Getting Started

The student is unable to correctly perform translations.

The student:

• Only identifies the coordinates of the vertices of the preimage.
  
  
  
  
• Attempts to translate the figure but does so incorrectly. The student:
  
  • Only translates the y-coordinate.
  • Is unable to correctly calculate the coordinates of the vertices of the image.
  
  

What is a translation? How are the vertices of the original image related to the vertices of the translated image?

Can you describe the translation in words? What happens to the vertices of the figure?

I see that you listed the coordinates of the vertices of triangle IJK as the coordinates of the vertices of triangle ? Would you expect these to be the same after the translation?

How can you calculate the coordinates of the vertices of the translated figure from the coordinates of the vertices of the original triangle?

What is the x-coordinate of point I? What is its y-coordinate?

Review the definition of translation as a transformation of the plane. Explain that a translation along a vector, v, assigns to each point, P, in the plane an image point, , so that the distance from to P corresponds to the magnitude (length) of vector, v, and the direction of  from P corresponds to the direction of vector, v. Guide the student to understand how the rule given in the first problem describes the translation in terms of a horizontal component (e.g., x – 3) and a vertical component (e.g., y + 5). Use grid paper to illustrate translations of points and to demonstrate the relationship between a point, its image, and the vector that defines the translation. Then illustrate translations of more complex figures such as segments, angles, and polygons.

Emphasize the basic properties of rigid motions:

• Lines are taken to lines, and line segments to line segments of the same length.
• Angles are taken to angles of the same measure.
• Parallel lines are taken to parallel lines.

Discuss how these properties ensure that the image of a figure under a rigid motion is always congruent to the preimage.

To develop an intuitive understanding of transformations, allow the student to experiment with a variety of transformations using transparent paper or interactive websites such as: http://www.mathopenref.com/translate.html.

Provide opportunities to perform translations of figures in the coordinate plane described algebraically and by vectors.

Making Progress

The student incorrectly labels or calculates a value of an ordered pair.

The student:

• Uses a counting method to locate the vertices of the translated figure but makes an error in identifying the coordinates.
  
  
  
  
• Reverses the order of the coordinates in one ordered pair but writes all other ordered pairs correctly.
• Incorrectly calculates one coordinate but all other coordinates are calculated correctly.

How did you translate the figure? Did you calculate the coordinates of the vertices of the image or did you use a counting method?

I think you may have made a small error when you determined the coordinates of the vertices of the translated figure. Can you check your work to find the error?

Provide specific feedback to the student concerning any error made. Allow the student to revise his or her work to correct any errors. Provide additional opportunities to perform translations of figures in the coordinate plane described algebraically and by vectors.

Got It

The student provides complete and correct responses to all components of the task.

The student correctly describes and labels the coordinates of the images of triangle IJK. The student may or may not plot the images.

1. (1, 5) (-1, 1), (-5, 9)
2. (-4, 6) (-6, 2), (-10, 10)

What would the image look like on the coordinate grid if you graphed it?

Can you determine a rule for the translation using the vector in the second problem?

Introduce the concept of congruence in terms of rigid motions. Ask the student to determine if two figures are congruent by describing a sequence of rigid motions that map one figure onto the other.

Consider implementing the MFAS tasks Dilation Coordinates, Rotation Coordinates, and Reflection Coordinates (8.G.1.3).