At a glance. One lesson of about 60 minutes. Rotating points and shapes by 90 and 180 degrees about the origin and about another centre, and describing a rotation fully.
Calculator: no calculator. The numbers are small on purpose: the work is in knowing which coordinate goes where, and in the direction.
What students will learn
Know
- A rotation has a centre, an angle and a direction.
- About the origin: 90° anticlockwise (x, y) → (−y, x); 180° (x, y) → (−x, −y); 90° clockwise (x, y) → (y, −x).
- A rotation is a congruence transformation; only the centre does not move.
Understand
- The same turn about different centres gives different images.
- A half turn has no direction; a quarter turn does.
Do
- Rotate a point or polygon through 90 or 180 degrees about the origin.
- Rotate about another centre by turning the vector from the centre.
- Describe a rotation fully.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: applying the three origin rules and a different centre (questions 1 to 8).
- Criterion B. Investigating patterns: the second figure. Ask students to say how the image moves when the centre moves one unit.
- Criterion C. Communicating: describing a rotation with centre, angle and direction, and checking an answer by quadrant.
- Criterion D. Applying mathematics in real-life contexts: pinwheels, signposts, clock hands and turning blades.
The one thing to do with the figure
Before pressing a button, ask for the image. Ask where B goes under the quarter turn before the working appears, and ask them to check it by quadrant.
On the second figure, ask: “Which corner does not move when the centre is at (1, 1)?” B, because it is the centre. Then ask which centre would fix A.
A possible order
| 5 min | Warm-up, then answers; dwell on warm-up question 2 (360 and 180). |
|---|---|
| 10 min | Hook, prediction for C, then the first figure. |
| 10 min | Key idea; the second figure, the centre. |
| 10 min | Worked examples; the faded example in pairs. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | (−5, −2); 360 and 180; −4. |
|---|---|
| Worked examples | (−2, 3); (−1, −1), (−4, −1), (−1, −3); (−2, −5); (−1, 4). |
| Faded example | S′ (3, 1). |
| 1. Quarter turn | A, (3, 4). B is clockwise; C is a half turn; D has the wrong sign on y. |
| 2. Half turn | −6. (6, 2) goes to (−6, −2). |
| 3. Quarter turn anticlockwise | −5. Image (−4, −5). |
| 4. Quarter turn clockwise | −4. Image (−4, −3). |
| 5. Rotate about another centre | 0. The image is (0, 6). |
| 6. Identify the turn | B. Test each rule on (4, 2): only (−y, x) gives (−2, 4). |
| 7. Two turns in a row | −7. Final image (−2, −5). |
| 8. Undo a turn | 3. The original point is (3, 6). |
| 9. Rule for a half turn | C. A is a quarter turn; B and D are reflections. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Used the clockwise rule, a half turn, or the wrong sign on y. | Sketch the quadrants and ask which one a quarter turn anticlockwise reaches. |
| 6 or −2 in question 2 | Left x alone, or swapped. | A half turn: plot (6, 2) and its opposite through the origin. |
| 5 or −4 in question 3 | Clockwise instead of anticlockwise, or the wrong half of the rule. | Write the rule (−y, x) above and read off the new y. |
| 4 or −3 in question 4 | Anticlockwise instead of clockwise, or the wrong half of the rule. | Write the rule (y, −x) above and read off the new x. |
| −3 or −2 in question 5 | Turned about the origin, or forgot the centre. | Mark the centre, then write the vector from the centre. |
| A, C or D in question 6 | Tested the wrong rule or no rule. | Substitute (4, 2) into each rule. |
| −3 or 7 in question 7 | Only one turn, or the second turn backwards. | Write the point after each turn. |
| −6 or −3 in question 8 | Took the image's x, or used the wrong sign. | Work forwards from a trial point to see which part of the rule is which. |
Questions worth asking
- Which point does not move in a rotation? The centre. Ask students to name the centre in each worked example.
- Why do 90° clockwise and 90° anticlockwise give different images but 180° does not? Listen for “180 either way goes to the same point”.
- How can you check a rotation by quadrant? A quarter turn anticlockwise moves a point from quadrant 1 to 2, 2 to 3, 3 to 4, 4 to 1.
Differentiation
A way in for anyone stuck
- Use tracing paper: trace the shape, pin the centre with a pencil, and turn.
- Draw the four quadrants with arrows showing an anticlockwise quarter turn from each one.
- Practise the origin rules with a single point before shapes.
Stretch
- Two quarter turns anticlockwise are the same as one half turn. Show it with the rules. ((x, y) to (−y, x) to (−x, −y).)
- A point at (4, 3) is rotated about (1, 1) to (−1, 4). Which rotation is it? (90° anticlockwise: the vector (3, 2) goes to (−2, 3).)
Students learning in English as an additional language
Rotation, centre, clockwise and anticlockwise are new in this sense. Pair each with a picture: a clock for clockwise, a hand drawn in the other direction for anticlockwise.
| Word | Plain meaning | Example |
|---|---|---|
| rotation | a turn about a fixed point | a wheel turning on its axle |
| centre of rotation | the point that does not move when a shape turns | the pin of a pinwheel |
| clockwise | in the direction a clock hand moves | from 12 towards 3 |
| anticlockwise | in the direction opposite to a clock hand | from 12 towards 9 |
| quarter turn | a turn of 90 degrees | from 12 to 3 on a clock |
Sentence starters for explaining aloud or in writing:
- The centre of rotation is …, the angle is … and the direction is …
- A quarter turn anticlockwise sends (x, y) to …
- I checked by quadrant: the point started in the … quadrant and ended in the …
Two things not to say
“Turn it 90.” Which way? Always say clockwise or anticlockwise, and about what.
“The rule is swap and change a sign.” It depends on the direction. Write the rule down for the turn.
“The shape rotates about its own corner.” The centre can be anywhere, inside or outside the shape.
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.
- RotateTurn (−3, 5) through 180° about the origin.
(3, −5). - Earlier work, reflectionsReflect (4, −2) in the x-axis.
(4, 2).
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- RotateTurn (2, −6) through 90° anticlockwise about the origin.
(6, 2). - Different centreTurn (3, 1) through 180° about the centre (1, 1).
(−1, 1): the vector (2, 0) becomes (−2, 0).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA point and its image under a 90° clockwise turn about the origin are (a, b) and (4, −3). Find a and b.
a = 3 and b = 4: the rule (x, y) to (y, −x) sends (3, 4) to (4, −3). - Earlier work, translationsTranslate (−1, −1) by (4, 6).
(3, 5).
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a button or slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Tracing paper and a pencil as the pin: trace the shape, turn the paper and mark the image.