8th Grade · 8.G.A.1

Rigid transformations in 8th grade

Rigid transformations are ways to move a shape without changing its size or shape. In 8th grade, your child learns the three kinds of rigid transformations: rotations, reflections and translations (turns, flips and slides). Your child moves shapes on the coordinate plane and checks what stays the same. The coordinate plane is the grid with an x-axis going across and a y-axis going up and down.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations:
8.G.A.1.a Lines are taken to lines, and line segments to line segments of the same length.
8.G.A.1.b Angles are taken to angles of the same measure.
8.G.A.1.c Parallel lines are taken to parallel lines.

What it means, in plain words

A transformation is a rule that moves every point of a shape to a new place. The shape you start with is the original, and the shape after the move is the image. The image of point $A$ is usually written $A'$ , read "A prime".

A translation slides the shape: every point moves the same distance in the same direction. A reflection flips the shape over a line, like a mirror image. A rotation turns the shape around a fixed point, often the origin $(0, 0)$ , by an angle such as $90^\circ$ or $180^\circ$ .

These three moves are called rigid transformations because the shape stays rigid, like a piece of cardboard. Children check this by experiment: they trace a shape, move the tracing and measure. Every side keeps its length and every angle keeps its size. Parallel lines (lines that stay the same distance apart and never meet) are still parallel after the move. Most work with rigid transformations in 8th grade happens on a coordinate grid, where your child moves single points and whole shapes.

Your child should be able to:

Tell a translation, a reflection and a rotation apart from a before-and-after picture
Translate a point on a grid, for example 4 units right and 2 units down
Reflect a point across the x-axis or the y-axis
Explain that side lengths, angle sizes and parallel lines stay the same after the move

Example

The point $(4, 2)$ is reflected across the y-axis. What is its image?

A reflection across the y-axis flips the point to the other side of that axis, at the same height.
The x-coordinate changes sign, so 4 becomes $-4$ . The y-coordinate stays 2.
The image is the point $(-4, 2)$ .

Example

Point $P$ is at $(-3, 1)$ . It is translated 5 units right and 3 units up. Where does it land?

Moving right adds to the x-coordinate: $-3 + 5 = 2$ .
Moving up adds to the y-coordinate: $1 + 3 = 4$ .
The image $P'$ is at $(2, 4)$ .

Where children usually get stuck

Asked to reflect $(4, 2)$ across the y-axis, your child writes $(4, -2)$ .
The two axes get mixed up. A reflection across the y-axis moves the point left or right, so the x-coordinate is the one that changes, which feels backwards at first.
What helps. Plot the point on graph paper and fold the paper along the y-axis. The fold shows where the point lands: on the left side, at the same height.
Your child translates "3 units down" by changing the x-coordinate, or by adding 3 to the y-coordinate.
The directions are not yet tied to the coordinates. Left and right change $x$ . Up and down change $y$ , and down means subtract.
What helps. Have your child count the moves on graph paper with a finger first, then write the coordinates of the point the finger stopped on.
Your child calls a turned shape a reflection, or thinks a flipped shape got bigger.
Flips and turns can look alike on paper, and a shape facing a new direction can look like a different size.
What helps. Trace the shape on thin paper and move the tracing onto the image. If you had to turn the paper over, it was a reflection. If you only turned it, it was a rotation. The tracing also shows the size did not change.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
The point below is reflected across the y-axis. Which ordered pair is the image?
$\displaystyle (3, 5)$
A
$\displaystyle (-3, 5)$
B
$\displaystyle (3, -5)$
C
$\displaystyle (-3, -5)$
D
$\displaystyle (3, 5)$
Show the answer and how to solve it
The answer is (-3, 5) because reflecting across the y-axis changes 3 to -3 and keeps 5 the same.
1. 1Look at the point shown.
2. 2A reflection across the y-axis changes the sign of the x-coordinate and keeps the y-coordinate the same.
3. 3So the image is (-3, 5). That is the correct choice.
Common mistake. A common mistake is changing the y-coordinate instead of the x-coordinate. That gives (3, -5), which is a reflection across the x-axis, not the y-axis.
The idea underneath. A reflection across an axis only changes the coordinate that measures distance from that axis; the other coordinate stays the same.
2On level
After translating point P 4 units right and 2 units down, which ordered pair gives the coordinates of its image?
A
$\displaystyle (-6,1)$
B
$\displaystyle (0,-1)$
C
$\displaystyle (2,1)$
D
$\displaystyle (2,5)$
Show the answer and how to solve it
The image of point P is (2, 1) because the translation adds 4 to x and subtracts 2 from y.
1. 1Look at point P on the grid. Its coordinates are (-2, 3).
2. 2Moving 4 units right changes only the x-coordinate.
3. 3Moving 2 units down changes only the y-coordinate.
4. 4Put the new x- and y-coordinates together.
Common mistake. A common mistake is moving down by adding 2 to the y-coordinate. On a coordinate grid, down means the y-value gets smaller, so 3 becomes 1, not 5.
The idea underneath. A translation changes the two coordinates independently: horizontal movement changes only x, and vertical movement changes only y.
3On level
Point P is translated 4 units right and 3 units down. A student claims the image is P'. Which statement best describes the student's work?
A
The horizontal change is wrong.
B
The vertical change is wrong.
C
The x- and y-changes were swapped.
D
Both directions were reversed.
E
The image is correct.
Show the answer and how to solve it
The correct choice is "The vertical change is wrong." The point moved 4 units right correctly from x = -3 to x = 1, but moving 3 units down from y = 2 should give y = -1, not 5.
1. 1Read the points on the grid. P is at (-3, 2) and the student's claimed image P' is at (1, 5).
2. 2The translation says 4 units right, so add 4 to the x-coordinate.
3. 3The translation says 3 units down, so subtract 3 from the y-coordinate.
4. 4The correct image is (1, -1). The x-coordinate in the student's claim is right, but the y-coordinate is 5 instead of -1. The student moved up 3 instead of down 3, so the vertical change is wrong.
Common mistake. A common mistake is to add 3 to the y-coordinate instead of subtracting 3. On a coordinate grid, moving down makes the y-value smaller.
The idea underneath. A translation changes x and y independently: horizontal motion affects only x, and vertical motion affects only y.
4Stretch
Use the work below. Which student is correct, and why?
$\displaystyle \begin{array}{l}\text{Original point: } P(5,0)\\\text{Reflect } P \text{ across the x-axis.}\\\text{Ava says } P' = (5,0)\\\text{Ben says } P' = (-5,0)\end{array}$
A
Ben is correct because reflecting across the x-axis changes only the x-coordinate.
B
Neither student is correct because a reflection always moves the point to a new location.
C
Neither student is correct because reflecting across the x-axis should swap the coordinates.
D
Ava is correct because a point on the x-axis stays fixed when reflected across the x-axis.
Show the answer and how to solve it
Ava is correct because P(5,0) lies on the x-axis, so reflecting across the x-axis leaves it fixed at P'(5,0).
1. 1The point is P(5,0) and it is being reflected across the x-axis. Notice that P lies directly on the x-axis because its y-coordinate is 0.
2. 2A reflection across the x-axis keeps the x-coordinate the same and changes the sign of the y-coordinate.
3. 3Apply the rule to P(5,0). The y-coordinate is 0, and the opposite of 0 is still 0. The image lands on the same point as the original.
4. 4Ava says P' = (5,0), which matches the computed image. Ben says P' = (-5,0), which negated the x-coordinate instead—that is the rule for reflecting across the y-axis, not the x-axis.
Common mistake. A common mistake is assuming every reflection must move the point. When a point lies on the reflecting axis, its distance from that axis is 0, so the image lands right back on the original point.
The idea underneath. The coordinate that measures distance from the reflecting axis changes sign, while the coordinate parallel to that axis stays the same.