Position and Displacement

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Overview

Unit Title: Vectors and Matrices Mastery
Lesson: 4 of 12
Total Duration: 50 minutes
Subject: Mathematics
Curriculum Reference: KS4 – GCSE Mathematics – AQA Specification (9–1), Higher Tier
Strand: Geometry and Measures – Vectors (GCSE content area: G15)
Topic Focus: Position Vectors and Displacement Vectors in 2D

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Learning Objectives

By the end of this lesson, students will be able to:

• Define and write a position vector of a point in 2D coordinate space.
• Determine and interpret displacement vectors between two points.
• Apply vector arithmetic to geometric problems involving movement from one point to another.

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Prior Knowledge Required

• Number operations (addition, subtraction)
• Coordinates in two dimensions
• Familiarity with the Cartesian plane
• Basic understanding of vector notation from previous lessons in the unit

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Success Criteria

Students will:

• Confidently represent position vectors in column vector form
• Accurately find the displacement vector between pairs of points
• Apply these concepts to at least one geometric problem (e.g., triangle or polygon movement)

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Key Vocabulary

• Position Vector
• Displacement Vector
• Column Vector
• Origin
• Magnitude
• Direction

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Materials Required

• Whiteboard markers (different colours)
• Mini-whiteboards and pens (for pair tasks)
• Printed worksheet with scaffolded tasks (provided by teacher or created via lessons below)
• Graph paper
• Vector Direction Cards (provided in lesson)

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Lesson Structure (50 Minutes)

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⏱️ Starter: “Mystery Map Moves” (5 minutes)

Purpose: Activate prior knowledge + create intrigue

• Display a rough “treasure map” on the board with five labelled points: A, B, C, D, E.
• Ask: “If you're standing at A and take 4 steps right and 3 steps up to reach B, how could we write that using maths?”
• Invite quick fire answers. Introduce the idea: That’s a displacement vector!

✨ WOW Factor: The treasure map visual and storytelling approach instantly draws students into thinking of vectors as real movements.

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🧠 Concept Introduction: Position Vectors (10 minutes)

Teaching Points:

1. Take a point P at (4, 2) on a grid.
   Ask: “How could I describe where P is, from the origin (0,0)?”
   Answer: Position vector → P =
   [ \begin{pmatrix} 4 \ 2 \end{pmatrix} ]

2. Generalise: The position vector of a point (x, y) is
   [ \vec{OP} = \begin{pmatrix} x \ y \end{pmatrix} ]

3. Model 3–4 quick-fire examples with increasing difficulty (e.g. P = (–3, 2); Q = (0, –5); R = (6, –4))

4. Students mirror on mini-whiteboards for rapid response check-ins.

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🧩 Activity: “Vector Snap!” (10 minutes)

Interactive Pair Task

• Each pair receives:

  • 6 location cards marked with coordinates
  • 6 matching position vector cards

• Objective: Snap match each coordinate with its correct position vector and sort them from shortest to longest vector (calculate magnitude as extension)

• Mini plenary: Ask a few pairs to explain any surprising pairings or to share the biggest displacement they found.

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🎯 Main Teaching: Displacement Vectors (10 minutes)

Teaching Points:

1. Define displacement vector as:
   [ \vec{AB} = \vec{OB} - \vec{OA} ]

2. Model Example:
   A =
   [ \begin{pmatrix} 1 \ 3 \end{pmatrix} ]
   B =
   [ \begin{pmatrix} 5 \ 6 \end{pmatrix} ]
   Then:
   [ \vec{AB} = \begin{pmatrix} 5 \ 6 \end{pmatrix}

   \begin{pmatrix} 1 \ 3 \end{pmatrix}

   \begin{pmatrix} 4 \ 3 \end{pmatrix} ]

3. Make it interactive: “You leave school (2, 1) and reach the library (7, 4). What was your displacement?”

📌 Note: Keep reinforcing the syntax – vector OB minus vector OA. Drill this by using hand signs (open hands to 'subtract').

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✍️ Guided Practice (10 minutes)

Students complete tasks on a custom worksheet including:

• Task 1: Match points with position vectors
• Task 2: Find the displacement between pairs of points
• Task 3: Identify whether vectors describe same movement (e.g., different points, same displacement?)
• Extension: Work with reversed displacements – e.g., calculate vector BA if vector AB is known.

Teacher circulates, checking understanding, probing with questions like:

• “How do you know you’ve done the subtraction the correct way around?"
• “Could there ever be a displacement vector that’s the same from two different origin points?”

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🗣️ Plenary: “Which Journey Am I?” (5 minutes)

• Challenge the class with riddle-style vector clues.

E.g.
“I start at (–1,2) and my displacement to reach my destination is
[ \begin{pmatrix} 3 \ –4 \end{pmatrix} ].
Where do I arrive?”

• Students solve on mini-whiteboards, hold up for quick assessment
• Reveal actual coordinate and relate back to visual journey on coordinate plane

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Assessment for Learning

• Use cold-calling & mini-whiteboard responses to assess understanding in conceptual and guided sections.
• Peer discussion ensures misconceptions are addressed (especially with vector direction and order of subtraction).
• Mark a sample of worksheets post-lesson to inform groupings and support for Lesson 5.

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Differentiation & Support

• Support Cards: With steps scaffolded — highlight direction using grid boxes
• Challenge Tasks: Involve identifying parallel or opposite vectors or calculating magnitude
• Visual Learners: Use graph paper and coloured vectors to represent movement
• Oral Learners: Pair discussions and verbal explanations of journeys between points

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Homework / Extension

Task: Students draw their own 4-point journey on a coordinate grid and:

• Write out the position vector of each point
• Find the displacement vectors for each leg
• Bonus: calculate total “journey vector”

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Teacher Reflection Prompts

To consider post-lesson:

• Which students thrived with the physical/visual support?
• Were there misconceptions around direction?
• Do students understand what subtraction really means in vector terms?

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Summary

This lesson makes vector concepts tangible with storytelling, movement-based tasks, and accessible visuals. It bridges prior knowledge to new learning while meeting GCSE specification G15 demands — ensuring relevance, rigour, and creativity all in one 50-minute package.