At a glance. One lesson of about 60 minutes. It assumes Pythagoras and the inverse tangent. Students find a vector from two points, its magnitude and its direction, and use a sketch to choose the quadrant.
Calculator: scientific calculator. Scientific calculator: square roots of non-perfect squares and inverse tangent. Several questions are Pythagorean triples so that the structure shows without the calculator.
What students will learn
Know
- A column vector lists the across and up movement.
- AB = B − A, in coordinates.
- Magnitude is the square root of the sum of the squares; direction is an angle from the positive x-axis.
Understand
- Why a vector has no fixed position.
- Why the inverse tangent alone cannot give the direction.
- Why the magnitude is not the sum of the entries.
Do
- Write the vector between two points.
- Find a magnitude with and without a calculator.
- Find a direction, using a sketch to choose the quadrant.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: vector notation and the magnitude formula (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe which sign changes keep the magnitude and which change the direction.
- Criterion C. Communicating: a sketch beside each direction calculation, with the acute angle labelled.
- Criterion D. Applying mathematics in real-life contexts: drone flights, boats and hikes (hook and questions 2, 3 and 5).
The one thing to do with the figure
Ask for the prediction first. Then set (3, 4), then change one sign at a time, and ask students to say the magnitude before they read it.
Ask: “Which sign changes leave the magnitude the same, and which change the direction?” Listen for: every sign change keeps the magnitude, and every sign change moves the vector into a different quadrant.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the drone; collect predictions. |
| 12 min | Figure, key idea and warning card; set (3, 4) and change signs. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask which operation goes from start to end. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 10; 45 degrees; 8. |
|---|---|
| Hook | The distance is √(602 + 802) = 100 metres, less than the 140 metres flown, and the direction is 53.1 degrees north of east. |
| Worked examples | (4, 3); 10; 126.9 degrees. |
| Faded example | (6, −4) with magnitude 7.2. |
| 1. Displacement between two points | D. A adds, B is the reverse, C swaps the entries. |
| 2. Magnitude | 15. √225. |
| 3. Magnitude of a displacement | 13. (5, 12). |
| 4. Finding a missing entry | 8. 289 − 225 = 64. |
| 5. Magnitude with a calculator | 8.1. √65. |
| 6. Direction in the third quadrant | 200.6. 180 + 20.6. |
| 7. A magnitude equation | 20. (20, 21) has magnitude 29. |
| 8. Same magnitude | B. A adds the entries, C subtracts, D uses the sum of squares. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Added the coordinates, reversed the journey, or swapped the entries. | Draw both points on a grid and count across and then up. |
| 21 or 225 in question 2 | Added the entries, or did not take the square root. | Ask: what shape do across and down make, and which side is the movement? |
| 17 or 169 in question 3 | Added the entries of the vector, or stopped before the square root. | Ask for the vector first, then for the right-angled triangle. |
| 2 or 64 in question 4 | Subtracted the magnitudes, or gave the square. | Write the equation x2 + 152 = 172. |
| 11 or 65 in question 5 | Added the entries, or did not take the square root. | Ask which side of the triangle the speed is. |
| 20.6 or 159.4 in question 6 | Gave the acute angle without the quadrant, or used the second-quadrant adjustment for a vector pointing down. | Sketch the vector and turn anticlockwise from the positive x-axis, past 180. |
| 14 or 21 in question 7 | Added the entries, or gave the up entry k + 1. | Write k2 + (k + 1)2 = 292 and check the answer in the column. |
| A, C or D in question 8 | Added the entries, subtracted them, or used the sum of the squares. | Ask what shape the entries make, then which side is the magnitude. |
Questions worth asking
- Does the starting point change a vector? Listen for: no, the same column drawn from anywhere is the same vector.
- Why is the magnitude not 3 + 4? Listen for: the movements are at right angles, so it is the hypotenuse.
- How do you know which quadrant a vector is in? Listen for: from the signs of the entries.
- How are (3, 4) and (−3, −4) related? Listen for: same magnitude, opposite direction, 180 degrees apart.
Differentiation
A way in for anyone stuck
- Give squared paper and ask students to draw each vector and count the entries before calculating.
- Provide a table of the four quadrants with the sign of x and y and the adjustment to the angle.
- Give the frame: “The vector goes … across and … up, so its magnitude is √(…).”
Stretch
- Find all column vectors with whole-number entries and magnitude 25.
- Show that the vectors (a, b) and (−b, a) have the same magnitude and are at 90 degrees to each other.
Students learning in English as an additional language
“Magnitude” means size or length. “Direction” means which way it points. “Column” here means numbers written one above the other. “Displacement” means the straight movement from a start to an end.
| Word | Plain meaning | Example |
|---|---|---|
| vector | a quantity with a size and a direction | a movement of 3 across and 4 up |
| column vector | a vector written as two numbers one above the other | across over up |
| magnitude | the length of a vector | how long the arrow is |
| direction | the angle a vector makes with the positive x-axis | measured anticlockwise |
| displacement | the vector from a start point to an end point | from home to school |
Sentence starters for explaining aloud or in writing:
- The vector goes … across and … up.
- The magnitude is the square root of … plus …, which is …
- It is in the … quadrant, so the direction is …
Two things not to say
“The length is the entries added together.” Say instead: the entries are the two shorter sides of a right-angled triangle, and the length is the hypotenuse.
“Just use the calculator for the angle.” Say instead: the calculator gives the acute angle, and the sketch tells you what to add or subtract.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- FindThe vector from (1, 1) to (5, 4). What is its magnitude?
(4, 3), so 5. - FindWhat is the magnitude of (5, −12)?
13.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA bird flies 8 km west and 15 km south. How far is it from where it started?
17 km. - Earlier work (lesson 1)A graph has 10 edges. What do the degrees add to?
20.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningTwo vectors have the same magnitude. Must they have the same direction?
No. (3, 4) and (4, 3) have the same magnitude but different directions. - Working backwardsA vector (k, 2k) has magnitude 10. Find k to one decimal place, with k positive.
k √5 = 10, so k = 10 ÷ √5 = 4.5 to one decimal place.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
A scientific calculator is used in several questions; set it to degrees.