065. Translations of Shapes and Points

Learning Intentions

• To understand that a shape can be translated left, right, up or down
• draw the result of a point or shape being translated in a given direction
• describe a translation given an original point/shape and an image point/shape

Pre-requisite Summary

• Understand that a point can be located on a grid or coordinate plane
• Be able to count horizontal and vertical movements accurately
• Know that left and right describe horizontal movement
• Know that up and down describe vertical movement
• Understand that a shape is translated by sliding, not turning or flipping
• Know that the size, shape and orientation stay the same during a translation
• Be able to match corresponding points on an original shape and its image
• Be able to describe movement using words such as “$3$ units right” or “$2$ units down”

Worked Examples

Worked Example 1

a) Explain what a translation is.
b) Describe what happens to a shape when it is translated $4$ units right.
c) State what stays the same during the translation.

Worked Example 2

A point $A$ is translated $3$ units right and $2$ units up.
a) Describe the movement.
b) Draw the image point $A^{\prime }$.
c) Explain how you know the new position is correct.

Worked Example 3

A triangle is translated $5$ units left.
a) Describe how each vertex moves.
b) Draw the image of the triangle.
c) State whether the orientation of the triangle changes.

Worked Example 4

A rectangle is translated $2$ units down and $4$ units right.
a) Describe the translation.
b) Draw the image of the rectangle.
c) Explain why the translated image is congruent to the original.

Worked Example 5

Point $B$ and its image $B^{\prime }$ are shown on a grid.
a) Describe the translation from $B$ to $B^{\prime }$.
b) State the horizontal movement.
c) State the vertical movement.

Worked Example 6

A shape and its image are shown on a grid.
a) Choose one vertex and compare it with its image.
b) Describe the translation using horizontal and vertical movement.
c) Check that the same translation works for every vertex.

Problems

Problem 1

a) Explain what a translation is.
b) Describe what happens to a shape when it is translated $6$ units left.
c) State what stays the same during the translation.

Problem 2

A point $P$ is translated $4$ units left and $3$ units up.
a) Describe the movement.
b) Draw the image point $P^{\prime }$.
c) Explain how you know the new position is correct.

Problem 3

A square is translated $2$ units right.
a) Describe how each vertex moves.
b) Draw the image of the square.
c) State whether the orientation of the square changes.

Problem 4

A parallelogram is translated $3$ units up and $5$ units right.
a) Describe the translation.
b) Draw the image of the parallelogram.
c) Explain why the translated image is congruent to the original.

Problem 5

Point $C$ and its image $C^{\prime }$ are shown on a grid.
a) Describe the translation from $C$ to $C^{\prime }$.
b) State the horizontal movement.
c) State the vertical movement.

Problem 6

A shape and its image are shown on a grid.
a) Choose one vertex and compare it with its image.
b) Describe the translation using horizontal and vertical movement.
c) Check that the same translation works for every vertex.

Exercises

Understanding and Fluency

1. Describe each translation in words:
   a) $3$ units right
   b) $5$ units left
   c) $2$ units down

2. State what happens in a translation:
   a) to the size of the shape
   b) to the orientation of the shape
   c) to the position of the shape

3. Translate each point as directed:
   a) $2$ units right
   b) $4$ units up
   c) $3$ units left and $1$ unit down

4. Translate each point as directed:
   a) $5$ units right and $2$ units down
   b) $1$ unit left and $4$ units up
   c) $6$ units down

5. Draw the image of each shape after translation:
   a) a triangle translated $3$ units right
   b) a square translated $2$ units up
   c) a rectangle translated $4$ units left

6. Draw the image of each shape after translation:
   a) a pentagon translated $2$ units down
   b) a trapezium translated $5$ units right and $1$ unit up
   c) a triangle translated $3$ units left and $2$ units down

7. Describe the translation from the original to the image:
   a) $A$ to $A^{\prime }$
   b) $B$ to $B^{\prime }$
   c) $C$ to $C^{\prime }$

8. Describe the translation from the original shape to the image shape:
   a) horizontal movement only
   b) vertical movement only
   c) both horizontal and vertical movement

Reasoning

9. Explain why a translation is sometimes called a “slide”.

10. A student says a translated shape can turn as it moves. Explain the mistake.

11. Explain why corresponding vertices must all move the same distance and in the same direction in a translation.

12. A student describes a translation as “$4$ units across” without saying left or right. Explain why this description is incomplete.

Problem-solving

13. A logo is moved $6$ units right on a grid for a poster design. Describe the translation and state what stays unchanged.

14. A point on a treasure map is moved $3$ units left and $5$ units up. Describe the translation.

15. A triangle is copied and slid $4$ units down to make a pattern border. Describe how each vertex moves.

16. A classroom floor plan shows a desk shape translated $2$ units right and $3$ units down. Describe the image position relative to the original.

17. A shape and its image are shown on a coordinate grid. One vertex moves from $(2,3)$ to $(6,1)$. Describe the translation.

18. A point moves from $(-1,4)$ to $(3,4)$. Describe the translation and explain how you know there is no vertical movement.

Potential Misunderstandings

• Students may think a translation changes the size of a shape
• Students may think a translation can include turning or flipping
• Students may move different vertices by different amounts
• Students may confuse left and right on a grid
• Students may confuse up and down on a grid
• Students may describe only the distance moved and forget to include the direction
• Students may compare non-corresponding vertices when describing a translation
• Students may not recognise that the orientation of the shape remains unchanged in a translation

Next: 066. Drawing 3D Solids Accurately