math-2017:8.G.1 Verify experimentally the properties of rotations, reflections, and translations (include examples both with and without coordinates).
math-2017:8.G.1a Lines are taken to lines, and line segments are taken to line segments of the same length.
math-2017:8.G.1b Angles are taken to angles of the same measure.
math-2017:8.G.1c Parallel lines are taken to parallel lines.
Provisional targets. Clause-by-clause educational review is still pending.
📘Section 1 · LearnLearn the idea
Triangle ABC with A(1, 1), B(5, 1), C(2, 4) is translated by the rule (x, y) → (x + 3, y + 2) to A′(4, 3), B′(8, 3), C′(5, 6). AB and A′B′ are both 4 units; AC and A′C′ both run 1 right and 3 up; the angle at A and at A′ is about 72°. Three parallel arrows of equal length join A to A′, B to B′ and C to C′, each 3 units right and 2 up, showing that every point moves the same way. Axis ticks are one unit apart.
Learning goal: Perform translations, reflections and rotations on paper with tracing paper or a transparency and on a coordinate grid, and verify experimentally that each one takes lines to lines, takes line segments to line segments of the same length, takes angles to angles of the same measure and takes parallel lines to parallel lines.
A transformation is a rule that moves every point of the plane to a new position. The original figure is the pre-image and the result is the image. Three transformations are called rigid motions because they move a figure without stretching, shrinking or bending it. Each one needs its own information to be fully described, and an instruction that leaves that information out is not a transformation at all.
A translation slides every point the same distance in the same direction. To describe one you need a direction and a distance, which on a grid is a pair of numbers such as "3 units right and 2 units up". Every point of the figure moves by exactly that pair, not only the vertices you happen to label; the points along each side move too, which is why the sides arrive intact.
A reflection flips the figure across a line called the line of reflection. Each point and its image are the same distance from that line, on opposite sides, and the segment joining a point to its image is perpendicular to the line. A point that lies on the line does not move. "Reflect the triangle" without naming the line is incomplete: reflecting across the x-axis and reflecting across the y-axis give different images.
A rotation turns the figure about a fixed point, the center, through a given angle in a given direction, clockwise or counterclockwise. The center stays put; every other point travels along a circle centered there. A rotation of 90° clockwise about the origin and a rotation of 90° clockwise about a vertex of the figure produce different images, so all three pieces of information, center, angle and direction, are required.
What "verify experimentally" means. The standard does not ask you to accept these properties on trust. Trace a figure on tracing paper or a transparency, move the copy by sliding, flipping or turning it, and then measure the pre-image and the image with a ruler and a protractor. Then repeat the same transformation with coordinates, where lengths and slopes can be computed exactly. Doing both matters: the physical experiment shows the properties hold for shapes you can draw, and the coordinate calculation shows why the measurements agree.
Property (a): lines are taken to lines, and line segments to line segments of the same length. Take triangle ABC with A(1, 1), B(5, 1) and C(2, 4). Translate it 3 units right and 2 units up. The coordinate rule is (x, y) → (x + 3, y + 2). Check one point completely: A(1, 1) → (1 + 3, 1 + 2) = A′(4, 3). In the same way B′ = (8, 3) and C′ = (5, 6). Side AB runs from (1, 1) to (5, 1); it is horizontal, with length 5 − 1 = 4 units. Side A′B′ runs from (4, 3) to (8, 3); it is also horizontal, with length 8 − 4 = 4 units. Side AC is not horizontal, so count its run and rise: from A to C is 1 unit right and 3 units up. From A′(4, 3) to C′(5, 6) is also 1 right and 3 up. Same run and same rise means the same length, because both segments are the diagonal of the same 1-by-3 box, and the same steepness. Laid over its image, the tracing of the triangle matches edge for edge. A translation cannot bend a segment into a curve, because every point of the segment moves by the same shift and so the points stay in one straight row.
Property (b): angles are taken to angles of the same measure. The angle at A is formed by the ray AB, heading 4 right and 0 up, and the ray AC, heading 1 right and 3 up. At A′ the rays A′B′ and A′C′ head in exactly the same two directions, so the opening between them is the same angle. A protractor on the drawing gives about 72° at both vertices. Rotations do change the direction each ray points, but a rotation turns both rays of the angle by the same amount, so the opening between them does not change. That is what "the angle is preserved" means.
Property (c): parallel lines are taken to parallel lines. Two distinct lines are parallel when they never meet; on a coordinate grid that means they have the same slope (or are both vertical). Take line m through (0, 0) and (4, 2), slope 2/4 = 1/2, and line n through (1, 3) and (5, 5), slope 2/4 = 1/2. Rotate both lines 90° counterclockwise about the origin. The rule is (x, y) → (−y, x); check it on one point: (4, 2) → (−2, 4). The image of m passes through (0, 0) and (−2, 4), slope 4/(−2) = −2. The image of n passes through (−3, 1) and (−5, 5), slope (5 − 1)/(−5 − (−3)) = 4/(−2) = −2. Both slopes changed from 1/2 to −2, but they changed together, so the images are still parallel. Reflecting the original lines across the x-axis, (x, y) → (x, −y), sends both slopes to −1/2, and again the images are parallel.
Why the properties hold, informally. In a translation every point shifts by the same amount, so the distance between any two points is unchanged. In a reflection, folding the paper along the line of reflection lays each point exactly on its image, and folded paper does not stretch. In a rotation, a transparency pinned at the center turns as one rigid sheet, so no two points on it move closer together or farther apart. Whatever keeps all distances fixed also keeps straightness, lengths, angle measures and parallelism fixed, because each of those is built from distances.
The misconception. Many students believe a reflection reverses lengths or that a rotation shrinks the parts far from the center. Neither happens. What a reflection changes is orientation: reading the vertices A, B, C around the pre-image counterclockwise, you read A′, B′, C′ around the image clockwise. Orientation is not one of the three preserved properties, which is why a reflected figure can look "backwards" while every length and angle in it is unchanged.
Without coordinates. Cut a scalene triangle from card. Trace it, then slide the card, flip it over and turn it, tracing it each time. Measure each side of every tracing with a ruler and each angle with a protractor. Within ordinary measuring precision, about one millimeter and one degree, the measures match. Small discrepancies of that size are measurement error, not evidence against the property; a discrepancy of a centimeter or ten degrees would mean the tracing slipped.
✏️Section 2 · ModelWork through the examples
Worked reflection with coordinates: Segment PQ has P(2, 1) and Q(6, 4). Reflect it across the y-axis, rule (x, y) → (−x, y). P′ = (−2, 1) and Q′ = (−6, 4). Length of PQ: run 4, rise 3, so the segment is the diagonal of a 3-by-4 box and has length 5. Length of P′Q′: from x = −2 to x = −6 is a run of 4 (leftward), rise 3, again length 5. The slope changed from 3/4 to 3/(−4) = −3/4, so the direction flipped, but the length did not. A segment is taken to a segment of the same length.
Worked rotation of an angle: An angle has its vertex at the origin and rays through (4, 0) and (3, 3). The second ray goes equally far right and up, so the angle is 45°. Rotate 90° counterclockwise about the origin with (x, y) → (−y, x): (4, 0) → (0, 4) and (3, 3) → (−3, 3). The image rays point straight up and up-and-left at 45° from vertical, so the image angle is still 45°. The angle is taken to an angle of the same measure even though both rays now point somewhere else.
Worked parallel lines under a translation: Line p passes through (0, 1) and (2, 5), slope 4/2 = 2. Line q passes through (3, 0) and (4, 2), slope 2/1 = 2. Translate both by (x, y) → (x − 1, y + 4). The image of p passes through (−1, 5) and (1, 9), slope 4/2 = 2. The image of q passes through (2, 4) and (3, 6), slope 2. Same slopes, so the images are parallel; a translation does not even change the slope, only the position.
Worked tracing-paper check, no coordinates: A triangle has sides measured at 4.0 cm, 5.5 cm and 7.0 cm and angles 35°, 52° and 93° (to the nearest degree, and the three sum to 180°). Its tracing is turned a quarter turn about one vertex and traced again. The new tracing measures 4.0, 5.5 and 7.0 cm and 35°, 52° and 93°. The vertex used as the center is the one point that did not move; every other point moved along a circle around it, and no length or angle changed.
🧩Section 3 · DoPractice, demonstrate, and explain
Recording task: On grid paper draw the trapezoid with vertices (−3, 1), (0, 1), (−1, 3) and (−3, 3). Rotate it 180° about the origin with the rule (x, y) → (−x, −y) and record a two-column table of pre-image and image vertices: (3, −1), (0, −1), (1, −3), (3, −3). Measure the two horizontal sides of the pre-image, 3 units and 2 units, and the corresponding sides of the image, and record that they are still horizontal, still 3 and 2 units, and still parallel to each other. Then trace the pre-image on tracing paper, pin it at the origin, turn it half a turn and confirm that it lands on the image you computed.
Independent performance: Choose your own triangle with no two sides equal and vertices on grid points, and draw one extra line through the third vertex parallel to one side, so that property (c) can be checked (a triangle on its own has no parallel sides). Apply one translation, one reflection across an axis and one rotation about the origin, stating the full description of each (direction and distance; line of reflection; center, angle and direction). For every transformation, record all three image vertices, compute one side length from its run and rise before and after, measure one angle with a protractor before and after, and state whether the orientation (clockwise or counterclockwise reading of the vertices) changed. Success criteria: every image coordinate is correct when checked against the rule; every recorded length is equal to its pre-image length; every recorded angle matches within one degree; the image of the extra line has the same slope as the image of the side it was parallel to (or both are vertical); the orientation column shows a change only for the reflection; and a closing sentence names which of properties (a), (b) and (c) each measurement verified.
✅Section 4 · Check8-question lesson practice test
An adult may read the prompts aloud. Answer independently with words, objects, drawings or a demonstration before opening the answer guide. Award 1 point for a correct result or claim and 1 point for a matching explanation or demonstration; where a question asks only for a result (a corrected sentence, a spelling, a count), a fully correct result earns both points, and the answer guide shows what a complete response contains. Record prompting separately. The total (two points per question) is local practice feedback, not an Ohio proficiency level.
Problem 1
Triangle with vertices (0, 0), (4, 0) and (0, 3) is translated by the rule (x, y) → (x − 2, y + 5). Give the image vertices, the length of the longest side before and after, and check whether the right angle at (0, 0) is still a right angle in the image.
Show worked solution
Apply the rule to each vertex: (0, 0) → (−2, 5), (4, 0) → (2, 5), (0, 3) → (−2, 8). The longest side of the pre-image joins (4, 0) and (0, 3): run 4, rise 3, length 5. Its image joins (2, 5) and (−2, 8): run 4, rise 3, length 5. At the image vertex (−2, 5) one side goes to (2, 5), which is horizontal, and the other to (−2, 8), which is vertical, so the angle there is still a right angle. A translation takes segments to segments of the same length and angles to angles of the same measure, and the arithmetic confirms both.
Problem 2
Reflect the point (3, −2) across the x-axis and, separately, across the y-axis. Explain how you know each image is correct.
Show worked solution
Across the x-axis the x-coordinate stays and the y-coordinate changes sign: (3, 2). The point is 2 units below the axis and its image is 2 units above, on the perpendicular through x = 3, which is what a reflection requires. Across the y-axis the y-coordinate stays and the x-coordinate changes sign: (−3, −2). The point is 3 units right of the y-axis and its image 3 units left of it, at the same height.
Problem 3
The segment from (1, 2) to (5, 2) is rotated 90° clockwise about the origin. Find the image of its endpoints and of its midpoint (3, 2), show that the three images still lie on one line, and show that the length is unchanged.
Show worked solution
A 90° clockwise rotation about the origin is (x, y) → (y, −x). So (1, 2) → (2, −1), (3, 2) → (2, −3) and (5, 2) → (2, −5). All three images have x = 2, so they lie on the vertical line x = 2: a line is taken to a line, and the midpoint lands between the image endpoints. The original segment is horizontal with length 5 − 1 = 4; the image runs from y = −1 down to y = −5, length 4. The direction turned a quarter turn but the length did not change.
Problem 4
A student traces a parallelogram on tracing paper, flips the paper over and traces it again. Which of the properties (a), (b), (c) can be checked with a ruler, and which needs a protractor?
Show worked solution
A ruler checks (a), that each side of the image has the same length as its pre-image side, and can help check (c) by showing the two pairs of opposite sides are still the same distance apart everywhere, so they are still parallel. A protractor checks (b), that each angle of the image has the same measure as the corresponding pre-image angle. The flip is a reflection, so all three properties should hold within measuring precision.
Problem 5
A student says, "A reflection changes the angles, because the image faces the other way." What is wrong with this reasoning?
Show worked solution
Facing the other way is a change of orientation, not of angle measure. The reflected figure reads its vertices in the opposite turning direction, but each angle in it is the opening between two rays, and the fold that produces the reflection does not stretch the paper, so every opening is the same size. Measuring the reflected angle with a protractor gives the same number. Reflections preserve angle measure; they reverse orientation.
Problem 6
Line r passes through (0, 0) and (1, 3). Line s passes through (2, 0) and (3, 3). Reflect both across the y-axis and show the images are parallel.
Show worked solution
Both lines have slope 3, so they are parallel. Under (x, y) → (−x, y) line r passes through (0, 0) and (−1, 3), slope 3/(−1) = −3. Line s passes through (−2, 0) and (−3, 3), slope 3/(−1) = −3. The slopes both changed sign, but they are still equal, so the image lines are parallel. Parallel lines are taken to parallel lines.
Problem 7
Triangle ABC has vertices A(3, 1), B(5, 1) and C(3, 2). Why is "rotate the triangle 90°" not a complete instruction? Give two different images of A that it could produce.
Show worked solution
A rotation needs a center and a direction as well as an angle. Rotating 90° clockwise about the origin, (x, y) → (y, −x), sends A(3, 1) to (1, −3); rotating 90° counterclockwise about the origin, (x, y) → (−y, x), sends it to (−1, 3); rotating 90° about A itself leaves A where it is. Different images come from the same incomplete instruction. Any two fully described rotations that give different images are a correct answer.
Problem 8
In a tracing-paper experiment, a side measures 5.0 cm before a rotation and 5.1 cm after. Does this show that rotations change lengths?
Show worked solution
No. One millimeter is within the precision of a ruler and a hand-traced line, so the two readings agree as well as the experiment can tell. For a quarter or half turn about the origin a coordinate check shows the lengths are exactly equal: the run and rise of the segment are only exchanged or negated, not changed in size. For a rotation through any other angle the informal reason from the lesson applies: a transparency pinned at the center turns as one stiff sheet, so no two points on it move closer together or farther apart. A real counterexample would need a difference far larger than the measuring error.