and marking notes
Pythagoras.
R = √(3.0² + 4.0²) = √(9 + 16) = √25R = 5.0 N
Magnitude.
R = √(12² + 5.0²) = √(144 + 25) = √169R = 13 N
Direction.
θ = tan⁻¹(5.0 ÷ 12) = 23°above the horizontal (12 N) force. ✓
- The forces are perpendicular, so they combine by Pythagoras, not by simple addition. ✓
- R = √(3² + 4²) = 5 N. ✓ (adding to 7 N would only be right if they pointed the same way)
- Choose a scale (for example 1 cm to 1 N) and draw the two forces accurately at right angles. ✓
- Complete the rectangle and draw the diagonal from the starting point. ✓
- Measure the diagonal's length (convert back using the scale) for the magnitude, and measure its angle with a protractor for the direction. ✓
Magnitude.
v = √(8.0² + 6.0²) = √(64 + 36) = √100v = 10 m/s
Direction.
θ = tan⁻¹(6.0 ÷ 8.0) = 37°north of east. So 10 m/s at 37° north of east. ✓
- The direction is missing (for example the angle to one of the forces). ✓
- A resultant is a vector, so it is only fully described by both a magnitude and a direction. ✓
1 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.
Two forces act at a point and are at right angles: 3.0 N due east and 4.0 N due north. What is the magnitude of the resultant force?
- A1.0 N
- B5.0 N
- C7.0 N
- D12 N
Quiz key
resultant = √(3² + 4²) = √25 = 5.0 N. Option C (7.0 N) adds them as if they were parallel; A subtracts them; D multiplies them.
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.