The unit opens with trigonometry that cannot be done in one triangle: two sightings of one tower give its height, and a journey of several legs is added up as east and north components. Then right-angled triangles are found inside solids, including the angle between a line and a plane and between two planes, and the angles 30°, 45° and 60° get exact values from two special triangles, with a proof that tan 15° is 2 minus root 3.
A sector becomes a fraction of a circle with an arc, an area, a perimeter and, by subtraction, a segment. The last three lessons prove the circle angle theorems in every case, run them backwards to decide when points lie on a circle, and show that crossing chords and a tangent with a secant keep the product of two lengths fixed.
A graphic display calculator (GDC) is optional all the way through. Lessons 1, 2, 4 and 6 have a closed “With a GDC” section to open if you have one; lessons 3, 5 and 7 are done by hand.
The lessons, in order
- Lesson 1 · Two sightings and bearings as components Height from two angles of elevation or depression, bearings split into components, distance and bearing of a journey, back bearings. Change the two angles of elevation and see the height and the distance to the foot, then set two legs of a journey and read the distance and bearing home. GDC optional
- Lesson 2 · Lines and planes in 3D Space diagonals, the angle between a line and a plane, the angle between two planes, and the pyramid's face angle against its edge angle. Change the height of a container and read the longest bar and its angle, then change a pyramid's base and height and compare a face's angle with an edge's. GDC optional
- Lesson 3 · Exact values and proof with surds Exact values of sine, cosine and tangent for 30, 45 and 60 degrees, proofs with surds, and tan 15 degrees from a constructed triangle. Change the size and angle of a special triangle and read its exact sides, then change a 30, 60, 90 triangle's size and see tan 15 degrees stay at 2 minus root 3. No calculator
- Lesson 4 · Arcs, sectors and segments Arc length, sector area and perimeter, working back to a radius or an angle, exact answers with π, and the area of a segment. Change a sector's radius and angle and read its arc, area and perimeter, then take the triangle away from a sector to read a segment's area. GDC optional
- Lesson 5 · The angle theorems, proved in every case The angle at the centre proved in the three cases, the angle in a semicircle, angles in the same segment, cyclic quadrilaterals and the alternate segment theorem. Walk a point right round a circle and see its angle on a chord switch from half the centre angle to its supplement, then fix the chord and watch the centre move in and out of the angle. No calculator
- Lesson 6 · Running the circle theorems backwards Converse of the angle in a semicircle, of angles in the same segment and of the cyclic quadrilateral theorem, with proofs, and using them to show points are concyclic. Move a point round and off a circle through two fixed points and watch the angle it sees, then see a right angle put its vertex on the circle with the hypotenuse as diameter. GDC optional
- Lesson 7 · Crossing chords, tangents and secants A tangent is perpendicular to the radius (proof by contradiction), intersecting chords, tangent and secant, and angle chasing with similar triangles. Swing a chord round a point inside a circle and see the product of its pieces stay fixed, then swing a secant from a point outside and compare it with the tangent. No calculator
Assessing this unit
Your teacher chooses which of these tasks to set, and when: nobody does all of them. Each sheet says where it fits in the unit, how long it takes and how your work is judged. To print a sheet, open it first and then print the page.
| Task | Criteria | When | Time |
|---|---|---|---|
| A1: Quick check: exact values and surds | A | After lesson 3 | 20 minutes |
| A2: End-of-unit test | A | End of unit, after lesson 7 | 50 minutes |
| B1: Polygons inside a circle | B | Mid-unit, after lesson 4 | 50 minutes |
| C1: Mali's proof | C | After lesson 5 | 35 minutes |
| D1: Survey boat and the fuel | D and C | End of unit, after lesson 7 | 50 minutes |
| D2: Turf and edging for a flower bed | D and C | End of unit, after lesson 7 | 50 minutes |
D1 and D2 are alternatives. Both assess criteria D and C, in two different real-life contexts: D1 is a journey and bearings (lesson 1), D2 is a circular segment (lessons 1 and 4). Set one, not both.
Task A1: Quick check: exact values and surds (criterion A)
When: After lesson 3. Time: 20 minutes. Calculator: no calculator. Criterion: A, knowing and understanding.
No calculator: every answer is exact, so leave surds as surds. Show your working: a correct method earns credit even if the final answer slips. Questions 6 and 7 are meant to make you think, so try something rather than leaving them blank.
Part 1. Exact values
- Find the exact value of sin 60° cos 30° − sin 30° cos 60°. (2 marks)
- A right-angled triangle has a hypotenuse of 12 cm and an angle of 30°. Find the exact lengths of the other two sides. (2 marks)
- Write 4 ÷ (√3 + 1) with a rational denominator, in its simplest form. (2 marks)
Part 2. Special triangles
- A right-angled isosceles triangle has a hypotenuse of 10 cm. Find the exact length of each short side and the exact area. (2 marks)
- An equilateral triangle has sides of 6 cm. Find its exact height and its exact area. (3 marks)
Part 3. Think it through
- Noor says: “sin 30° + sin 30° = sin 60°, because 30 + 30 = 60.” Use exact values to decide whether she is right. (2 marks)
- A rectangle has a shorter side of 5 cm. Its diagonal makes an angle of 30° with the longer side. Find the exact length of the diagonal and the exact area of the rectangle. (3 marks)
How your work is judged
| Level | Criterion A in task A1: what it looks like |
|---|---|
| 1–2 | You give some of the exact values correctly, for example the sides in question 2, but struggle to carry them through a calculation. |
| 3–4 | You answer most of questions 1 to 5 correctly using exact values, with some slips in surds or in rationalizing. |
| 5–6 | You answer questions 1 to 5 correctly with exact values and clear working, and make a good start on question 6 or 7. |
| 7–8 | You answer questions 1 to 5 correctly and solve question 6 or 7 with reasoning that shows your answer works. |
Task A2: End-of-unit test (criterion A)
When: End of unit, after lesson 7. Time: 50 minutes. Calculator: scientific calculator. Criterion: A, knowing and understanding.
Three sections, from familiar questions to ones you have not met in this form. Show your working everywhere: it earns credit even when an answer slips. Give lengths to 3 significant figures and angles to 1 decimal place unless the question asks for an exact value.
Section 1. Familiar questions
- A tower stands on level ground. Looking up at its top from a point A, the elevation is 25°. From a point B, 50 m nearer along the same line, the elevation is 40°. Find the height of the tower. (3 marks)
- A sector of a circle has radius 9 cm and angle 140°. Find the arc length, the area and the perimeter of the sector. (3 marks)
- O is the centre of a circle and A, B, C and D are on it. Angle AOB is 118°. C is on the major arc AB and D is on the minor arc AB. Find angle ACB and angle ADB, giving a reason for each. (3 marks)
Section 2. More complex questions
- A cuboid is 8 cm by 6 cm by 5 cm. Find the angle between its space diagonal and the base. (3 marks)
- Find the exact value of (sin 60° − cos 60°)2, giving your answer in the form a + b√3 with a and b fractions. (3 marks)
- Chords AB and CD of a circle cross at P, with AP = 6, PB = 4 and CP = 3. Find PD. Separately, a tangent from an outside point T has length 12 and a secant from T meets the circle at A then B, with TA = 8. Find AB. (4 marks)
- A chord subtends an angle of 90° at the centre of a circle of radius 10 cm. Find the exact area of the minor segment cut off by the chord, then give it to 3 significant figures. (3 marks)
Section 3. Unfamiliar questions
- A circle has radius 13 cm. Two parallel chords of lengths 24 cm and 10 cm lie on opposite sides of the centre. How far apart are they? Is there another possible distance if the chords are allowed on the same side? (4 marks)
- A ladder 5 m long leans against a vertical wall on level ground, and its foot slides slowly away from the wall. Its midpoint M is marked. Show that M stays the same distance from the corner where the wall meets the ground, and say what that distance is. (3 marks)
How your work is judged
| Level | Criterion A in task A2: what it looks like |
|---|---|
| 1–2 | You answer some Section 1 questions correctly, for example one length or one angle, with the method for finding the others unclear. |
| 3–4 | You answer most of Section 1 correctly and make progress in Section 2, for example finding the space diagonal or an arc. |
| 5–6 | You answer Section 1 and most of Section 2 correctly, with clear methods and exact values where asked, and make a start in Section 3. |
| 7–8 | You answer Sections 1 and 2 correctly and solve most of Section 3, explaining why your method works. |
Task B1: Polygons inside a circle (criterion B)
When: Mid-unit, after lesson 4. Time: 50 minutes. Calculator: scientific calculator. Criterion: B, investigating patterns.
A regular polygon with n sides is drawn so that all of its corners lie on a circle of radius 1 cm. Joining each corner to the centre cuts the polygon into n equal triangles. This task is about what happens to the polygon as n gets larger.
Part 1. Collect results
- Find a formula for the area of the polygon, and one for its perimeter, in terms of n. Each triangle has two sides of 1 cm. Show how you got them.
- Use your formulas to find the area and the perimeter for n = 3, 4, 6, 8 and 12. Record them, to 4 decimal places, in a table you design yourself.
Part 2. Describe what you find
- Compare each polygon with the circle it sits in. Choose your own way to measure how different they are, and add that measure to your table.
- Describe what you notice, as a rule another student could use. Say what you would expect for n = 24.
Part 3. Test your rule
- Find the area and the perimeter for n = 24 and n = 48. Did your rule predict them? Show how close you were.
- Use your rule, not the formulas, to predict the area for n = 96. Then check with the formula.
Part 4. Explain
- Is the area of the polygon ever larger than the area of the circle? Explain, using the diagram, why or why not.
- Does your rule work for the perimeter as well? Give evidence, and say how you decided what to test.
How your work is judged
| Level | Criterion B in task B1: what it looks like |
|---|---|
| 1–2 | With help, you work out some areas or perimeters, record them, and say something you notice about the numbers. |
| 3–4 | With help, you complete the table for the area and the perimeter and describe a pattern that fits most of it. |
| 5–6 | You organize your results in a clear table, describe a rule that fits the area and the perimeter, and use it to predict n = 24 and check it. |
| 7–8 | You organize your results, describe a general rule that fits all your findings, use it to make an accurate prediction for n = 96, and justify that the rule holds for the perimeter and why the area stays below the circle. |
Task C1: Mali's proof (criterion C)
When: After lesson 5. Time: 35 minutes. Calculator: no calculator. Criterion: C, communicating.
Mali missed lesson 5. She wrote her own working for two questions and brings it to you.
Mali’s working
(a) Prove that the angle at the centre of a circle is twice the angle at the circumference. Angle ACB = x. Angle AOB must be double angle ACB, so angle AOB = 2x. Triangle AOB is isosceles, so its other two angles are (180 − 2x) ÷ 2 = 90 − x. This is proved.
(b) PQRS is a cyclic quadrilateral. Angle P = 70°. Find angle Q. The angles of a cyclic quadrilateral add to 180°, so angle Q = 110°.
Write a one-page explanation for Mali. She should be able to follow it on her own, without asking anyone.
Part 1. Put it right
- Find what is wrong with each of Mali’s answers. For each, say what she did, why it does not work and what she should have done.
Part 2. Teach the idea
- Write a correct proof for the case where the centre O is inside triangle ABC. Draw and label a diagram, and give a reason for every line.
- Say what changes in your diagram and your proof when O is outside triangle ABC.
Part 3. Check your explanation
- Write one question about a cyclic quadrilateral that would catch out a student who makes Mali’s mistake in (b). Give its correct answer.
- Read your page as if you were Mali. Check that every angle you name has three letters or a clear label, that each step has its reason, and that you use the words centre, circumference, chord, isosceles and opposite correctly.
How your work is judged
| Level | Criterion C in task C1: what it looks like |
|---|---|
| 1–2 | Some of your explanation can be followed, but a step or a word is missing, and the diagram or the angle names are hard to follow. |
| 3–4 | You explain both of Mali’s answers with some correct words and angle names; your steps are in a sensible order, so Mali could follow most of it. |
| 5–6 | Each step of your proof has a reason and a clear angle name; your diagram matches your words, and Mali could follow all of it with little effort. |
| 7–8 | Your proof reads as a chain of reasons with no gaps, the diagram is labelled to match every line, the notation is always correct, and the page is ordered so Mali could learn from it alone. |
Task D1: Survey boat and the fuel (criteria D and C)
When: End of unit, after lesson 7. Time: 50 minutes. Calculator: scientific calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A survey boat leaves a harbour H and sails 12 km on a bearing of 060°. It then sails 9 km on a bearing of 150° to a buoy B. Its fuel tank held 32 litres when it left the harbour, and the boat uses 0.9 litres for each kilometre.
Part 1. Where is the boat?
- Draw the journey, and find how far east and how far north of the harbour the boat is when it reaches the buoy.
- Find the straight-line distance from the harbour to the buoy, and the bearing of the buoy from the harbour.
- State the bearing the boat would sail on, from the buoy, to go straight back to the harbour.
Part 2. Fuel
- The boat now sails straight back to the harbour. How many litres of fuel will the whole trip use, and will 32 litres be enough?
Part 3. Does it make sense?
- The 0.9 litres per kilometre is a rough figure: it could be anything from 0.85 to 0.95. Does your answer to question 4 still hold across that range?
- Write a short note to the harbour office saying what you recommend and why, so that someone who has not seen your working can understand it.
How your work is judged
| Level | Criterion D in task D1: what it looks like |
|---|---|
| 1–2 | You find some of the right numbers, for example how far east the boat is, but do not reach the distance or the fuel. |
| 3–4 | You find the distance and bearing back to the harbour and say whether 32 litres is enough, and whether your answer makes sense. |
| 5–6 | You choose suitable methods (components, Pythagoras, inverse tangent) for every part, reach correct answers, and discuss how the fuel range affects the decision. |
| 7–8 | You choose and use suitable methods successfully throughout, reach correct answers, explain how the range in fuel use changes the decision and why, and judge whether your recommendation is sensible for a real boat. |
| Level | Criterion C in task D1: what it looks like |
|---|---|
| 1–2 | Some working is shown, but it is hard to see how you reached the distance or the bearing. |
| 3–4 | Your working for the components and the bearing can be followed, with some correct notation, bearings in three figures and units. |
| 5–6 | Your working is clear and complete, with bearings in three figures and units (km, litres), and your note to the harbour office is mostly well organized. |
| 7–8 | Your working is clear, complete and well ordered, bearings and units are always correct, and your note to the harbour office makes sense on its own. |
Task D2: Turf and edging for a flower bed (criteria D and C)
When: End of unit, after lesson 7. Time: 50 minutes. Calculator: scientific calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A community centre has a circular flower bed with a radius of 6.0 m. A straight path will be laid across it along a chord of length 8.0 m, and the small segment between the chord and the circle will be covered in turf. The curved edge of the segment needs a strip of edging. Turf is sold in rolls that cover 1.5 m2 each, and edging comes in lengths of 2.5 m.
Part 1. The segment
- Draw the bed and the chord, and find the angle the chord makes at the centre of the circle.
- Find the area of the segment. Show the sector and the triangle you used.
- Find the length of the curved edge.
Part 2. Buying
- How many rolls of turf and how many lengths of edging must be bought? Explain how you decided each.
Part 3. Does it make sense?
- The chord was measured to the nearest 0.1 m. Between which values could it have been? Does the number of rolls and of lengths you chose still hold across that range?
- Write a short note to the caretaker with what to buy and how sure you are, so that someone who has not seen your working can understand it.
How your work is judged
| Level | Criterion D in task D2: what it looks like |
|---|---|
| 1–2 | You find some of the right numbers, for example the angle at the centre, but do not reach the area or the numbers to buy. |
| 3–4 | You find the area of the segment and the edge length, work out what to buy, and say whether your answer makes sense. |
| 5–6 | You choose suitable methods (inverse sine, sector minus triangle, arc length) for every part, reach correct answers, and discuss whether the purchase still holds across the measuring range. |
| 7–8 | You choose and use suitable methods successfully throughout, reach correct answers, explain how the measuring range affects the area and the edge, and why the purchase does or does not change. |
| Level | Criterion C in task D2: what it looks like |
|---|---|
| 1–2 | Some working is shown, but it is hard to see how you found the angle or the area. |
| 3–4 | Your working for the angle and the areas can be followed, with some correct notation and units. |
| 5–6 | Your working is clear and complete, usually with correct units (m, m²) and notation, and your note to the caretaker is mostly well organized. |
| 7–8 | Your working is clear, complete and well ordered, units and notation are always correct, and your note to the caretaker makes sense on its own. |
Teachers: the assessment teacher page has the markschemes, exemplar answers, the MYP descriptors for this year and support for each task.
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.