Back to the lesson plan builder
Lesson length30 min45 min60 min
Questions🔀 New questions

Print the plan for yourself; the student sheet prints on its own page after it.

Math lesson plan · 10th grade (Year 11 in England) · 60 minutes

Express vector paths in terms of a and b

Geometry

CCSS HSN.VM.B.4England: GCSE Higher Geometry

Objective and success criteria

Students will: Express vector paths in terms of a and b.

  • I can answer questions on today's skill on my own, without help.
  • I can explain each step and say why it works.
  • I can check my answer and spot a mistake.

Before and after

Needs first: Vectors

Leads to: Show that vectors are parallel or points are collinear

Materials: Rulers and protractors, Squared or dotted paper, Shape cut-outs, Mini whiteboards and pens, Printed student sheet (below).

Warm-up · 6 min

  • Quick retrieval from earlier learning (Vectors). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. a = (−6, 0) and b = (−3, 2). Find 3a + b. Type it like (2, −1). → (−21, 2)Prompt: Multiply each component by the scalar.
  2. Find the magnitude of the vector (6, −8). → 10Prompt: Magnitude = √(x² + y²).
  3. a = (−4, 3) and b = (1, 4). Find a − 2b. Type it like (2, −1). → (−6, −5)Prompt: Multiply each component by the scalar.

Hook · 6 min

  • Show a photo of a building, a tile pattern or a road sign and ask what shapes and angles students can spot.
  • Share the objective: Express vector paths in terms of a and b. Read the success criteria together.

I do: model it · 12 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: OA = a and OB = b. M is the midpoint of AB. Find OM in terms of a and b.
  • Step 1: OM = OA + ½AB = a + ½(b − a)
  • Step 2: ½a + ½b
  • Answer: ½a + ½b
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 12 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. OA = a and OB = b. M is the midpoint of AB. Find AM in terms of a and b. (½b − ½a / b − a / ½a + ½b / ½a − ½b) → ½b − ½aPrompt: Go along a route of vectors you know, e.g. A → O → B.
  2. OA = a and OB = b. P lies on AB with AP : PB = 3 : 1. Find OP in terms of a and b. ((1/4)a + (3/4)b / (3/4)b − (3/4)a / a + (3/4)b / (3/4)a + (1/4)b) → (1/4)a + (3/4)bPrompt: OP = OA + AP, and AP is 3/4 of AB.

You do: independent practice · 17 min

  • Students work on the student sheet. Circulate and check the first two answers of every student early.
  • Students who finish: write a question of their own for a partner, with the answer.

The 8 questions are on the student sheet below, with answers in the key.

Exit ticket · 7 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. OA = a and OB = b. M is the midpoint of AB. Find OM in terms of a and b. (b − a / 2b − a / ½a + ½b / a + b) → ½a + ½bPrompt: Go along a route of vectors you know, e.g. A → O → B.
  2. OA = a and OB = b. P lies on AB with AP : PB = 3 : 1. Find OP in terms of a and b. ((1/2)a + (1/2)b / a + (3/4)b / (1/4)a + (3/4)b / (3/4)a + (1/4)b) → (1/4)a + (3/4)bPrompt: OP = OA + AP, and AP is 3/4 of AB.

Watch out for

  • Thinking a square turned on its corner is no longer a square.
  • Reading the wrong scale on a protractor.

Support and stretch

Support

  • Review first: Vectors.
  • Use concrete materials: rulers and protractors.
  • Give the first step of each question, or a partly worked example to finish.
  • Pair with a confident partner for the guided practice.

Stretch

  • Move on to: Show that vectors are parallel or points are collinear.
  • Explain a common mistake in writing and how to fix it.
  • Write a word problem that uses today's skill, then solve a partner's.

Homework and more practice

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-10.

Free lesson plan from eurekami.com/lesson-plan-builder

eurekami

Express vector paths in terms of a and b

Name

Date

Score/ 8

  1. Question 1

    OA = a and OB = b. M is the midpoint of AB. Find AB in terms of a and b.

    • A. b − a
    • B. ½a + ½b
    • C. a + b
    • D. 2b − a

  2. Question 2

    OA = a and OB = b. P lies on AB with AP : PB = 1 : 3. Find OP in terms of a and b.

    • A. (3/4)a + (1/4)b
    • B. (1/2)a + (1/2)b
    • C. a + (1/4)b
    • D. (1/4)a + (3/4)b

  3. Question 3

    OA = a and OB = b. P lies on AB with AP : PB = 2 : 1. Find OP in terms of a and b.

    • A. a + (2/3)b
    • B. (1/3)a + (2/3)b
    • C. (2/3)b − (2/3)a
    • D. (1/2)a + (1/2)b

  4. Question 4

    OA = a and OB = b. M is the midpoint of AB. Find MB in terms of a and b.

    • A. 2b − a
    • B. ½b − ½a
    • C. ½a − ½b
    • D. a − b

  5. Question 5

    OA = a and OB = b. P lies on AB with AP : PB = 1 : 2. Find OP in terms of a and b.

    • A. (1/2)a + (1/2)b
    • B. (1/3)b − (1/3)a
    • C. (1/3)a + (2/3)b
    • D. (2/3)a + (1/3)b

  6. Question 6

    OA = a and OB = b. M is the midpoint of AB. Find OM in terms of a and b.

    • A. 2b − a
    • B. ½a − ½b
    • C. ½a + ½b
    • D. a + b

  7. Question 7

    OA = a and OB = b. M is the midpoint of AB. Find AM in terms of a and b.

    • A. ½b − ½a
    • B. b − a
    • C. 2b − a
    • D. a − b

  8. Question 8

    OA = a and OB = b. P lies on AB with AP : PB = 3 : 1. Find OP in terms of a and b.

    • A. a + (3/4)b
    • B. (1/4)a + (3/4)b
    • C. (3/4)a + (1/4)b
    • D. (1/2)a + (1/2)b

Free from eurekami.com

for the adult only

Answer key: Express vector paths in terms of a and b

  1. 1. A. b − a

  2. 2. A. (3/4)a + (1/4)b

  3. 3. B. (1/3)a + (2/3)b

  4. 4. B. ½b − ½a

  5. 5. D. (2/3)a + (1/3)b

  6. 6. C. ½a + ½b

  7. 7. A. ½b − ½a

  8. 8. B. (1/4)a + (3/4)b