Working through the lesson below will help your child to understand the effects of transformations (translations, rotations and reflections) on coordinates in a Cartesian plane.
Learning Takeaways: After this lesson, students will be able to understand the effects of:
Important Vocabulary
Isometry: An isometry is a transformation that maintains congruency. This means that the transformation does not change the figure's size or shape.
Make sure your child is familiar with the Cartesian coordinate system including the horizontal xaxis, the vertical yaxis, and the (x,y) convention used for locating points.
If you have not already done so, you may wish to review this congruency lesson with your child.
This section will help your child to translate congruent figures on a Cartesian plane. It will also introduce translation vectors which show the distance and direction of the translation.
Translations
Figure J is the preimage. Figure S is translated.
A figure is a translation if it is moved without rotation.
A Translation Vector is a vector that gives the length and direction of a particular translation. Vectors in the Cartesian plane can be written (x,y) which means a translation of x units horizontally and y units vertically.
Vectors translations can be written as shown in either of these two ways:
This vector can be said to be ray AB or vector D.
You translate a figure according to the numbers indicated by the vector. So if one point on a figure has coordinates of (3,3) and the translation vector is (1,3), the new coordinate is (4,6). You add or subtract according to the signs in the numbers in the vector.
Work through the two examples below with your child to see how to apply translation vectors.
The red square's coordinates are:
(4,3); (4,8); (9,3); (9,8).
What are the coordinate pairs under a translation vector of (1,1) as shown by the blue square?
The new coordinates are:
(3,2); (3,7); (8,2); (8,7).
You can see that the translation did not move the figure far because the vector translation is small.
We will try a different vector translation of (3,9) on the same red square
The new coordinates are:
(1,6); (1,1); (6,6); (6,1).
Get you child to try the next example. Do not look at the answer until he or she has tried to work it through the task.
Task: Translate ΔABC with a vector translation of (6,2)
Steps:
Answer and Explanation
Coordinates of ΔABC are:
(5,1); (2.5,7); (0,1)
Add 6 and subtract 2
A^{1} = (5 + 6, 1  2) = (1,1)
B^{1} = (2.5 + 6, 7  2) = (3.5,5)
C^{1} = (0 + 6, 1  2) = (6,1)
Plot ΔA^{1}B^{1}C^{1}
This section will show your child how to rotate a figure about the origin on a Cartesian plane. Work through the two examples below:
Note: Make sure your child knows that the origin is the intersection of the X and Y axes; (0,0) on the Cartesian plane.
Look at the four triangles on the Cartesian plane below. Each one is rotated about the origin as shown in the table.
Triangle B  Triangle C  Triangle D 
90° rotation of Triangle A about the origin  90° rotation of Triangle B about the origin  90° rotation of Triangle C about the origin 
180° rotation of Triangle A about the origin  180° rotation of Triangle B about the origin  

270° rotation of Triangle A about the origin 
The coordinates are:  
Triangle A  (2,2)  (10,2)  (6,8) 
Triangle B  (2,2)  (2,10)  (8,6) 
Triangle C  (2,2)  (10,2)  (6,8) 
Triangle D  (2,2)  (2,10)  (8,6) 
Note the pattern in the coordinates for corresponding vertices on the triangles. 
The coordinates of Pentagon ABCDE are: (0,4); (7,4); (9,2); (7,0); (0,0)
Pentagon ABCDE is rotated 180° about the origin making the coordinates of Pentagon A^{1}B^{1}C^{1}D^{1}E^{1:}
(0,4); (7,4); (9,2); (7,0); (0,0)
Note how the coordinates of corresponding vertices are opposite integers (just the +/ sign is different). This is always the case with 180° rotations about the origin.
This section will help your child to reflect a figure over an axis on a Cartesian plane
Work through the two reflection examples below.
The coordinates of LMNO are:
(7,5); (0,5); (2,1); (5,1)
LMNO is reflected over the Xaxis making the coordinates of L^{1}M^{1}N^{1}O^{1}:
(7,5); (0,5); (2,1); (5,1)
Note how the xcoordinates remain the same but the ycoordinates change to their opposite integer (i.e. the sign changes). This is always the case with reflections over the Xaxis.
The coordinates of PQSR are:
(8,3); (2,3); (2,6); (8,6)
PQSR is reflected over the Yaxis making the coordinates of P^{1}Q^{1}S^{1}R^{1}:
(8,3); (2,3); (2,6); (8,6)
Note how the ycoordinates remain the same but the xcoordinates change to their opposite integer (i.e. the sign changes). This is always the case with reflections over the Yaxis.
Discuss with your child how, when reflecting a figure over the xaxis, the figure remains the same distance from the yaxis and the ycoordinate therefore does not change. Similarly, when reflecting a figure over the yaxis, the figure remains the same distance from the xaxis and the xcoordinate therefore does not change.
Click the links below and get your child to try the transformation worksheets that will allow for practice with questions based on what is shown above.