Math lesson plan · 10th grade (Year 11 in England) · 30 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 · 4 min
- Quick retrieval from earlier learning (Vectors). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- a = (2, 3) and b = (−4, −4). Find a + 3b. Type it like (2, −1). → (−10, −9)Prompt: Multiply each component by the scalar.
- Find the magnitude of the vector (−6, 8). → 10Prompt: Magnitude = √(x² + y²).
- a = (−3, 3) and b = (−4, −2). Find 3a − 2b. Type it like (2, −1). → (−1, 13)Prompt: Multiply each component by the scalar.
Hook · 3 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 · 7 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 · 6 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- OA = a and OB = b. M is the midpoint of AB. Find AM in terms of a and b. (2b − a / b − a / ½a − ½b / ½b − ½a) → ½b − ½aPrompt: Go along a route of vectors you know, e.g. A → O → B.
- OA = a and OB = b. P lies on AB with AP : PB = 3 : 1. Find OP in terms of a and b. (a + (3/4)b / (1/2)a + (1/2)b / (3/4)b − (3/4)a / (1/4)a + (3/4)b) → (1/4)a + (3/4)bPrompt: OP = OA + AP, and AP is 3/4 of AB.
You do: independent practice · 7 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 6 questions are on the student sheet below, with answers in the key.
Exit ticket · 3 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- OA = a and OB = b. P lies on AB with AP : PB = 2 : 1. Find OP in terms of a and b. ((2/3)a + (1/3)b / (1/3)a + (2/3)b / a + (2/3)b / (1/2)a + (1/2)b) → (1/3)a + (2/3)bPrompt: OP = OA + AP, and AP is 2/3 of AB.
- OA = a and OB = b. M is the midpoint of AB. Find BA in terms of a and b. (2b − a / a + b / b − a / a − b) → a − bPrompt: Go along a route of vectors you know, e.g. A → O → B.
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