Learning Translations
Explore how shapes move across a coordinate plane
Welcome to Translations!
In this lesson, you'll learn how to translate shapes on a coordinate grid.
Translation means moving every point of a shape the same distance and in the same direction.
By the end of this activity, you will be able to:
- Understand what translation means in mathematics
- Translate shapes using coordinate notation
- Describe translations using vectors
- Solve problems involving translations
Step 1: Understanding Coordinates
Before we start translating shapes, let's review coordinate points.
A point on a coordinate grid is written as (x, y):
- x is the horizontal position (left to right)
- y is the vertical position (bottom to top)
Interactive Simulation
Try this interactive coordinate explorer:
- Click anywhere on the grid to place a blue point
- Drag the blue point to see how coordinates change
- Try to place it at specific coordinates
- Find and note the coordinates of the red target point
The coordinates of your blue point will display in real-time.
Blue point: Click on grid to place
Your turn! Let's identify some points:
Count how many units right (x) and up (y) from the origin (0,0).
Step 2: How Points Move in a Translation
Before we look at translating shapes, let's understand how individual points move.
In a translation:
- Every point moves the same distance
- Every point moves in the same direction
- The shape doesn't rotate or change size
For example, if we move a point 3 units right and 2 units up:
Point (x, y) → Point (x + 3, y + 2)
Interactive Simulation
Try this interactive point translator:
- Adjust the x-translation (horizontal) and y-translation (vertical) sliders
- Watch how the blue point moves to create a new translated red point
- Notice the vector arrow showing the direction and distance of movement
- Try different combinations to see their effects
This simulation helps you visualize how coordinates change during a translation.
Your turn! Complete the following:
Left means subtract from x, down means subtract from y.
Step 3: Translation Vectors
A translation moves every point of a shape the same distance and in the same direction.
We describe translations using vectors. A vector is written as:
Where:
- a is the horizontal movement (+ for right, - for left)
- b is the vertical movement (+ for up, - for down)
For example, a translation by the vector (3, 2) means:
- Move 3 units to the right
- Move 2 units up
Interactive Simulation
Try this vector translation simulator:
- Use the controls to set the vector components
- Click "Apply Vector" to see the translation
- Watch how the entire triangle moves according to the vector
- Notice that all points in the shape move the same distance and direction
Try to find the vector that matches the question below.
Your turn! Watch the animation and answer:
Find how many units right/left and up/down each point moves from its original position.
Step 4: Applying Translations
Now, let's try applying a translation to a shape ourselves.
Remember, when we translate a shape, we move every point by the same amount and in the same direction.
For each point (x, y) in the original shape, the new coordinates will be:
New position = (x + a, y + b) where (a, b) is the translation vector
Interactive Simulation
Try this interactive triangle translator:
- See the original triangle with vertices at A(-1,1), B(1,1), and C(0,3)
- Use the "Translate Step by Step" button to move through the translation process
- Watch as each vertex is calculated and the new triangle appears
- The vector for this translation is (4, -3)
Use the simulation to help you calculate the new coordinates in the exercise below.
Translation Vector: (4, -3)
Click "Translate Step by Step" to begin
Your turn! Translate the triangle by the vector (4, -3):
The original triangle has vertices at A(-1,1), B(1,1), and C(0,3).
Calculate the new coordinates:
Add the vector (4, -3) to each point. For point A(-1,1), calculate (-1+4, 1+(-3)).
Step 5: Translation Challenge
Now let's try a more challenging problem.
This time, you'll need to find the translation vector that moves a shape from one position to another.
Interactive Simulation
Try this interactive vector finder:
- Look at the blue rectangle (A) and pink rectangle (B)
- Use the sliders to adjust your guess for the translation vector
- Watch how the orange rectangle (your prediction) moves
- Try to perfectly overlay the orange rectangle with the pink rectangle (B)
- When they match, you've found the correct vector!
Click the vertices to see their exact coordinates.
Find a matching point on both rectangles. Subtract the first position coordinates from the second position coordinates.
Step 6: Create Your Own Translation
Now it's your turn to create and apply your own translation!
You'll see a shape on the grid. Choose a translation vector to move it to a new position.
Interactive Simulation
Create your own translations:
- Enter x and y values for your translation vector
- Click "Apply" to see the shape move
- Click "Show Path" to see the translation path
- Try multiple translations to create interesting patterns
- Click "Reset" to start over
Challenge: Create a sequence of translations that returns the shape to its starting position!
Well Done!
Congratulations on completing this lesson on translations!
Here's what you've learned:
- A translation moves every point of a shape the same distance in the same direction
- Individual points move according to the translation vector
- Translations are described using vectors (a, b)
- To translate a point (x, y) by vector (a, b), the new coordinates are (x+a, y+b)
- Translations preserve the size and shape of the original figure
Key Points to Remember:
- In a translation, the original shape and the translated shape are congruent (same shape and size)
- The vector (a, b) means: move a units horizontally and b units vertically
- Positive a means move right, negative a means move left
- Positive b means move up, negative b means move down