The unit begins with a drawing that looks like a map but ignores distance and shape: dots joined by lines. What matters is only which things are joined to which. That is enough to prove that a party cannot have an odd number of guests who each shake an odd number of hands, to decide when a postal round can use every street exactly once, and to find the cheapest way to link a set of villages.
The second half replaces a route with a single movement. A column of two numbers says how far across and how far up, and once movements are numbers they can be added, reversed and scaled. Position vectors fix a point from an origin, so the midpoint of a line, the point a third of the way along it and the fourth corner of a parallelogram are all arithmetic.
The last lessons use the same tools to prove that the lines joining midpoints of a triangle are parallel to the third side, and to combine velocities and forces, where a bearing is an angle from north and the resultant is the single vector that has the same effect as the others together.
The lessons, in order
- Lesson 1 · Graphs, degree and the handshake result Vertices, edges and degree; drawing a graph from a description; the handshake result (the degrees add to twice the number of edges); which lists of degrees are possible. Add edges one at a time and watch the degrees and their sum. No calculator
- Lesson 2 · Euler and Hamilton routes Euler trails and circuits and the odd-vertex test, Hamilton paths and cycles, and counting the Hamilton cycles of a complete graph. Choose a network and compare the test for an Euler route with the search for a Hamilton cycle. No calculator
- Lesson 3 · Shortest paths and minimum spanning trees Weighted graphs, a shortest path found by labelling outwards, trees and spanning trees, and a minimum spanning tree found by taking the cheapest edge that does not make a cycle. Choose a destination and see the shortest route from A, then switch to the cheapest tree that joins every vertex. No calculator
- Lesson 4 · Column vectors, magnitude and direction Column vector notation, displacement between two points, equal vectors, magnitude by Pythagoras, and direction as an angle measured anticlockwise from the positive x-axis with quadrants. Set the across and up entries and read the magnitude and the direction. Scientific calculator
- Lesson 5 · Adding, subtracting and scaling vectors Adding and subtracting column vectors, the triangle rule, scalar multiples, negative vectors and parallel vectors. Choose how many of each vector to take and see that the order does not matter. No calculator
- Lesson 6 · Position vectors and midpoints Position vectors, the vector between two points, the midpoint and a point dividing a line in a given ratio, and finding a missing vertex of a parallelogram. Move B and a dividing point and read its position vector. No calculator
- Lesson 7 · Proving geometric results with vectors A vector proof of the midpoint theorem, testing for parallel and collinear points, diagonals of a parallelogram, and the difference between an example and a proof. Move the corners of a triangle and see that the midpoint line is always half the third side. No calculator
- Lesson 8 · Navigation and force contexts Bearings, components of velocity and force, the resultant by adding components, speed and bearing of the resultant, and steering across a current. Set a boat's speed and heading and a current, and read the actual speed and bearing. Scientific 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: degree, routes and cheapest networks | A | After lesson 3 | 20 minutes |
| A2: End-of-unit test | A | End of unit, after lesson 8 | 50 minutes |
| B1: Grids of vertices | B | Mid-unit, after lesson 3 | 50 minutes |
| C1: Explain it to someone who missed it | C | After lesson 6 | 35 minutes |
| D1: Solar lamps for the island paths | D and C | End of unit, after lesson 8 | 50 minutes |
| D2: Crossing the channel to the island | D and C | End of unit, after lesson 8 | 50 minutes |
D1 and D2 are alternatives. Both assess criteria D and C. D1 is a network problem (lessons 1 to 3) and D2 is a vector problem (lessons 4 to 8): set the one that matches the half of the unit you want to assess, not both.
Task A1: Quick check: degree, routes and cheapest networks (criterion A)
When: After lesson 3. Time: 20 minutes. Calculator: no calculator. Criterion: A, knowing and understanding.
Answer every question you can and show your working: a correct method earns credit even if the final answer slips. Questions 7 and 8 are meant to make you think, so try something rather than leaving them blank.
Part 1. Degree and routes
- A graph has vertices P, Q, R, S and T, and edges PQ, PR, QR, QS, RS and ST. Write down the degree of each vertex. (2 marks)
- How many edges does that graph have? (1 mark)
- Does the graph have an Euler route? Give a reason. (2 marks)
- Is there a graph on 5 vertices in which every vertex has degree 3? Explain how you know. (2 marks)
Part 2. Weighted graphs
- The edge lengths are AB 5, AC 3, BC 1, BD 6, CD 4, CE 8 and DE 2. Find the shortest route from A to E and its length. (2 marks)
- For the same graph, find a minimum spanning tree. List its edges and give its total weight. (3 marks)
Part 3. Think it through
- Zoe says: “Join the shortest routes from A to every other vertex and you always get the minimum spanning tree.” Three vertices A, B and C have edges AB 4, AC 4 and BC 1. Does this triangle show Zoe is wrong? Show why. (2 marks)
- A tree has 10 vertices. Find its number of edges, and the sum of all its degrees. Could every vertex have degree 3? (3 marks)
How your work is judged
| Level | Criterion A in task A1: what it looks like |
|---|---|
| 1–2 | You find some degrees and answer some of questions 1 to 3 correctly, for example the count of edges. |
| 3–4 | You answer most of questions 1 to 6 correctly: degrees, an Euler route decision, a shortest route and a spanning tree with a few slips. |
| 5–6 | You answer questions 1 to 6 correctly with working that shows your method, and make a good start on question 7 or 8. |
| 7–8 | You answer questions 1 to 6 correctly and settle question 7 or 8 with reasoning that shows why your answer must hold. |
Task A2: End-of-unit test (criterion A)
When: End of unit, after lesson 8. Time: 50 minutes. Calculator: scientific calculator. Criterion: A, knowing and understanding.
Show your working. Give angles and bearings to 1 decimal place. Section 3 contains problems you have not seen before: choose a method and try it, because a correct method earns credit.
Section 1. Vectors
- Let p = (6, −8) and q = (−2, 5). Find (a) the magnitude of p; (b) p + q and its magnitude; (c) 2p − 3q. (5 marks)
- Relative to an origin O, A is (1, 2) and B is (7, −2). Find (a) the vector AB; (b) the position vector of the midpoint of AB; (c) the point C so that ABCD is a parallelogram with D at (−1, 5). (5 marks)
- A vector is (4, 3). Find its angle with the horizontal, and its bearing (the angle clockwise from north). (2 marks)
Section 2. Networks
- A graph has 6 vertices with degrees 3, 3, 2, 2, 2 and 2. How many edges does it have? Does it have an Euler route, and if so where must the route start and end? (3 marks)
- A weighted graph on vertices A to F has edges AB 7, AC 9, BC 5, BD 8, CD 3, CE 6, DE 4, DF 10 and EF 2. Find (a) a minimum spanning tree, listing its edges and total weight; (b) the shortest route from A to F and its length. (5 marks)
Section 3. Unfamiliar problems
- O, A and B are points with OA = a and OB = b. P is the point on OA with OP equal to one third of OA, and Q is the point on OB with OQ equal to one third of OB. Prove that PQ is parallel to AB, and find the ratio of their lengths. (3 marks)
- A boat can travel at 6 km/h through still water, heading due north, in a river that flows due east at 8 km/h. Find (a) the boat's speed over the ground; (b) the bearing of its actual path; (c) its position relative to the start after 3 hours. (3 marks)
How your work is judged
| Level | Criterion A in task A2: what it looks like |
|---|---|
| 1–2 | You do some of the routine steps in sections 1 and 2 correctly, for example a magnitude or a list of edges. |
| 3–4 | You complete most of sections 1 and 2 correctly, using a method that is the right one even where a slip appears. |
| 5–6 | You complete sections 1 and 2 correctly with your method shown, and make a good start on question 6 or 7. |
| 7–8 | You complete sections 1 and 2 correctly and carry out question 6 or 7 to a correct conclusion, choosing your own method. |
Task B1: Grids of vertices (criterion B)
When: Mid-unit, after lesson 3. Time: 50 minutes. Calculator: no calculator. Criterion: B, investigating patterns.
Picture a rectangle of dots, with a rows of b dots in each row. Every dot is joined to the dot next to it in its row and to the dot below it in its column, and to nothing else (no diagonals). Call this an a by b grid. A 2 by 3 grid has six dots and one edge between each neighbouring pair. This task asks what you can say about grids of any size, without drawing the large ones.
Part 1. Collect results
- Draw the grids 2 by 2, 2 by 3, 2 by 4, 3 by 3, 3 by 4 and 4 by 4. For each, record the number of edges and the number of vertices whose degree is odd. Design the table yourself.
- Add the 4 by 5 grid to your table, using its drawing or a method you have found.
Part 2. Describe what you find
- Describe, in words and then with letters, a rule that gives the number of edges of an a by b grid.
- Describe a rule that gives the number of vertices of odd degree. Say for which grids it works.
Part 3. Use and test your rules
- Predict the number of edges and the number of odd vertices in a 5 by 6 grid, and in a 10 by 10 grid. Show how your rules give each answer. Does the 10 by 10 grid have an Euler route?
- Test your rules on a single row of 5 dots (a 1 by 5 grid). Say what is the same and what is different, and what that tells you about the limits of your rules.
Part 4. Explain
- Explain why your rule for the edges must be right, without counting edges one by one.
- Find all the grids with at least two rows and two columns that have an Euler route, and explain how you know there are no others.
How your work is judged
| Level | Criterion B in task B1: what it looks like |
|---|---|
| 1–2 | With help, you draw some grids, count edges and odd vertices and say something you notice. |
| 3–4 | You complete the table and suggest a rule for the number of edges or for the number of odd vertices. |
| 5–6 | You organize your results clearly, describe a rule for each count that fits all your findings, and check both on the 5 by 6 grid and the 1 by 5 row. |
| 7–8 | You find rules for both counts, use them correctly for large grids, test them on the single row, and prove or justify why they hold, including where the odd vertex rule stops working. |
Task C1: Explain it to someone who missed it (criterion C)
When: After lesson 6. Time: 35 minutes. Calculator: no calculator. Criterion: C, communicating.
Dara was away for lesson 6 and tried the question below from the lesson notes. Dara's working is shown. Your job is to write a clear explanation Dara can follow: what went wrong, why, and how to do it properly.
The question. P is the point (2, −3) and Q is the point (8, 9). Find (a) the midpoint of PQ and (b) the point R on PQ that is one third of the way from P to Q.
Dara's working. “PQ = (8 − 2, 9 − (−3)) = (6, 12). Half of PQ is (3, 6), so the midpoint is (3, 6). One third of PQ is (2, 4), so R is (2, 4).”
Your explanation
- Say which of Dara's steps is correct and what each number in them stands for (a movement or a point).
- Explain why (3, 6) and (2, 4) are not the answers, using a diagram.
- Show Dara a correct method, using position vectors, for the midpoint and for R, with each step labelled.
- Finish by giving Dara one short sentence to remember that stops the same mistake happening again.
How your work is judged
| Level | Criterion C in task C1: what it looks like |
|---|---|
| 1–2 | You use a few correct words, such as vector or point, and your explanation is hard for Dara to follow in places. |
| 3–4 | You use some correct words and notation, and your steps are in a sensible order, so Dara could follow most of it even if some parts are unclear. |
| 5–6 | You usually use correct notation, move between a diagram, symbols and words, and your explanation is complete and in a logical order. |
| 7–8 | You consistently use correct notation, move smoothly between a diagram, symbols and words, and your explanation is complete, brief and organized so that each step leads to the next. |
Task D1: Solar lamps for the island paths (criteria D and C)
When: End of unit, after lesson 8. Time: 50 minutes. Calculator: no calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
Six hamlets, A to F, on a small island in the Andaman Sea are to be linked by one cable that powers the solar lamps along the paths. Each possible path between two hamlets has a cost to lay the cable, in thousands of coins. The costs are estimates, correct to about 10%. The planner has listed them: AB 12, AC 18, BC 9, BD 15, CD 11, CE 20, DE 7, DF 16 and EF 10. The planner asks you for the cheapest cable layout that reaches every hamlet, and for advice on whether it is a good plan.
Your task
- Say what the vertices, the edges and the weights are in this model, and one thing the model leaves out.
- Find the cheapest layout that reaches every hamlet. Say which method you used, list the paths chosen and give the total cost.
- Check that your layout has the right number of paths for six hamlets and explain what that tells you.
- A technician must walk from hamlet A to hamlet F along the paths in the planner’s list. Find the shortest route and its length. How much longer is the route that follows only your cheapest layout?
- The costs are only correct to about 10%. Give the total cost with a range, and say whether the planner should quote it as exact.
- The planner would like every hamlet to stay linked if any single path is damaged by a storm. Show whether adding the cheapest extra path achieves that, and say what you recommend.
- Write a short reply to the planner: your layout, its cost, the one thing you are least sure of and your recommendation.
How your work is judged
| Level | Criterion D in task D1: what it looks like |
|---|---|
| 1–2 | You pick out some of the hamlets and paths and find a layout, but it is not clear that it is the cheapest or that it reaches everyone. |
| 3–4 | You identify the hamlets, paths and costs, apply a spanning tree method to reach a total, and say whether the answer is sensible. |
| 5–6 | You choose a suitable method, reach a valid layout and total, find the shortest route, describe the accuracy of the cost and discuss whether the plan makes sense for the island. |
| 7–8 | You model the situation, reach a correct layout, route and total, justify the range on the cost, and give a justified recommendation that deals with storms and what the model leaves out. |
| Level | Criterion C in task D1: what it looks like |
|---|---|
| 1–2 | Your working is hard to follow and you use few correct terms. |
| 3–4 | Your steps can be followed and you use some correct terms such as vertex, edge and weight. |
| 5–6 | Your explanation is mostly logical, with a clear list of paths and a clearly set out reply. |
| 7–8 | Your explanation is complete and concise, with the method, the numbers and the recommendation all easy to find. |
Task D2: Crossing the channel to the island (criteria D and C)
When: End of unit, after lesson 8. Time: 50 minutes. Calculator: scientific calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A ferry crosses a channel to an island that is 26 km due north of its harbour. Through still water the ferry travels at 12 km/h. A current flows due east at 5 km/h. The ferry has fuel for 3 hours. The captain has asked you to plan the crossing and tell her what to steer. The speeds are measured to the nearest km/h.
Your task
- Write the ferry's velocity through still water, steering due north, and the current's velocity, as column vectors. Find the resultant velocity and the ferry's speed over the ground.
- Find the bearing of the ferry's actual path, to 1 decimal place, when it steers due north.
- If the captain steers due north the whole way, how long does it take to be 26 km north of the harbour, and how far east of the island is the ferry then?
- Find the bearing the captain should steer, to 1 decimal place, to travel directly to the island. How long does the crossing take? Is there enough fuel?
- The current is only correct to the nearest km/h, so it could be anything from 4.5 to 5.5 km/h. Find the range of bearings the captain should be ready to steer.
- Write a short reply to the captain: what to steer, how long, how far you trust the figures and one thing that could make the plan fail.
How your work is judged
| Level | Criterion D in task D2: what it looks like |
|---|---|
| 1–2 | You pick out some of the speeds and directions and find one correct quantity, such as the resultant, with little support for why it makes sense. |
| 3–4 | You identify the velocities, add them correctly, find the drift or the heading, and say whether the answer is sensible for the crossing. |
| 5–6 | You choose suitable vector methods, find a valid heading and time, describe the accuracy using the range of the current and discuss whether the plan is practical. |
| 7–8 | You model the crossing, find a correct heading and time, justify the range of bearings from the uncertainty in the current and give a justified recommendation to the captain. |
| Level | Criterion C in task D2: what it looks like |
|---|---|
| 1–2 | Your working is hard to follow; you use few correct terms such as velocity or bearing. |
| 3–4 | Your steps can be followed, with some correct terms and a sketch of the vectors. |
| 5–6 | Your explanation is mostly logical with a clear vector diagram and a clear reply to the captain. |
| 7–8 | Your explanation is complete and concise, with a labelled vector diagram, each result in context and a reply the captain could act on. |
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.