Lesson 9 — Coordinates and Vectors: Position and Direction

Summary

Learners use coordinates and vectors to describe position and direction: a vector has a magnitude and a direction, written as components, and two vectors add tip-to-tail into a resultant. They connect this to navigation, mapping, and the everyday sense of “three blocks east, four north,” and set up the vector proofs of Lesson 10.

Objectives

• Use coordinates and vectors to analyze geometric situations. (D05.S3.12.01)

Connection

“Walk three blocks east, then four north” is a vector: a distance and a direction. Add a second instruction — “then two west” — and you are adding vectors, the same arithmetic a pilot, a delivery driver, or a hiker uses to know where they will end up. A coordinate grid turns the instruction into two numbers, so position and movement become something you can calculate, not just feel.

Materials

• Vector diagram sheet
• Math journal
• Low-tech: grid paper and ruler

Preparation

• Copy or draw the coordinate grid with vectors.
• Retrieval: from Grade 11 (D05.S1.11.02), vectors as a column of components. Today we draw and add them in the plane.
• Prepare the worked example (two displacements added tip-to-tail) to model first (S-011).

Facilitator note

This lesson is written to the learner (“you”). The idea to land: a vector = magnitude + direction, written as components (x, y); adding vectors means adding components, which matches the tip-to-tail picture. Teach the drawing and the component arithmetic together — picture and symbol must agree — with a worked example and guided practice (S-011). Keep the connection to real movement strong, because that is what makes components intuitive rather than abstract.

The ethics lens: an arrow that states its direction and size is an honest claim — a vector does not pretend to be more or less than it is. The egalitarianism lens: the same two numbers describe a route for anyone, in any place, with or without a device. The global lens: locating a place by two distances (latitude and longitude, grid squares, “three east and four north”) is far older than the coordinate names we inherit — grid ideas appear across many mapping traditions (S-240). The technology lens: coordinates and vectors are the language of every screen, map, drone, and flight path. The environment lens: a vector describes the wind, a current, a migrating animal’s path, or the flow of water off a field — direction and size are how we model the moving Earth (S-005). Preview: Lesson 10 uses vectors to prove geometric facts.

Procedure

1. Recall (5 min). From Grade 11, a vector is a column of numbers. Today we draw it as an arrow and add arrows.
2. Draw a vector (10 min). On a grid, draw “three east, four north” as an arrow from the origin to (3, 4). Its components are (3, 4); its magnitude is its length (by Pythagoras, √(3²+4²) = 5); its direction is the angle it makes. One arrow carries all three ideas.
3. Worked example — add tip-to-tail (15 min). Add v = (3, 4) and w = (−2, 1): draw v, then start w at v’s tip. The resultant runs from the start of v to the tip of w. In components: (3 + (−2), 4 + 1) = (1, 5). Check the picture matches the numbers.
4. Scalar multiples (8 min). Doubling v gives (6, 8): same direction, twice the length. Multiplying by −1 reverses the arrow. A scalar multiple stretches, shrinks, or reverses a vector without turning it.
5. Guided practice (12 min). With a partner, add (2, −3) and (5, 4) two ways — draw tip-to-tail and add components — and confirm they agree. Then describe — aloud or in writing, sign, gesture, or AAC — a real route (e.g., across a town, down a river) as a sum of vectors.
6. Retrieve and connect (5 min). Write one sentence on why “add the components” is the same as “walk the second arrow from the first arrow’s tip.”
7. Close (2 min). Say — or write, sign, gesture, or use AAC to express — what a vector is, in one sentence, using the words magnitude and direction.

Differentiation

• Support: Walk the route on a floor grid first, then draw it, then write the components.
• Extension: Given a vector’s magnitude and direction, recover its components (with trigonometry), and prove that reversing both components reverses the vector.
• Number access (dyscalculia): Provide the step-2 coordinate grid pre-printed so drawing the arrows (3, 4) and (−2, 1) is placing rather than computing; offload the component addition (3 + (−2), 4 + 1) and the magnitude squaring (3² + 4²) to a partner or calculator; and take the verbal route — walk “three east, four north, then two west” on a floor grid and say where you end up, describing tip-to-tail in words or gesture without writing coordinates.
• Communication access: Every spoken step — saying, naming, reading aloud, discussing, or closing — can be done in writing, sign, gesture, or AAC instead. Learners who are non-speaking, d/Deaf, hard-of-hearing, or who use AAC complete every task in their preferred mode; no step requires producing or hearing sound.

Assessment

• Formative (peer + self): Can the learner draw a vector from components, add two vectors tip-to-tail and by components, and check that the two agree?
• Portfolio artifact (unit): A drawn vector sum with its component arithmetic, in the math journal.

Home connection

Describe one trip you took today (or one animal’s movement you saw) as a sum of vectors — “two blocks east, then one north.” Write the components and the resultant, and one sentence on where you ended up.

Resources

• On worked examples and guided practice: Kirschner, Sweller & Clark (2006) (S-011).
• On coordinates and grids across traditions: MacTutor History of Mathematics (S-240).