A translation slides every point of a shape the same distance in the same direction. Right adds to x, left takes away; up adds to y, down takes away. The shape does not change size or turn.
A chess piece slides across the board, a lift goes up, a character steps left on a screen. Each is a translation: every point moves the same way.
A translation slides a shape without turning or flipping it. Every point moves the same distance in the same direction, so the shape keeps its size and shape. The shape is called the object and its new position the image; a point A becomes A′ (said "A prime").
On the coordinate plane, a move right or up adds to a coordinate and a move left or down takes away. A translation of 3 right and 2 down changes (x,y) to (x+3,y−2).
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.
Where does A land?
Add to x for right, take from y for down.
TryTry it on the board and compare the result with your prediction.
Slide the triangle
The orange triangle is the image. The panel lists every corner and where it goes.
Try it yourselfYour edits are saved in this browser.
Slide the triangle until A′ is at (4, −1). Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Reading the move
Apply the same move to B.
Read Ravi's working. Which way does x go when you move left?
Ravi translates A (2, 3) by 4 left and 1 up. Click the line where his reasoning goes wrong.
ModifyChange the problem on purpose and see what happens.
Choose the move
Work out the change in x, then in y.
Need a hint?
Compare the x-coordinates of A and A′.
Another hint
Then compare the y-coordinates.
Need a hint?
The top corner C needs a y below 0.
Another hint
A′ must be to the right of the y-axis.
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Slide the triangle so that A′ is at (−3, 2).
Use the board. This ticks when you get there.
Slide the triangle into the fourth quadrant (right of the y-axis, below the x-axis).
Use the board. This ticks when you get there.
PractiseA few on your own, to check you have it.
Translations
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C is at (3, 2). After a translation of 3 right and 2 down, what is the x-coordinate of C′?
With the same translation, what is the y-coordinate of C′?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A move to the origin
Then write the translation in words, and as a rule for (x,y).
Try it yourselfYour edits are saved in this browser.
Need a hint?
Corner C is at (3, 2).
Another hint
Both coordinates must become 0.
Slide the triangle so that corner C′ lands exactly on the origin (0, 0).
Use the board. This ticks when you get there.
For teachers
Students slide a triangle on the coordinate plane and describe the move by the change in each coordinate (AC9M7SP03, node 1). About 35 minutes.
I can translate a shape and write the new coordinates.
I can describe a translation as so many units left or right and up or down.
I can say what changes and what stays the same.
Before class
Do the Run board yourself first. Students need to plot and read coordinates in all four quadrants (Year 6).
The one idea: every point moves the same way, so the same amount is added to every x and every y. The common mistake is adding for left or down (Ravi's mistake).
This is the hinge for the transformations node: a coordinate slip here carries into reflections and rotations, so check the Practise answers.
Combining two translations (the last Modify question) is a wider idea.
Running the lesson
Predict: ask for the direction of the change in x and in y separately.
Run: ask what stays the same for all three corners.
Investigate: ask Ravi's question again with 3 left and 2 down.
Modify: for the undo question, ask what the opposite of each part is.
Make: ask for the translation as a rule for (x, y).
The hard part (on purpose)
Left and down mean taking away. Ask students to say, before moving, whether x and y get bigger or smaller.
When students get stuck
A student adds when moving left or down. Ask: On the x-axis, does a number get bigger or smaller as you go left? Fix: Left and down take away; right and up add.
A student swaps the x and y changes. Ask: Which way does x measure, across or up? Fix: x is across, y is up; write the horizontal move first.
A student moves only corner A. Ask: Where does corner B go? Fix: Every corner moves the same way.
Stretch and support
Stretch: Two translations are done one after the other. Write the single translation that does the same, and show that the order does not matter.
Support: Count squares on the grid for each move before using the sliders.
Where it leads
This starts AC9M7SP03.
Where it leads: reflections (next lesson), rotations, then combining transformations and the vector notation for translations in Year 8.
What each step is for
where does A land?: Committing to the image of one point under 3 right and 2 down shows whether they add to the right coordinate in the right direction.
slide the triangle: Sliding the triangle and reading each corner's change shows that the same amount is added to every x and to every y.
reading the move: Right and up add, left and down take away (Ravi adds for left); every corner moves the same; the shape stays the same size.
choose the move: Working backwards from where a corner should end up, and from the image back to the object, shows a translation is a reversible pair of changes.
translations: Reading images, the move between two points and the quadrant, mixed; left and down not taking away is the planted slip; the quadrant review keeps coordinates fresh.
a move to the origin: Finding the translation that puts a chosen corner at the origin makes them reason from the target back to the move.
Now you can
Translate a shape and write the new coordinates.
Describe a translation as so many units left or right and up or down.