Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Placing figures on a coordinate grid turns geometric moves into number patterns. Shifting a shape 5 right and 2 down adds 5 to every x-coordinate and subtracts 2 from every y-coordinate, written (x, y) → (x + 5, y - 2). Reflecting over the x-axis keeps x and flips the sign of y, (x, y) → (x, -y). A 90° counterclockwise rotation about the origin sends (x, y) to (-y, x), and a 180° rotation sends it to (-x, -y).
Dilations join the list here. A dilation centered at the origin with scale factor k multiplies both coordinates by k, so with k = 3 the point (2, -1) moves to (6, -3), and the image is three times as large. Students use these rules to find images, to identify which transformation produced a given image, and to describe the effect on the figure: rigid moves change only position or orientation, while a dilation changes size but keeps angles and shape. Working with coordinates also lets students check their drawings algebraically rather than by eye.
Reflecting over the x-axis changes y, not x. Plotting one point before and after quickly shows which coordinate flips.
Students often negate a coordinate without swapping, writing (x, y) → (-x, y), which is actually a reflection. A 90° turn swaps x and y as well.
A dilation by 2 multiplies coordinates by 2. Adding 2 to each coordinate is a translation and leaves the size unchanged.
Only angles and shape are preserved by a dilation. All lengths are multiplied by the scale factor.
Triangle DEF has vertices D(1, 2), E(3, 2) and F(1, 5). Rotate it 90° counterclockwise about the origin, then dilate the image by a scale factor of 2 centered at the origin. Give the final coordinates.
Answer: D''(-4, 2), E''(-4, 6) and F''(-10, 2).
Let students discover each rule: they plot a triangle, perform the move with tracing paper, read off the new coordinates, and look for the pattern in a table. Rules they find themselves are remembered far better than rules handed out. A quick check routine, testing the rule on the point (1, 0) and (0, 1), helps with rotations.
Assessments commonly give a figure and its image and ask for the rule, or give a rule and ask for one vertex of the image. Encourage students to track one vertex carefully rather than redrawing the whole figure.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: B) (x, y) → (-x, y)
Reflecting over the y-axis flips left and right, so x changes sign and y stays the same.
Answer: -4
A 180° rotation about the origin maps (x, y) to (-x, -y), so (4, -3) goes to (-4, 3).
Answer: 3
Multiply both coordinates by 1/2: (-2, 6) maps to (-1, 3).
Answer: A) A translation 4 left and 1 up
Adding or subtracting the same amounts from every coordinate slides the figure: 4 units left and 1 unit up.
Answer: B) (-2, 5)
The rule for 90° counterclockwise is (x, y) → (-y, x), so (5, 2) maps to (-2, 5).
Answer: 12
A dilation multiplies every length by the scale factor: 4 × 3 = 12.
A full lesson with slides, activities and an exit ticket on transformations on the coordinate plane, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.G.A.3 with an answer key, ready in about a minute.
Make a worksheet →Turn transformations on the coordinate plane into a quiz students answer online that marks itself, with a class summary for you.
Build a test →It helps to know the 90°, 180° and 270° rules about the origin, but students should also be able to rediscover them by rotating a simple point such as (1, 0).
Coordinate work in 8.G.A.3 usually uses the origin as the center, which keeps the rule (x, y) → (kx, ky) simple.