How to explain it
The anchor students hold onto: For a translation by (dx, dy), every point (x, y) maps to (x + dx, y + dy). Image vertices take primed names: A maps to A-prime, B to B-prime.
Students extend translation thinking into reflections, rotations, and dilations, then compose multiple rigid motions to prove congruence and similarity in the transformations strand.
Worked examples
Example 1
— Translate a point
Translate (1, 2) by (4, -3).
Step 1Add dx to x: 1 + 4 = 5.
Step 2Add dy to y: 2 + (-3) = -1.
AnswerImage: (5, -1).
Example 2
— Translate a triangle
Translate ABC by (2, 4).
Step 1Apply (2, 4) to each vertex.
Step 2A'(2, 4); B'(5, 4); C'(3, 6).
AnswerA'(2, 4); B'(5, 4); C'(3, 6).
Example 3
— Find the vector
A(1,3) to A'(5,1). Find vector.
Step 1dx = 5 - 1 = 4.
Step 2dy = 1 - 3 = -2.
AnswerTranslation vector: (4, -2).
Common mistakes
What students write
Treating negative dy as moving left.
The fix
In (3, -2), dx = 3 moves right and dy = -2 moves DOWN. dx is horizontal; dy is vertical.
Try this
A student translated A(-2, 3) by the vector (4, -5) and recorded the image as A'(2, 8). Identify the error and find the correct image.
What students write
Forgetting to apply the vector to every vertex.
The fix
Translate each vertex individually using (dx, dy). All vertices shift the same way — image is congruent to the preimage.
Try this
A student translated B(5, -1) by the vector (-3, 2) and recorded the image as B'(8, 1). Identify the error and find the correct image.
Teacher tip
Head off the two predictable errors before they happen. First: In (3, -2), dx = 3 moves right and dy = -2 moves DOWN. dx is horizontal; dy is vertical. Second: Translate each vertex individually using (dx, dy). All vertices shift the same way — image is congruent to the preimage.