In plain English: 8.G.A.3 is the Common Core grade 8 math standard that asks students to use coordinates to describe what translations, reflections, rotations and dilations do to a figure. Students apply and write rules such as (x, y) → (x + 3, y - 2) and (x, y) → (2x, 2y), and explain which moves keep a figure's size and shape. It is taught in Grade 8 Math.
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software. Also written as 8.G.3 · Official standard
In 8.G.A.1, students moved figures with tracing paper and saw that slides, flips and turns keep lengths and angles the same. This lesson puts those moves on the coordinate plane. A transformation is a rule that moves or resizes every point of a figure. The starting figure is the pre-image, and the result is the image. We name image points with a prime mark: the image of A is A′, read "A prime." Students learn a coordinate rule for each move, such as (x, y) → (x + 5, y - 2), and use it to find images without tracing.
The standard asks students to describe the effect of each move. So for every transformation, students say what happens to the coordinates, and also what happens to the figure: does it change position, size or orientation (the order its vertices, or corner points, go around, clockwise or counterclockwise)? Translations, reflections and rotations keep every length and angle, while a dilation multiplies every length by its scale factor and keeps the angles. Rotations and dilations in this lesson are centered at the origin, the point (0, 0) where the axes cross, and reflections are over the x-axis or the y-axis.
Learning Objectives
By the end of this lesson, students will be able to:
Find the image of a figure under a translation, and write the rule (x, y) → (x + a, y + b) for a given slide
Find the image of a figure reflected over the x-axis or the y-axis, and describe how the signs of the coordinates change
Find the image of a figure rotated 90° or 180° about the origin, using the rules (x, y) → (-y, x), (x, y) → (y, -x) and (x, y) → (-x, -y)
Find the image of a figure dilated from the origin, and explain that lengths are multiplied by the scale factor while angles stay the same
Identify a transformation from the coordinates of a figure and its image, and describe what stays the same and what changes
Prior Knowledge Required
Students should already be comfortable with:
Plotting points in all four quadrants (the four regions the x-axis and y-axis cut the plane into) and finding horizontal and vertical distances 6.NS.C.8
Seeing that points whose coordinates differ only in sign, such as (2, 3) and (2, -3), are mirror images across an axis 6.NS.C.6
Knowing from hands-on work that rotations, reflections and translations keep lengths and angle measures the same 8.G.A.1
Using a scale factor, the number every length is multiplied by in a scale drawing 7.G.A.1
Hand out grid paper. Students draw an x-axis and a y-axis from -6 to 6 and plot the point (4, 1).
Warm-Up Prompt
"Slide the point 5 units to the left. Where does it land? Now go back to (4, 1) and flip it over the x-axis, as if the x-axis were a mirror. Then flip (4, 1) over the y-axis. Write the three new points. Which numbers changed each time, and which stayed the same?"
The slide gives (-1, 1): only the x-coordinate changed, by 5. The flip over the x-axis gives (4, -1), and the flip over the y-axis gives (-4, 1): only the sign of one coordinate changed. Ask: "Could you find the new point without drawing, just from the numbers?" That is the goal of the lesson: a rule for each move.
Direct Instruction20 minutes
Introduce one move at a time. For each, students write the rule, draw one example on grid paper and say what stays the same.
Words to know: a transformation moves or resizes a figure. The pre-image is the figure before, the image is the figure after, and A′ ("A prime") is the image of point A.
Translation (a slide): every point moves the same distance in the same direction. Moving a units right and b units up is the rule (x, y) → (x + a, y + b). A move left or down uses a negative a or b.
Reflection (a flip) over a line, called the line of reflection: each image point is the same distance from the line as its pre-image point, on the other side. Over the x-axis, (x, y) → (x, -y). Over the y-axis, (x, y) → (-x, y).
Rotation (a turn) about a point, called the center of rotation: every point turns through the same angle. Counterclockwise means the opposite way to a clock's hands. About the origin: 90° counterclockwise is (x, y) → (-y, x), 180° is (x, y) → (-x, -y), and 90° clockwise is (x, y) → (y, -x).
Dilation (a resize) from a center point with a scale factor k: every point moves along the line from the center, to k times its distance from the center. From the origin, (x, y) → (kx, ky). A scale factor greater than 1 makes the figure larger, and one between 0 and 1 makes it smaller.
What stays the same: translations, reflections and rotations are rigid motions, moves that keep every length and every angle, so the image is congruent to the pre-image (same size and same shape). A dilation keeps the angles but multiplies every length by k.
Orientation: read the vertices in order, such as A, B, C. If they go counterclockwise in the pre-image, they still go counterclockwise after a translation, a rotation or a dilation. A reflection reverses the order to clockwise.
Translation
Triangle PQR has vertices P(-3, 1), Q(0, 1) and R(-3, 5). Translate it 5 units right and 2 units down.
Equation: The rule is (x, y) → (x + 5, y - 2). P′(2, -1), Q′(5, -1), R′(2, 3). Every point moved the same way, so PQ = P′Q′ = 3, PR = P′R′ = 4 and QR = Q′R′ = 5. Only the position changed.
Reflections over the axes
Triangle JKL has vertices J(2, 1), K(5, 3) and L(4, 6). Reflect it over the x-axis, and separately over the y-axis.
Equation: Over the x-axis, (x, y) → (x, -y): J′(2, -1), K′(5, -3), L′(4, -6). Over the y-axis, (x, y) → (-x, y): J′(-2, 1), K′(-5, 3), L′(-4, 6). J, K, L go counterclockwise, but both images go clockwise: a reflection reverses the orientation.
Rotations about the origin (Diagram 1)
Triangle ABC has vertices A(2, 1), B(5, 1) and C(5, 3). Rotate it 90° counterclockwise about the origin, and then rotate the original 180°.
Equation: 90° counterclockwise, (x, y) → (-y, x): A′(-1, 2), B′(-1, 5), C′(-3, 5). 180°, (x, y) → (-x, -y): A″(-2, -1), B″(-5, -1), C″(-5, -3). Segments OC and OC′ (O is the origin) are the same length and meet at a right angle, so C really turned 90°.
Dilation from the origin (Diagram 2)
Triangle DEF has vertices D(1, 1), E(3, 1) and F(1, 2). Dilate it by a scale factor of 3, centered at the origin.
Equation: The rule is (x, y) → (3x, 3y): D′(3, 3), E′(9, 3), F′(3, 6). DE = 2 becomes D′E′ = 6, and DF = 1 becomes D′F′ = 3. The right angle at D is still a right angle at D′. Same shape, 3 times as long.
Two moves in a row
Segment GH has endpoints G(-4, 2) and H(-1, 6). Reflect it over the x-axis, then translate it 6 units right.
Equation: After the reflection: (-4, -2) and (-1, -6). After the translation: G′(2, -2) and H′(5, -6). One rule does both: (x, y) → (x + 6, -y). GH and G′H′ both have length 5.
Use Diagram 1 to show that a rotation about the origin keeps each point the same distance from the origin. Point out that the rotation rules swap the two coordinates, and a reflection rule never does. Use Diagram 2 to show that each image vertex lies on the ray (a half-line) from the origin through its pre-image vertex.
Guided Practice15 minutes
Pairs solve each problem on grid paper. One partner applies the rule to the numbers, and the other draws both figures to check. They switch roles for each problem.
Guided practice problems with answers
Problem
Answer
Translate the triangle with vertices (-2, -1), (1, -1) and (1, 3) 3 units left and 4 units up.
Rule (x - 3, y + 4): (-5, 3), (-2, 3), (-2, 7). Same size and shape
Reflect the triangle with vertices (-3, 2), (-1, 4) and (2, 3) over the y-axis.
Rule (-x, y): (3, 2), (1, 4), (-2, 3). The orientation is reversed
Rotate the triangle with vertices (1, 2), (4, 2) and (1, 6) 90° clockwise about the origin.
Rule (y, -x): (2, -1), (2, -4), (6, -1)
Dilate the rectangle with vertices (-6, 2), (4, 2), (4, 8) and (-6, 8) by a scale factor of 1/2, centered at the origin.
Rule (x/2, y/2): (-3, 1), (2, 1), (2, 4), (-3, 4). It was 10 by 6 and is now 5 by 3; the angles are still right angles
Listen for students who mix up the two reflection rules, or who forget to swap the coordinates in a 90° rotation. Ask them to test their rule on one vertex with tracing paper.
Independent Practice15 minutes
Students work alone, then compare answers with a partner.
Independent practice problems with answers
Problem
Answer
A translation takes A(-2, 5) to A′(3, 1). Write the rule, and find the image of B(0, 2).
5 right and 4 down: (x + 5, y - 4). B′(5, -2)
Reflect the triangle with vertices (-5, -1), (-2, -1) and (-2, -4) over the x-axis.
(-5, 1), (-2, 1), (-2, 4)
Rotate the triangle with vertices (1, -3), (4, -3) and (4, -1) 180° about the origin.
(-1, 3), (-4, 3), (-4, 1)
A segment goes from (3, -2) to (6, 2) and has length 5. Dilate it by a scale factor of 3, centered at the origin. Find the new endpoints and length.
(9, -6) and (18, 6); length 15
A figure is dilated by a scale factor of 3, centered at the origin. Name one thing about the figure that changes and one thing that stays the same.
Every length becomes 3 times as long (the size changes); every angle stays the same, so the shape stays the same
Closure5-10 minutes
Exit ticket: (1) Rotate (-2, 5) 90° counterclockwise about the origin. ((-5, -2).) (2) A figure is dilated by a scale factor of 4. One side was 3 units long. How long is it now? (12 units.) (3) Name the transformation (x, y) → (x, -y), and say what it does to the point (7, 2). (A reflection over the x-axis; (7, -2).)
Differentiation Strategies
For Struggling Students
Give a reference card with the four kinds of rule and one sketch of each
Let students check every image with tracing paper: trace the figure and the axes, then slide, flip or turn the paper
Use a two-column table (pre-image, image) and fill in one vertex at a time
For Advanced Students
Ask: which single move has the same effect as a reflection over the x-axis followed by a reflection over the y-axis? Have students test it on a triangle
Have students find a rule that undoes each transformation, such as the rule that undoes (x, y) → (3x, 3y)
Extension (beyond this standard): reflect a triangle over the line y = x and find its rule, which previews transformations in high school geometry
Assessment Guidance
What to Look For
A complete answer gives the image of every vertex, names the transformation with all its details (the direction and distance of a slide, the line of a reflection, the angle and direction of a turn, the scale factor and center of a dilation) and says what stays the same. Watch for students who use the x-axis rule for a y-axis reflection, who turn the wrong way in a 90° rotation, who add the scale factor instead of multiplying by it, and who think a dilation changes the angles.
02
Classroom Activities
3 Activities
1
Rule Match Card Sort
15 minPairs
Each pair gets 18 cards: 6 rule cards, 6 description cards and 6 point cards. Every point card shows the image of the same point, Z(-4, 3). Pairs sort the cards into 6 sets of 3.
The 18 Cards
Set 1: rule card (x, y) → (x + 6, y - 2); description card "Translate 6 units right and 2 units down"; point card (2, 1)
Set 2: rule card (x, y) → (x, -y); description card "Reflect over the x-axis"; point card (-4, -3)
Set 3: rule card (x, y) → (-x, y); description card "Reflect over the y-axis"; point card (4, 3)
Set 4: rule card (x, y) → (-y, x); description card "Rotate 90° counterclockwise about the origin"; point card (-3, -4)
Set 5: rule card (x, y) → (-x, -y); description card "Rotate 180° about the origin"; point card (4, -3)
Set 6: rule card (x, y) → (2x, 2y); description card "Dilate by a scale factor of 2, centered at the origin"; point card (-8, 6)
Procedure
Shuffle all 18 cards face up.
Match each rule card with its description card first.
Apply the rule to Z(-4, 3) to find its point card. Plot Z and the six images on one grid to check.
The list above is the answer key; hand out the cards without the set numbers.
Discussion Questions
Z is 5 units from the origin. Exactly four of the images are also 5 units from the origin. Which four, and why do those moves keep the distance to the origin?
Which two rules only change signs, and which rule swaps the coordinates?
How could you tell the dilation card apart from the others without plotting?
Modification for Distance Learning
Put the cards in a shared slide deck and have pairs drag them into groups of three while they talk it through on a call.
2
Trace, Move, Measure
20 minPairs
Pairs draw triangle KLM with K(1, 1), L(5, 1) and M(1, 4) on grid paper with axes from -10 to 10. They move it four ways, first with the coordinate rule and then with tracing paper, and record what changes.
The Four Moves
Move 1: translate 6 units left and 2 units up
Move 2: reflect over the x-axis
Move 3: rotate 90° counterclockwise about the origin (hold the tracing paper at the origin with a pencil tip and turn it)
Move 4: dilate by a scale factor of 2, centered at the origin (use a ruler to draw rays from the origin through each vertex)
Recording Table
For each image, students record the three vertices, the three side lengths (measured with a ruler in grid units) and whether K, L, M go counterclockwise or clockwise.
Only one image has different side lengths. Which one, and by how much did they change?
Only one image goes clockwise. Which move caused it?
Did the tracing paper always agree with the rule? If not, find the mistake.
3
Mystery Transformations
15 minGroups of 3
Each group gets 6 mystery cards. A card lists the vertices of a triangle and of its image. Groups name the transformation, write its rule and check it on every vertex.
Card 1: translation 3 units left and 3 units down, (x - 3, y - 3)
Card 2: rotation 90° counterclockwise about the origin, (-y, x)
Card 3: reflection over the x-axis, (x, -y)
Card 4: dilation by a scale factor of 1/3, centered at the origin, (x/3, y/3)
Card 5: rotation 180° about the origin, (-x, -y)
Card 6: reflection over the y-axis, then translation 2 units up, (-x, y + 2)
Discussion Questions
On Card 5, one vertex alone also fits a reflection over the x-axis. Which vertex, and how do the other two vertices rule that reflection out?
On Card 4, how can you find the scale factor from one pair of vertices?
Why is it not enough to check a rule on only one vertex?
Challenge Variation
Each group writes its own mystery card with a two-step move, and another group has to find both steps.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Rotating a Triangle About the Origin
Triangle ABC with A(2, 1), B(5, 1) and C(5, 3), drawn to scale. Turned 90° counterclockwise about the origin, it becomes A′(-1, 2), B′(-1, 5), C′(-3, 5), using (x, y) → (-y, x). Turned 180°, it becomes A″(-2, -1), B″(-5, -1), C″(-5, -3), using (x, y) → (-x, -y). The dashed segments OC and OC′ have the same length and meet at a right angle.
Diagram 2: Dilating a Triangle From the Origin
Triangle DEF with D(1, 1), E(3, 1) and F(1, 2), dilated by a scale factor of 3 from the origin, drawn to scale. The image D′(3, 3), E′(9, 3), F′(3, 6) lies on the dashed rays from the origin. The sides 2 and 1 become 6 and 3, and the right angle stays a right angle.
04
Homework Assignment
~30 min
8.G.A.3 Homework: Transformations on the Coordinate Plane
Directions: Draw every figure and its image on grid paper. Label each image vertex with a prime mark, write the rule you used, and say what stayed the same.
Part 1: Slides, Flips and Turns (Problems 1-3)
Triangle ABC has vertices A(-4, -2), B(-1, -2) and C(-1, 2). (a) Translate it 7 units right and 3 units up, and write the rule. (b) Give the coordinates of A′, B′ and C′. (c) Find the length of AB, BC and AC and of the matching image sides.
A trapezoid has vertices (2, -1), (5, -1), (6, -4) and (1, -4). (a) Reflect it over the x-axis. (b) Reflect the original trapezoid over the y-axis. (c) In which quadrant does each image lie, and what stayed the same in both reflections?
Triangle STU has vertices S(-1, 4), T(2, 4) and U(2, 6). (a) Rotate it 90° counterclockwise about the origin. (b) Rotate the original 180° about the origin. (c) Explain why both images have the same side lengths as STU.
Part 2: Dilations and Describing the Effect (Problems 4-6)
A triangle has vertices (-8, 4), (0, 4) and (0, -2), and its sides are 8, 6 and 10 units long. Dilate it by a scale factor of 1/2, centered at the origin. Give the new vertices and the new side lengths.
Name each transformation and write its rule. (a) (3, -1) → (1, 3) and (5, 2) → (-2, 5). (b) (-2, 4) → (-6, 12) and (1, -3) → (3, -9). (c) (4, 7) → (4, -7) and (-1, 2) → (-1, -2).
Triangle PQR has vertices P(1, 2), Q(4, 2) and R(4, 4). It is reflected over the y-axis and then dilated by a scale factor of 2, centered at the origin. (a) Give the vertices after each step. (b) Write one rule for both steps together. (c) Say which of these changed: the side lengths, the angles, the orientation.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Coordinates
Every image vertex correct
One or two vertices wrong
Many vertices wrong or missing
Rules
Each rule is correct and matches the move
Rule has a sign or order error
No rule written
Drawings
Figures and images drawn to scale and labeled with prime marks
Drawn, but some labels missing
Missing or incorrect
Describing the Effect
Says correctly what stays the same (lengths, angles, orientation) and what changes
Partly correct description
No description
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
What is the image of the point (8, -3) under the translation (x, y) → (x - 4, y + 3)?
Answer: C
Subtract 4 from x and add 3 to y: (8 - 4, -3 + 3) = (4, 0). Choice A adds 4 to x instead of subtracting it. Choice B subtracts 3 from y instead of adding it. Choice D makes both sign errors.
Question 2 of 20 · Multiple Choice
Which rule moves every point 3 units left and 5 units up?
Answer: B
Left is the negative x-direction and up is the positive y-direction, so subtract 3 from x and add 5 to y. Choice A moves right instead of left. Choice D puts the 5 with x and the 3 with y. Choice C swaps the numbers too, and also turns left into right and up into down.
Question 3 of 20 · Multiple Choice
The point (-7, 4) is reflected over the x-axis. What is its image?
Answer: D
A reflection over the x-axis keeps x and changes the sign of y: (x, y) → (x, -y) gives (-7, -4). Choice A is the reflection over the y-axis. Choice C changes both signs, which is a 180° rotation. Choice B swaps the coordinates, which no reflection over an axis does.
Question 4 of 20 · Multiple Choice
A point is reflected over the y-axis. What happens to its coordinates?
Answer: A
The y-axis is the mirror, so each point moves to the same distance on the other side, left to right: (x, y) → (-x, y). Choice B describes a reflection over the x-axis. Choice C describes a 180° rotation about the origin.
Question 5 of 20 · Multiple Choice
The point (6, -5) is rotated 90° counterclockwise about the origin. What is its image?
Answer: A
Use (x, y) → (-y, x): -y = 5 and x = 6, so the image is (5, 6). Choice B is the 90° clockwise image, (y, -x). Choice C is the 180° image. Choice D swaps the coordinates but forgets to change the sign.
Question 6 of 20 · Multiple Choice
The point (-4, -9) is rotated 180° about the origin. What is its image?
Answer: D
A 180° rotation changes the sign of both coordinates: (x, y) → (-x, -y) gives (4, 9). Choice A only changes y (a reflection over the x-axis), and choice B only changes x (a reflection over the y-axis). Choice C is the 90° counterclockwise image.
Question 7 of 20 · Multiple Choice
The point (-1, 3) is dilated by a scale factor of 5, centered at the origin. What is its image?
Answer: C
Multiply both coordinates by 5: (5 × -1, 5 × 3) = (-5, 15). Choice A adds 5 instead of multiplying. Choice B multiplies only the x-coordinate. Choice D divides by 5, which would be a scale factor of 1/5.
Question 8 of 20 · Multiple Choice
A rectangle has vertices (2, 2), (10, 2), (10, 16) and (2, 16), so it is 8 units by 14 units. It is dilated by a scale factor of 1/2, centered at the origin. What are the side lengths of the image?
Answer: B
The image vertices are (1, 1), (5, 1), (5, 8) and (1, 8), so every length is multiplied by 1/2: 4 by 7. Choice A treats the dilation as a rigid motion. Choice C multiplies by 2 instead of 1/2. Choice D subtracts 1/2 from each length.
Question 9 of 20 · Multiple Choice
Which kind of transformation can change the side lengths of a figure?
Answer: D
A dilation multiplies every length by the scale factor, so any scale factor other than 1 changes the lengths. Translations, reflections and rotations are rigid motions: the image is congruent to the pre-image.
Question 10 of 20 · Multiple Choice
Triangle ABC has its vertices in counterclockwise order. After which move are the image vertices A′, B′, C′ in clockwise order?
Answer: B
A reflection flips the figure over, so the order of the vertices reverses. Rotations and translations move the figure without flipping it, so the order stays counterclockwise. Choice D is a common mix-up: a 180° turn looks upside down, but it is not flipped.
Question 11 of 20 · Multiple Choice
A transformation takes (2, 5) to (-5, 2) and (4, 1) to (-1, 4). Which transformation is it?
Answer: A
Both pairs fit (x, y) → (-y, x): (2, 5) → (-5, 2) and (4, 1) → (-1, 4). Choice B would send (2, 5) to (5, -2). Choice C would give (-2, -5), and choice D would give (-2, 5).
Question 12 of 20 · Multiple Choice
A transformation takes (-3, 1) to (-3, -1) and (2, 6) to (2, -6). Which transformation is it?
Answer: C
Each x-coordinate stays the same and each y-coordinate changes sign, which is (x, y) → (x, -y). Choice D fits the first point only: the second point moved down 12 units, and a translation moves every point the same distance. Choice A would change the sign of x instead.
Question 13 of 20 · Multiple Choice
A dilation centered at the origin takes (4, -10) to (6, -15). What is the scale factor?
Answer: C
Divide an image coordinate by the matching pre-image coordinate: 6 ÷ 4 = 3/2 and -15 ÷ -10 = 3/2. Choice B divides the other way, pre-image by image. Choice A uses the change in x (6 - 4 = 2), and choice D uses the size of the change in y (15 - 10 = 5); a dilation multiplies, it does not add.
Question 14 of 20 · Multiple Choice
The point (1, 5) is translated 4 units right and then reflected over the y-axis. What is its final image?
Answer: A
Translate first: (1, 5) → (5, 5). Then reflect over the y-axis: (5, 5) → (-5, 5). Choice B does the steps in the other order, which shows that order matters. Choice C reflects over the x-axis instead. Choice D forgets the translation.
Question 15 of 20 · Short Answer
Triangle XYZ has vertices X(-6, 0), Y(-3, 0) and Z(-3, 4). Translate it using (x, y) → (x + 2, y - 5). Give the image vertices, and compare the side lengths of XYZ and X′Y′Z′.
X′(-4, -5), Y′(-1, -5), Z′(-1, -1). XY = X′Y′ = 3, YZ = Y′Z′ = 4 and XZ = X′Z′ = 5. A translation is a rigid motion, so the side lengths do not change; only the position does.
Question 16 of 20 · Short Answer
Reflect the triangle with vertices (1, 3), (6, 3) and (1, 5) over the y-axis. Give the image vertices and describe what happens to the orientation.
Use (x, y) → (-x, y): (-1, 3), (-6, 3), (-1, 5). The side lengths (5, 2 and the slanted side) stay the same. Read in order, the vertices went counterclockwise before and go clockwise after: the reflection reversed the orientation.
Question 17 of 20 · Short Answer
Rotate the rectangle with vertices (3, 0), (7, 0), (7, 2) and (3, 2) 90° counterclockwise about the origin. Give the image vertices and its length and width.
Use (x, y) → (-y, x): (0, 3), (0, 7), (-2, 7), (-2, 3). The image is still a rectangle, 4 units by 2 units, now standing on its short side, just left of the y-axis.
Question 18 of 20 · Short Answer
Dilate the triangle with vertices (-2, 1), (2, 1) and (2, 4) by a scale factor of 3, centered at the origin. Give the image vertices and side lengths, and say what happens to the right angle.
Use (x, y) → (3x, 3y): (-6, 3), (6, 3), (6, 12). The sides 4, 3 and 5 become 12, 9 and 15, each 3 times as long. The right angle at (2, 1) is still a right angle at (6, 3): a dilation keeps every angle.
Question 19 of 20 · Short Answer
Leo says: "A 90° counterclockwise rotation about the origin takes (6, 1) to (-6, 1)." Is he right? Explain, and give the correct image.
No. (-6, 1) is the reflection of (6, 1) over the y-axis: Leo changed a sign but did not swap the coordinates. The rule (x, y) → (-y, x) gives the correct image, (-1, 6).
Question 20 of 20 · Short Answer
A figure is dilated by a scale factor of 2, centered at the origin, and then translated 1 unit left and 3 units up. Write one rule for both steps, and find the image of (3, -2).
Dilating gives (2x, 2y); translating then gives (2x - 1, 2y + 3). For (3, -2): (6, -4) after the dilation, then (5, -1).
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.G.A.3 mean?
8.G.A.3 means students use coordinates to describe what translations, reflections, rotations and dilations do to a figure. They find image points from a rule, write the rule for a given move, and say what changes (position, size, orientation) and what stays the same (lengths, angles).
What are the coordinate rules for 8.G.A.3?
There is one rule for each kind of move. Translation a units right and b units up: (x, y) → (x + a, y + b). Reflection over the x-axis: (x, y) → (x, -y); over the y-axis: (x, y) → (-x, y). Rotation about the origin: 90° counterclockwise (x, y) → (-y, x), 180° (x, y) → (-x, -y), 90° clockwise (x, y) → (y, -x). Dilation from the origin with scale factor k: (x, y) → (kx, ky).
How can students remember the rotation rules?
They can test the rule on a point on an axis. A point on the positive x-axis, such as (1, 0), turns 90° counterclockwise to (0, 1) on the positive y-axis, and only (-y, x) does that. Tracing paper turned about the origin is also a quick check.
What is the difference between a rigid motion and a dilation?
A rigid motion keeps every length and angle, and a dilation keeps the angles but changes the lengths. Translations, reflections and rotations are rigid motions, so the image is congruent to the pre-image. A dilation with scale factor k multiplies every length by k, so the image has the same shape but a different size.
What does orientation mean for a transformation?
Orientation is the order in which the vertices go around the figure, clockwise or counterclockwise. Translations, rotations and dilations keep it. A reflection reverses it, which is how students can tell a reflection from a rotation when both images look "upside down."
Do rotations in 8.G.A.3 have to be about the origin?
No, but coordinate rules are usually taught for rotations about the origin. Turns about other points are usually explored with tracing paper or geometry software in grade 8. Writing rules for any center is part of high school geometry (HSG.CO.A.2 and HSG.CO.A.5).
Why does the order of two transformations matter?
Doing the same two moves in a different order can give a different image. A slide followed by a reflection over an axis can land in a different place than the reflection followed by the slide. Students should always apply the steps in the order given and check each step.
What mistakes do students make with 8.G.A.3?
A common mistake is using the x-axis rule for a y-axis reflection, or the other way around. Others turn the wrong way in a 90° rotation, forget to swap the coordinates, or add the scale factor in a dilation instead of multiplying by it. Checking one vertex with tracing paper catches most of these.
How does 8.G.A.3 connect to 8.G.A.2 and 8.G.A.4?
8.G.A.3 gives the coordinate tools for the next two standards. In 8.G.A.2, students use sequences of rigid motions to show that two figures are congruent. In 8.G.A.4, they add dilations to show that two figures are similar, meaning same shape but possibly a different size.
How can parents help with 8.G.A.3 at home?
Draw a small triangle on graph paper and ask your child to slide it, flip it over an axis, turn it about the origin or enlarge it, and to predict the new corners before drawing. Then ask what stayed the same. Explaining why the side lengths did or did not change is the heart of this standard.
07
Related Standards
6 standards
These standards connect to 8.G.A.3: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.NS.C.6Prerequisite
Understand signs of coordinates, and points that are reflections across the axes
