{
  "format": "stapler-pack",
  "version": 1,
  "kind": "quiz",
  "meta": {
    "title": "Transformations & Dilations",
    "author": "Staples Education",
    "description": "Rigid motions and dilations on the coordinate plane, with the length/area/volume scaling consequences that follow. Transcribed from the worked Transformations section of the tutoring whiteboard archive.",
    "created": "2026-08-06T04:14:56.152Z",
    "license": "Proprietary - Staples Education"
  },
  "payload": {
    "quiz": {
      "id": "transformations-dilations",
      "title": "Transformations & Dilations",
      "blurb": "Reflections, rotations and translations as rigid motions, plus dilations and how length, area and volume scale with k.",
      "meta": "19 questions · easy → hard",
      "categories": [
        "congruence-transformations",
        "dilations",
        "similar-figures"
      ],
      "questions": [
        {
          "id": "trn-001",
          "topic": "Reflections",
          "difficulty": "easy",
          "stem": "Reflect the point (3, −5) over the x-axis. What are its coordinates?",
          "choices": [
            "(3, 5)",
            "(−3, −5)",
            "(−3, 5)",
            "(−5, 3)"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Reflecting over the x-axis keeps x and negates y: (x, y) → (x, −y). So (3, −5) → (3, 5).",
            "(−3, −5) negates x instead, which is a reflection over the y-axis.",
            "(−3, 5) negates both, which is a reflection through the origin (equivalently a 180° rotation).",
            "(−5, 3) swaps the coordinates and negates one — that is a 90° rotation, not a reflection."
          ],
          "hint": "The x-axis is horizontal, so reflecting across it flips the point vertically. Which coordinate measures vertical position?"
        },
        {
          "id": "trn-002",
          "topic": "Rotations",
          "difficulty": "medium",
          "stem": "Rotate the point (2, 7) by 90° counterclockwise about the origin. What are its coordinates?",
          "choices": [
            "(7, −2)",
            "(−7, 2)",
            "(−2, −7)",
            "(2, −7)"
          ],
          "correctIndex": 1,
          "explanations": [
            "(7, −2) is the 270° counterclockwise rule (equivalently 90° clockwise), (x, y) → (y, −x). It turns the point the wrong way.",
            "Correct. A 90° counterclockwise rotation about the origin is (x, y) → (−y, x). Applying it to (2, 7) gives (−7, 2).",
            "(−2, −7) is the 180° rule, (x, y) → (−x, −y). That is a half-turn, twice as far as asked.",
            "(2, −7) is a reflection over the x-axis, not a rotation. A rotation of a point not on an axis changes both coordinates here."
          ],
          "hint": "Sketch it: (2, 7) is up and slightly right. A quarter turn counterclockwise should land it up and to the LEFT."
        },
        {
          "id": "trn-003",
          "topic": "Dilations",
          "difficulty": "easy",
          "stem": "Dilate the point (4, −6) by a factor of k = ½ centred at the origin.",
          "choices": [
            "(2, −3)",
            "(8, −12)",
            "(2, −6)",
            "(3.5, −6.5)"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A dilation centred at the origin multiplies both coordinates by k: (x, y) → (kx, ky). So (4, −6) → (2, −3).",
            "(8, −12) multiplies by 2 instead of by ½. A factor below 1 shrinks the figure toward the centre.",
            "(2, −6) scales only x. A dilation acts on both coordinates, or the shape would distort rather than stay similar.",
            "(3.5, −6.5) subtracts ½ from each coordinate. Dilation multiplies; translation adds."
          ],
          "hint": "Dilation is multiplication by k, applied to both coordinates."
        },
        {
          "id": "trn-004",
          "topic": "Scale and Area",
          "difficulty": "medium",
          "stem": "A figure has area 20. After a dilation by k = 3, what is the new area?",
          "choices": [
            "60",
            "180",
            "23",
            "20"
          ],
          "correctIndex": 1,
          "explanations": [
            "60 multiplies the area by k rather than by k². Lengths scale by k, but area is a product of two lengths, so it scales by k².",
            "Correct. Area scales by the square of the factor: 20 × 3² = 20 × 9 = 180.",
            "23 adds k to the area. Dilation multiplies; it does not add.",
            "20 leaves the area unchanged. Only a rigid motion — translation, reflection, rotation — preserves area. A dilation with k ≠ 1 does not."
          ],
          "hint": "Area is length × length. If every length grows by a factor of 3, what happens to a product of two of them?"
        },
        {
          "id": "trn-005",
          "topic": "Scale and Volume",
          "difficulty": "medium",
          "stem": "A solid of volume 16 is scaled by k = 2. Find its new volume.",
          "choices": [
            "32",
            "64",
            "128",
            "18"
          ],
          "correctIndex": 2,
          "explanations": [
            "32 multiplies by k. Volume scales by k³, not k.",
            "64 multiplies by k², the area rule. Volume needs one more factor of k.",
            "Correct. Volume scales by the cube of the factor: 16 × 2³ = 16 × 8 = 128.",
            "18 adds 2. Scaling multiplies."
          ],
          "hint": "Volume is length × length × length. Count how many factors of k that gives you."
        },
        {
          "id": "trn-006",
          "topic": "Reflections",
          "difficulty": "medium",
          "stem": "Reflect the point (5, 2) over the line y = x.",
          "choices": [
            "(2, 5)",
            "(−5, −2)",
            "(5, −2)",
            "(−2, −5)"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Reflecting over y = x swaps the coordinates: (x, y) → (y, x). So (5, 2) → (2, 5). The line y = x is exactly the set of points where the two coordinates are already equal, which is why it acts as the mirror that exchanges them.",
            "(−5, −2) negates both, which is a reflection through the origin.",
            "(5, −2) reflects over the x-axis instead.",
            "(−2, −5) swaps AND negates, which is a reflection over the line y = −x."
          ],
          "hint": "Points on y = x have their coordinates equal. What does the mirror do to a point where they differ?",
          "diagram": "<svg viewBox=\"0 0 170 148\" width=\"100%\" style=\"max-width:170px; height:auto; display:block; margin:0 auto;\" role=\"img\"><line x1=\"25\" y1=\"16\" x2=\"25\" y2=\"130\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"44\" y1=\"16\" x2=\"44\" y2=\"130\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"63\" y1=\"16\" x2=\"63\" y2=\"130\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"82\" y1=\"16\" x2=\"82\" y2=\"130\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"101\" y1=\"16\" x2=\"101\" y2=\"130\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"120\" y1=\"16\" x2=\"120\" y2=\"130\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"139\" y1=\"16\" x2=\"139\" y2=\"130\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"158\" y1=\"16\" x2=\"158\" y2=\"130\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"130\" x2=\"158\" y2=\"130\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"111\" x2=\"158\" y2=\"111\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"92\" x2=\"158\" y2=\"92\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"73\" x2=\"158\" y2=\"73\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"54\" x2=\"158\" y2=\"54\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"35\" x2=\"158\" y2=\"35\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"16\" x2=\"158\" y2=\"16\" stroke=\"#eceaf2\" stroke-width=\"1\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"130\" x2=\"158\" y2=\"130\" stroke=\"#6b6b76\" stroke-width=\"1.4\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"16\" x2=\"25\" y2=\"130\" stroke=\"#6b6b76\" stroke-width=\"1.4\" stroke-linecap=\"round\"/><line x1=\"25\" y1=\"130\" x2=\"139\" y2=\"16\" stroke=\"#6b6b76\" stroke-width=\"1.6\" stroke-linecap=\"round\"/><circle cx=\"120\" cy=\"92\" r=\"3.2\" fill=\"#7c3aed\"/><circle cx=\"63\" cy=\"35\" r=\"3.2\" fill=\"#c0392b\"/><line x1=\"120\" y1=\"92\" x2=\"63\" y2=\"35\" stroke=\"#c0392b\" stroke-width=\"1.5\" stroke-dasharray=\"5 4\" stroke-linecap=\"round\"/><text x=\"122\" y=\"100\" fill=\"#6b6b76\" font-size=\"10.5\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">(5,2)</text><text x=\"48\" y=\"40\" fill=\"#c0392b\" font-size=\"10.5\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">(2,5)</text><text x=\"120\" y=\"34\" fill=\"#6b6b76\" font-size=\"10.5\" font-family=\"system-ui, sans-serif\" font-weight=\"600\" paint-order=\"stroke\" stroke=\"#ffffff\" stroke-width=\"3.5\" stroke-linejoin=\"round\">y = x</text></svg>",
          "diagramCaption": "The mirror is the line y = x. Not drawn to scale"
        },
        {
          "id": "trn-007",
          "topic": "Rotations",
          "difficulty": "easy",
          "stem": "Rotate the point (4, 1) by 180° about the origin.",
          "choices": [
            "(−4, −1)",
            "(−1, 4)",
            "(1, −4)",
            "(4, −1)"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. A 180° rotation about the origin is (x, y) → (−x, −y), so (4, 1) → (−4, −1). A half-turn sends every point to the opposite side of the origin, the same distance away.",
            "(−1, 4) is the 90° counterclockwise rule.",
            "(1, −4) is the 270° counterclockwise rule.",
            "(4, −1) reflects over the x-axis. That is not a rotation — it reverses orientation."
          ],
          "hint": "A half-turn puts the point directly opposite the centre, at the same distance."
        },
        {
          "id": "trn-008",
          "topic": "Rotations",
          "difficulty": "medium",
          "stem": "Rotate the point (3, 5) by 270° counterclockwise about the origin.",
          "choices": [
            "(−5, 3)",
            "(5, −3)",
            "(−3, −5)",
            "(3, −5)"
          ],
          "correctIndex": 1,
          "explanations": [
            "(−5, 3) is the 90° counterclockwise rule. 270° counterclockwise is three quarter-turns, which lands somewhere else.",
            "Correct. The 270° counterclockwise rule is (x, y) → (y, −x), so (3, 5) → (5, −3). Equivalently: rotate 180° to get (−3, −5), then a further 90° counterclockwise to reach (5, −3).",
            "(−3, −5) is only the 180° rotation, one quarter-turn short.",
            "(3, −5) reflects over the x-axis rather than rotating."
          ],
          "hint": "270° counterclockwise is the same as 90° clockwise. Sketch which quadrant the point should end up in."
        },
        {
          "id": "trn-009",
          "topic": "Scale and Area",
          "difficulty": "medium",
          "stem": "A rectangle of area 24 is dilated by k = ½. Find the new area.",
          "choices": [
            "12",
            "6",
            "48",
            "24"
          ],
          "correctIndex": 1,
          "explanations": [
            "12 halves the area, applying k rather than k². Halving every length quarters the area, not halves it.",
            "Correct. Area scales by k²: 24 × (½)² = 24 × ¼ = 6.",
            "48 doubles rather than shrinks. A factor below 1 makes the figure smaller.",
            "24 is unchanged, which would require k = 1."
          ],
          "hint": "Halve the width and halve the height. What fraction of the original rectangle is left?"
        },
        {
          "id": "trn-010",
          "topic": "Translations",
          "difficulty": "easy",
          "stem": "Translate the point (7, −2) by the rule (x, y) → (x − 3, y + 5).",
          "choices": [
            "(4, 3)",
            "(10, −7)",
            "(4, −7)",
            "(−3, 5)"
          ],
          "correctIndex": 0,
          "explanations": [
            "Correct. Apply the rule coordinate by coordinate: (7 − 3, −2 + 5) = (4, 3).",
            "(10, −7) applies both operations backwards, adding 3 and subtracting 5.",
            "(4, −7) gets x right but subtracts 5 from y instead of adding it. Watch the sign in the rule.",
            "(−3, 5) is the translation vector itself, not the image of the point. The vector says how far to move, not where you end up."
          ],
          "hint": "Substitute directly and mind the sign on the −2."
        },
        {
          "id": "trn-011",
          "topic": "Similar Solids",
          "difficulty": "hard",
          "stem": "Two similar solids have a scale factor of 2 : 3. What is the ratio of their volumes?",
          "choices": [
            "2 : 3",
            "4 : 9",
            "8 : 27",
            "6 : 9"
          ],
          "correctIndex": 2,
          "explanations": [
            "2 : 3 is the ratio of corresponding LENGTHS. Volumes grow faster than lengths.",
            "4 : 9 is the ratio of their surface AREAS, which is the scale factor squared. Volume needs the cube.",
            "Correct. Volume scales by the cube of the linear ratio: 2³ : 3³ = 8 : 27.",
            "6 : 9 multiplies both parts by 3, which reduces back to 2 : 3 — the length ratio again, just written differently."
          ],
          "hint": "Length ratio, area ratio, volume ratio — first power, second power, third power."
        },
        {
          "id": "trn-012",
          "topic": "Rigid Motions",
          "difficulty": "medium",
          "stem": "Which transformation does NOT always produce a figure congruent to the original?",
          "choices": [
            "Translation",
            "Reflection",
            "Rotation",
            "Dilation"
          ],
          "correctIndex": 3,
          "explanations": [
            "A translation slides every point the same distance in the same direction. Distances and angles are untouched, so the image is congruent.",
            "A reflection reverses orientation but preserves every distance and angle, so the image is congruent to the original.",
            "A rotation turns the figure about a fixed point, again preserving all distances and angles.",
            "Correct. A dilation multiplies every distance by k, so unless k = 1 the image is a different size. It preserves ANGLES but not lengths, which makes the image similar rather than congruent. Translation, reflection and rotation are the three rigid motions; dilation is the odd one out."
          ],
          "hint": "Congruent means same size and same shape. Which of these four changes the size?"
        },
        {
          "id": "trn-013",
          "topic": "Rotation Rules",
          "difficulty": "medium",
          "stem": "Rotate the point (3, 5) by 270° counterclockwise about the origin. What are its new coordinates?",
          "choices": [
            "(−5, 3)",
            "(5, −3)",
            "(−3, −5)",
            "(3, −5)"
          ],
          "correctIndex": 1,
          "explanations": [
            "(−5, 3) is the image under a 90° counterclockwise rotation, which uses (x, y) → (−y, x). A 270° CCW turn is three quarter-turns, landing somewhere different.",
            "Correct. A 270° counterclockwise rotation about the origin uses (x, y) → (y, −x). So (3, 5) → (5, −3). Equivalently, 270° CCW is the same as 90° clockwise.",
            "(−3, −5) is the 180° image, from (x, y) → (−x, −y). That is only two quarter-turns, not three.",
            "(3, −5) is a reflection over the x-axis, which negates y and leaves x alone. Reflections and rotations produce different images here."
          ],
          "hint": "Count quarter-turns. 270° counterclockwise is the same destination as 90° clockwise — sketch it if the rule is hard to recall."
        },
        {
          "id": "trn-014",
          "topic": "Reflection over y = x",
          "difficulty": "easy",
          "stem": "Reflect the point (5, 2) over the line y = x.",
          "choices": [
            "(−5, 2)",
            "(5, −2)",
            "(2, 5)",
            "(−2, −5)"
          ],
          "correctIndex": 2,
          "explanations": [
            "(−5, 2) is the reflection over the y-axis, which negates x. Reflecting over y = x is a different operation entirely.",
            "(5, −2) is the reflection over the x-axis, which negates y.",
            "Correct. Reflecting over the line y = x swaps the coordinates: (x, y) → (y, x). So (5, 2) → (2, 5). The line y = x is the mirror, and swapping is what mirroring across it does.",
            "(−2, −5) swaps the coordinates and then negates both, which combines a reflection over y = x with a 180° rotation. Only the swap is called for here."
          ],
          "hint": "The mirror line y = x treats the two axes symmetrically. What single operation on the coordinates reflects that symmetry?"
        },
        {
          "id": "trn-015",
          "topic": "Volume and Scale Factor",
          "difficulty": "medium",
          "stem": "A solid with volume 16 is scaled by a factor of k = 2. What is the new volume?",
          "choices": [
            "32",
            "64",
            "128",
            "48"
          ],
          "correctIndex": 2,
          "explanations": [
            "32 is 16 × 2, which scales volume by k. Only LENGTHS scale by k; volume is three-dimensional and picks up the factor three times.",
            "64 is 16 × 2², which uses the AREA scale factor. k² applies to surface areas; volume needs k³.",
            "Correct. Volume scales by k³. So the new volume is 16 × 2³ = 16 × 8 = 128.",
            "48 is 16 × 3, which multiplies by the exponent rather than raising the scale factor to it. The relationship is k³, not 3k."
          ],
          "hint": "Match the exponent on k to the number of dimensions the quantity measures."
        },
        {
          "id": "trn-016",
          "topic": "Similar Solids",
          "difficulty": "hard",
          "stem": "Two similar solids have a scale factor of 2:3. What is the ratio of their volumes?",
          "choices": [
            "2:3",
            "4:9",
            "8:27",
            "6:9"
          ],
          "correctIndex": 2,
          "explanations": [
            "2:3 is the ratio of corresponding LENGTHS — edges, radii, heights. Volume ratios are not the same as length ratios.",
            "4:9 is the ratio of SURFACE AREAS, since areas scale by k². It is the right idea applied one dimension short.",
            "Correct. Volumes of similar solids are in the ratio k³, so 2³:3³ = 8:27.",
            "6:9 multiplies both terms by 3, which simplifies right back to 2:3. Scaling both parts of a ratio equally does not change the ratio at all."
          ],
          "hint": "Cube each part of the length ratio — and check that your answer does not simplify back to the original."
        },
        {
          "id": "trn-017",
          "topic": "Rigid Motions",
          "difficulty": "medium",
          "stem": "Which transformation does NOT necessarily produce an image congruent to the original figure?",
          "choices": [
            "Translation",
            "Reflection",
            "Rotation",
            "Dilation"
          ],
          "correctIndex": 3,
          "explanations": [
            "A translation slides every point by the same vector. Distances and angles are unchanged, so it is a rigid motion and the image is congruent.",
            "A reflection flips the figure across a line. It reverses orientation but preserves all distances and angle measures, so the image is still congruent.",
            "A rotation turns the figure about a fixed point. Distances from the center are preserved and so are all internal measurements, making it rigid.",
            "Correct. A dilation multiplies all distances by the scale factor k, so unless k = ±1 the image is a different size. Angles are preserved, which makes the image SIMILAR to the original but not congruent."
          ],
          "hint": "Three of these preserve size and are called rigid motions. The fourth preserves shape only."
        },
        {
          "id": "trn-018",
          "topic": "Reasoning: Properties",
          "difficulty": "medium",
          "stem": "Which single property separates a dilation from a translation, reflection and rotation?",
          "choices": [
            "A dilation is the only one that changes orientation.",
            "A dilation is the only one that does not preserve distance, so it changes size while keeping shape.",
            "A dilation is the only one that has a fixed point.",
            "A dilation is the only one that preserves angle measure."
          ],
          "correctIndex": 1,
          "explanations": [
            "Reflection changes orientation; dilation with a positive scale factor does not.",
            "Correct. Translations, reflections and rotations are RIGID motions — they preserve every distance. A dilation scales distances by k, which changes size while preserving shape and angle measure.",
            "Rotations and reflections have fixed points too (the centre, and every point on the mirror line).",
            "All four preserve angle measure, so this cannot be what distinguishes one of them."
          ],
          "hint": "Sort the four into those that preserve distance and those that do not.",
          "kind": "reasoning"
        },
        {
          "id": "trn-019",
          "topic": "Reasoning: Error Analysis",
          "difficulty": "hard",
          "stem": "A figure is dilated by a scale factor of 3. A student says the area is also 3 times larger. What is the correct factor, and why?",
          "choices": [
            "The area is 3 times larger; area scales the same way as length.",
            "The area is 9 times larger, because area involves two dimensions and each is scaled by 3.",
            "The area is 27 times larger, matching the cube of the scale factor.",
            "The area is unchanged; dilation preserves area."
          ],
          "correctIndex": 1,
          "explanations": [
            "Length scales by k. Area does not, because it is built from two lengths multiplied together.",
            "Correct. Area scales by k² = 9. Each of the two dimensions is stretched by 3, so the product grows by 3 × 3.",
            "k³ = 27 is how VOLUME scales, for a three-dimensional figure.",
            "A dilation with k ≠ 1 changes size by definition, so area cannot stay the same."
          ],
          "kind": "reasoning"
        }
      ]
    }
  }
}