At a glance. One lesson of about 60 minutes. It uses column vectors, Pythagoras and trigonometry. Students find components from a bearing, resultants of velocities and forces, and a heading to cross a current.
Calculator: scientific calculator. Scientific calculator: sines, cosines and inverse tangents of non-special angles. Students should set degree mode and keep full values until the end.
What students will learn
Know
- A bearing is clockwise from north, three digits.
- East = r sin t and north = r cos t.
- The resultant is the vector sum of the components.
Understand
- Why a bearing swaps sine and cosine compared with a graph.
- Why a heading is not the path over the ground.
- Why steering across a current needs the speed through water as the hypotenuse.
Do
- Resolve a vector given by a bearing into components.
- Find the speed and bearing of a resultant.
- Find the heading that cancels a cross-current.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: bearing, components and resultant (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe how the current changes the speed and the bearing of the resultant.
- Criterion C. Communicating: labelled vector diagrams with a compass direction and units.
- Criterion D. Applying mathematics in real-life contexts: ferries, aircraft and forces (hook and questions 2 to 7).
The one thing to do with the figure
Ask for the prediction first. Then set no current and compare with a current of 4, with the boat pointing north, and ask for the new bearing before it is read.
Ask: “When the boat points west and the current is east, what happens to the speed?” Listen for: the current opposes the heading, so the resultant is slower and may point the other way.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the ferry captain; collect predictions. |
| 12 min | Figure, key idea and warning card; set up a boat with and without a current. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask where a bearing is measured from. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 13; (5, −6); 5. |
|---|---|
| Hook | The captain should steer a little west of north, at about 17.5 degrees, so that the boat's westward component cancels the 3 km/h current. Pointing straight north lands the boat downstream (east) of the jetty. |
| Worked examples | 17.3 east and 10.0 north; 13 km/h on 22.6; 25 N on 73.7. |
| Faded example | 206.2 km/h on bearing 076.0. |
| 1. Direction of a resultant | D. A measures from east; B is one force alone; C turns south of east. |
| 2. Distance from components | 130. 5, 12, 13 scaled by 10. |
| 3. Forces along one line | 3. 12 − 9, towards the west. |
| 4. Bearing from components | 61.9. tan−1(15 ÷ 8). |
| 5. Bearing in the fourth quadrant | 302.0. 360 − 58.0. |
| 6. Bearing of a path from a heading | 67.4. East 10.79, north 4.5. |
| 7. Steering across a current | 19.5. sin−1(1 ÷ 3). |
| 8. Heading and path in a current | A. B adds the speeds and keeps the heading; C turns north; D adds the angle from south. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Measured the angle from east, gave one force alone, or turned south of east. | Sketch the two forces from a point, mark north, and ask which angle is the bearing. |
| 170 or 70 in question 2 | Added the legs, or subtracted them. | Draw the two legs and ask which side is the distance from the start. |
| 21 or 15 in question 3 | Added the sizes, or used Pythagoras for forces on one line. | Ask: are the forces at right angles? Sketch them from one point, with east as positive. |
| 28.1 or 118.1 in question 4 | Used the angle from east, or the wrong quadrant. | Sketch the point and mark north; ask which angle is a bearing. |
| 58.0 or 32.0 in question 5 | Gave the acute angle from north without the quadrant, or the angle from west. | Sketch the path, mark north, and turn clockwise from north until you reach the path. |
| 60 or 22.6 in question 6 | Gave the heading, or measured the angle from east. | Ask: does the current change where the boat points, or where it goes? Then sketch and mark north. |
| 18.4 or 70.5 in question 7 | Used tangent with the boat's speed as an adjacent side, or took the complement. | Draw the triangle with the speed through water as the hypotenuse. |
| B, C or D in question 8 | Added the speeds and kept the heading, turned the wrong way, or added the wrong angle to 090. | Sketch east and south from a point, mark north, and ask where the path lies. |
Questions worth asking
- Why is the east component r sin t for a bearing? Listen for: the angle is measured from north, so the side opposite to it is the east component.
- Does the boat go where it points? Listen for: no, the current adds a component, so the path is different from the heading.
- What is the least and the greatest resultant of two forces? Listen for: the difference when opposite, the sum when in the same direction.
- Why must the speed through the water be the hypotenuse when steering across? Listen for: it is the whole speed of the boat in the water, of which one part cancels the current.
Differentiation
A way in for anyone stuck
- Give a compass diagram with north marked and a bearing triangle template for each question.
- Provide a table with columns for east, north, speed and bearing for each vector in a problem.
- Give the frame: “The east component is … and the north component is …, so the speed is …”
Stretch
- A boat can travel at 10 km/h through still water and wants to cross a river 120 m wide flowing at 3 km/h as fast as possible. How long does it take, and how far downstream does it land?
- Two forces of 20 N and 30 N have a resultant of 40 N. Find the angle between them.
Students learning in English as an additional language
“Bearing” means a direction measured as an angle from north, clockwise. “Current” is the movement of the water. “Heading” is the direction the boat points. “Resultant” is the single vector with the same effect as the others together.
| Word | Plain meaning | Example |
|---|---|---|
| bearing | an angle clockwise from north | 090 for east |
| component | the part of a vector in one direction | the east part |
| resultant | the single vector that has the same effect as several | the actual path |
| heading | the direction a vehicle points | the way the bow faces |
| current | the steady movement of water or air | a river flowing east |
Sentence starters for explaining aloud or in writing:
- The east component is … and the north component is ….
- The resultant has size … and bearing ….
- The boat must steer … degrees west of north because …
Two things not to say
“The boat goes where it points.” Say instead: the boat's path is the resultant of its velocity through the water and the current.
“Use tan of north over east.” Say instead: for a bearing, which is measured from north, the east component goes on top.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- FindA boat goes 6 km east and 8 km north. How far is it from the start?
10 km. - FindForces of 5 N north and 12 N east. What is the resultant?
13 N.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA plane flies 100 km on bearing 090 and then 200 km on bearing 000. What is its bearing from the start, to the nearest degree?
027. - Earlier work (lesson 7)Show that (2, 3) and (6, 9) are parallel.
(6, 9) = 3 × (2, 3).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA boat's speed through still water is 5 km/h and the current is 5 km/h. Can it go straight across? Explain.
No: it would need sin t = 1, which means it points directly upstream and makes no crossing. - Working backwardsTwo perpendicular forces have a resultant of 25 N and one of them is 15 N. Find the other force.
20 N, since 252 − 152 = 400.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
A scientific calculator is used in several questions; set it to degrees.