A rotation turns a shape about a centre by an angle in a direction. About the origin, a quarter turn anticlockwise changes (x, y) to (−y, x) and a half turn changes it to (−x, −y).
The hands of a clock, a Ferris wheel, a spinning fan. Each turns about a centre by a certain angle, in a certain direction.
A rotation turns a shape about a fixed point, the centre of rotation. To describe one, give the centre, the angle and the direction (clockwise or anticlockwise). The shape keeps its size and shape.
About the origin, an anticlockwise turn of 90° changes (x,y) to (−y,x); 180° (a half turn) changes it to (−x,−y); 270° changes it to (y,−x).
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
A quarter turn
Picture a clock hand pointing right, turning up.
TryTry it on the board and compare the result with your prediction.
Turn the triangle
The black dot is the centre of rotation. The sliders turn the shape anticlockwise.
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Turn the triangle 90° anticlockwise. Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
How to turn a point
Both signs change.
Read Lee's working. Which way does P turn?
Lee turns P (3, 1) by 90° anticlockwise about the origin. Click the line where his reasoning goes wrong.
ModifyChange the problem on purpose and see what happens.
Change the turn and the centre
Need a hint?
A is at (2, 1).
Another hint
Three quarter turns anticlockwise is 270°.
Need a hint?
The centre is now (1, 0), not the origin.
Another hint
A is one right and one up from the centre. A quarter turn takes it to one left and one up.
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Turn the triangle so that A′ lands at (1, −2).
Use the board. This ticks when you get there.
The centre of rotation is (1, 0). Turn the triangle 90° anticlockwise. Where does A land?
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Rotations
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B is at (4, 1). After a 180° turn about the origin, what is the x-coordinate of B′?
The point (3, 1) turns 90° anticlockwise about the origin. What is the y-coordinate of the image?
Review: the point (3, 5) is reflected in the x-axis. What is the y-coordinate of the image?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
Turn to a target
Then describe the rotation fully: the centre, the angle and the direction. How else could you say it, turning clockwise?
Try it yourselfYour edits are saved in this browser.
Need a hint?
Corner B is at (4, 1).
Another hint
Try each turn and read B′.
Turn the triangle so that B′ is at (1, −4).
Use the board. This ticks when you get there.
For teachers
Students turn a shape about a centre and find the image of each corner for quarter and half turns (AC9M7SP03, node 3). About 35 minutes.
I can rotate a shape 90°, 180° or 270° about the origin.
I can describe a rotation by its centre, angle and direction.
I can use the pattern in the coordinates to check an answer.
Before class
Do the Run board yourself first. Students need translations and reflections (previous lessons).
The one idea: a rotation needs a centre, an angle and a direction. About the origin, an anticlockwise quarter turn is (x,y)→(−y,x); the half turn changes both signs. The common mistake is only swapping the coordinates (Lee's mistake).
The board turns anticlockwise only; clockwise turns are the questions in Modify and Make (90° clockwise is 270° anticlockwise).
Rotation is the hardest transformation to reason about: let students trace and turn before using the rule. The rule is a check on the answer, not the method.
Turning about another centre (the second Modify task) is done by the picture: the rule does not apply.
Running the lesson
Predict: ask students to point a pencil and turn it a quarter.
Run: ask what stays the same after each turn (the distance to the centre).
Investigate: ask how Lee could test his answer.
Modify: for the centre (1, 0), ask where A is measured from.
Make: ask for the description with the direction.
The hard part (on purpose)
Picturing the turn. Use tracing paper pinned at the centre, or have students point a pencil at the shape and turn it.
When students get stuck
A student swaps the coordinates without a sign. Ask: Which quadrant should the image be in? Fix: Picture the turn first, then use the rule to check.
A student turns clockwise. Ask: Which way do clock hands go? Which way do we turn here? Fix: The board turns anticlockwise; clockwise 90° is 270° anticlockwise.
A student forgets the centre. Ask: What point does the shape turn about? Fix: Always name the centre, the angle and the direction.
Stretch and support
Stretch: Show that reflecting in the x-axis and then the y-axis gives the same image as a 180° rotation about the origin. Which two reflections (one in the line y = x) make a 90° rotation?
Support: Use tracing paper: trace the shape, pin the centre and turn the paper a quarter at a time.
Where it leads
This completes AC9M7SP03.
Where it leads: combining transformations and tessellations (Year 8); rotational symmetry.
What each step is for
a quarter turn: Committing to the image of a point after a quarter turn shows whether they swap coordinates or picture the turn.
turn the triangle: Turning the triangle in quarter steps and reading the corners shows how each coordinate changes for 90°, 180° and 270°.
how to turn a point: A quarter turn swaps and changes one sign, depending on the direction (Lee only swaps); a rotation needs a centre, an angle and a direction.
change the turn and the centre: Choosing an angle to reach a target, then moving the centre, shows the image depends on the centre; clockwise and anticlockwise turns are linked.
rotations: Half turns, finding the turn between two points and a reflection review, mixed; swapping without a sign is the planted slip; the review keeps mirrors fresh.
turn to a target: Finding the turn that puts a chosen corner on a target applies the rule and then states the full description: centre, angle and direction.
Now you can
Rotate a shape 90°, 180° or 270° about the origin.
Describe a rotation by its centre, angle and direction.
Use the pattern in the coordinates to check an answer.