At a glance. One lesson of about 60 minutes. It assumes lessons 1 and 5, Pythagoras and bearings as clockwise angles from north. Students use the cosine rule for a side and for an angle, the area formula, bearings with a turn, and decide between the sine and cosine rules. Radians are not used.
Calculator: scientific calculator. A scientific calculator is needed for cosines and inverse cosines. Working out the whole of 2ab cos C before subtracting, and keeping unrounded values, are the two practical points.
What students will learn
Know
- c² = a² + b² − 2ab cos C, with C between a and b.
- Area = ½ ab sin C.
- A bearing is measured clockwise from north.
Understand
- The cosine rule is Pythagoras with a correction term from the height.
- An obtuse angle makes the correction positive.
- The inverse cosine gives the right angle with no second case.
Do
- Find a side or an angle using the cosine rule.
- Find an area from two sides and an angle.
- Solve bearing problems with a turn.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the rule, its rearrangement, the area formula and bearings (check and questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask students to explain where the correction term comes from.
- Criterion C. Communicating: a labelled sketch, the rule used and its substitution, with units.
- Criterion D. Applying mathematics in real-life contexts: ships and a field, where distances cannot be measured directly.
The one thing to do with the figure
Make students commit before the height appears. In the first figure, ask “What will c be at C = 90°?” and take values before the slider moves.
In the second figure, ask “For which bearing of Q is PQ largest?” before moving the slider, and ask why it cannot be more than 21 km.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: two ships. Collect predictions of more or less. |
| 10 min | First figure and key idea: the derivation. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose B or C, ask which side is opposite the angle. |
| 18 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 | 12; 15; −0.5. |
|---|---|
| Hook | Less than 12 + 9 = 21 for every bearing but one, where the ships are in a line on opposite sides. Whether it is more than 12 depends on the angle. |
| Worked examples | c = 7.9; d = 26.3 km. |
| Faded example | C = 97.9°. |
| 1. Setting up the cosine rule for an angle | D. A adds all three; B is the angle at P, opposite the longest side; C is the angle at R. |
| 2. Cosine rule for a side | 13.1. √170.5. |
| 3. Area from two sides and an angle | 48.3. 65 × 0.7431. |
| 4. Cosine rule for an angle | 115.4. cos⁻¹(−0.4286). |
| 5. A journey with a turn | 20.5. √419.8. |
| 6. Area from three sides | 53.4. 54 × sin 98.3°. |
| 7. Angle from an area | 51.1. sin⁻¹(0.778). |
| 8. Choosing the rule | C. The angle at A is between the two known sides; A, B and D each give a side with its opposite angle. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 (the check) | Adds all the squares, or subtracts the wrong side (often the longest, from habit). | Ask: which side is opposite the angle? That one is subtracted. |
| 16.6, 170.5 or 15 in question 2 | Uses Pythagoras, forgets the root, or leaves out the 2. | Write c squared on its own line before the root. |
| 96.6, 43.5 or 65 in question 3 | Leaves out the half, uses the cosine, or leaves out the sine. | Say the formula aloud: half, a, b, sine. |
| 64.6, −0.43 or 149 in question 4 | Drops the sign, stops at the cosine, or leaves out the 2. | Write the top of the fraction on its own first. |
| 28.8, 34.5 or 25 in question 5 | Uses the turn, uses the bearing, or Pythagoras. | Sketch the journey and mark the angle inside the triangle. |
| 54, 106.9 or 95.0 in question 6 | Takes sine as 1, leaves out the half, or uses an angle not between the sides. | Mark the two sides and the angle between them. |
| 128.9, 38.9 or 22.9 in question 7 | Gives the obtuse angle, the complement, or leaves out the 2. | Write half, 120, 150, sin P equals 7000. |
| A, B or D in question 8 | Chooses the rule from the numbers, not from what is known. | Ask: is there a side and its opposite angle? |
Questions worth asking
- Why is Pythagoras a special case of the cosine rule? cos 90° = 0, so the correction term is zero.
- Why is the third side longer than Pythagoras gives for an obtuse angle? cos is negative, so the correction is added.
- How do you find the angle inside a triangle at a turn? 180° minus the angle through which the boat turned.
- How do you know the largest angle is obtuse without a calculator? If the square of the longest side is bigger than the sum of the other two squares.
Differentiation
A way in for anyone stuck
- Give a labelled triangle template with the cosine rule written beside it and the included angle marked.
- Use a bearing diagram with north lines at each point already drawn.
- Provide a checklist: sketch; mark the known; choose the rule; substitute; check the size.
Stretch
- Prove the cosine rule for an obtuse angle C, using the foot of the height outside the triangle.
- A triangle has sides 7, 9 and 12. Find all three angles, and check that they add to 180°.
Students learning in English as an additional language
“Bearing” is a direction measured clockwise from north, with three digits. “Included angle” is the angle between two given sides. “Obtuse” means bigger than 90 degrees. “Turn” is a change of direction.
| Word | Plain meaning | Example |
|---|---|---|
| cosine rule | a rule linking the three sides and one angle of any triangle | c² = a² + b² − 2ab cos C |
| bearing | a direction measured clockwise from north, with three digits | 090° is due east |
| included angle | the angle between two given sides | the angle at C between a and b |
| area formula | half the product of two sides and the sine of the angle between them | ½ ab sin C |
| obtuse | an angle between 90° and 180° | an angle of 130° |
Sentence starters for explaining aloud or in writing:
- I know …, so I use the … rule.
- The angle inside the triangle is … because ….
- The cosine is negative, so ….
Two things not to say
“Use sin, cos or tan, whichever you like.” The rule depends on what is known.
“Bearings are measured from the east.” They are measured clockwise from north.
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.
- BearingState the angle between the bearings 070° and 135° from one point.
65°. - AreaTwo sides are 6 and 8 with 90° between them. Find the area.
24. ½ × 6 × 8.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextTwo wires of length 5 m and 8 m make 35° at the top of a mast. How far apart are their lower ends?
c² = 89 − 80 cos 35° = 23.5, so 4.8 m. - Earlier work (Lesson 5)In triangle ABC, A = 50°, B = 60° and a = 10. Find b.
11.3. 10 × sin 60° ÷ sin 50°.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningSides 5, 12 and 13: what is the largest angle, and how do you know without the calculator?
90°, because 25 + 144 = 169. - Working backwardsTwo sides are 6 and 8 and the third is 12. Is the angle between the first two acute or obtuse?
Obtuse, because 36 + 64 = 100 is less than 144.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each figure, “Show me the steps” gives the same states in words.
No device? Give each pair a protractor and a ruler: draw the two courses to scale and measure the distance between the ships.